4.3 fourier series periodic · 4.3 fourier series de nition 4.41.exponential fourier series: let...

4.3 Fourier Series

Definition 4.41. Exponential Fourier series: Let the (real or complex)signal r (t) be a periodic signal with period T0.Suppose the following Dirichlet conditions are satisfied:

(a) r (t) is absolutely integrable over its period; i.e.,∫ T00 |r (t)|dt <∞.

(b) The number of maxima and minima of r (t) in each period is finite.

(c) The number of discontinuities of r (t) in each period is finite.

Then r (t) can be “expanded” into a linear combination of the complexexponential signals

(ej2π(kf0)t

)∞k=−∞ as

r (t) =∞∑

k=−∞

ckej2π(kf0)t = c0 +

∞∑k=1

(cke

j2π(kf0)t + c−ke−j2π(kf0)t

)(37)

where

f0 =1

T0and

ck =1

T0

α+T0∫

α

r (t) e−j2π(kf0)tdt, (38)

for some arbitrary α.We give some remarks here.

• r (t) =

{r (t) , if r (t) is continuous at tr(t+)+r(t−)

2 , if r (t) is not continuous at t

Although r (t) may not be exactly the same as r(t), for the purposeof our class, it is sufficient to simply treat them as being the same (toavoid having two different notations). Of course, we need to keep inmind that unexpected results may arise at the discontinuity points.

• The parameter α in the limits of the integration (38) is arbitrary. Itcan be chosen to simplify computation of the integral. Some referencessimply write ck = 1

T0

∫T0

r (t) e−jkω0tdt to emphasize that we only need

to integrate over one period of the signal; the starting point is notimportant.

54

• The coefficients ck are called the (kth) Fourier (series) coefficientsof (the signal) r (t). These are, in general, complex numbers.

• c0 = 1T0

∫T0

r (t) dt = average or DC value of r(t)

• The quantity f0 = 1T0

is called the fundamental frequency of thesignal r (t). The kth multiple of the fundamental frequency (for positivek’s) is called the kth harmonic.

• ckej2π(kf0)t + c−ke−j2π(kf0)t = the kth harmonic component of r (t).

k = 1 ⇒ fundamental component of r (t).

4.42. Being able to write r (t) =∑∞

k=−∞ cnej2π(kf0)t means we can easily

find the Fourier transform of any periodic function:

r (t) =∞∑

k=−∞

ckej2π(kf0)t F−−⇀↽−−

F−1R(f) =

The Fourier transform of any periodic function is simply a bunch ofweighted delta functions occuring at multiples of the fundamental frequencyf0.

4.43. Formula (38) for finding the Fourier (series) coefficients

ck =1

T0

α+T0∫

α

r (t) e−j2π(kf0)tdt (39)

is strikingly similar to formula (5) for finding the Fourier transform:

R(f) =

∞∫−∞

r(t)e−j2πftdt. (40)

There are three main differences.We have spent quite some effort learning about the Fourier transform of

a signal and its properties. It would be nice to have a way to reuse thoseconcepts with Fourier series. Identifying the three differences above lets ussee their connection.

55

4.44. Getting the Fourier coefficients from the Fourier transform:

Step 1 Consider a restricted version rT0(t) of r(t) where we only consider r(t)for one period.

1

Step 2 Find the Fourier transform RT0(f) of rT0(t)

Step 3 The Fourier coefficients are simply scaled samples of theFourier transform :

ck =1

T0RT0(kf0).

Example 4.45. Train of Impulses: Find the Fourier series expansion forthe train of impulses

δ(T0)(t) =∞∑

n=−∞δ (t− nT0)

drawn in Figure 21. This infinite train of equally-spaced -functions is usuallydenoted by the Cyrillic letter (shah).

1

3 ‐2 ‐ 0 2 3

1 1 1 1 1 1 1

Figure 21: Train of impulses

56

4.46. The Fourier series derived in Example 4.44 gives an interestingFourier transform pair:

∞∑n=−∞

δ (t− nT0) =∞∑

k=−∞

1

T0ej2π(kf0)t F−−⇀↽−−

F−1(41)

1

‐3 ‐2 ‐ 0 2 3

1 1 1 1 1 1 1

1

‐3 ‐2 ‐ 0 2 3

A special case when T0 = 1 is quite easy to remember:

∞∑n=−∞

δ (t− n)F−−⇀↽−−F−1

∞∑k=−∞

δ (f − k) (42)

1

‐3‐2 ‐1 01 2 3

1 1 1 1 1 1 1

1

‐3‐2 ‐1 01 2 3

1 1 1 1 1 1 1

Once we remember (42), we can easily use the scaling properties of theFourier transform (21) and the delta function (18) to generalize the specialcase (42) back to (41):

∞∑n=−∞

δ (at− n) = x (at)F−−⇀↽−−F−1

1

|a|X

(f

a

)=

1

|a|

∞∑k=−∞

δ

(f

a− k)

1

|a|

∞∑n=−∞

δ(t− n

a

) F−−⇀↽−−F−1

1

|a||a|

∞∑k=−∞

δ (f − ka)

∞∑n=−∞

δ(t− n

a

) F−−⇀↽−−F−1|a|

∞∑k=−∞

δ (f − ka)

At the end, we plug-in a = f0 = 1/T0.

57

Example 4.47. Find the Fourier coefficients of the square pulse periodicsignal [6, p 57].

1

44

: the scaled

Fourier transform of the restricted (one period) version of .

period

0

0

ck =1

T0RT0 (kf0) =

1

T0

(T0

2sinc

(2π

(T0

4

)(f)

)∣∣∣∣f=kf0

)=

1

2sinc

(kπ

2

)=

1

2

sin(kπ2)

kπ2=

sin(kπ2)

kπ

k k × π2 sin

(k × π

2

)ck =

sin(k π2 )kπ

0

1

2

3

4

5

58

Remarks:

(a) Multiplication by this signal is equivalent to a switching (ON-OFF)operation. (Same as periodically turning the switch on (letting anothersignal pass through) for half a period T0.

44

OFF ON OFF ON OFF ON OFF ON OFF ON OFF

(b) This signal can be expressed via a cosine function with the same period:

r (t) = 1 [cos (2πf0t) ≥ 0] =

{1, cos (2πf0t) ≥ 0,0, otherwise.

1

44

(c) A duty cycle is the percentage of one period in which a signal is“active”. Here,

duty cycle = proportion of the “ON” time =width

period.

In this example, the duty cycle is T0/2T0

= 50%. When the duty cycle is 1n ,

the nth harmonic (cn) along with its nonzero multiples are suppressed.

59

4.48. Parseval’s Identity: Pr =⟨|r (t)|2

⟩= 1

T0

∫T0

|r (t)|2 dt =∞∑

k=−∞|ck|2.

4.49. Fourier series expansion for real valued function: Supposer (t) in the previous section is real-valued; that is r∗ = r. Then, we havec−k = c∗k and we provide here three alternative ways to represent the Fourierseries expansion:

r (t) =∞∑

k=−∞

cnej2πkf0t = c0 +

∞∑k=1

(cke

j2πkf0t + c−ke−j2πkf0t

)(43a)

= c0 +∞∑k=1

(ak cos (2πkf0t)) +∞∑k=1

(bk sin (2πkf0t)) (43b)

= c0 + 2∞∑k=1

|ck| cos (2πkf0t+ ∠ck) (43c)

where the corresponding coefficients are obtained from

ck =1

T0

α+T0∫

α

r (t) e−j2πkf0tdt =1

2(ak − jbk) (44)

ak = 2Re {ck} =2

T0

∫T0

r (t) cos (2πkf0t) dt (45)

bk = −2Im {ck} =2

T0

∫T0

r (t) sin (2πkf0t) dt (46)

2 |ck| =√a2k + b2

k (47)

∠ck = − arctan

(bkak

)(48)

c0 =a0

2(49)

The Parseval’s identity can then be expressed as

Pr =⟨|r (t)|2

⟩=

1

T0

∫T0

|r (t)|2dt =∞∑

k=−∞

|ck|2 = c20 + 2

∞∑k=1

|ck|2

60

4.50. To go from (43a) to (43b) and (43c), note that when we replace c−kby c∗k, we have

ckej2πkf0t + c−ke

−j2πkf0t = ckej2πkf0t + c∗ke

−j2πkf0t

= ckej2πkf0t +

(cke

j2πkf0t)∗

= 2 Re{cke

j2πkf0t}.

• Expression (43c) then follows directly from the phasor concept:

Re{cke

j2πkf0t}

= |ck| cos (2πkf0t+ ∠ck) .

• To get (43b), substitute ck by Re {ck}+ j Im {ck}and ej2πkf0t by cos (2πkf0t) + j sin (2πkf0t).

Example 4.51. For the train of impulses in Example 4.44,

∞∑n=−∞

δ (t− n) =∞∑

k=−∞

1

T0ej2π(kf0)t =

1

T0+

2

T0

∞∑k=1

cos kω0t (50)

Example 4.52. For the rectangular pulse train in Example 4.46,

1

44Fourier series expansion:12

1 13

15 ⋯

1 13

15 ⋯

12

2cos 2

23 cos 2 3

25 cos 2 5 ⋯

Trigonometric Fourier series expansion: 2cos

61

1 [cosω0t ≥ 0] =1

2+

2

π

(cosω0t−

1

3cos 3ω0t+

1

5cos 5ω0t−

1

7cos 7ω0t+ . . .

)(51)

Example 4.53. Bipolar square pulse periodic signal [6, p 59]:

sgn(cosω0t) =4

π

(cosω0t−

1

3cos 3ω0t+

1

5cos 5ω0t−

1

7cos 7ω0t+ . . .

)

1

-1

0T 0T− t

1

0T 0T− t

Figure 22: Bipolar square pulse periodic signal

62

Sirindhorn International Institute of Technology

Thammasat University

School of Information, Computer and Communication Technology

ECS332 2017/1 Part II.3 Dr.Prapun

4.4 Classical DSB-SC Modulators

To produce the modulated signal Ac cos(2πfct)m(t), we may use the follow-ing methods which generate the modulated signal along with other signalswhich can be eliminated by a bandpass filter restricting frequency contentsto around fc.

4.54. Multiplier Modulators [6, p 184] or Product Modulator[3, p180]: Here modulation is achieved directly by multiplying m(t) by cos(2πfct)using an analog multiplier whose output is proportional to the product oftwo input signals.

• Such a multiplier may be obtained from

(a) a variable-gain amplifier in which the gain parameter (such as thethe β of a transistor) is controlled by one of the signals, say, m(t).When the signal cos(2πfct) is applied at the input of this amplifier,the output is then proportional to m(t) cos(2πfct).

(b) two logarithmic and an antilogarithmic amplifiers with outputsproportional to the log and antilog of their inputs, respectively.

◦ Key equation:A×B = e(lnA+lnB).

63

4.55. When it is easier to build a squarer than a multiplier, we may use asquare modulator shown in Figure 23.

m t +

cos 2 cc f t

2 BPH f d t x t

Figure 23: Block dia-gram of a square modu-lator

Note that

d (t) = (m (t) + c cos (2πfct))2

= m2 (t) + 2cm (t) cos (2πfct) + c2cos2 (2πfct)

= m2 (t) + 2cm (t) cos (2πfct) +c2

2+c2

2cos (2π (2fc) t)

2

22 0

Using a band-pass filter (BPF) whose frequency response is

HBP (f) =

g, |f − fc| ≤ B,g, |f − (−fc)| ≤ B,

0, otherwise,(52)

we can produce 2cgm(t) cos(2πfct) at the output of the BPF. In particular,choosing the gain g to be (c

√2)−1, we get m(t)×

√2 cos(2πfct).

• Alternative, can use(m(t) + c cos

(ωc2 t))3

.

64

4.56. Another conceptually nice way to produce a signal of the formAcm(t) cos(2πfct) is to

(1) multiply m(t) by “any” periodic and even signal r(t) whose periodis Tc = 1

fc

and then

(2) pass the result though a BPF used in (52).

BPF cos 2×

To see how this works, recall that because r(t) is an even function, weknow that

r (t) = c0 +∞∑k=1

ak cos (2π(kfc)t) for some c0, a1, a2, . . ..

Therefore,

m(t)r (t) = c0m(t) +∞∑k=1

akm(t) cos (2π(kfc)t).

0 2

12

12

2

ℱ

See also [5, p 157]. In general, for this scheme to work, we need

• a1 6= 0 period of r;

• fc > 2B (to prevent overlapping).

Note that if r(t) is not even, then by (43c), the resulting modulated signalwill have the form x(t) = a1m(t) cos(2πfct+ φ1).

65

4.57. Switching modulator : An important example of a periodic andeven function r(t) is the square pulse train considered in Example 4.47.Recall that multiplying this r(t) to a signal m(t) is equivalent to switchingm(t) on and off periodically.

1

BPF2

cos 2

12

2cos 2

23 cos 2 3

25 cos 2 5 ⋯

12

2cos 2

23 cos 2 3

25 cos 2 5 ⋯

1


0

2

53

355 3

ℱ

4.58. Switching Demodulator : The switching technique can also beused at the demodulator as well.

1

LPFcos 2

1


We have seen that, for DSB-SC modem, the key equation is given by(34). When switching demodulator is used, the key equation is

LPF{m(t) cos(2πfct)× 1[cos(2πfct) ≥ 0]} =1

πm(t) (53)

66

[5, p 162].

1 2 2 2cos 2 cos 2 3 cos 2 52 3 512

2 cos 2

2 cos 2 332

cos 2

cos 2

cos 2

co cos 55

s 2 2

c c c

c

c

c c

c c

c c

c cc

y t y t y t y t y t

A m t f t

A

f t f t f t

f t

f t

m t f t

A m t f t

A m t

r

f

t

f tt

1 cos 2 2

cos 2 2 cos 2 4

c

cos 212

1

1315

os 2 4 cos 2 6

c c

c

c c c

cc

c

c

f t

f t

A m t f t

A m t

A m t

A m t

f t

f t f t

12

1 1 cos 2 2

cos 2 2 cos 2 4

co

c

s 2 4 cos 2 6

1 13 31 1

5

2

5

osc c

c c

c c

c cc

c

c c

c

f t

f t

A m t f t

A m t A m t

A m t A m t

A m

f t

f t f tt A m t

cos cos12 cos

12 cos

12

2cos 2

23 cos 2 3

25 cos 2 5 ⋯

Note that this technique still requires the switching to be in sync with theincoming cosine as in the basic DSB-SC.

67

4.5 (Standard) Amplitude modulation: AM

4.59. DSB-SC amplitude modulation (which is summarized in Figure 24)is easy to understand and analyze in both time and frequency domains.However, analytical simplicity is not always accompanied by an equivalentsimplicity in practical implementation.

1

×

2 cos 2 cf t

Modulator

Message(modulating signal)

Figure 24: DSB-SC modulation.

Problem: The (coherent) demodulation of DSB-SC signal requires thereceiver to possess a carrier signal that is synchronized with the incomingcarrier. This requirement is not easy to achieve in practice because themodulated signal may have traveled hundreds of miles and could even sufferfrom some unknown frequency shift.

4.60. If a carrier component is transmitted along with the DSB signal,demodulation can be simplified.

(a) The received carrier component can be extracted using a narrowbandbandpass filter and can be used as the demodulation carrier. (There isno need to generate a carrier at the receiver.)

(b) If the carrier amplitude is sufficiently large, the need for generating ademodulation carrier can be completely avoided.

