Chapter 4

Bandpass Signaling

In this chapter, we consider the situations where the information from a source is transmitted at its non-natural frequency (i.e., shifted frequency). This process is called the modulation.

1. Representation of modulated signals

2. Spectra

3. Distortions (linear and non-linear)

4. Functional blocks in bandpass communication systems

Basic Model for Bandpass Communication

Source Destination

Source can be analog or digital. The use of channel is restricted around certain frequency, fc (>> 0). For example, a radio station may be given this frequency range for commercial broadcasting. The goal is to recover the original information, m, exactly or in the minimum, as closely as possible.

m(t)tm

g(t)tg

tr

ts

tg

tm

ingapproximat signal tedreconstruc :)(~ ingapproximat signal ddemodulate :)(~

frequency) (shifted channel in the corrupted signal modulated :)(

frequency) (shifted ed transmittbe tosignal modulated :)(

(baseband) modulated be n toinformatio processed :)(

(baseband) sourceby sent be n toinformatio :)(

Definition. A baseband waveform has a spectral magnitude (and thus its power) concentrated around f=0 and zero elsewhere.

Definition. A bandpass waveform has a spectral magnitude concentrated around f=±fc (fc >> 0) and zero elsewhere. (fc: carrier frequency)

Definition. Modulation translates the baseband waveform from a source to a bandpass waveform with carrier frequency, fc. baseband waveform: modulating signal bandpass waveform: modulated signal

Examples of Frequency Spectrum

300 Hz – 20K Hz human voice / sound50 kHz navigation (ships, submarines, etc)1 MHz AM radio (20 k Hz channels)10 MHz CB, short wave100 MHz FM radio, TV1 GHz UHF TV, mobile telephony10 GHz amateur satellite100 GHz upper microwave 10 T Hz Infrared1015 Hz Visible light1018 Hz X-rays

sinusoid. pure a is then constant, a is If

signal) baseband(complex envelopecomplex :)(

signal) baseband (real modulation (PM) phase :)(

signal) baseband positive and (real modulation (AM) amplitude :)(

signal) baseband (real modulation (Q) quadrature :)(

signal) baseband (real modulation (I) phase-in :)(

.)()()()( where

sin)(cos)()(cos)()( ely,Alternativ

.2

frequency,carrier a tod)heterodyne(or shifted equals I.e.,

.)(Re)(

byrepresent becan ,)( waveform,bandpass physicalAny Theorem.

)(

v(t)g(t)

tg

t

tR

ty

tx

etRtjytxtg

twtytwtxttwtRtv

)π

w(fg(t)v(t)

etgtv

tv

tj

ccc

cc

tjwc

corruption little with recovered becan -

efficentlybandwidth channel theuses -

receivers and rstransmitteimplement easy to -

such that develop toision communicat bandpass of goalmain The

. offunction a is where)(Re)( can write weThus

signal) (bandpass )( signal) (baseband :Modulation

1)( if180

1)( if0)( and )(1)(

0)( and )(1)(

)(1

(AM), Modulation AmplitudeFor :Example

m(t)

s(t)

)g(

m(t))g(etgts

tsm(t)

tm

tmttmAtR

tytmAtx

tmAg(t)

tjw

oc

c

c

c

.4

1 Density SpectrumPower

2

1 Spectrum

, and )(Re)( waveform,bandpass aFor Theorem.

*

*

cgcgv

cc

tjw

ffff(f)

ffGffGV(f)

G(f)g(t)etgtv c

PPP

.real is )( and )()( Also

real. is because )()(fact theused wewhere

)()()()(2

1

)()()()(2

1

Then, .cos)(1)( Thus

).( where )()()()(1

(AM), Modulation AmplitudeFor :Example

*

**

fff

m(t) fMfM

ffMffffMffA

ffMffffMffAS(f)s(t)

twtmAts

fMm(t)fMfAfGtmAg(t)

ccccc

ccccc

cc

cc

22

*

2

22

)(2

1)(Re

2

1)0

)0()(Re2

1)0(Re

2

1)0

Then, ).( offunction ation autocorrel

theis where)(Re2

1) seen that becan It

).( signal, any timefor trueis 0)( Proof.

