ee456 – digital communications · ee456 – digital communications professor ha nguyen september...

Chapter 3: Frequency Modulation (FM)

EE456 – Digital CommunicationsProfessor Ha Nguyen

September 2016

EE456 – Digital Communications 1

Angle Modulation

In AM signals the information content of message m(t) is embedded as amplitudevariation of the carrier.

Two other parameters of the carrier are frequency and phase. They can also bevaried in proportion to the message signal, which results in frequency-modulated

and phase-modulated signals.

Frequency modulation (FM) and phase modulation (PM) are closely related andcollectively known as angle modulation. In our study, we will mainly focus on FM.

Instantaneous Frequency

Consider a generalized sinusoidal signal c(t) = A cos θ(t), where θ(t) is thegeneralized angle and is a function of t.

Over the infinitesimal duration of ∆t between [t1, t2], draw a tangential line ofθ(t), which can be described by equation ωct+ θ0.

It is clear from the figure that, over the interval t1 < t < t2 one has:

c(t) = A cos θ(t) = A cos(ωct+ θ0), t1 < t < t2.

This means that, over the small interval ∆t, the angular frequency of c(t) is ωc,which is the slope of the tangential line of θ(t) over this small interval.

For a conventional sinusoid A cos(ωct+ θ0), the generalized angle is a straightline ωct+ θ0 and the angular frequency is fixed.

For a generalizes sinusoid, the angular frequency is not fixed but varies with time.At every time instant t, the instantaneous frequency is the slope of angle θ(t) attime t:

ωi(t) =dθ(t)

The equivalent relationship between angle θ(t) and the instantaneous frequencyωi(t) is:

θ(t) =

−∞

ωi(α)dα

Phase Modulation (PM) and Frequency Modulation (FM)

In PM, the angle θ(t) is varied linearly with the message signal m(t):

θ(t) = ωct+ kpm(t), (assuming θ0 = 0)

sPM(t) = A cos[ωct+ kpm(t)], (where kp is a constant)

The instantaneous angular frequency ωi(t) of the PM signal is

ωi(t) =dθ(t)

dt= ωc + kp

which varies linearly with the derivative of the message.

If the instantaneous angular frequency ωi(t) varies linearly with the message,then we have frequency-modulated (FM) signal:

ωi(t) = ωc + kfm(t), (where kf is a constant)

θ(t) =

−∞

ωi(α)dα =

−∞

[ωc + kfm(α)]dα = ωct+ kf

−∞

m(α)dα

sFM(t) = A cos

ωct + kf

−∞

m(α)dα

Relationship Between FM and PM

FM ( )s t

PM ( )s t

PM and FM are very much related. It is not possible to tell from the time waveform

whether a signal is FM or PM. This is because either m(t),dm(t)

dt, or

∫m(α)dα can

be treated as a message signal.

PM and FM Circuits (Analog)

Note: RFC stands for radio-frequency choke

Example 3

The figure below shows a message signal m(t) and its derivative. Suppose that theconstants kf and kp are 2π × 105 and 10π, respectively, and the carrier frequency fcis 100 MHz.

(a) Write an expression of the instantaneous frequency of the FM signal. Determinethe minimum and maximum values of the instantaneous frequency.

(b) Write an expression of the instantaneous frequency of the PM signal. Determinethe minimum and maximum values of the instantaneous frequency.

(c) Sketch the FM and PM signals and offer your comments.

Solution:

(a) For FM, we have:

fi(t) =ωi(t)

2π= fc +

2πm(t) = 108 + 105m(t)

[fi(t)]min = 108 + 105[m(t)]min = 99.9 MHz

[fi(t)]max = 108 + 105[m(t)]max = 100.1 MHz

(b) For PM, we have:

fi(t) =ωi(t)

2π= fc +

2πm(t) = 108 + 5m(t)

[fi(t)]min = 108 + 5[m(t)]min = 99.9 MHz

[fi(t)]max = 108 + 5[m(t)]max = 100.1 MHz

(c) Sketches of the FM and PM signals are shown below.