• This will be the focus of this section.

68

Definition 4.61. For AM, the transmitted signal is typically defined as

xAM (t) = (A+m (t)) cos (2πfct) = A cos (2πfct)︸︷︷︸carrier

+m (t) cos (2πfct)︸︷︷︸sidebands

Assumptions for m(t):

(a) Band-limited to B; that is, |M(f)| = 0 for |f | > B.

(b) Bounded between −mp and mp; that is, |m(t)| ≤ mp.

4.62. Spectrum of xAM (t):

• Basically the same as that of DSB-SC signal except for the two addi-tional impulses (discrete spectral component) at the carrier frequency±fc.

◦ This is why we say the DSB-SC system is a suppressed carriersystem.

Definition 4.63. Consider a signal A(t) cos(2πfct). If A(t) varies slowly incomparison with the sinusoidal carrier cos(2πfct), then the envelope E(t)of A(t) cos(2πfct) is |A(t)|.

4.64. Envelope of AM signal : For AM signal, A(t) ≡ A+m(t) and

E(t) = |A+m(t)| .

See Figure 25.

Case (a) If ∀t, A(t) > 0, then E(t) = A(t) = A+m(t)

• The envelope has the same shape as m(t).

• Enable envelope detection: Extract m(t) from the envelope.

69

Case (b) If ∃t, A(t) < 0, then E(t) 6= A(t).

• The envelope shape differs from the shape of m(t) because thenegative part of A+m(t) is rectified.

◦ This is referred to as phase reversal and envelope distortion.

t

t

t

t

t

AA

Case (a) Case (b)

≡ 0 forall ≡ 0 forsome

AM cos 2

Figure 25: AM signal and its envelope [6, Fig 4.8]

Definition 4.65. The positive constant

µ ≡maxt

(envelope of the sidebands)

maxt

(envelope of the carrier)=

maxt|m (t)|

maxt|A|

=mp

A

is called the modulation index.

• The quantity µ×100% is often referred to as the percent modulation.

70

Example 4.66. Consider a sinusoidal (pure-tone) messagem(t) = Am cos(2πfmt).Suppose A = 1. Then, µ = Am. Figure 26 shows the effect of changing themodulation index on the modulated signal.

1

Time

50% Modulation

0

−1.5

1.5

−0.5

0.5

Time

100% Modulation

0

−2

2

EnvelopeModulated Signal

Time

150% Modulation

0

−2.5

2.5

−0.5

0.5

Figure 26: Modulated signal in standard AM with sinusoidal message

4.67. It should be noted that the ratio that defines the modulation indexcompares the maximum of the two envelopes. In other references, the nota-tion for the AM signal may be different but the idea (and the correspondingmotivation) that defines the modulation index remains the same.

• In [3, p 163], it is assumed that m(t) is already scaled or normalized tohave a magnitude not exceeding unity (|m(t)| ≤ 1) [3, p 163]. There,

xAM (t) = Ac (1 + µm (t)) cos (2πfct) = Ac cos (2πfct)︸︷︷︸carrier

+Acµm (t) cos (2πfct)︸︷︷︸sidebands

.

71

◦ mp = 1

◦ The modulation index is then

maxt


maxt


maxt|Acµm (t)|

maxt|Ac|

=|Acµ||Ac|

= µ.

• In [15, p 116],

xAM (t) = Ac

(1 + µ

m (t)

mp

)cos (2πfct) = Ac cos (2πfct)︸︷︷︸

carrier

+Acµm (t)

mpcos (2πfct)︸︷︷︸

sidebands

.

◦ The modulation index is then

maxt


maxt


maxt

∣∣∣Acµm(t)mp

∣∣∣maxt|Ac|

=|Ac|µmpmp|Ac|

= µ.

4.68. Power of the transmitted signals.

(a) In DSB-SC system, recall, from 4.39, that, when

x(t) = m(t) cos(2πfct)

with fc sufficiently large, we have

Px =1

2Pm.

Therefore, all transmitted power are in the sidebands which containmessage information.

(b) In AM system,

xAM (t) = A cos (2πfct)︸︷︷︸carrier

+m (t) cos (2πfct)︸︷︷︸sidebands

.

If we assume that the average of m(t) is 0 (no DC component), then thespectrum of the sidebandsm(t) cos(2πfct+θ) and the carrierA cos(2πfct+θ) are non-overlapping in the frequency domain. Hence, when fc is suf-ficiently large

Px =1

2A2 +

1

2Pm.

72

• Efficiency:

• For high power efficiency, we want smallm2p

µ2Pm.

◦ By definition, |m(t)| ≤ mp. Therefore,m2p

Pm≥ 1.

◦ Want µ to be large. However, when µ > 1, we have phasereversal. So, the largest value of µ is 1.

◦ The best power efficiency we can achieved is then 50%.

• Conclusion: at least 50% (and often close to 2/3[3, p. 176]) ofthe total transmitted power resides in the carrier part which isindependent of m(t) and thus conveys no message information.

4.69. An AM signal can be demodulated using the same coherent demod-ulation technique that was used for DSB. However, the use of coherentdemodulation negates the advantage of AM.

• Note that, conceptually, the received AM signal is the same as DSB-SC signal except that the m(t) in the DSB-SC signal is replaced byA(t) = A + m(t). We also assume that A is large enough so thatA(t) ≥ 0.

• Recall the key equation of switching demodulator (53):

LPF{A(t) cos(2πfct)× 1[cos(2πfct) ≥ 0]} =1

πA(t) (54)

We noted before that this technique requires the switching to be insync with the incoming cosine.

73

4.70. Demodulation of AM Signals via rectifier detector: The receiverwill first recover A+m(t) and then remove A.

• When ∀t, A(t) ≥ 0, we can replace the switching demodulator bythe rectifier demodulator/detector . In which case, we suppressthe negative part of y(t) = xAM(t) using a diode (half-wave rectifier:HWR).

◦ Here, we define a HWR to be a memoryless device whose input-output relationship is described by a function fHWR(·):

fHWR (x) =

{x, x ≥ 0,0, x < 0.

• Surprisingly, this is mathematically equivalent to a switching demodu-lator in (53) and (54).

• It is in effect synchronous detection performed without using a localcarrier [5, p 167].

• This method needs A(t) ≥ 0 so that the sign of A(t) cos(2πfct) will bethe same as the sign of cos(2πfct).

• The dc term Aπ may be blocked by a capacitor to give the desired output

m(t)/π.

74

196 AMPLITUDE MODULATIONS AND DEMODULATIONS

Figure 4.10 Rectifier detector for AM.

[a+ m(t)] cos wet

'

' /

[A + m(l)] cos wet

VR(t) /[A + m(t)]

-_f I " rr [A + 111(1)]

Low-pass filter

-·

I -;-[A + m(1)]

~

signal is multiplied by w(t). Hence, the half-wave rectified output vR(t) is

VR(t) ={[A+ m(t)] COS Wet) w(t) (4.12)

=[A+ m(t)] cos Wet [ ~ + ~ (cos (Vet- ~cos 3wet + ~cos Swet- · · ·)] (4.13)

l = -[A+ m(t)] +other terms of higher frequencies (4.14)

][

When vR(t) is applied to a low-pass filter of cutoff B Hz, the output is [A+ m(t)]jn, and all the other terms of frequencies higher than B Hz are suppressed. The de term Ajn may be blocked by a capac itor (Fig. 4.10) to give the desired output m(t) j n. The output can be doubled by using a full-wave rectifi er.

It is interesting to note that because of the multip lication with ll '(l), rectifier detection is in effect synchronous detection performed without using a local carrier. The high carrier content in AM ensures that its zero crossings are periodic and the information about frequency and phase of the carrier at the transmitter is built in to the AM signal itself.

Envelope Detector: fn an enve lope detector, the output of the detector follows the envelope of the modulated signal. The simple circuit show n in Fig. 4. lla functions as an envelope detector. On the positive cycle of the input signal, the input grows and may exceed the charged vo ltage on the capacity vc(t), turning on the diode and allowing the capacitor C to charge up to the peak voltage of the input signal cycle. As the input signal fall s below this peak value, it falls quickly below the capacitor voltage (which is very nearly the peak voltage), thus caus ing the diode to open. The capacitor now di scharges through the resi stor R at a slow rate (with a time constant RC). During the next positive cycle, the same drama repeats . As the input signal rises above the capacitor voltage, the diode conducts again. The capacitor again charges to the peak value of this (new) cycle. The capacitor discharges slowly during the cutoff period.

During each positive cycle, the capacitor charges up to the peak voltage of the input signal and then decays slowly until the next positive cycle, as shown in Fig. 4 . ll b. Thus, the output voltage vc(t), close ly follows the (rising) envelope of the input AM signal. Equally important, the slow capacity discharge via the resistor R a llows the capacity vo ltage to follow

Figure 27: Rectifier detector for AM [6, Fig. 4.10].

Figure 4.11 Envelope detector for AM.

4 .4 Bandwidth-Efficient Amplitude Modulations 197

AM signal c

(a)

Envelope detector output

RC too large \

····· K'f<K~--~. . Enve lop~.--· ... ·· · '( K I"' I"" ~-~ . , .. · < i"" !'--, ., ~ ··· ···~"

W'~

.... ... -·· '

.. ··· · ... ·· .. ..

(b) ······

a declining envelope. Capacitor d ischarge between positi ve peaks causes a ripple s ignal of freque ncy We in the output. Thi s rip ple can be reduced by choosing a larger time constant RC so that the capac itor discharges very littl e between the positive peaks (RC » I /eve) . If RC were made too large, however, it would be imposs ible for the capac itor voltage to follow a fast declining e nvelope (Fig. 4 .11 b). Because the max imum rate of AM envelope dec line is do minated by the bandw idth B of the message signal m (r ) , the des ign criterion of RC should be

I /eve « RC < I / (2Jr8) or I

2Jr8 < - « (t!c RC

The envelope detector output is vc(t ) = A+ m(r) with a ripple o f frequency W e . The de term A can be blocked oul by a capacitor or a s imple RC high-pass filte r. The ripple may be reduced further by another (low-pass) RC filter.

4.4 BANDWIDTH-EFFICIENT AMPLITUDE MODULATIONS

As seen from Fig. 4.12, the DSB spectrum (including suppressed carrier and AM) has two sidebands: the upper sideband (USB) and the lower sideband (LSB~both containing complete informatinn about the baseband signal m (r ). As a result , for a baseband signal m (t) with bandwidth B Hz, DSB modulations require twice the radio-frequency bandwidth to transmit.

Figure 28: Envelope detector for AM [6, Fig. 4.11].

75

4.71. Demodulation of AM signal via envelope detector :

• Design criterion of RC:

2πB � 1

RC� 2πfc.

• The envelope detector output is A+m(t) with a ripple of frequency fc.

• The dc term can be blocked out by a capacitor or a simple RC high-passfilter.

• The ripple may be reduced further by another (low-pass) RC filter.

4.72. AM Trade-offs:

(a) Disadvantages :

• Higher power and hence higher cost required at the transmitter

• The carrier component is wasted power as far as information trans-fer is concerned.

• Bad for power-limited applications.

(b) Advantages :

• Coherent reference is not needed for demodulation.

• Demodulator (receiver) becomes simple and inexpensive.

• For broadcast system such as commercial radio (with a huge num-ber of receivers for each transmitter),

◦ any cost saving at the receiver is multiplied by the number ofreceiver units.

◦ it is more economical to have one expensive high-power trans-mitter and simpler, less expensive receivers.

(c) Conclusion: Broadcasting systems tend to favor the trade-off by mi-grating cost from the (many) receivers to the (fewer) transmitters.

4.73. References: [3, p 198–199], [6, Section 4.3] and [14, Section 3.1.2].

76

4.6 Bandwidth-Efficient Modulations

4.74. We are now going to define a quantity called the “bandwidth” of asignal. Unfortunately, in practice, there isn’t just one definition of band-width.

Definition 4.75. The bandwidth (BW) of a signal is usually calculatedfrom the differences between two frequencies (called the bandwidth limits).Let’s consider the following definitions of bandwidth for real-valued signals[3, p 173]

(a) Absolute bandwidth: Use the highest frequency and the lowest fre-quency in the positive-f part of the signal’s nonzero magnitude spec-trum.

• This uses the frequency range where 100% of the energy is confined.

• We can speak of absolute bandwidth if we have ideal filters andunlimited time signals.

(b) 3-dB bandwidth (half-power bandwidth): Use the frequencieswhere the signal power starts to decrease by 3 dB (1/2).

• The magnitude is reduced by a factor of 1/√

2.

(c) Null-to-null bandwidth: Use the signal spectrum’s first set of zerocrossings.

(d) Occupied bandwidth: Consider the frequency range in which X%(for example, 99%) of the energy is contained in the signal’s bandwidth.

(e) Relative power spectrum bandwidth: the level of power outsidethe bandwidth limits is reduced to some value relative to its maximumlevel.

• Usually specified in negative decibels (dB).

• For example, consider a 200-kHz-BW broadcast signal with a max-imum carrier power of 1000 watts and relative power spectrumbandwidth of -40 dB (i.e., 1/10,000). We would expect the sta-tion’s power emission to not exceed 0.1 W outside of fc± 100 kHz.

77

Example 4.76. Message bandwidth and the transmitted signal bandwidth

1

f

f

f

f

B-B

fc-fc

(a) Baseband

(b) DSB-SC

(c) USB

(d) LSB

LSB LSBU

SB

USB

USB

USB

LSB LSB

Figure 29: SSB spectra from suppressing one DSB sideband.

4.77. BW Inefficiency in DSB-SC system: Recall that for real-valued base-band signal m(t), the conjugate symmetry property from 2.30 says that

M(−f) = (M(f))∗ .

The DSB spectrum has two sidebands: the upper sideband (USB) and thelower sideband (LSB), each containing complete information about the base-band signal m(t). As a result, DSB signals occupy twice the bandwidthrequired for the baseband.

4.78. Rough Approximation: If g1(t) and g2(t) have bandwidths B1 andB2 Hz, respectively, the bandwidth of g1(t)g2(t) is B1 +B2 Hz.

This result follows from the application of the width property18 of con-volution19 to the convolution-in-frequency property.

Consequently, if the bandwidth of g(t) is B Hz, then the bandwidth ofg2(t) is 2B Hz, and the bandwidth of gn(t) is nB Hz. We mentioned thisproperty in 2.42.

18This property states that the width of x ∗ y is the sum of the widths of x and y.19The width property of convolution does not hold in some pathological cases. See [5, p 98].

78

4.79. To improve the spectral efficiency of amplitude modulation, thereexist two basic schemes to either utilize or remove the spectral redundancy:

(a) Single-sideband (SSB) modulation, which removes either the LSB orthe USB so that for one message signal m(t), there is only a bandwidthof B Hz.

(b) Quadrature amplitude modulation (QAM), which utilizes spectral re-dundancy by sending two messages over the same bandwidth of 2BHz.

4.7 Single-Sideband Modulation

4.80. Transmitting both upper and lower sidebands of DSB is redundant.Transmission bandwidth can be cut in half if one sideband is suppressedalong with the carrier.