).( offunction ation autocorrel

theis where)(2

10)( power,

normalized average total the),( waveform,bandpass aFor Theorem.

tgtg(R

tgtgR(R

tg

)(ReR(τR

tv)(R(f)dftvP

tv

)(Rtg)(R(f)dftvP

tv

v

gv

gjw

gv

vvv

vvvv

c

P

P

).( n,informatioith w

associatedpower :2

1

powercarrier :2

1

12

1

)(12

1

coding, goodand/or signal of

nature todue 0)(With

)()(212

1)()(21

2

1)(1

2

1

)()()()(2

1

. spectrum,ngular with triasignal AM :Example

2

2

2

22

222222

tm

PA

A

PA

tmAP

tm

tmtmAtmtmAtmAP

ffMffffMffAS(f)

M(f)

mc

c

mc

cs

cccs

ccccc

2PEP

PEP

)(max2

1

by given is , PEP, nomarlized The Theorem.

.peak value

itsat constant held be to were if obtained be that would

power average theis (PEP)power enveloppeak The .Definition

tgP

P

g(t)

signals? bandpassfor ion approximat baseband a thereIs -

answer? therate samplingNyquist theIs

n?informatio original theoftion reconstruc

perfectfor sampled be waveformbandpass amust fast How -

power. computing intense requires

0frequency carrier at waveformbandpass a Simulating -

difficult. is waveformsbandpassfor analysis alMathematic -

Questions and nsObservatio

) (fc

Bandpass Signals over Bandpass Channel

Out of Transmitter Into ReceiverChannel

cffK *

2

1 cffK 2

1 fH

Can we translate this into a baseband model? YES!

)(

2

1)(

2

1)()(

)(Re)( :response impulse bandpass

*cc

tjw

ffKffKfHth

etkth c

)( n,informatio

corruptedy potentiall containing envelopecomplex :

)(Re)(

)( input, todue channel theofout waveformbandpass :)(

)( signal, bandpassfor envelopecomplex :

)(Re)(

frequency, bandpass thearound channel for the response impulse :)(

)( n,informatio thecontaining envelopecomplex :

)(Re)(

channel theinto ed transmittbe to waveformbandpass :)(

Notation

2

22

12

1

11

1

tm

(t)g

etgtv

tvtv

thk(t)

etkth

fth

tm(t)g

etgtv

tv

tjw

tjw

c

tjw

c

c

c

)()(

2

1)()(

2

1

)()(2

1

thatimplies )()( )( Thus,

)()(2

1)(

)()(2

1)( )()(

2

1)(

thatNote .assumptionlinearity from )()()( that trueisIt Proof.

)(2

1)(

2

1)(

2

1

)(2

1)(

2

1)(

2

1

Theorem.

**11

*22

12

*

*111

*222

12

12

12

cccc

cc

cc

cccc

ffKffKffGffG

ffGffG

fHfVfV

ffKffKfH

ffGffGfVffGffGfV

fHfVfV

fKfGfG

tktgtg

)(2

1)(

2

1)(

2

1

.)()(4

1)(

2

1 that truebemust it Thus

)()()()(4

1

)()()()(

)()()()(4

1

)()(2

1)()(

2

1

)()(2

1

).(Continued Proof

12

12

**11

**1

*1

*11

**11

*22

fKfGfG

ffKffGffG

ffKffGffKffG

ffKffGffKffG

ffKffGffKffG

ffKffKffGffG

ffGffG

ccc

cccc

cccc

cccc

cccc

cc

Equivalent Baseband Model for Bandpass Signals

Out of Transmitter Into ReceiverChannel

Equivalent baseband impulse response

fK2

1

We can now decouple the complexity of shifted frequency.

less.distortiony essentiall is distortionlinear Thus,

. from t reconstruc then to trivialisIt

.input, thefromdelay time

constant has and offactor aby changed amplitude its has channel

thefrom , output, thesatified, are conditions twoabove theIf

delay) group :( 2)(

.)10Typically change. magnitude(Constant )(

:conditions following the

satisfymust ,)()( function, transfer channel The

DistortionLinear

21

1

2

)(

(t)v(t)v

(t)v

A

(t)v

TTdf

fd

AAfH

efHfH

gg

fj

Distortionless Bandpass Channel

TT

TtwTtAy

TtwTtAx(t)v

twtytwtx(t)v

dg

dcg

dcg

cc

. general,In

sin)(

cos)(

then

,sin)(cos)( If

Example.