FM ( )s t

PM ( )s t

Observations:

Because m(t) increases and decreases linearly with time, the instantaneousfrequency of the FM signal increases linearly from 99.9 to 100.1 MHz over ahalf-cycle, and then decreases linearly from 100.1 MHz to 99.9 MHz over theremaining half-cycle.

Because m(t) switches back and forth from a value of −20, 000 to 20, 000, thecarrier frequency switches back and forth from 99.9 to 100.1 MHz everyhalf-cycle of m(t).

Comparison of AM, FM and PM Signals with the same massage m(t)

Can you tell which signals on the right are AM, FM and PM, respectively?

0 5 10−0.5

Messagem(t)

0 5 10−1

0 5 10−0.5

∫t−∞m(α

0 5 10−2

tEE456 – Digital Communications 11

Comparison of AM, FM and PM Signals with the same massage m(t)

0 5 10−0.5

Messagem(t)

0 5 10−1

0 5 10−0.5

∫t−∞m(α

0 5 10−2

s AM(t)

0 5 10−2

s PM(t)

0 5 10−2

s FM(t)

Comparison of AM, FM and PM Signals under the same amount of noiseCompared to AM, FM and PM signals are much less susceptible to additive noise andinterference. This is because of two reasons: (i) Additive noise/interference acts onamplitude, and (ii) the message is embedded in amplitude in AM, while is isembedded in frequency/phase in FM/PM.

0 1 2 3 4 5−0.5

Messagem(t)

0 1 2 3 4 5−1

0 1 2 3 4 5−0.5

∫t−∞m(α

0 1 2 3 4 5−2

s AM(t)

0 1 2 3 4 5−2

s FM(t)

Power and Bandwidth of Angle-Modulated Signals

Since the amplitude of either PM or FM signal is a constant A, the power of anangle-modulated (i.e., PM or FM) signal is always A2/2, regardless of the valueof kp, kf , and power of m(t).

Unlike AM, angle modulation is nonlinear and hence its spectrum/bandwidthanalysis is not as simple as for AM signals.

To determine the bandwidth of an FM signal, define

a(t) =

−∞

m(α)dα

sFM(t) = Aej[ωct+kfa(t)] = Aejkfa(t)ejωct ⇒ sFM(t) = ℜ{sFM(t)}

Expanding the exponential ejkfa(t) in power series gives:

sFM(t) = A

1 + jkfa(t) −k2f

2!a2(t) + · · ·+ jn

n!an(t) + · · ·

ejωct

sFM(t) = ℜ{sFM(t)}

cos(ωct) − kfa(t) sin(ωct)−k2f

2!a2(t) cos(ωct) +

3!a3(t) sin(ωct) + · · ·

Observations:

The FM signal consists of an unmodulated carrier and variousamplitude-modulated terms, such as a(t) sin(ωct), a2(t) cos(ωct), a3(t) sin(ωct),etc.

Since a(t) is an integral of m(t), if M(f) is band-limited to [−B,B], then A(f)is also band-limited to [−B,B].

The spectrum of a2(t) is the spectrum of A(f) ∗A(f) (where ∗ is the integralconvolution operation) and is band-limited to [−2B, 2B]. Similarly, the spectrumof an(t) is band-limited to [−nB, nB].

The spectrum of sFM(t) consists of an unmodulated carrier, plus spectra of a(t),a2(t), . . . , an(t), . . . , centered at ωc.

Clearly, the bandwidth of sFM(t) is theoretically infinite!

For practical message signals, because n! increases much faster than |kfa(t)|n,

we haveknf an(t)

n!≈ 0 for large n. Hence most of the modulated-signal power

resides in a finite bandwidth.