Definition 4.81. Conceptually, in single-sideband (SSB) modulation,a sideband filter suppresses one sideband before transmission. [3, p 185–186]

(a) If the filter removes the lower sideband, the output spectrum consistsof the upper sideband (USB) alone. Mathematically, the time domainrepresentation of this SSB signal is

xUSB(t) = m(t)√

2 cos(2πfct)−mh(t)√

2 sin(2πfct). (55)

where mh(t) is the Hilbert transform of the message:

mh(t) = H{m(t)} =1

π

∫ ∞−∞

m(τ)

t− τdτ = m(t) ∗ 1

πt. (56)

(b) If the filter removes the upper sideband, the output spectrum consistsof the lower sideband (LSB) alone. Mathematically, the time domainrepresentation of this SSB signal is

xLSB(t) = m(t)√

2 cos(2πfct) +mh(t)√

2 sin(2πfct). (57)

Derivation of the time-domain representation is given in Section 4.9. Morediscussion on SSB can be found in [3, Sec 4.4], [14, Section 3.1.3] and [5,Section 4.5].

79

4.8 Quadrature Amplitude Modulation (QAM)

Definition 4.82. In quadrature amplitude modulation (QAM ) orquadrature multiplexing , two baseband real-valued signals m1(t) andm2(t) are transmitted simultaneously via the corresponding QAM signal:

xQAM (t) = m1 (t)√

2 cos (2πfct) +m2 (t)√

2 sin (2πfct) .

1m t

Transmitter (modulator) Receiver (demodulator)

1v t LPH f 1m t

2m t 2v t LPH f 2m t

2 cos 2 cf t

2 sin 2 cf t

2 h t y t QAMx t

Channel

2 cos 2 cf t

2 sin 2 cf t

2

Figure 30: QAM Scheme

• QAM operates by transmitting two DSB signals via carriers of the samefrequency but in phase quadrature.

• Both modulated signals simultaneously occupy the same frequencyband.

• The “cos” (upper) channel is also known as the in-phase (I ) channeland the “sin” (lower) channel is the quadrature (Q) channel.

4.83. Demodulation : Under the usual assumption (B < fc), the twobaseband signals can be separated at the receiver by synchronous detection:

LPF{xQAM (t)

√2 cos (2πfct)

}= m1 (t) (58)

LPF{xQAM (t)

√2 sin (2πfct)

}= m2 (t) (59)

80

To see (58), note that

v1 (t) = xQAM (t)√

2 cos (2πfct)

=(m1 (t)

√2 cos (2πfct) +m2 (t)

√2 sin (2πfct)

)√2 cos (2πfct)

= m1 (t) 2cos2 (2πfct) +m2 (t) 2 sin (2πfct) cos (2πfct)

= m1 (t) (1 + cos (2π (2fc) t)) +m2 (t) sin (2π (2fc) t)

= m1 (t) +m1 (t) cos (2π (2fc) t) +m2 (t) cos (2π (2fc) t− 90◦)

• Observe that m1(t) and m2(t) can be separately demodulated.

Example 4.84. (1)√

2 cos (2πfct) + (1)√

2 sin (2πfct)

Example 4.85. 3√

2 cos (2πfct) + 4√

2 sin (2πfct)

4.86. Suppose, during a time interval, the messages m1(t) and m2(t) areconstant. Consider the signal m1

√2 cos (2πfct) +m2

√2 sin (2πfct)

4.87. Sinusoidal form (envelope-and-phase description [3, p. 165]):

xQAM (t) =√

2E(t) cos(2πfct+ φ(t)),

where

envelope: E(t) = |m1(t)− jm2(t)| =√m2

1(t) +m22(t)

phase: φ(t) = ∠ (m1(t)− jm2(t))

81

Example 4.88. In a QAM system, the transmitted signal is of the form

xQAM (t) = m1 (t)√

2 cos (2πfct) +m2 (t)√

2 sin (2πfct) .

Here, we want to express xQAM(t) in the form

xQAM (t) =√

2E(t) cos(2πfct+ φ(t)),

where E(t) ≥ 0 and φ(t) ∈ (−180◦, 180◦].Consider m1(t) and m2(t) plotted in the figure below. Draw the corre-

sponding E(t) and φ(t).

1

1

-1

18090

-90-180

2

1

t

t

t

1

-1

t

4.89. m1

√2 cos (2πfct) +m2

√2 sin (2πfct)

82

4.90. Complex form:

xQAM (t) =√

2Re{

(m(t)) ej2πfct}

where20 m(t) = m1(t)− jm2(t).

• We refer to m(t) as the complex envelope (or complex basebandsignal) and the signals m1(t) and m2(t) are known as the in-phaseand quadrature(-phase) components of xQAM (t).

• The term “quadrature component” refers to the fact that it is in phasequadrature (π/2 out of phase) with respect to the in-phase component.

• Key equation:

LPF

(

Re{m (t)×

√2ej2πfct

})︸︷︷︸

xQAM(t)

×(√

2e−j2πfct) = m (t) .

4.91. Three equivalent ways of saying exactly the same thing:

(a) the complex-valued envelope m(t) complex-modulates the complex car-rier ej2πfct,

• So, now you can understand what we mean when we say that acomplex-valued signal is transmitted.

(b) the real-valued amplitude E(t) and phase φ(t) real-modulate the am-plitude and phase of the real carrier cos(2πfct),

(c) the in-phase signal m1(t) and quadrature signal m2(t) real-modulatethe real in-phase carrier cos(2πfct) and the real quadrature carriersin(2πfct).

20If we use − sin(2πfct) instead of sin(2πfct) for m2(t) to modulate,

xQAM (t) = m1 (t)√

2 cos (2πfct)−m2 (t)√

2 sin (2πfct)

=√

2 Re{m (t) ej2πfct

}where

m(t) = m1(t) + jm2(t).

83




ECS332 2017/1 Part II.4 Dr.Prapun

5 Angle Modulation: FM and PM

5.1. We mentioned in 4.2 that a sinusoidal carrier signal

A cos(2πfct+ φ)

has three basic parameters: amplitude, frequency, and phase. Varying theseparameters in proportion to the baseband signal results in amplitude mod-ulation (AM), frequency modulation (FM), and phase modulation (PM),respectively.

5.2. As in 4.61, we will again assume that the baseband signal m(t) is

(a) band-limited to B; that is, |M(f)| = 0 for |f | > B

and

(b) bounded between −mp and mp; that is, |m(t)| ≤ mp.

Definition 5.3. Phase modulation (PM ):

xPM (t) = A cos (2πfct+ φ+ kpm (t))

• max phase deviation:

87

Definition 5.4. The main characteristic22 of frequency modulation (FM)is that the carrier frequency f(t) would be varied with time so that

f(t) = fc + km(t), (65)

where k is an arbitrary constant.

• The arbitrary constant k is sometimes denoted by kf to distinguish itfrom a similar constant in PM.

• f(t) is varied from fc − kmp to fc + kmp.

• fc is assumed to be large enough such that f(t) ≥ 0.

Example 5.5. Figure 31 illustrates the outputs of PM and FM modulatorswhen the message is a unit-step function.158 Chapter 4 ∙ Angle Modulation and Multiplexing

m(t)

1

t0t

t

t

t

(a)

t0(b)

t0(c)

t0(d)

Frequency = fc + fdFrequency = fc

Figure 4.1Comparison of PM and FM modulatoroutputs for a unit-step input.(a) Message signal. (b) Unmodulatedcarrier. (c) Phase modulator output(!" =

12#). (d) Frequency modulator

output.

where Re(⋅) implies that the real part of the argument is to be taken. Expanding $%&(') in apower series yields

()(') = Re{*)

[1 + %&(') − &2(')

2! −⋯]$%2#+)'

}(4.11)

If the peak phase deviation is small, so that the maximum value of |&(')| is much less thanunity, the modulated carrier can be approximated as

()(') ≅ Re[*)$%2#+)' + *)&(')%$%2#+)']

Taking the real part yields

()(') ≅ *) cos(2#+)') − *)&(') sin(2#+)') (4.12)

The form of (4.12) is reminiscent of AM. The modulator output contains a carrier com-ponent and a term in which a function of ,(') multiplies a 90◦ phase-shifted carrier. Thefirst term yields a carrier component. The second term generates a pair of sidebands. Thus,if &(') has a bandwidth - , the bandwidth of a narrowband angle modulator output is 2- .The important difference between AM and angle modulation is that the sidebands are pro-duced by multiplication of the message-bearing signal, & ('), with a carrier that is in phase

Figure 31: Comparison of PM and FMmodulator outputs for a unit-step input.(a) Message signal. (b) Unmodulatedcarrier. (c) Phase modulator output (d)Frequency modulator output. [15, Fig4.1 p 158]

22Treat this as a practical definition. The more rigorous definition will be provided in 5.15.

88

• For the PM modulator output,

◦ the (instantaneous) frequency is fc for both t < t0 and t > t0

◦ the phase of the unmodulated carrier is advanced by kp = π2 radians

for t > t0 giving rise to a signal that is discontinuous at t = t0.

• For the FM modulator output,

◦ the frequency is fx for t < t0, and the frequency is fc+fd for t > t0

◦ the phase is, however, continuous at t = t0.

Example 5.6. With a sinusoidal message signal in Figure 32a, the frequencydeviation of the FM modulator output in Figure 32d is proportional tom(t). Thus, the (instantaneous) frequency of the FM modulator output ismaximum when m(t) is maximum and minimum when m(t) is minimum.

4.1 Phase and Frequency Modulation Defined 159

(a)

(b)

(c)

(d)

Figure 4.2Angle modulation with sinusoidal messsage signal. (a) Message signal. (b) Unmodulated carrier. (c)Output of phase modulator with !("). (d) Output of frequency modulator with !(").

quadrature with the carrier component, whereas for AM they are not. This will be illustrated inExample 4.1.

The generation of narrowband angle modulation is easily accomplished using the methodshown in Figure 4.3. The switch allows for the generation of either narrowband FM or narrow-

m(t)

(.)dt2 fd

kp

Ac

π

ω ω

FM

PM

(t) ×

−sin ct cos ct

Σxc(t)

Carrieroscillator

90° phaseshifter

Figure 4.3Generation of narrowband angle modulation.

Figure 32: Different modulations of sinu-soidal message signal. (a) Message signal. (b)Unmodulated carrier. (c) Output of phasemodulator (d) Output of frequency modula-tor [15, Fig 4.2 p 159 ]

The phase deviation of the PM output is proportional to m(t). However,because the phase is varied continuously, it is not straightforward (yet) tosee how Figure 32c is related to m(t). In Figure 36, we will come back tothis example and re-analyze the PM output.

89

AM

FM

PM

Modulatingsignal

Figure 5.1–2 Illustrative AM, FM, and PM waveforms.

212 CHAPTER 5 • Angle CW Modulation

carrier amplitude, we modulate the frequency by swinging it over a range of, say,�50 Hz, then the transmission bandwidth will be 100 Hz regardless of the messagebandwidth. As we’ll soon see, this argument has a serious flaw, for it ignores the dis-tinction between instantaneous and spectral frequency. Carson (1922) recognizedthe fallacy of the bandwidth-reduction notion and cleared the air on that score.Unfortunately, he and many others also felt that exponential modulation had noadvantages over linear modulation with respect to noise. It took some time to over-come this belief but, thanks to Armstrong (1936), the merits of exponential modula-tion were finally appreciated. Before we can understand them quantitatively, wemust address the problem of spectral analysis.

Suppose FM had been defined in direct analogy to AM by writing xc(t) � Ac cos vc(t) twith vc(t) � vc[1 � mx(t)]. Demonstrate the physical impossibility of this definition byfinding f(t) when x(t) � cos vmt.

Narrowband PM and FMOur spectral analysis of exponential modulation starts with the quadrature-carrierversion of Eq. (1), namely

(9)

where

(10)xci1t 2 � Ac cos f1t 2 � Ac c1 �1

2! f21t 2 � p d

xc1t 2 � xci1t 2 cos vct � xcq1t 2 sin vct

EXERCISE 5.1–1

car80407_ch05_207-256.qxd 12/8/08 10:49 PM Page 212

Confirming Pages

Figure 33: Illustrative AM, FM, and PM waveforms. [3, Fig 5.1-2 p 212]

Example 5.7. Figure 33 illustrates the outputs of AM, FM, and PM mod-ulators when the message is a triangular (ramp) pulse.

AM

PM

Sudden drop in the value of

Sudden change in the phase

In this region, is increasing

Higher but constant frequency.

cos 2

cos 2

> 0

cos 2 2

,

Figure 34: ExplainingPM waveform in Figure33.

To understand more about FM, we will first need to know what it actuallymeans to vary the frequency of a sinusoid.

90

5.1 Instantaneous Frequency

Definition 5.8. The generalized sinusoidal signal is a signal of the form

x(t) = A cos (θ(t)) (66)

where θ(t) is called the generalized angle.

• The generalized angle for conventional sinusoid is θ(t) = 2πfct+ φ.

• In [3, p 208], θ(t) of the form 2πfct + φ(t) is called the total instan-taneous angle.

Definition 5.9. If θ(t) in (66) contains the message information m(t), wehave a process that may be termed angle modulation.

• The amplitude of an angle-modulated wave is constant.

• Another name for this process is exponential modulation.

◦ The motivation for this name is clear when we write x(t) asARe{ejθ(t)

}.

◦ It also emphasizes the nonlinear relationship between x(t) andm(t).

• Since exponential modulation is a nonlinear process, the modulatedwave x(t) does not resemble the message waveform m(t).

5.10. Suppose we want the frequency fc of a carrier A cos(2πfct) to varywith time as in (65). It is tempting to consider the signal

A cos(2πg(t)t), (67)

where g(t) is the desired frequency at time t.

Example 5.11. Consider the generalized sinusoid signal of the form 67above with g(t) = t2. We want to find its frequency at t = 2.

(a) Suppose we guess that its frequency at time t should be g(t). Then,at time t = 2, its frequency should be t2 = 4. However, when com-pared with cos (2π(4)t) in Figure 35a, around t = 2, the “frequency”of cos(2π

(t2)t) is quite different from the 4-Hz cosine approximation.

Therefore, 4 Hz is too low to be the frequency of cos(2π(t2)t) around

t = 2.

91

1

(a) (b)

Figure 35: Approximating the frequency of cos(2π (t2) t) by (a) cos (2π(4)t) and (b)cos (2π(12)t).

(b) Alternatively, around t = 2, Figure 35b shows that cos (2π(12)t) seemsto provide a good approximation. So, 12 Hz would be a better answer.

Definition 5.12. For generalized sinusoid A cos(θ(t)), the instantaneousfrequency 23 at time t is given by

f(t) =1

2π

d

dtθ(t). (68)

Example 5.13. For the signal cos(2π(t2)t) in Example 5.11,

θ (t) = 2π(t2)t

and the instantaneous frequency is

f (t) =1

2π

d

dtθ (t) =

1

2π

d

dt

(2π(t2)t)

= 3t2.

In particular, f (2) = 3× 22 = 12.

5.14. The instantaneous frequency formula (68) implies

θ(t) = 2π

∫ t

−∞f(τ)dτ = θ(t0) + 2π

∫ t

t0

f(τ)dτ. (69)

23Although f(t) is measured in hertz, it should not be equated with spectral frequency. Spectral frequencyf is the independent variable of the frequency domain, whereas instantaneous frequency f(t) is a time-dependent property of waveforms with exponential modulation.

92

First-order (straight-line) approximation/linearization

23

How does the formula work?

Technique from Calculus: first-order (tangent-line) approximation/linearization


24

How does the formula work?