2

1

. wheresamplest independen2by

interval over time specified completely becan

tolimitedbandwidth with waveformbandpassA Theorem.

baseband.

at thenecessary frequency sampling the twiceis thisy,Essentiall

).( oftion reconstrucfor 4 and 2 Thus, .

and ,2

Assuming . todbandlimite

signal baseband a is )( wherecos)()( Suppose Example.

. where

2at signal sampled from tedreconstruc becan

tolimitedbandwidth with waveformbandpassA Theorem.

12

21

2

1

12

21

ffB TBN

Tfff

txBfBBBff

BffBw

fHzB

txtwtxtv

ffB

Bffff

ToT

o

sTc

cc

c

c

T

Ts

Definition. An filter is a linear time-invariant system used for frequency

discrimination. (Filters modify the frequency spectrum of input waveform

to produce output waveform.)

Quality Factor: ofQB

: center (or resonant) frequency, : 3 bandwidthof B - dB

Types of (Analog) Filter

Suited for PCM type signals

Implemented typically in software algorithms (e.g., DFT, DCT)

Practical for inherently digital signals

Advantages

- Flexible

- Ada

Digital Filters

ptive

- Finely tunable (i.e., can be idealized)

A/DDigital Filter D/A

analog x(t) analog y(t)

manipulate digital data

21 2

21 2

Filter Characterstics

( ) ( ) ... ( )Transfer function: ( )

( ) ( ) ... ( )

: order of the filter

Design goal: deter

ko k

no n

b b jw b jw b jwH f

a a jw a jw a jw

n

mine 's, 's, , and to result in a desired ( ).

There are many different types of filters (according to different classes of ( )).

i ia b n k H f

H f

Signals may be linearly or non-linearly distorted (by the filters, channels, etc).

- Distortion-less transfer function:

- Linear distortion: linear input-output relationship, but no

c djw TKe

21 2

0

0

t in form

(linearity is defined by ( ) ( ) ( ).)

- Non-linear distortion: ...

1 where

!

Fo

c d

i

jw T

o i

no o i i n i

n

no

n ni v

Ke

v t K t v t

v K K v K v K v

d vK

n dv

r non-linear distortion, there exists 0 for 2.

nK n

Example of Non-Linear Distortion by Output Saturation

Harmonic Distortion

1 1 2 2

This is a non-linear distortion at multiple harmonic frequencies of the input.

Test: Use a single frequency input ( ) sin and measure output ( ).

In general, ( ) cos cos 2i o o o

o o o o

v t A w t v t

v t V V w t V w t

3 3

2

2

1

22 2

2

22

cos 3 ...

Total Harmonic Distortion (THD%) 100

Example. Second Harmonic Distortion for ( ) sin

The second order term of the output = sin 1 cos 22

: DC bi2

o

nn

i o o

oo o o

o

V w t

V

V

v t A w t

K AK A w t w t

K A

22as distortion cos 2 : second harmonic distortion2

oo

K Aw t

2

2

Example. Output from filter: ( ) cos

1Power in each harmonic term =

2Since the harmonic terms are orthogonal to each other,

1 total power of output signal = .

2

Suppose that the

o n on

n

nn

v t V nw t

V

V

1

22 2

12

2

2

2

1

4input and output each has total power of 1 and .