Carson’s rule for Bandwidth Approximation of an FM Signal (captures 98% of total power):

BFM = 2(∆f +B) = 2B(β + 1)

where ∆f = kfmmax −mmin

2 · 2πis defined as the peak frequency deviation

β =∆f

Bis the deviation ratio

Spectral Analysis of Tone FM

When the message m(t) is a sinusoid, namely m(t) = Am cos(ωmt), and withthe initial condition a(−∞) = 0, one has

a(t) =Am

ωmsin(ωmt)

β =∆f

sFM(t) = Ae(jωct+jkfAm/ωm sin(ωmt))

= Ae(jωct+jβ sin(ωmt)) = Aejωct(

ejβ sin(ωmt))

Since ejβ sin(ωmt) is a periodic signal with period T = 2π/ωm, it can beexpanded by the exponential Fourier series:

ejβ sin(ωmt) =∞∑

n=−∞

Dnejnωmt

where Dn =ωm

∫ π/wm

−π/ωm

ejβ sin(ωmt)e−jnωmtdt

∫ π

−πej(β sinx−nx)dx = Jn(β)

︸︷︷︸

nth-order Bessel function of the first kind

It then follows that

sFM(t) = A∞∑

n=−∞

Jn(β)ej(ωct+nωmt)

sFM(t) = A∞∑

n=−∞

Jn(β) cos((ωc + nωm)t)

Observations:

The tone-modulated FM signal has a carrier component and an infinite numberof sidebands of frequencies ωc ± ωm, ωc ± 2ωm,. . . ,ωc ± nωm. This is verydifferent from DSB-SC spectrum of tone-modulated AM signal!

The strength of the nth sideband at ωc + nωm is A2Jn(β), which quickly

decreases with n. In fact, there are only a finite number of significant sidebandspectral lines.

In general, Jn(β) is negligible for n > β + 1, hence the bandwidth oftone-modulated FM signal is approximated as:

BFM = 2(β + 1)fm = 2(∆f + B)

Plot of Jn(β)

0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4 4.4 4.8 5.2 5.6 6 6.4 6.8 7.2 7.6 8 8.4 8.8 9.2 9.6 10−0.4

−0.3

−0.2

−0.1

J0(β)

J1(β)

J2(β)J3(β) J4(β) J5(β) J6(β)

Two important properties: J−n(β) = (−1)nJn(β)∞∑

n=−∞

J2n(β) = 1

Table of Jn(β)

Illustration of Tone FM Spectrum

A FM()

Example 4

The figure below shows a message signal m(t) and its derivative. Suppose that theconstant kf = 2π × 105.

(a) Since m(t) is periodic with a fundamental frequency f0 = 12×10−4

, it can be

represented as m(t) =∑

k=−∞ake

j2πkf0t. Show that a0=0 and

π2k2, k odd

0, k even

(b) Assume that the essential bandwidth of m(t) to be the frequency of its thirdharmonic, estimate the bandwidth of the FM signal when the modulating signal ism(t).

(c) Repeat Part (b) if the amplitude of m(t) is doubled (i.e., if m(t) is multiplied by2).

(d) Repeat Part (b) if m(t) is time-expanded by a factor of 2 (i.e., if the period ofm(t) is 4× 10−4).

Narrow-Band FM (NBFM)

sFM(t) = A

cos(ωct)− kfa(t) sin(ωct)−k2f

2!a2(t) cos(ωct) +

3!a3(t) sin(ωct) + · · ·

When kf is very small such that |kfa(t)| ≪ 1, then all higher order terms in theabove expression are negligible, except for the first two terms. We then have agood approximation of an FM signal:

sFM(t) ≈ A[cos(ωct) − kfa(t) sin(ωct)

The above approximation is a linear modulation similar to that of an AM signalwith the message signal being a(t).

Because the bandwidth of a(t) is the same as the bandwidth of m(t), which is BHz, the bandwidth of the narrowband FM signal in (1) is 2B Hz.

It is pointed out that the sideband spectrum for a NBFM signal has a phase shiftof π/2 with respect to the carrier, whereas the sideband spectrum of an AMsignal is in phase with the carrier.