Technique from Calculus: first-order (tangent-line) approximation/linearization

When we consider a function near a particular time, say, , the value of the function is approximately

Therefore, near ,

Now, we can directly compare the terms with .

t t t t t t t t t t

t t t t t t


25

For example, for t near t = 2,

ttt t t t t


26


ttt t t t t


27


ttt t t t t

Same idea

28

Suppose we want to find .

Let .

Note that .

Approximation:

15.9 is near 16.

MATLAB: >> sqrt(15.9)ans =

3.987480407475377

5.2 FM and PM

Definition 5.15. Frequency modulation (FM ):

xFM (t) = A cos

2πfct+ φ+ 2πkf

t∫−∞

m (τ)dτ

. (70)

Its instantaneous frequency is

f (t) = fc + kfm (t) .

5.16. Phase modulation (PM ): The phase-modulated signal is definedin Definition 5.3 to be

xPM (t) = A cos (2πfct+ φ+ kpm (t))

When m(t) is differentiable, the instantaneous frequency of xPM(t) is

(71)

Therefore, the instantaneous frequency of the PM signal varies in pro-portion to the slope of m(t).

1

cos 2

PM

FM

Figure 36: A revisit ofthe PM signal in Figure32.

93

In particular, the instantaneous frequency of the PM signal is maximumwhen the slope of m(t) is maximum and minimum when the slope of m(t)is minimum.

Example 5.17. Sketch FM and PM waves for the modulating signal m(t)shown in Figure 37a.

1

FMx t PMx t

Figure 37: FM and PM waveforms generated from the same message.

5.18. The “indirect” method of sketching xPM(t) (using m(t) to frequency-modulate a carrier) works as long as m(t) is a continuous signal. If m(t)is discontinuous, this indirect method fails at points of discontinuities. Insuch a case, a direct approach should be used to specify the sudden phasechanges. This is illustrated in Example 5.20.

5.19. Summary: To sketch xPM(t) from m(t),

(a) in the region where m(t) is differentiable, vary the the instantaneousfrequency of xPM(t) in proportion to the slope of m(t)

(b) at the location where m(t) is discontinuous (has a jump), calculate theamount of phase shift from the jump amount:

∆θ = θ(t+0 )− θ(t−0 ) = kp(m(t+0 )−m(t−0 )

)= kp∆m.

94

Example 5.20. Sketch FM and PM waves for the modulating signal m(t)shown in Figure 38a.

1

FMx t PMx t

Figure 38: FM and PM waveforms generated from the same message.

5.21. Generalized angle modulation (or exponential modulation):

x(t) = A cos (2πfct+ θ0 + (m ∗ h)(t))

where h is causal.

(a) Frequency modulation (FM ): h(t) = 2πkf1[t ≥ 0]

(b) Phase modulation (PM ): h(t) = kpδ(t).

95

5.22. Relationship between FM and PM:

• Equation (70) implies that one can produce frequency-modulated signalfrom a phase modulator.

• Equation (71) implies that one can produce phase-modulated signalfrom a frequency modulator.

• The two observations above are summarized in Figure 39.

( )FMx t ( )m t

( )t

m dτ τ−∞∫

Phase Modulator ∫

Frequency modulator

( )PMx t ( )m t

( )m t′ Frequency Modulator

ddt

Phase modulator

Figure 39: With the helpof integrating and dif-ferentiating networks, aphase modulator can pro-duce frequency modula-tion and vice versa [5, Fig5.2 p 255].

• By looking at an angle-modulated signal x(t), there is no way of tellingwhether it is FM or PM.

◦ Compare Figure 32c and 32d in Example 5.6.

◦ In fact, it is meaning less to ask an angle-modulated wave whetherit is FM or PM. It is analogous to asking a married man withchildren whether he is a father or a son. [6, p 255]

96

5.23. So far, we have spoken rather loosely of amplitude and phase modula-tion. If we modulate two real signals a(t) and φ(t) onto a cosine to producethe real signal x(t) = a(t) cos(ωct + φ(t)), then this language seems unam-biguous: we would say the respective signals amplitude- and phase-modulatethe cosine. But is it really unambiguous?

The following example suggests that the question deserves thought.

Example 5.24. [9, p 15] Let’s look at a “purely amplitude-modulated”signal

x1(t) = a(t) cos(ωct).

Assuming that a(t) is bounded such that 0 ≤ a(t) ≤ A, there is a well-defined function

θ(t) = cos−1

(1

Ax1(t)

)− ωct.

Observe that the signal

x2(t) = A cos (ωct+ θ(t))

is exactly the same as x1(t) but x2(t) looks like a “purely phase-modulated”signal.

5.25. Example 5.24 shows that, for a given real signal x(t), the factorizationx(t) = a(t) cos(ωct+φ(t)) is not unique. In fact, there is an infinite numberof ways for x(t) to be factored into “amplitude” and “phase”.

97

5.3 Bandwidth of FM Signals

5.26. FM: The “Holy Grail” Technique for BW Saving?In the 1920s, the idea of frequency modulation (FM) was naively proposed

very early as a method to conserve the radio spectrum. The argument waspresented as follows:

• If m(t) is bounded between −mp and mp, then the maximum and mini-mum values of the (instantaneous) carrier frequency would be fc+kfmp

and fc − kfmp, respectively. (Think of this as a delta function shiftingto various location between fc + kfmp and fc − kfmp in the frequencydomain.)

• Hence, the spectral components would remain within this band with abandwidth 2kfmp centered at fc.

• Conclusion: By using an arbitrarily small k, we could make the infor-mation bandwidth arbitrarily small (much smaller than the bandwidthof m(t).

In 1922, Carson argued that this is an ill-considered plan. We will illustratehis reasoning later. In fact, experimental results shows that

As a result of his observation, FM temporarily fell out of favor.

5.27. Armstrong (1936) reawakened interest in FM when he realized ithad a much different property that was desirable. When the kf is large, theinverse mapping from the modulated waveform xFM(t) back to the signalm(t) is much less sensitive to additive noise in the received signal than isthe case for amplitude modulation. FM then came to be preferred to AMbecause of its higher fidelity. [1, p 5-6]

98

Finding the “bandwidth” of FM Signals turns out to be a difficult task.Here we present a few approximation techniques.

5.28. First, from 5.21, we see that both FM and PM can be viewed as

x(t) = A cos (2πfct+ θ0 + φ(t)) (72)

where φ(t) = (m ∗ h)(t) if h(t) is selected properly.The Fourier transform of φ(t) is Φ(f) = M(f)H(f). So, if M(f) is

band-limited to B, we know that Φ(f) is also band-limited to B as well.Now, let us rewrite (72) as

x(t) = ARe{ej(2πfct+θ0+φ(t))

}= ARe

{ej(2πfct+θ0)ejφ(t)

}(73)

Recall that Taylor series expansion of ez around z = 0 is

ez =∞∑k=0

zk

k!= 1 + z +

z2

2!+z3

3!+ · · · .

Plugging in z = jφ(t) gives

ejφ(t) = 1 + jφ(t) +(jφ(t))2

2!+

(jφ(t))3

3!+ · · · = 1 + jφ(t)− φ2(t)

2!+ (−j) φ

3(t)

3!+ · · · (74)

Applying the Euler’s formula

ej(2πfct+θ0) = cos (2πfct+ θ0) + j sin (2πfct+ θ0)

and (74) to (73) gives

x (t) = A

(cos (2πfct+ θ0)− φ(t) sin (2πfct+ θ0)−

φ2(t)

2!cos (2πfct+ θ0) +

φ3(t)

3!sin (2πfct+ θ0) + · · ·

).

Recall that if φ(t) is band-limited to B, then φn(t) is band-limited to nB. With such series, there

is no bound for the value of n and therefore, we conclude that the absolute bandwidth would be

infinite.

5.29. Narrowband Angle Modulation: When φ(t) is small, we mayapproximate ez by z + 1. Therefore,

ejφ(t) ≈ 1 + jφ(t). (75)

Applying the Euler’s formula

ej(2πfct+θ0) = cos (2πfct+ θ0) + j sin (2πfct+ θ0)

99

and (75) to (73) gives

x(t) = ARe{ej(2πfct+θ0)ejφ(t)

}≈ ARe {(cos (2πfct+ θ0) + j sin (2πfct+ θ0)) (1 + jφ(t))}= A cos (2πfct+ θ0)− Aφ(t) sin (2πfct+ θ0)

• The “approximated” expression of x(t) is similar to AM.

◦ The first term yields a carrier component. The second term gen-erates a pair of sidebands. Thus, if φ(t) has a bandwidth B, thebandwidth of x(t) is 2B.

• The important difference between AM and angle modulation is thatthe sidebands are produced by multiplication of the message-bearingsignal, φ(t), with a carrier that is in phase quadrature with the carriercomponent, whereas for AM they are not.

• The FM signal whose

∣∣∣∣2πkf t∫−∞

m (τ)dτ

∣∣∣∣ � 1 is called narrowband

FM (NBFM). The PM signal whose |kpm(t)| � 1 is called narrow-band PM (NBPM). Note that these conditions are satisfied whenkf � 1 or kp � 1, respectively. [6, p 260]

• For larger values of |φ(t)| the terms φ2(t), φ3(t), . . . in (74) cannot be ignored and will

increase the bandwidth of x(t).

• Recall, from (32) that

g(t) cos(2πfct+ φ)F−−−⇀↽−−−F−1

1

2

(ejφG(f − fc) + e−jφG(f + fc)

).

Therefore, when

x (t) ≈ A cos (2πfct+ θ0)−Aφ (t) cos (2πfct+ θ0 − 90◦) ,

100

we have

X (f) ≈ A

2

(ejθ0δ(f − fc) + e−jθ0δ(f + fc)− ej(θ0−90

◦)Φ(f − fc)− e−j(θ0−90◦)Φ(f + fc)

)=A

2

(ejθ0δ(f − fc) + e−jθ0δ(f + fc) + jejθ0Φ(f − fc)− je−jθ0Φ(f + fc)

).

5.30. Wideband FM (WBFM): For potentially wideband m(t), here,we present a technique to roughly estimate the bandwidth of xFM(t).

To do this, we consider m(t) that is a piecewise constant function (alsoknown as step function or staircase function); this implies that the instan-taneous frequency f(t) = fc+kfm(t) of xFM(t) is also piecewise constant asshown in Figure 40.

1

t

t

Figure 40: FM fordiscrete-valued (digital)message

For example, we can consider the transmitted signal xFM(t) constructedfrom five different tones. Its instantaneous frequency is increased from f1

to f5.

Five Frequencies

1

0 0.05 0.1 0.15 0.2 0.25-1

-0.5

0

0.5

1

Seconds

-6000 -4000 -2000 0 2000 4000 60000

0.01

0.02

0.03

Frequency [Hz]

Magnitude

cos 2𝜋𝑓1𝑡 cos 2𝜋𝑓2𝑡 cos 2𝜋𝑓5𝑡cos 2𝜋𝑓4𝑡cos 2𝜋𝑓3𝑡

Rate = Rs frequency-changes per second

Each tone lasts

1/Rs sec.

Figure 41: xFM(t) for discrete-valued (digital) message in Figure 40.

101

Assume that each tone lasts Ts = 1Rs

[s] where Rs is called the “(symbol)rate” of the data transmission. The value of Rs indicates how fast the valuesof m(t) is changed. Increasing the value of Rs reduces the time to completethe transmission.

Recall that the Fourier transform of a cosine contains simply (two shiftedand scaled) delta functions at the (plus and minus) frequency of the cosine.However, recall also that when we consider the cosine pulse, which is time-limited, its Fourier transform contains (two) sinc functions. In particular,the cosine pulse

p (t) =

{cos (2πf0t) , t1 ≤ t < t2,0, otherwise,

can be viewed as the pure cosine function cos (2πf0t) multiplied by a rect-angular pulse r (t) = 1 [t1 ≤ t < t2]. By (31), we know that multiplicationby cos (2πf0t) will shift the spectrum R(f) of the rectangular pulse to ±fcand scaled its values by a factor of 1

2 : P (f) = 12R (f − f0) + 1

2R (f + f0)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.5

0

0.5

1

t [s]

x(t)

-200 -150 -100 -50 0 50 100 150 2000

0.01

0.02

0.03

0.04

0.05

f [Hz]

|X(f)

|

Cos Pulse

1

cos 2 100 , 0.5 0.6,0, otherwise.

Figure 42: Cosine pulseand its spectrum whichcontains two sinc func-tions at ± freqeuncy ofthe cosine (which is 100Hz in the figure). Whenthe pulse only lasts fora short time period, thesinc pulses in the fre-quency domain are wide.

where the Fourier transform24 R(f) of the rectangular pulse is given by

R (f) = (t2 − t1) e−jπf(t1+t2) sinc (πf (t2 − t1)) .24To get this, first consider the rectangular pulse of width t2 − t1 centered at t = 0. From (15), the

corresponding Fourier transform is 2(t2−t1

2

)sinc

(2π(t2−t1

2

)f). Finally, by time-shifting the rectangular

pulse in the time domain by t2+t12 , we simply multiply the Fourier transform by e−2πf(

t2−t12 ) in the

frequency domain.

102

See Figure 42 for an example.When m(t) is piecewise constant, xFM(t) is a sum of cosine pulses. There-

fore, its spectrum X(f) will be the sum of the sinc functions centered at thefrequencies of the pulses as shown in Figure 43.

1

cos 2 cos 2 cos 2cos 2cos 2300 Hz100 Hz 200 Hz 500 Hz400 Hz

0 0.05 0.1 0.15 0.2 0.25-1

-0.5

0

0.5

1

Seconds

-1000 -800 -600 -400 -200 0 200 400 600 800 10000

0.01

0.02

0.03

Frequency [Hz]

Mag

nitude

Figure 43: A digital version of FM: xFM(t) and the corresponding XFM(f).

• X(f) extends to ±∞. It is not band-limited.

• One may approximate its bandwidth by assuming that “most” of theenergy in the sinc function is contained in its main lobe which is at± 1Ts

= ±Rs from its peak. Therefore, the bandwidth of xFM(t) becomes

BWFM ≈ Rs + (fmax − fmin) +Rs = (fmax − fmin) + 2Rs

103




ECS332 2017/1 Part III.1 Dr.Prapun

6 Sampling and Reconstruction

6.1 Sampling

Definition 6.1. Sampling is the process of taking a (sufficient) number ofdiscrete values of points on a waveform that will define the shape of waveform.

1

∞ ∞Sampling … , 2 , 1 , 0 , 1 , 2 , …

Continuous-time signal(analog)

Discrete-time signal(sequence of numbers)

Ts 2Ts 3Ts 4Ts-Ts-2Ts-3Ts-4Tst

m[0]m[1]

m[3]m[4]

m[-1]m[-2]m[-4]

m[2]m[-3]Ts 2Ts 3Ts 4Ts-Ts-2Ts-3Ts-4Ts

t

m(t)

Figure 44: The Sampling Process

• In this class, the signal is sampled at a uniform rate, once every Tsseconds.

m[n] = m(nTs) = m(t)|t=nTs.

• We refer to Ts as the sampling period, and to its reciprocal fs = 1/Tsas the sampling rate which is measured in samples/sec [Sa/s].

• At this stage, we assume “infinite” precision (no quantization) for eachvalue of m[n].

104

• The reverse process is called “reconstruction”.

6.2. Sampling = loss of information? If not, how can we recover the originalwaveform back.

• The more samples you take, the more accurately you can define a wave-form.