1 1 4 2 1 2 1

2 2

1 42 1

2THD (%) 100 100 = 48.3%

4

nn

nn

V

V V

V

V

Intermodulation Distortion (IMD)

1 1 2 2

2 1

This is a non-linear distortion caused by the interference among multiple signal

Test: Use input with two sinusoidal frequencies: ( ) sin sin

In general, the second order output term:

s

iv t A w t A w t

K A

2 2 2 2 21 2 2 2 1 1 1 2 1 2 2 2

2 1 2 1 2

2 1 2 1 2 2 1 2 1 2 1 2

in sin sin 2 sin sin sin

2 sin sin : Intermodulation Distortion (IMD)

2 sin sin cos cos

If the bandpass region is narrow eno

w t A w t K A w t A A w t w t A w t

K A A w t w t

K A A w t w t K A A w w t w w t

1 2 1 2

1 2 1 2

ugh, both and

components fall outside the bandpass region (since , 0, >>0 ).

Thus the IMD caused by the second order outputs (i.e., the square terms)

is not significant.

w w w w

w w w w

3 3 3 2 23 1 1 2 2 3 1 1 1 2 1 2

2 2 3 31 2 1 2 2 2

3 1

Now consider the third order output term. In general,

sin sin sin 3 sin sin

3 sin sin sin

3

K A w t A w t K A w t A A w t w t

A A w t w t A w t

K A

2 2 22 1 2 3 1 2 2 1 2 1 2

2 2 23 1 2 1 2 3 1 2 1 2 1 2 1

1 2 2

3 1sin sin sin sin 2 sin 2

2 2

3 13 sin sin sin sin 2 sin 2

2 2

The IMD terms correspond to those with the 2 , 2

A w t w t K A A w t w w t w w t

K A A w t w t K A A w t w w t w w t

w w w

1 1 2

2 1 1 2 1

2 1 2 2 1

1 2

, 2 , and

2 . Again, if the bandpass region is narrow and , 0, and

0, then, the 2 and 2 frequency components fall outside the

bandpass region. However, 2 and 2

w w w

w w w w w

w w w w w

w w

2 1

1 2

1IMD 2

3

frequency components are

likely to be within the bandpass region. If , we define,

4: ratio of desired output to IMD output

3

w w

A A A

KR

K A

IMD Analysis for Filter Output

1 3

IMD

1IMD 10 2

3

12

3

12

3

2

Example. For a given filter let 1 and 0.01.

Suppose we want 40 .

Find maximum allowed input magnitude.

440 20log

3

4100

3

75

100 75

2

K K

R dB

KR dB

K A

K

K A

K

K A

A

3

A

Cross Modulation (Distortion)

This is a non-linear distortion caused by the modulating signal of one carrier

signal generating interferences to other carrier signals.

Test: Use input with two sinusoidal frequencies and one AM modul

1 1 2 2

223 1 2 1 2

ated.

( ) 1 ( ) sin sin .

Again, the second order output terms fall outside the bandpass region.

( ) contains the third order output terms in the form of

31 ( ) sin 2

2

i

o

v t A m t w t A w t

v t

K A m t A w w t

23 1 2 2 1

231 2 1 2 3 1

33 1

3 and 1 ( ) sin 2 .

23

If and , the two terms become 1 ( ) sin and 2

3 1 ( ) sin . These are the cross modulation distortion terms.

2

K A m t A w w t

w w A A A K A A m t w t

K A m t w t

LimiterA limiter is a non-linear circuit that compares the input to a certain threshold value.

The output indicates either comparison is true or false (i.e., binary results).

Typically the output is a saturated minimum or maximum value.

Mixer

This is a non-linear device whose output is the product of two inputs.

The result is a mathematical multiplication of real time values of the two inputs.

(Different from the audio mixers that combine multiple audio signals into

a single signal.)

input1(t)

input2(t)

output(t) = input1(t) x input2(t)

In communication, mixers are used with bandpass or low-pass filters to shift

the frequency of incoming signal and filter out the undesired components of

the multiplication.

in in in

in

1 in in

Let ( ) be a bandpass signal with ( ) =Re g ( ) .

Thus ( ) is a baseband signal containing the message.