The expression of the NBFM signal in (1) suggests a method of generating aNBFM signal by using a DSB-SC modulator (see Fig. 1-(a) on the next slide).

The output of the NBFM modulator in Fig. 1-(a) has some amplitude variations(distortion). Such distortion can be removed by using a hard-limiter and abandpass filter as shown in Fig. 1-(b).

The analysis of Fig. 1-(b) shall be explored in Assignment 2.

( )m t∫ ∑

( )a t

cos( )c

sin( )c

A tω−

( )sin( )f cAk a t tω−

NBFM signal

( )cos[ ( )]c

A t t tω ϕ+

( )cos[ ( )]c

A t t tω ϕ+4

cos[ ( )]ct tω ϕπ

Figure 1: Generating a NBFM signal.

Demodulation of FM Signals

Signal at point b : sFM(t) = A cos[

ωct+ kf∫ t−∞

m(α)dα]

Signal at point c :

sFM(t) =d

ωct+ kf

−∞

m(α)dα

= A[ωc + kfm(t)] sin

ωct+ kf

−∞

m(α)dα − π

Signal at point d : A[ωc + kfm(t)]

Signal at point e : kfm(t)

A Practical (Continuous-Time) Differentiator

Recall that the frequency response of an ideal differentiator is H(f) = j2πf .

A differentiator can be approximated by a linear system whose frequency responsecontains a linear segment of a positive slope.

One simple device would be an RC high-pass filter. The RC frequency responseis simply

H(f) =j2πfRC

1 + j2πfRC≈ j2πfRC, if 2πfRC ≪ 1.

Thus, if the parameter RC is very small such that its product with the carrierfrequency ωcRC ≪ 1, the RC filter approximates a differentiator.

FCC FM Standards

FM Stations in Saskatoon

Stereo FM

Review of Discrete-Time Processing of Continuous-Time Signals

( )cH jω

ˆ( )jdH e ω

The frequency response of the discrete-time LTI system, Hd(ejw) is periodic with

period 2π. Over −π ≤ w ≤ π, it is simply a frequency-scaled version of Hc(ω):

Hd(ejw) = Hc (ωfs) , −π ≤ w ≤ π

where fs = 1T

is the sampling frequency.

Example: Discrete-Time Low-Pass Filter

( )cH jω

ˆ( )jdH e ω

Discrete-Time Integrator

An illustration of the backward difference, forward difference, and trapezoid rule forapproximating the integral of a continuous-time signal using discrete-time processing.

Realization of discrete-time integrators: (a) A realization of the discrete-time integrator based on

the trapezoid rule. (b) A realization of the discrete-time integrator based on the backward

difference. (c) A rearrangement of (b) to produce the more traditional system block diagram of an

accumulator.

Freq. Responses: Ideal Integrator, Accumulator, Trapezoid-Rule Integrator

Ideal Integrator: Hideal(ejw) = 1

Accumulator: Hacc(z) =1

1−z−1, Hacc(ejw) = 1

1−e−jw .

Trapezoid-Rule Integrator: Htrap(z) = 0.5 1+z−1

1−z−1, Htrap(ejw) = 0.5 1+e−jw

1−e−jw .

−3 −2 −1 0 1 2 30

ω (radians/sample)

Magnituderespon

−π π−π/2 π/2

|Hideal(ejω)|

|Hacc(ejω)|

|Htrap(ejω)|

The accumulator works very well as a DT integrator, especially forsmall-bandwidth signals.

Discrete-Time Differentiator

ˆ( )jdH e ω

H(ω) =

{jω, |ω| ≤ Wc

0, otherwise⇒ Hd(e

{j ωT, |ω| ≤ WcT

0, WcT < |ω| ≤ π

hd[n] =1

∫ WcT

−WcTjω

Tejwndω =

cos(WcTn)

sin(WcTn)

The impulse response has infinite support ⇒ The discrete-time system is an IIR filter.For the special case of full-bandwidth, i.e., when WcT = π, the impulse response is

hd[n] =

(−1)n

n, n 6= 0

0, n = 0

The first few samples of the impulse response for the full-bandwidth differentiator areshown below.