• Obviously, if the sampling rate is too low, you may experience distortion(aliasing).

• The sampling theorem, to be discussed in the section, says that whenthe waveform is band-limited, if the sampling rate is fast enough, we canreconstruct the waveform back and hence there is no loss of information.

◦ This allows us to replace a continuous time signal by a discretesequence of numbers.

◦ Processing a continuous time signal is therefore equivalent to pro-cessing a discrete sequence of numbers.

◦ In the field of communication, the transmission of a continuoustime message reduces to the transmission of a sequence of numbers.

Example 6.3. Mathematical functions are frequently displayed as contin-uous curves, even though a finite number of discrete points was used toconstruct the graphs. If these points, or samples, have sufficiently closespacing, a smooth curve drawn through them allows us to interpolate in-termediate values to any reasonable degree of accuracy. It can therefore besaid that the continuous curve is adequately described by the sample pointsalone.

Example 6.4. Plot y = x2.

105

Example 6.5. In Figure 45, we plot the function g(t) = sin(100πt) from0 to 1 by connecting the values of the function at fifty uniformly-spacedpoints.

Example: sin(100t) (1/4)

1

This is the plot of sin(100t). What’s wrong with it?

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

t

[AliasingSin_2.m]

Figure 45: Plot of the function g(t) =sin(100πt) from 0 to 1 by connecting the val-ues at fifty uniformly-spaced points.

Although the plot shows the correct shape of the sine wave, the perceivedfrequency is just 1 Hz.

Theorem 6.6. Sampling Theorem: In order to (correctly and com-pletely) represent an analog signal, the sampling frequency, fs, must beat least twice the highest frequency component of the analog signal.

6.7. If the conditions of the sampling theorem are not satisfied, we expe-rience an effect called aliasing in which different signals become indistin-guishable (or aliases of one another) when sampled.

• The term “aliasing” also refers to the distortion or artifact that resultswhen the signal reconstructed from samples is different from the originalcontinuous signal.

Example 6.8. In Example 6.5, the frequency of the sine wave is 50 Hz.Therefore, we need the sampling frequency to be at least 100.

6.9. For now, instead of trying to infer the “perceived” frequency by an-alyzing the plot of the function in the time domain, it is easier to use ourplotspect function to visualize the location of the peaks (of the delta func-tions) in the frequency domain.

106

Example 6.10. Suppose the sampling frequency is 200 samples/sec. Theanalog signal should not have the frequency higher than 100 Hz. This isillustrated in Figure 46 in which cosine functions of different frequencies aresampled with fs = 200.

cos 2 50

-100 -80 -60 -40 -20 0 20 40 60 80 1000

2

4

6

Frequency [Hz]

Mag

nitu

de

cos 2 70

-100 -80 -60 -40 -20 0 20 40 60 80 1000

2

4

6

Frequency [Hz]

Mag

nitu

de

-100 -80 -60 -40 -20 0 20 40 60 80 1000

2

4

6

8

Frequency [Hz]

Mag

nitu

de

cos 2 100

plotspect sees 50 Hz signal (correct)

plotspect sees 70 Hz signal (correct)

plotspect sees 100? Hz signal (correct?)

cos 2 110

cos 2 130

cos 2 190

-100 -80 -60 -40 -20 0 20 40 60 80 1000

1

2

3

4

Mag

nitu

de

-100 -80 -60 -40 -20 0 20 40 60 80 1000

2

4

6

Mag

nitu

de

-100 -80 -60 -40 -20 0 20 40 60 80 1000

2

4

6

Mag

nitu

de

plotspect sees 90 Hz signal (wrong!)



Figure 46: Our plotspect function can be used to visualize the “perceived” frequency ofa sampled signal.

6.11. Steps to find the “perceived” frequency of the sampled signal whenthe sampling rate is fs:

107

(a) For cos (2π(f0)t), we may use the “folding technique”:

(i) Consider the window of frequency from 0 to fs2 .

(ii) Start from 0, increase the frequency to f0.Fold back at 0 and fs

2 if necessary.

Remark: By the symmetry in the spectrum of cosine, we can alwaysgive a nonnegative answer for the perceived frequency.

(b) For ej2π(f0)t, we use the “tunneling technique”:

(i) Consider the window of frequency from −fs2 to fs

2 .

(ii) Start from 0.

i. If f0 > 0, increase the frequency to f0 (going to the right).Restart at −fs

2 when fs2 is reached.

ii. If f0 < 0, decrease the frequency to f0 (going to the left).Restart at +fs

2 when −fs2 is reached.

(c) We will study a more general analysis in Section 6.3.

Example 6.12. Find the perceived frequency of cos (300πt) when the sam-pling rate is 200 [Sa/s].

Example 6.13. Find the perceived frequency of e300πt when the samplingrate is 200 [Sa/s].

6.14. A cosine function at frequency f0 can also be thought of as a combi-nation of two complex exponential at frequency f0 and −f0. Therefore, wecan also use the tunneling technique to analyze the cosine function as wellby looking at its individual complex-exponential components.

108

Example 6.15. Let’s consider a signal that is closer to Example 6.5. Sup-pose we consider cos (100πt). The sampling rate used is 49 [Sa/s]. Find theperceived frequency.

Example 6.16. Now, let’s consider the signal sin (100πt) discussed in Ex-ample 6.5. Again, the sampling rate used is 49 [Sa/s]. Find the perceivedsignal.

Example 6.17. Application of the sampling theorem: In telephony, theusable voice frequency band ranges from approximately 300 Hz to 3400 Hz.The bandwidth allocated for a single voice-frequency transmission channelis usually 4 kHz, including guard bands, allowing a sampling rate of 8 kHzto be used as the basis of the pulse code modulation system used for thedigital PSTN.

Definition 6.18.

(a) Given a sampling frequency, fs, the Nyquist frequency is fs/2.

(b) Given the highest (positive-)frequency component fmax of an analogsignal,

(i) the Nyquist sampling rate is 2fmax and

(ii) the Nyquist sampling interval is 1/(2fmax).

109

6.19. For the remaining analysis in this section, we will use g(t) to denotethe signal under consideration. You may replace g(t) below by m(t) if youwant to think of it as an analog message to be transmitted by a communi-cation system. We use g(t) here because the results provided here work inbroader setting as well.

6.2 Ideal Sampling

Definition 6.20. In ideal sampling, the (ideal instantaneous) sampledsignal is represented by a train of impulses whose area equal the instanta-neous sampled values of the signal

gδ (t) =∞∑

n=−∞g [n]δ (t− nTs) .

6.21. The Fourier transform Gδ(f) of gδ (t) can be found by first rewriting gδ (t) as

gδ (t) =

∞∑n=−∞

g (nTs)δ (t− nTs) =

∞∑n=−∞

g (t)δ (t− nTs)

= g (t)∞∑

n=−∞δ (t− nTs).

Multiplication in the time domain corresponds to convolution in the frequency domain. Therefore,

Gδ (f) = F {gδ (t)} = G (f) ∗ F

{ ∞∑n=−∞

δ (t− nTs)

}.

For the last term, the Fourier transform can be found by applying what we found in Example4.4525:

∞∑n=−∞

δ (t− nTs)F−−−⇀↽−−−F−1

fs

∞∑k=−∞

δ (f − kfs).

This gives

Gδ (f) = G (f) ∗ fs∞∑

k=−∞δ (f − kfs) = fs

∞∑k=−∞

G (f) ∗ δ (f − kfs).

Hence, we conclude that

gδ (t) =∞∑

n=−∞g [n]δ (t− nTs)

F−−⇀↽−−F−1

Gδ (f) = fs

∞∑k=−∞

G (f − kfs). (76)

In words, Gδ (f) is simply a sum of the scaled and shifted replicas of G(f).25We also considered an easy-to-remember pair and discuss how to extend it to the general case in 4.46.

110

6.22. As usual, we will assume that the signal g(t) is band-limited to BHz ((G(f) = 0 for |f | > B)).

(a) When B < fs/2 as shown in Figure 47, the replicas do not overlap andhence we do not need to spend extra effort to find their sum.Ideal Sampling: MATLAB Exploration

1

-3 -2 -1 0 1 2 30

0.5

1

f [fs]

[A]

G(f)

-3 -2 -1 0 1 2 30

0.5

1

f [fs]

[A f s]

G(f)

The Fourier transform of the original signal

The Fourier transform of the (ideal) sampled signal

Figure 47: The Fouriertransform Gδ(f) of gδ (t)when B < fs/2

(b) When B > fs/2 as shown in Figure 48, overlapping happens in thefrequency domain. This spectral overlapping of the signal is (also)commonly referred to as “aliasing” mentioned in 6.7. To find Gδ(f),dont forget to add the replicas

-3 -2 -1 0 1 2 30

0.5

1

f [fs]

[A]

G(f)

-3 -2 -1 0 1 2 30

0.5

1

f [fs]

[A f s]

G(f)

Ideal Sampling: MATLAB Exploration

1

B-B

12

12

The Fourier transform of the original signal

The Fourier transform of the (ideal) sampled signal

Figure 48: The Fouriertransform Gδ(f) of gδ (t)when B > fs/2

111

6.23. Remarks:

(a) Gδ (f) is “periodic” (in the frequency domain) with “period” fs.

• So, it is sufficient to look at Gδ (f) between ±fs2

(b) The MATLAB script plotspect that we have been using to visualizemagnitude spectrum also relies on sampled signal. Its frequency domainplot is between ±fs

2 .

(c) Although this sampling technique is “ideal” because it involves the useof the δ-function. We can extract many useful conclusions.

(d) One can also study the discrete-time Fourier transform (DTFT) to lookat the frequency representation of the sampled signal.

6.3 Reconstruction

Definition 6.24. Reconstruction (interpolation) is the process of re-constructing a continuous time signal g(t) from its samples.

6.25. From (76), we see that when the sampling frequency fs is largeenough, the replicas of G(f) will not overlap in the frequency domain. Insuch case, the original G(f) is still intact and we can use a low-pass filterwith gain Ts to recover g(t) back from gδ (t).

6.26. To prevent aliasing (the corruption of the original signal because itsreplicas overlaps in the frequency domain), we need

Theorem 6.27. A baseband signal g whose spectrum is band-limited toB Hz (G(f) = 0 for |f | > B) can be reconstructed (interpolated) exactly(without any error) from its sample taken uniformly at a rate (samplingfrequency/rate) fs > 2B Hz (samples per second).[6, p 302]

112

6.28. Ideal Reconstruction: Continue from 6.25. Assuming that fs >2B, the low-pass filter that we should use to extract g(t) from Gδ(t) shouldbe

HLP (f) =

|f | ≤ B,B < |f | < fs −B,|f | ≥ fs −B,

In particular, for “brick-wall” LPF, the cutoff frequency fcutoff should bebetween B and fs −B.

6.29. Reconstruction Equation: Suppose we use fs2 as the cutoff fre-

quency for our “brick-wall” LPF in 6.28,

1

1

The impulse response of the LPF is hLP (t) = sinc(

2π(fs2

)t)

= sinc(πfst).

The output of the LPF is

g(t) = gδ (t) ∗ hLP (t) =

( ∞∑n=−∞

g [n]δ (t− nTs)

)∗ hLP (t)

=∞∑

n=−∞g [n]hLP (t− nTs) =

∞∑n=−∞

g [n] sinc (πfs (t− nTs)) .

When fs > 2B, this output will be exactly the same as g(t):

g (t) =∞∑

n=−∞g [n] sinc (πfs (t− nTs)) (77)

113

• This formula allows perfect reconstruction the original continuous-timefunction from the samples.

• At each sampling instant t = nTs, all sinc functions are zero exceptone, and that one yields g(nTs).

• Note that at time t between the sampling instants, g(t) is interpolatedby summing the contributions from all the sinc functions.

• The LPF is often called an interpolation filter, and its impulse responseis called the interpolation function.

Example 6.30. In Figure 49, a signal gr(t) is reconstructed from the sam-pled values g[n] via the reconstruction equation (77).

1

Figure 49: Application of the reconstruction equation

114

Example 6.31. We now return to the sampling of the cosine function (si-nusoid).

1

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-1

0

1

2

t

g[n]g(t)

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-1

-0.5

0

0.5

1

t

g[n]g(t)

Figure 50: Reconstruction of thesignal g(t) = cos(2π(2)t) by itssamples g[n]. The upper plot usesTs = 0.4. The lower plot usesTs = 0.2.

Theorem 6.32. Sampling theorem for uniform periodic sampling: Ifa signal g(t) contains no frequency components for |f | ≥ B, it is completelydescribed by instantaneous sample values uniformly spaced in time withsampling period Ts ≤ 1

2B . In which case, g(t) can be exactly reconstructedfrom its samples (. . . , g[−2], g[−1], g[0], g[1], g[2], . . .) by the reconstructionequation (77).

6.33. Remarks:

• Need a lot of g[n] for the reconstruction.

• Practical signals are time-limited.

◦ Filter the message as much as possible before sampling.

6.34. The possibility of fs = 2B:

• If the spectrum G(f) has no impulse (or its derivatives) at the highestfrequency B, then the overlap is still zero as long as the sampling rateis greater than or equal to the Nyquist rate, that is, fs ≥ 2B.

• If G(f) contains an impulse at the highest frequency ±B, then fs = 2Bwould cause overlap. In such case, the sampling rate fs must be greaterthan 2B Hz.

115

Example 6.35. Consider a sinusoid g(t) = sin (2π(B)t). This signal isbandlimited to B Hz, but all its samples are zero when uniformly taken ata rate fs = 2B, and g(t) cannot be recovered from its (Nyquist) samples.Thus, for sinusoids, the condition of fs > 2B must be satisfied.

Let’s check with our formula (76) for Gδ(f). First, recall that

sinx =ejx − e−jx

2j=

1

2jejx − 1

2je−jx.

Therefore,

g (t) = sin (2π (B) t) =1

2jej2π(B)t − 1

2je−j2π(B)t = −1

2jej2π(B)t +

1

2jej2π(−B)t

and

Note that G(f) is pure imaginary. So, it is more suitable to look at theplot of its imaginary part. (We do not look at its magnitude plot becausethe information about the sign is lost. We also do not consider the real partbecause we know that it is 0.)

116

6.36. The big picture:

• g(t) is a continuous-time signal.

• gδ(t) is also a continuous signal.

◦ However, gδ(t) is 0 almost all the time except at nTs where we haveweighted δ−function.

◦ We define gδ(t) so that we can have an easy way to analyze g[n]below.(Another approach is to use DTFT.)

◦ It provides an intermediate step that leads to the sampling theo-rem, the Nyquist sampling rate requirement, and the reconstruc-tion equation.It also provides a way to “visualize” aliasing.

• g[n] is a discrete-time signal.

◦ This is simply a sequence of numbers.

◦ The reconstruction equation says that we can recover g(t) backfrom g[n] under appropriate condition.

◦ So, there is no need to transmit the whole signal g(t). We onlyneed to transmit g[n].

6.37. A maximum of 2B independent pieces (samples or symbols) of infor-mation per second can be transmitted, errorfree, over a noiseless channel ofbandwidth B Hz [5, p 260].

• Start with 2B pieces of information per second. Denote the sequenceof such information by mn.

• Construct a signalm(t) whose (Nyquist) sample valuesm[n] = m(n 1

2B

)agrees with mn by the reconstruction equation (77).