( ) = Re g ( ) Re g ( )2 2

If the upper frequency is selected (u

c

c o c o

jw t

j w w t j w w to o

v t v t t e

g t

A Av t t e t e

2 in

2 in

2 in

2

p converter),

( ) = Re g ( ) 2

If the lower frequency is selected (down converter),

( ) = Re g ( ) if 2

or ( ) = Re g ( ) if 2

or

c o

c o

o c

j w w to

j w w toc o

j w w too c

Av t t e

Av t t e w w

Av t t e w w

v

in( ) = g ( ) if 2

oc o

At t w w

Mixer implementation

- Solid state devices (e.g., FET)

- Non linear devices

- Switching devices

The nonlinear device generates “undesired” effects of product term between vin(t) and vLO(t).

Mixer Implementation through Switching

Double-Balanced Mixer

1

Non-linear device that multiplies the input signal frequency by a factor of .

is a positive integer.

( ) ( ) cos ( )

( )

Frequency Multiplier

in c

n

n

v t R t w t t

v t

out

( ) cos ( ) other terms

( ) ( ) cos ( )

nc

nc

CR t nw t n t

v t CR t nw t n t

More on Frequency Multiplier

If the message ( ) is contained in ( ), then it is distorted in a non-linear

way, i.e., ( ). (Hence, this is not good for AM modulated signals.)

If the message ( ) is contained in ( ), then it i

n

m t R t

R t

m t t s NOT distorted, rather it is

amplified by a factor of . This may be a desired property for FM or PM

modulated signal. However, the amplification of the angle makes the required

bandwidth larger.

n

Detector Circuits

Source Destination

Suppose that the sent signal ( ) (which becomes ( ) and ( )) was received

(as ( )) and frequency translated to a baseband signal ( ( )). Now the task

is tp extract ( ) (approximating ) fro

m t g t s t

r t g t

m t m(t)

m ( ). This task is called the

"detection." I.e., the detector circuits generate ( ), ( ), ( ), and/or ( )

from ( ).

There are different ways to implement the detector circuits. Common ones a

g t

R t t x t y t

g t

re:

Envelop Detector (for AM signals)

Product Detector (for AM and PM signals)

Frequency Modulation Detector (for FM signals)

Envelop Detector ( ) ( )cos ( )

( ) ( )

The time constant must

be chosen so that the output of

the low pass filter follows the

envelop of the input signal, ( ).

The bandwidth of the low pass

filt

in c

out

in

v t R t w t t

v t KR t

RC

v t

er must be much smaller than

and much larger than the bandwidth

of ( ).

1

2

c

c

f

m t

B f

low pass filter

The envelop detector is suted for AM signals. For AM signals,

( ) 1 ( ) . ( 0 is the strength of the signal. ( ) 1.)

( ) ( ) ( ) 1 ( ) ( )

can be used for automatic gai

c c

out c c c

c

g t A m t A m t

v t KR t K g t KA m t KA KA m t

KA

n control.

( ) is the detected signal.

1For AM radios, the cutoff frequency of the low pass filter, 20 ,

210 , and 500 1600 .

c

c

KA m t

kHzRC

B kHz kHz f kHz

Product Detector

Product detector uses a mixer and manipulates the input signal frequency.

1

- ( )

( ) ( )cos ( ) cos

1 1 ( )cos ( ) ( )cos 2 ( )

2 21

After low pass filtering, ( ) ( )cos ( ) .2

1If we write ( ) Re ( ) , then ( ) ( ) (

2o

c o c o

o o o c o

out o o

j j tout o

v t R t w t t A w t

A R t t A R t w t t

v t A R t t

v t A g t e g t R t e x

o

) ( ).

With the oscillator signal frequency equalling the input signal frequency,

the detector is "frequency synchronized" (or "tuned") with the incoming signal.

If 0, then the detector output, o

t jy t

v

o

1( ) ( ).

21

If 90 , then the detector output, ( ) ( ).2

ut o

oout o

t A x t

v t A y t

o

1If ( ) 0, ( ) ( ). (AM demodulation)

21

If ( ) (constant) and 90 , ( ) sin ( ).2

1This becomes ( ) ( ) if ( ) is small. (Phase demodulation)

2Thus, the product det

out o

oc out o c

out o c

t v t A R t

R t A v t A A t

v t A A t t

ector is sensitive to AM and PM.