An Approximate Discrete-Time Differentiator

By truncating the impulse response to n = −1, 0, 1, the differentiator consists ofthe three center coefficients. The output of such a differentiator is

y[n] =1

T(x[n+ 1]− x[n− 1])

The above system is non-causal. It can be made causal by introducing a delay of1 sample:

y[n] =1

T(x[n]− x[n− 2])

[ ]x n1z− 1

−∑ [ ] [ ] [ 2]y n x n x n= − −

approximate a differentiator

with a delay of 1 sample

Ignoring the scaling factor 1T, the impulse response of the above approximate

differentiator is h[n] = δ[n]− δ[n− 2] .

The system function H(z) = 1− z2 has 2 zeros at 0 and π.

The frequency response is

H(ejw) = 1− z−2

∣∣∣∣z=ejw

= 1− e−j2w

= e−jw(ejw − e−jw) = e−jw︸︷︷︸

delay of 1 sample

(2j sin ω︸︷︷︸

≈ω for ω small

) ≈ 2jωe−jw

−3 −2 −1 0 1 2 30

ω (radians/sample)

Magnituderespon

−π π−π/2 π/2

|Happrox(ejω)|

|Hideal(ejω)|

The above length-3 FIR approximation to a differentiator works reasonably wellfor small-bandwidth signal, about |ω| ≤ 0.2π

Better Approximations of a Discrete-Time Differentiator

Use a Blackman window (Matlab command blackman) to approximate an idealdifferentiator as an FIR filter.

0 0.1 0.2 0.3 0.4 0.50

frequency (cycles/sample)

Frequen

cyrespon

N=3, 7, 11, 15, 19, 23, 27, 31

0 0.1 0.2 0.3 0.4 0.5−60

frequency (cycles/sample)

Frequen

cyrespon

se(dB)

N=3, 7, 11, 15, 19, 23, 27, 31

Building FM Transmitter and Receiver in Lab # 3

Transmitter

+ FM[ ] cos( [ ])

cs n n nω θ= +

[ ] [ 1] 2 [ ]fn n k m nθ θ π= − + ⋅ ⋅ (cycles/sample)

[ ]m n

Receiver

cos[( ) ]c

nω ω+ ∆

sin[( ) ]c

nω ω+ ∆

cf f+ ∆

− 1z

' [ 1]c

y n −

' [ 1]s

y n −

'[ 1]nθ −

y n −

FM[ ]s n

Analysis of the FM Demodulator

sFM(t) = cos

ωct+ kf

−∞

m(α)dα

= cos [ωct + θ(t)]

sFM[n] = cos [ωcnTs + θ(nTs)] = cos (ωcn+ θ[n])

yc[n] = cos(∆ωn− θ[n]); where ∆ω = 2π∆f

ys[n] = sin(∆ωn− θ[n]);

y′c[n− 1] ≈d

dtyc(t)

∣∣∣∣t=(n−1)Ts

, where yc(t) = cos(∆ωt− θ(t)), ∆ω =∆ω

= −(∆ω − θ′(t)) sin(∆ωt − θ(t))

∣∣∣∣t=(n−1)Ts

= −(∆ω − θ′[n− 1]) sin(∆ω(n− 1) − θ[n− 1])

Similarly,

y′s[n− 1] ≈ (∆ω − θ′[n− 1]) cos(∆ω(n− 1) − θ[n− 1])

Finally,

y′c[n− 1]ys[n− 1]− y′s[n− 1]yc[n− 1]

= (θ′[n− 1]−∆ω)[cos2(∆ω(n− 1)− θ[n− 1]) + sin2(∆ω(n− 1)− θ[n− 1])]

= (θ′[n− 1]−∆ω) = θ′[n− 1]−∆ω

In the above ∆ω is the DC offset due to error in the receiver’s local oscillator, whileθ′[n− 1] is proportional to m[n− 1].

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