◦ The reconstruction equation uses linear combination of the (scaledand time-shifted) sinc function that are all band-limited to B. So,m(t) will also be band-limited to B.

117

6.38. A bandpass signal whose spectrum exists over a frequency bandfc − B

2 < |f | < fc + B2 has a bandwidth B Hz. Such a signal is also

uniquely determined by samples taken at above the Nyquist frequency 2B.The sampling theorem is generally more complex in such case. It uses twointerlaced sampling trains, each at a rate of fs > B samples per second(known as second-order sampling). [6, p 304]

118

6.4 Triangular (Linear) Interpolation

Here, we use triangular waveform instead of the sinc function for interpola-tion.

1

Ts 2Ts 3Ts 4Ts-Ts-2Ts-3Ts-4Ts

t

tTs-Ts

1

Figure 51: Triangular (Linear) Interpolation

6.39. When linear interpolation is used, high frequency content of G(f) isattenuated and (small part of) the replicas at even higher freqencies (whichdo not exist before) are also introduced.

-3 -2 -1 0 1 2 30

0.5

1

t [Ts]

-2 -1.5 -1 -0.5 0 0.5 1 1.5 20

0.5

1

f [fs]

[Ts]

sinc vs. triangular interpolation

1

-3 -2 -1 0 1 2 3-0.5

0

0.5

1

t [Ts]

-2 -1.5 -1 -0.5 0 0.5 1 1.5 20

0.5

1

f [fs]

[Ts]

Figure 52: Triangular (Linear) Interpolation: Effects on G(t)

119





7 Pulse Modulation

In Section 6.1 we saw that bandlimited continuous-time signals can be rep-resented by a sequence of discrete-time samples. Moreover, in Section 6.3,we saw that the continuous-time signal can be reconstructed if the samplingrate is sufficiently high.

Because the sequence m[n] completely contains the information aboutm(t), instead of trying to send m(t), we may consider transmitting ourmessage via m[n] in the form of pulse modulation.

7.1 Analog Pulse Modulation

7.1. In this section, we start with a sequence of numbers (discrete-timemessage):

· · · ,m[−3],m[−2],m[−1],m[0],m[1],m[2],m[3], · · ·

as shown in Figure 53.

1

t

m[0]m[1]

m[2]

m[3]m[4]

m[-1]m[-2]

m[-3]

m[-4]


Figure 53: Sequence of Numbers for PAM

120

• These numbers may come from the process of sampling a continuous-time signal m(t).

• Alternatively, it may directly represent (digital) information that in-trinsically available in discrete-time. See Example 7.2.

• Because the m[n] may not come from sampling, we call each m[n] asymbol.

Example 7.2. Naturally digital information is an ordered sequence ofsymbols (or characters). Each symbol is drawn from an alphabet of M ≥ 2different symbols.

• English text: 26 (a to z) + 26 (A to Z) + 10 (0 to 9) + Punctuationand Other Signs (. , ! @ ( ))

◦ Text is commonly encoded using ASCII, and MATLAB automati-cally represents any string file as a list of ASCII numbers.

• Thai text: 44 (Consonants) + 15 (Vowel Symbols) + 4 (Tone Marks)+. . .

• A typical computer terminal has an alphabet of M ≈ 90 symbols (thenumber of character keys multiplied by two to account for the shiftkey)

Definition 7.3. In analog pulse modulation, some attribute of a pulsevaries continuously in one-to-one correspondence with a sample value.

• Example of a pulse:

• Three attributes can be readily varied: amplitude, width, and position.

• These lead to pulse-amplitude modulation (PAM), pulse-width modu-lation (PWM), and pulse-position modulation (PPM) as illustrated inFigure 55.

121

Definition 7.4. Unmodulated pulse train:∞∑

n=−∞p (t− nTs)

• The pulse is repeated every Ts seconds.

• This replaces the role of a sinusoidal carrier.

tt

A

Ts

Pulse

Ts 3Ts 4Ts-Ts-2Ts-3Ts-4Ts 2Ts

Unmodulated pulse train

Figure 54: Unmodulated pulse train

7.5. Figure 55 compares three types of analog pulse modulation.

;rJ0r

;equence of : error if the p ic of pulse )Ll[se varies 1hich some

PWM signal

PPM signal

0 T, 2T, 9T s

3.5 Analog Pulse Mo dul a ti o n 183

Figure 3.56 Illustration of PAM, PWM, and PPM.

attribu te of a pu lse can take on a certa in va lue from a set of allowable va lues. In thi s section we examine analog pulse mod ulation . In the fo ll owing secti on we exami ne a coup le of examples of dig ital pul se modulat ion.

As mentioned, analog pulse modularion results when some attr ibute of a pu lse varies conti nuou sly in one-to-one cotTespondence wit h a samp le value . T hree attributes can be readi ly varied: amplitude, width , and position . T hese lead to pu lse ampl itude mod ul at ion (PAM), pul se-width modul ati on (PWM), and pulse-positio n mod ul ation (PPM) as ill ustrated in Figure 3.56.

3.5.1 Pulse-Amplitude M odulation

A PAM waveform consists of a sequence of fl at-topped pul ses des ignat ing sample va lues. The ampli tude of each pulse corresponds to the value of the message s ig nal m( t ) at the leading edge of the pulse. The essential di fference between PAM and the sampling operati on discussed in the previous chapter is that in PAM we allow the sampling pulse to have fi nite width . The fin itewidth pulse can be generated from the impul se-train sampling functio n by passing the impulsetrain samples through a holding circuit as shown in Figure 3.57. The impulse response of the ideal hold ing circuit is given by

Figure 55: Illustration of PAM, PWM, andPPM. [14, Fig. 3.56]

Definition 7.6. In Pulse-Amplitude Modulation (PAM), the samplevalues modulate the amplitude (height) of a pulse train. We will focus onthis type of modulation in Section 7.2.

122

Definition 7.7. Pulse-Width Modulation (PWM): A PWM waveformconsists of a sequence of pulses with the width of the nth pulse is propor-tional to the value of m[n].

• Seldom used in modern communications systems.

• Used extensively for DC motor control in which motor speed is propor-tional to the width of the pulses . Since the pulses have equal amplitude,the energy in a given pulse is proportional to the pulse width.

Definition 7.8. Pulse-Position Modulation (PPM): A PPM signalconsists of a sequence of pulses in which the pulse displacement from a spec-ified time reference is proportional to the sample values of the information-bearing signal.

• Have a number of applications in the area of ultra-wideband commu-nications.

7.9. Pulse-modulation scheme are really baseband coding schemes, andthey yield baseband signal.

7.2 Pulse-Amplitude Modulation

Definition 7.10. In Pulse-Amplitude Modulation (PAM), the sam-ple values modulate the amplitude (height) of a pulse train. The pulse-modulated signal has the form

xPAM (t) =∞∑

n=−∞m [n] p (t− nTs)

Example 7.11. An example of a PAM signal is shown in Figure 56.

Example 7.12. Another example of a PAM signal is shown in Figure 57.Note that the discrete-time sequence m[n] is the same as in Example 7.11.However, the pulses used in these examples are different.

123

t

m[0]m[1]

m[2]

m[3]m[4]

m[-1]m[-2]

m[-3]

m[-4]


p t

t-.5Ts

1

.5Ts

Pulse

A sequence m[n] of symbols (numbers)

The height (amplitude) of each pulse is scaled by the corresponding m[n].

t

0 01 1

2 2

3 34 4

1 1

2 2

3 3

4 4


PAMn

x t m n p t nT

Figure 56: An example of a PAM signal

t

m[0]m[1]

m[2]

m[3]m[4]

m[-1]m[-2]

m[-3]

m[-4]


A sequence m[n] of symbols (numbers)

The height (amplitude) of each pulse is scaled by the corresponding m[n].

PAMn

x t m n p t nT

t

p t

t

1

Ts

Pulse

m[2]m[-3]

m[0]m[1]

m[3]m[4]

m[-1]m[-2]m[-4]


Figure 57: Another Example of a PAM signal

124

7.13. One advantage of using pulse modulation is that it permits the si-multaneous transmission of several signals on a time-sharing basis.

• When a pulse-modulated signal occupies only a part of the channeltime, we can transmit several pulse-modulated signals on the samechannel by interleaving them.

• One User: TDM (time division multiplexing).

◦ Transmit/multiplex multiple streams of information simultaneously.

• Multiple Users: TDMA (time division multiple access).

Example 7.14.

t

t

1

Ts

Pulse

m[2]m[-3]


m[0]m[1]

m[3]m[4]

m[-1]m[-2]m[-4]

Figure 58: Time sharing in PAM

7.15. Frequency-Domain Analysis of PAM:We start with

xPAM (t) =∞∑

n=−∞m [n] p (t− nTs)

(a) Method 1: By the time-shift property,

p (t− nTs)F−−⇀↽−−F−1

P (f) e−j2πfnTs.

Applying the linearity property of Fourier transform to xPAM (t) =∑nm [n] p (t− nTs) , we have

XPAM (f) =∑n

m [n]P (f) e−j2πfnTs = P (f)∑n

m [n] e−j2πfnTs.

125

-5 -4 -3 -2 -1 0 1 2 3 4 50

2

4

6

8

10

f [Hz]

-5 -4 -3 -2 -1 0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

f [Hz]

1

1

=1t

t

1

-1

= [-1,-1,1,-1,-1,1,1,-1,-1,-1,1,-1,-1,1,-1,1,1,-1,-1,-1,-1,1,-1,-1,-1,-1,-1,1,-1,1]

1 0, sp t t T P f

PAMx t

Figure 59: Frequency-Domain Analysis of PAM

The oscillation in the complex-exponential function

e−j2π(nTs)f = cos (2π (nTs) f)− j sin (2π (nTs) f)

adds the oscillation on top of the spectrum P (f) of the pulse. Thelarger the value of n, the faster XPAM (f) oscillates in the frequencydomain.

(b) Method 2: Alternatively, one may express

xPAM (t) =∞∑

n=−∞m [n] p (t− nTs) =

∞∑n=−∞

m [n] p (t) ∗ δ (t− nTs)

= p (t) ∗

( ∞∑n=−∞

m [n] δ (t− nTs)

).

Recall, from Definition 6.20, that∞∑

n=−∞m [n] δ (t− nTs) can be denoted

by mδ (t) if we think of the sequence m[n] as originally comes from acontinuous-time signal m(t). In which case

xPAM (t) = p (t) ∗mδ (t)

andXPAM (f) = P (f)Mδ (f) .

126

7.3 Inter-symbol Interference and Pulse Shaping

7.16. Continue from the previous section. The generated PAM signal x (t)would be transmitted via the communication channel which usually corruptsit. At the receiver, the received signal is y(t). A more advanced receiverwould try to first cancel the effect of the channel. However, for simplicity,let’s assume that our receiver simply samples y(t) every T seconds to get

y[n] = y(t)|t=nT

and we will take this to be the estimate m[n] of our m[n].

• In this section, we drop the subscript s from Ts.

• If m[n] is the sampled version of m(t), then at the receiver, after werecover m[n], we can reconstruct m(t) by using the reconstructing equa-tion (77).

Because our assumed receiver is so simple, we are going to also assume26

that y(t) = x(t).

7.17. Our goal is to design a “good” pulse p(t) that satisfies two importantproperties

(a) m[n] = m[n] for all n. Under our assumptions above, this means wewant x[n] ≡ x(nT ) = m[n] for all n.

(b) P (f) is band-limited and hence X(f) is band-limited.

We will first give examples of “poor” p(t).

26Alternatively, we may assume that there is an earlier part of the receiver that (perfectly) eliminatesthe effect of the channel for us.

127

Example 7.18. Let’s consider the rectangular pulse used in Figure 60 inwhich

p(t) = 1[|t| ≤ T/2].

p t

T 2T 3T 4T-T-2T-3T-4Tt

t

ˆn

t nTt nTm n y t

x t m n p t nT

x t

-.5T

1

.5T

m n m n

No ISI

ˆ 1 1

ˆ 0 0

ˆ 1 1

m m

m m

m m

Pulse

y t x t

Transmitted signalReceived signal

Assume

Also assume

Figure 60: PAM

(a) m[n] = m[n] for all n.

(b) The Fourier transform of the rectangular pulse is a sinc function. So,it is not band-limited.

Example 7.19. Let’s try a wider rectangular pulse:

p(t) = 1[|t| ≤ 1.5T ].

Figure 61 illustrates that we face a problem called inter-symbol in-terference (ISI) in our sequence m[n] at the receiver. The pulses are toowide; they interfere with other pulses at the sampling time instants (decisionmaking instants), making m[n] 6= m[n].

128

p t


tT-T

m n m n

Suffer ISI

1

1.5T-1.5T

2ˆ 1 1

ˆ 0 0

ˆ 1 1

0

1 1

0 2

m mm m

m m m

m

m

m mm

ˆn

t nTt nTm n y t

x t m n p t nT

x t

y t x t


Figure 61: ISI in PAM

p t


t

1

m n m n

No ISI

ˆ 1 1

ˆ 0 0

ˆ 1 1

m m

m m

m m

Pulse

44

y t x t


ˆn

t nTt nTm n y t

x t m n p t nT

x t

Figure 62: PAM using narrow pulses

129

Example 7.20. p(t) = 1[|t| ≤ T/4] is used in Figure 62.

• When the pulse p(t) is narrower than T , we know that the pulses inPAM signal will not overlap and therefore we won’t have any ISI prob-lem.

Example 7.21. Even when the pulses are wider than T , if they do notinterfere with other pulses at the sampling time instants (decision makinginstants), we can still have no ISI.

1

p t

tT-T

1

2T-2T


p t

t

1

34

34 T 2T 3T 4T-T-2T-3T-4T

t

p t

tT-T

1


Figure 63: Examples of pulses that do not cause ISI.

7.22. We can now conclude that a “good” pulse satisfying condition (a)in 7.17 must not cause inter-symbol interference (ISI): at the receiver,the nth symbol m[n] should not be affected by the preceding or succeedingtransmitted symbol m[k], k 6= n. This requirement means that a “good”pulse should have the following property:

p (t)|t=nT =

{1, n = 0,0, n 6= 0.

(78)

Combining this with condition (b) in 7.17, we then want “band-limitedpulses specially shaped to avoid ISI (by satisfying (78))” [3, p 506].

7.23. An obvious choice for such p(t) would be the sinc function that weused in the reconstruction equation (77):

130

Recall Figure 49, repeated here (with modified labels) as Figure 64.

1 2 3 4 5 6 7 80

2

4

6

8

n

m[n

]

1 2 3 4 5 6 7 8-2

0

2

4

6

8

t [Ts]

x(t)

Figure 64: Using the sinc pulse inPAM

Practically, there are problems that force us to seek better pulse shape.

(a) Infinite duration

(b) Steep slope at each 0-intercept.

(c) maxt{x (t)} could be a lot larger than max

n{m [n]}.

2 4 6 8 10 12 14 16-2

-1

0

1

2

n

m[n

]

2 4 6 8 10 12 14 16-2

-1

0

1

2

3

t [Ts]

x(t)

Figure 65: Using the sinc pulse inPAM can cause high peak.

131

7.24. Because the sinc function may not be a good choice, we now have toconsider other pulses that are band-limited and also satisfy (78). To checkthat a signal is band-limited, we need to look in the frequency domain.However, condition (78) is specified in the time domain. Therefore, we willtry to translate condition (78) into a requirement in the frequency domain.

7.25. Note that condition (78) considers p (t)|t=nT which can be thought ofas the samples p[n] of the pulse p(t) where the sampling period is Ts = T .Recall, from (76), that

gδ (t) =∞∑

n=−∞g [n]δ (t− nTs)

F−−⇀↽−−F−1

Gδ (f) = fs

∞∑k=−∞

G (f − kfs).

Therefore,

pδ (t) =∞∑

n=−∞p [n]δ (t− nT )

F−−⇀↽−−F−1

Pδ (f) =1

T

∞∑k=−∞

P

(f − k

T

). (79)

On the LHS, by condition (78), the only nonzero term in the sum is theone with n = 0. Therefore, condition (78) is equivalent to pδ (t) = δ(t).

However, recall that δ(t)F−−⇀↽−−F−1

1. Therefore, we must have Pδ (f) ≡ 1.

Hence, to check condition (78), we can equivalently check that the RHS of(79) must be ≡ 1.

Note that Pδ(f) is “periodic” (in the freq. domain) with “period” 1T .

(Recall that Gδ(f) is “periodic” (in the freq. domain) with “period” fs.)Therefore, the checking does not need to be performed across all frequencyf . We only need to focus on one period: |f | ≤ 1

2T .This observation is formally stated as the “Nyquist’s criterion” below.

7.26. Nyquist’s (first) Criterion for Zero ISI: A pulse p(t) whoseFourier transform P (f) satisfies the criterion

∞∑k=−∞

P

(f − k

T

)≡ T, |f | ≤ 1

2T(80)

has sample values satisfying condition (78):

p[n] = p (t)|t=nT =

{1, n = 0,0, n 6= 0.

132

• Using this pulse, there will be no ISI in the sample values of y(t) be-cause

y[n] = y (t)|t=nT =

∞∑k=−∞

m [n] p (t− kT )

∣∣∣∣∣t=nT

=

∞∑k=−∞

m [n] p (nT − kT )

=

∞∑k=−∞

m [n] p [n− k] = m [n]

Definition 7.27. A pulse p(t) is a Nyquist pulse if its Fourier transformP (f) satisfies (80) above.

Example 7.28. We know that the sinc pulse we used in Example 7.23 works(causing no ISI). Let’s check it with the Nyquist’s criterion:

Example 7.29.

Example 7.30.

133

Example 7.31.

Example 7.32. An important family of Nyquist pulses is called the raisedcosine family. Its Fourier transform is given by

PRC (f ;α) =

T, 0 ≤ |f | ≤ 1−α

2TT2

(1 + cos

(πTα

(|f | − 1−α

2T

))), 1−α

2T ≤ |f | ≤1+α2T

0, |f | ≥ 1+α2T

with a parameter α called the roll-off factor.

1

f

T

2

12

12

Figure 66: Raised cosinepulse (in the frequencydomain)

1

f

T

2

12

12

00.51

34

1

Figure 67: Raised co-sine pulse (in the fre-quency domain) with dif-ferent values of the roll-off factor

134

1

t

00.5

1

1

22

Figure 68: Raised cosinepulse (in the time do-main) with different val-ues of the roll-off factor

1

n

x t m n p t nT

0 0.5 1 1.5 2 2.5 3 3.5-1.5

-1

-0.5

0

0.5

1

1.5

t

0 0.5 1 1.5 2 2.5 3 3.5-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

t

0 0.5 1 1.5 2 2.5 3 3.5-1.5

-1

-0.5

0

0.5

1

1.5

t

;0RCp t p t

;1RCp t p t

;0.5RCp t p t

a)

b)

c)

Figure 69: Using the raised cosine pulses in PAM

135





8 PCM

8.1. Generally, analog signals are continuous in time and in range (ampli-tude); that is, they have values at every time instant, and their values canbe anything within the range. On the other hand, digital signals exist onlyat discrete points of time, and their amplitude can take on only finitely (orcountably) many values.

8.2. Suppose we want to convey an analog message m(t) from a source toour destination. We now have many options.

(a) Use m(t) to modulate a carrier A cos(2πfct) using AM (Defn. 4.61),FM (Defn. 5.15), or PM (Defn. 5.3) techniques studied earlier.

(b) Sample the continuous-time message m(t) to get a discrete-time mes-sage m[n]:

m (t)→ Sampler → m [n]

• May LPF m(t) before sampling to eliminate aliasing (and reduceout-of-band noise).

• Need to make sure that the sampling rate fs is fast enough.

(i) Send m[n] using analog pulse modulation techniques (PAM,PWM, PPM) illustrated in Example 7.5.

• Note that m[n] is a sequence of numbers. Even when m(t)(and hence m[n]) is bounded, there are uncountably many pos-sibilities for these numbers. They lie in a continuous dynamicrange.

136

• Therefore, information is transmitted basically in analog (notdigital) form but the transmission takes place at discrete times.

(ii) In digital pulse modulation, m[n] is represented by a (discrete)number (symbol) selected from a finite alphabet set.

i. In Pulse Code Modulation (PCM), we further quantizem[n] into mn which has finitely many levels. Then, convert mn

into binary sequence represented by two basic pulses.

1

LPF EncoderSampler QuantizerAnalog Signal

𝑚 𝑡Anti-aliasing

Filter

𝑚 𝑛 𝑚𝑛t

Encoded PulsesConversion to bits

P/S

Line Coding

S/H

(sample and hold)

Figure 70: PCM System Diagram

Time

Vertical lines are used for sampling

Horizontal lines are used for quantization

001

000

010

011

100

101

110

111111

100

100

111

011

100

101

010

000

001 001

100111111100001000010100011001

Figure 71: An overview of PCM

ii. There are also other forms of “source coding” such as DPCMand DM.

Definition 8.3. Pulse-code modulation (PCM) is a discrete-time, discrete-amplitude waveform-coding process, by means of which an analog signal isdirectly represented by a sequence of coded pulses.

8.4. Advantages of PCM

(a) Robustness to channel noise, distortion, and interference

137

(i) assuming these corruptions are within limits.

(ii) With analog messages, on the other hand, any distortion or noise,no matter how small, will distort the received signal.

(b) Efficient regeneration of the coded signal along the transmission pathby using regenerative repeaters.

• For analog communications,

◦ A message signal becomes progressively weaker as it travelsalong the channel, whereas the cumulative channel noise andthe signal distortion grow progressively stronger.

◦ Ultimately the signal is overwhelmed by noise and distortion.

◦ Amplification offers little help because it enhances the signaland the noise by the same proportion.

◦ Consequently, the distance over which an analog message canbe transmitted is limited by the initial transmission power.

• For digital communications,

◦ We can set up repeater stations along the transmission pathat distances short enough to be able to detect signal pulsesbefore the noise and distortion have a chance to accumulatesufficiently.

◦ At each repeater station the pulses are detected, and new, cleanpulses are transmitted to the next repeater station, which, inturn, duplicates the same process.

(c) Digital signals can be coded to remove redundancy, protect againstchannel corruption, and provide privacy.

8.5. PCM has emerged as the most favored scheme for the digital trans-mission of analog information-bearing signals (e.g., voice and video signals).[4, p 267]

• The method of choice for the construction of public switched telephonenetworks (PSTNs).

8.6. Technically, the term “modulation” used in PCM, DPCM, and DM isa misnomer. In reality, PCM, DPCM, and DM are different forms of sourcecoding. [4, p 277]

138

8.1 Uniform Memoryless Quantization

Definition 8.7. Through quantization, each sample value m[n] is trans-formed to (, e.g., approximated, or “rounded off,” to the nearest) quantizedlevel [6, p 320] or quantum level [3, p 545] (permissible number) takenfrom a finite alphabet set M.

8.8. Sampling vs. Quantization:

(a) Sampling operates in the time domain. Quantization operates in theamplitude domain.

(b) The sampling process is the link between an analog waveform and itsdiscrete-time representation. The quantization process is the link be-tween an analog waveform and its discrete-amplitude representation.

Definition 8.9. Suppose the range of the quantizer is (−mp,mp).

• Note that, here, mp is not necessarily the peak value of m(t). Theamplitudes beyond ±mp will be simply chopped off.

A simple (memoryless) quantizer partitions the range into L intervals. Eachsample value is approximated by the midpoint of the interval in which thesample value falls.

• Each sample is now represented by one of the L numbers.

• Such a signal is known as an L-ary digital signal.

Time

Quantized levelsDecision levels

2

Figure 72: Quantized levels

139

• The length of each interval is denoted by ∆ =2mp

L .

• Because the quantized levels are uniformly spaced, we say that thequantizer is uniform.

8.10. Quantization introduces permanent errors that appear at the receiveras quantization noise in the reconstructed signal.

280 Chapter 6 Conversion of Analog Waveforms into Coded Pulses

With the input M having zero mean and the quantizer assumed to be symmetric as inFigure 6.9, it follows that the quantizer output V and, therefore, the quantization error Qwill also have zero mean. Thus, for a partial statistical characterization of the quantizer interms of output signal-to-(quantization) noise ratio, we need only find the mean-squarevalue of the quantization error Q.

Consider, then, an input m of continuous amplitude, which, symmetrically, occupies therange [–mmax, mmax]. Assuming a uniform quantizer of the midrise type illustrated inFigure 6.9b, we find that the step size of the quantizer is given by

(6.25)

where L is the total number of representation levels. For a uniform quantizer, thequantization error Q will have its sample values bounded by –/2 q /2. If the step size is sufficiently small (i.e., the number of representation levels L is sufficiently large), it isreasonable to assume that the quantization error Q is a uniformly distributed randomvariable and the interfering effect of the quantization error on the quantizer input is similarto that of thermal noise, hence the reference to quantization error as quantization noise.We may thus express the probability density function of the quantization noise as

(6.26)

For this to be true, however, we must ensure that the incoming continuous sample does notoverload the quantizer. Then, with the mean of the quantization noise being zero, itsvariance is the same as the mean-square value; that is,

Figure 6.10Illustration of the quantization process.

1 Input wave

2 Quantized output

Mag

nitu

deD

iffe

renc

e be

twee

ncu

rves

1 &

2

Time

Error

2mmax

L----------------=

fQ q 1--- ,

2--- q

2---–

0, otherwise

=

Q2

Haykin_ch06_pp3.fm Page 280 Friday, January 4, 2013 5:18 PM

Figure 73:Error in thequantized out-put [4, Fig.6.10].

Let q[n] = m[n] − mn be the quantization error for the nth sample. Ifthe step size ∆ is sufficiently small (i.e., L is sufficiently large) and m[n]is bounded by (−mp,mp), it is reasonable to assume that the quantizationerror is uniformly distributed on

[−∆

2 ,∆2

].

Assuming that the sampling is done with sampling rate fs that is fastenough (> 2B). Then, we know, from the reconstruction equation (77),that

m (t) =∞∑

n=−∞m [n] sinc (πfs (t− nTs)) . (81)

With quantization, the sequence mn is transmitted instead of m[n]. Hence,at the receiver, the reconstructed signal is

m (t) =∞∑

n=−∞mn sinc (πfs (t− nTs)) . (82)

140

The distortion component in the reconstructed signal, which is referred toas the quantization noise, is

q(t) = m(t)− m(t).

One can then show that the average power of the quantization noise is

Pq = E[(m[n]−mn)

2]

= E[q2[n]

]=

∆2

12=m2p

3L2. (83)

So, the signal-to-(quantization)-noise power ratio (SNR or SQNR) is

SNR =PmPq

=12Pm∆2

=3L2Pmm2p

. (84)

This is an indication of the quality of the received signal. We want SNR tobe large.

8.11. Sinusoidal Modulating Signal: For sinusoidal signal m(t) withamplitude A, its power is Pm = 1

2A2 and therefore,

SNR =12Pm∆2

=12(

12A

2)

∆2=

6A2

∆2= 10log10

(6A2

∆2

)[dB] . (85)

8.12. Full-Load Sinusoidal Modulating Signal: If the sinusoidal m(t)fully covers the whole range of the quantizer, we have A = mp, Pm = 1

2m2p

and

SNR =3L2Pmm2p

=3L2

(12m

2p

)m2p

= 1.5L2 = 10log10

(1.5L2

)[dB] . (86)

8.13. As mentioned earlier, in PCM, the quantized samples are coded andtransmitted as binary pulses. At the receiver, some pulses may be detectedincorrectly. Hence, there are two sources of error in this scheme:

(a) quantization error

(b) pulse detection error

In almost all practical schemes, the pulse detection error is quite smallcompared to the quantization error and can be ignored. [6, p 322]

8.14. The quantization error can be reduced as much as desired by increasing the number of quantizing levels, the price of which is paid in anincreased required transmission bandwidth. See (88).

141

8.2 Pulse Coding

8.15. From practical viewpoint, a binary digital signal (a signal that cantake on only two values) is very desirable because of its simplicity, economy,and ease of engineering. We can convert an L-ary signal into a binary signalby using pulse coding.

• A binary digit is called a bit.

• L = 2` levels can be mapped into (represented by) ` bits.

Example 8.16. In Figure 71, L = 8. The binary code can be formed bythe binary representation of the 8 decimal digits from 0 to 7; that is weassign the word 000 to the lowest level and progresses upward to 111 inthe natural order of binary counting. This “natural” code is also known as“offset binary code”.

Example 8.17. Telephone (speech) signal:

• The components above 3.4 kHz are eliminated by a low-pass filter.

◦ For speech, subjective tests show that signal intelligibility is notaffected if all the components above 3.4 kHz are suppressed.

• The resulting signal is then sampled at a rate of 8,000 samples persecond (8 kHz).

◦ This rate is intentionally kept higher than the Nyquist samplingrate of 6.8 kHz so that realizable filters can be applied for signalreconstruction.

• Each sample is finally quantized into 256 levels (L = 256), which re-quires eight bits to encode each sample (28 = 256).

[6, p 320]

142

Example 8.18. Compact disc (CD) audio signal:

• High-fidelity: Require the audio signal bandwidth to be 20 kHz.

• The sampling rate of 44.1 kHz is used.

• The signal is quantized into L = 65, 536 of quantization levels, each ofwhich is represented by 16 bits (16-bit two’s complement integer) toreduce the quantizing error.

[6, p 321]

8.19. The SNR for the full-load sinusoidal modulating signal discussed in8.12 is then

SNR = 1.5(22`)

= 10log101.5 + 2` 10log102︸︷︷︸≈3

[dB] ≈ 1.76 + 6` [dB] . (87)

8.20. Recall, from 6.37, that a maximum of 2B independent pieces (sym-bols) of information per second can be transmitted, errorfree, over a noise-less channel of bandwidth B Hz. In other words, one can send at most 2[Sa/s/Hz]. This is achieved by using the sinc pulse train. Here, because weuse binary coding, one symbol is the same as one bit. Therefore, for PCM,one can send at most 2 [b/s/Hz]. Equivalently, if the bit rate is Rb [bps],the minimum required baseband transmission bandwidth is

Rb

2=fs`

2≥ 2B`

2= `B. (88)

143

8.3 Line Coding

8.21. The last signal-processing operation in the transmitter is that of linecoding, the purpose of which is to represent/convert sequence of bits by/intoa sequence of (electrical) pulses.

8.22. Figure 74 depicts various line codes for the binary message 10110100,taking rectangular pulses for clarity.

(a)

(b)

(c)

(d)

(e)

Tb Tb

A/2

– A/2

t

t

t

t

t

1

0

0

A

– A

0

A/2

A/2

– A/2

– A/2

– 3A/2

0

0

0 1 1 0 1 0

RZ

0

1 0 1 1 0 1 0 0

1 0 1 1 0 1 0 0

A

Ts

NRZ

3A/2

Figure 11.1–1 Binary PAM formats with rectangular pulses: (a) unipolar RZ and NRZ; (b)polar RZ and NRZ; (c) bipolar NRZ; (d) split-phase Manchester; (e) polar quaternary NRZ.

11.1 Digital Signals and Systems 483

Finally, Fig. 11.1–1e shows a quaternary signal derived by grouping the mes-sage bits in blocks of two and using four amplitude levels to prepresent the four pos-sible combinations 00, 01, 10, and 11. Thus, D � 2Tb and . Differentassignment rules or codes may relate the ak to the grouped message bits. Two suchcodes are listed in Table 11.1–1. The Gray code has advantages relative to noise-induced errors because only one bit changes going from level to level.

Quaternary coding generalizes to M-ary coding in which blocks of n messagebits are represented by an M-level waveform with

(4a)

Since each pulse now corresponds to n � log2 M bits, the M-ary signaling rate hasbeen decreased to

M � 2n

r � rb>2

car80407_ch11_479-542.qxd 12/17/08 6:35 PM Page 483

Confirming Pages

Figure 74: Line codes withrectangular pulses: (a) unipo-lar RZ and NRZ; (b) po-lar RZ and NRZ; (c) bipolarNRZ; (d) split-phase Manch-ester; (e) polar quaternaryNRZ. [3, Fig 11.1-1 p 483]

Definition 8.23. The simple on-off waveform in Figure 74a representseach 0 by an “off” pulse and each 1 by an “on” pulse.

(a) In the (unipolar) return-to-zero (RZ) format, the pulse duration issmaller than Tb after which the signal return to the zero level.

144

(b) A (unipolar) nonreturn-to-zero (NRZ) format has “on” pulses forfull bit duration Tb.

Definition 8.24. The polar signal in Figure 74b has opposite polaritypulses

• Its DC component will be zero if the message contains 1s and 0s inequal proportion.

Definition 8.25. Figure 74c, we have bipolar signal where successive 1sare represented by pulses with alternating polarity.

• Use three amplitude levels

• Also known as pseudo-ternary or alternate mark inversion (AMI)

Definition 8.26. The split-phase or Manchester format in Figure 74drepresents 1s with a positive half-interval pulse followed by a negative half-interval pulse, and vice versa for the representation of 0s.

• Also called twinned binary.

• Guarantee zero DC component regardless of the message sequence.

Definition 8.27. Figure 74e shows a quaternary signal derived by group-ing the message bits in blocks of two and using four amplitude levels toprepresent the four possible combinations 00, 01, 10, and 11.

• Quaternary coding can be generalized to M-ary coding in whichblocks of n message bits are represented by an M-level waveform withM = 2n.

145

8.4 Companding: Nonuniform Quantization

8.28. Recall, from (84) that SNR = 3L2Pmm2p

. So, the SNR is directly pro-

portional to the signal power which implies that the quality of the receivedsignal will deteriorate markedly when the person speaks softly. Statistically,it is found that smaller amplitudes predominate in speech and larger ampli-tudes are much less frequent. This means the SNR will be low most of thetime.

8.29. The problem can be solved by using smaller steps for smaller ampli-tudes (nonuniform quantizing). In other words,

(a) the weak (soft) passages needing more protection are favored at theexpense of the loud passages,

(b) the loud talkers and stronger signals are penalized with higher stepsizes to compensate the soft talkers and weaker signals.

This is illustrated in Figure 75.326 SAMPLING AND ANALOG-TO-DIGITAL CONVERSION

Figure 6.15 Nonuniform quantization .

Quantization levels

t y

,_

(a)

l ~-------------

Nonuniform

(b)

or f c = 255 has been used for all North American 8-bit (256-leve l) digita l terminals. and the eal"li e r value orr< is now almost ex tinct. For th e A-law, a va lue of A= 87.6 gi\·es comparable results and has been standardi zed by th e !TU-T.6

The compressed samples must be restored to the ir original \'a lues at the rece i\·e r by using an ex pander with a characteristic compl~ ~11 e ntary to that of the compressor. The compressor and the expander together are call ed the com pandor. Figure 6. 17 describes the use of compressor and expander along with a uniform quantizer to achieve nonuniform quantization.

Genera ll y speaking, time compression of a signal increases its bandwidth. But in PCM, v,re are compressing not the s ignalm (t) in time but its sample values. Because ne ither the time scale not the number of samples changes, the problem of bandwidth increase does not ari se here. It is shown in Sec. 10.4 that when a {1-law com pandor is used, the ou tput SNR is _

[In (1 + f-L) ]2 (6.36)

I I

I Figure 75: Nonuniform quantization. [6, Fig6.15a p. 326]

146

8.30. The same result is obtained by first compressing (with a compressor)signal samples and then using a uniform quantization.

m (t)→ Compressor → y (t)→ UniformQuantizer

→ CompressedOutput Signal

An example of the input-output characteristic of the compressor is illus-trated in Figure 76.

326 SAMPLING AND ANALOG-TO-DIGITAL CONVERSION

Figure 6.15 Nonuniform quantization .

Quantization levels

t y

,_

(a)

l ~-------------

Nonuniform

(b)

or f c = 255 has been used for all North American 8-bit (256-leve l) digita l terminals. and the eal"li e r value orr< is now almost ex tinct. For th e A-law, a va lue of A= 87.6 gi\·es comparable results and has been standardi zed by th e !TU-T.6

The compressed samples must be restored to the ir original \'a lues at the rece i\·e r by using an ex pander with a characteristic compl~ ~11 e ntary to that of the compressor. The compressor and the expander together are call ed the com pandor. Figure 6. 17 describes the use of compressor and expander along with a uniform quantizer to achieve nonuniform quantization.

Genera ll y speaking, time compression of a signal increases its bandwidth. But in PCM, v,re are compressing not the s ignalm (t) in time but its sample values. Because ne ither the time scale not the number of samples changes, the problem of bandwidth increase does not ari se here. It is shown in Sec. 10.4 that when a {1-law com pandor is used, the ou tput SNR is _

[In (1 + f-L) ]2 (6.36)

I I

I

Figure 76: An example of a compressor char-acteristic curve [6, Fig 6.15b p. 326]

8.31. Among several choices, two compression laws have been accepted asdesirable standards by the ITU-T:

(a) the µ-law used in North America and Japan

(b) the A-law used in Europe and the rest of the world and on internationalroutes

Example of their characteristic curves are plotted in Figure 77.

Example 8.32. In the standard audio file format used by Sun, Unix andJava, the audio in “au” files can be pulse-code-modulated or compressedwith the ITU-T G.711 standard through either the µ-law or the A-law. Inboth cases, the sampling is performed at 8000 [Sa/s] and the compressorconverts the linear PCM samples to 8-bit samples. Therefore, the resultingaudio bit stream is at

147

1 t

I

\ I

Figure 6.16 (a) 11-law characteristic. (b) A-Law characteristic.

Figure 6.17 Utilization of compressor and expander for nonuniform quantization .

t I '

Compressor nonlinearity

(a)

Uniform quantizer

Nonu ni form quanrizer

PCM channel

6.2 Pulse Code Modulation (PCM) 327

Expander nonlinearity

(b)

The output SNR for the cases of f.l = 255 and f.J., = 0 (uniform quanti zat ion ) as a function of

m 2 (1) (the message signal power) is shown in Fig. 6.1 8.

The Compandor A logarithmic compressor can be rea lized by a semiconductor diode. because the V-/ characteri stic of such a di ode is of the d~sired form in the first quadrant:

V = - In I +-KT ( l) q l ,

Two matched diodes in parall el with opposite polarity provide the approx imate characteristic in the first and third quadrants (ignoring the saturation current). In practi ce, adj ustabl e resistors are placed in series with each diode and a third variable resistor is added in parallel. By adjusting various res istors, the resulting characteristic is made to fit a finite number of points (usually seven) on the ideal characteristi cs.

An alternative approach is to use a piecewise linear approximation to the logarithmic characteristics. A IS-segmented approximation (Fig. 6 .19) to the eighth bit (L = 256) with f.l = 255 Ia w is widely used in the D2 channel bank that is used in conjunction with the T 1 carrier system. The segmented approximation is only marginally inferior in terms of SNR 8 The piecewise linear approximation has almost universally replaced earli er logarithmic approximations to the true f.J., = 255 characteristic and is the method of choice in North American standards.

Figure 77: (a) µ-law characteristic. (b) A-law characteristic. [6, Fig 6.16 p. 326]

Example 8.33. Microsoft WAV audio format also has compression optionsthat use µ-law and A-law.

8.34. At the receiver, to restore the signal samples to their correct relativelevel, we must, of course, use a device in the receiver with a characteristiccomplementary to the compressor. Such a device is called an expander.Ideally, the compression and expansion laws are exactly the inverse of eachother. The compressor and the expander together are called the compan-dor.

148

8.5 DPCM and DM

8.35. PCM is not a very efficient system because it generates so many bitsand requires so much bandwidth to transmit. DPCM and DM are two otheruseful forms of digital pulse modulation.

8.36. In differential pulse-code modulation (DPCM), the main ideais that instead of transmitting the sample values, we transmit the differencebetween the successive sample values.

• The difference between successive samples is generally much smallerthan the sample values. Thus, the peak amplitude of the transmittedvalues is reduced considerably. Therefore, our quantizer can considersmaller range (smaller mp).

• For a given L, from (84), we see that the SNR is improved.

• Alternatively, for a given SNR, we can reduce L which in turn reduce` (or transmission bandwidth).

• In general, we try to predict (estimate) m[n] from several previoussample values and transmit the difference (prediction error): d[n] =m[n]− m[n]

8.37. Figure 78 illustrates the tradeoffs between standard PCM, DPCM,and DM.

6.9 Delta Modulation 305

6.9 Delta Modulation

In choosing DPCM for waveform coding, we are, in effect, economizing on transmissionbandwidth by increasing system complexity, compared with standard PCM. In otherwords, DPCM exploits the complexity–bandwidth tradeoff. However, in practice, the needmay arise for reduced system complexity compared with the standard PCM. To achievethis other objective, transmission bandwidth is traded off for reduced system complexity,which is precisely the motivation behind DM. Thus, whereas DPCM exploits thecomplexity–bandwidth tradeoff, DM exploits the bandwidth–complexity tradeoff. We may,therefore, differentiate between the standard PCM, the DPCM, and the DM along the linesdescribed in Figure 6.20. With the bandwidth–complexity tradeoff being at the heart ofDM, the incoming message signal m(t) is oversampled, which requires the use of asampling rate higher than the Nyquist rate. Accordingly, the correlation between adjacentsamples of the message signal is purposely increased so as to permit the use of a simplequantizing strategy for constructing the encoded signal.

DM Transmitter

In the DM transmitter, system complexity is reduced to the minimum possible by using thecombination of two strategies:

1. Single-bit quantizer, which is the simplest quantizing strategy; as depicted in Figure6.21, the quantizer acts as a hard limiter with only two decision levels, namely, .

2. Single unit-delay element, which is the most primitive form of a predictor; in otherwords, the only component retained in the FIR predictor of Figure 6.17 is the front-endblock labeled z–1, which acts as an accumulator.

Thus, replacing the multilevel quantizer and the FIR predictor in the DPCM transmitter ofFigure 6.19a in the manner described under points 1 and 2, respectively, we obtain theblock diagram of Figure 6.21a for the DM transmitter.

From this figure, we may express the equations underlying the operation of the DMtransmitter by the following set of equations (6.95)–(6.97):

(6.95)

Figure 6.20 Illustrating the tradeoffs between standard PCM, DPCM, and DM.

en mn mn–=

mn mq n 1––=

Increasing

Systemcomplexity

DPCM

StandardPCM

DM

Transmissionbandwidth

Increasing

Haykin_ch06_pp3.fm Page 305 Monday, November 26, 2012 1:00 PM

Figure 78: Tradeoffs between PCM, DPCM,and DM [4, Fig. 305]

149

8.38. In delta modulation (DM),

• m(t) is oversampled (typically using four times the Nyquist rate [6, p346]), and

• system complexity is reduced to the minimum possible by using asingle-bit quantizer with only two levels: ±∆.

◦ DM is basically a 1-bit DPCM

8.39. Two types of error in DM:

308 Chapter 6 Conversion of Analog Waveforms into Coded Pulses

digital approximation to the quantizer input or, equivalently, as the inverse of the digitalintegration process carried out in the DM transmitter. If, then, we consider the maximumslope of the original message signal m(t), it is clear that in order for the sequence ofsamples {mq,n} to increase as fast as the sequence of message samples {mn} in a region ofmaximum slope of m(t), we require that the condition

(6.100)

be satisfied. Otherwise, we find that the step-size is too small for the staircaseapproximation mq(t) to follow a steep segment of the message signal m(t), with the resultthat mq(t) falls behind m(t), as illustrated in Figure 6.23. This condition is called slopeoverload, and the resulting quantization error is called slope-overload distortion (noise).Note that since the maximum slope of the staircase approximation mq(t) is fixed by thestep size , increases and decreases in mq(t) tend to occur along straight lines. For thisreason, a delta modulator using a fixed step size is often referred to as a linear deltamodulator.

In contrast to slope-overload distortion, granular noise occurs when the step size istoo large relative to the local slope characteristics of the message signal m(t), therebycausing the staircase approximation mq(t) to hunt around a relatively flat segment of m(t);this phenomenon is also illustrated in the tail end of Figure 6.23. Granular noise isanalogous to quantization noise in a PCM system.

Adaptive DM

From the discussion just presented, it is appropriate that we need to have a large step sizeto accommodate a wide dynamic range, whereas a small step size is required for theaccurate representation of relatively low-level signals. It is clear, therefore, that the choiceof the optimum step size that minimizes the mean-square value of the quantization error ina linear delta modulator will be the result of a compromise between slope-overloaddistortion and granular noise. To satisfy such a requirement, we need to make the deltamodulator “adaptive,” in the sense that the step size is made to vary in accordance with theinput signal. The step size is thereby made variable, such that it is enlarged duringintervals when the slope-overload distortion is dominant and reduced in value when thegranular (quantization) noise is dominant.

Figure 6.23 Illustration of the two different forms of quantization error in DM.

Ts----- max dm t

dt--------------

Ts

Δ

Slope-overloaddistortion

Granular noise

Staircaseapproximation

mq(t)

m (t)

Haykin_ch06_pp3.fm Page 308 Monday, November 26, 2012 1:00 PM

Figure 79: Two differentforms of error in DM

(a) Slope overload distortion

• Occur when mq(t) cannot follow a steep segment of m(t).

• During the sampling interval Ts, mq(t) is capable of changing bythe step size ∆. Hence, the maximum slope that mq(t) can followis ∆/Ts = ∆fs. Hence, no overload occurs if

max |m(t)| < ∆fs.

• The slope overload noise can be reduced by increasing ∆ (the stepsize). This unfortunately increases the granular noise.

(b) Granular noise

• Occur when the step size ∆ is too large relative to the local slopecharacteristics of the message signal m(t), thereby causing thestaircase approximation mq(t) to hunt around a relatively flat seg-ment of m(t).

150

4.3 fourier series periodic · 4.3 fourier series de nition 4.41.exponential fourier series: let...

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