Frequency Modulation Detector

The input signal, ( ), contains the message, ( ), as part of its

frequency information.

( ) ( )cos ( )

( )( ) ( )

( )If the detector is balanced, ( ) .

I

in

in c

out c c

out

v t m t

v t R t w t t

d d tv t K w t t K w

dt dt

d tv t K

dt

mplementation methods

FM-to-AM conversion

Phase-shift or quadrature detection

Zero-crossing detection

1

Let ( ) ( )cos ( ) , where ( ) ( ) s.

( ) has nothing to do with the message, ( ). It is a result of something

created during the transmission (e.g., interferences, noise, etc.)

( )

t

in c fv t A t w t t t K m s d

A t m t

v t

2

cos ( )

( )( ) sin ( )

( )( ) ( )

L c

L c c

out L c L c L f

V w t t

d tv t V w w t t

dt

d tv t V w V w V K m t

dt

Slope Detector (FM-to AM Conversion)

Slope Detector Circuit

Balanced Discriminator

Balanced Zero-Crossing Detector

A PLL is a device whose output is a periodic signal synchronized in phase

with an input reference (periodic or almost periodic) signal.

Phase Lock Loop (PLL)

2

VCO: oscillator that produces a period waveform with a frequency that may

be varied around free running frequency, .

VCO output frequency = when ( ) 0.

Phase Detector (PD): o

o

o

f

f v t utput is a function of the phase difference between

incoming signal ( ) and ( ).

PLL has two modes:

- Narrowband mode: tracks average frequency of ( ).

- Wideband mode: t

o in

in

v t v t

v t

racks instantaneous frequency of ( ).

Lock: When the PLL tracks the (average or instantaneous) frequency of ( ).

Hold-in range: When the PLL is in lock, the range of frequency of ( ) to remain

in

in

in

v t

v t

v t

in lock. (Also called the lock range.)

Pull-in range: When the PLL is not in lock, the range of frequency of ( ) to

capture a lock. (Also called the cainv t

pture range.)

Maximum locked sweep range: When the PLL is in lock, the maximum rate

of change of the frequency of ( ) to remain in lock.

A PLL can be made in analog (APLL) or dinv t

igital (DPLL) circuits.

Different Phase Detector Characteristics

2

1

Let input be ( ) sin ( )

Suppose ( ) cos ( ) where ( ) ( ) .

: VCO gain in rad / volt-second

( ) sin ( ) ( ) sin 2 ( ) ( ) 2 2

: multiplier gain,

in i o i

t

o o o o o v

v

m i o m i oi o o i o

m

v t A w t t

v t A w t t t K v d

K

K A A K A Av t t t w t t t

K

2

no unit

( ) sin ( ) ( )

Phase error: ( ) ( ) ( )

Equivalent PD gain: 2

Impulse response of LPF: ( )

d e

e i o

m i od

v t K t f t

t t t

K A AK

f t

The PLL is a negative feedback system such that the error ( ) ( ) ( )

is minimized instantaneously. This is true if is large enough (i.e., sensitive

to the incoming phase ( )).

Note: When

e i o

d

i

e

t t t

K

t

o( ) 0, ( ) and ( ) are 90 apart since they are sine and

cosine functions respectively. This is needed to use sin

approximation after the PD.

Then, the overall charatersi

in ot v t v t

x x

tics of the PLL becomes

( ) ( )( ) ( )

( ) ( )( ) : Transfer function of PLL

( ) 2 ( )

( ) ( ) and ( ) ( )

Note. ( ) is a transfer function for a linear system

e id v e

o d v

i d v

o o i i

d t d tK K t f t

dt dtf K K F f

H ff j f K K F f

t f t f

H f

. I.e., the PLL is linearized

based on approximations and assumptions used here.

Linearized PLL Model

( ) ( )

( ) 2 ( )o d v

i d v

f K K F f

f j f K K F f

Hold-in Range and Pull-in Range

Hysteresis indicatesstored energy (or inertia)in the PLL.

Hysteresis is useful againstnoises or unexpected interruptionsin received signals. This iscalled the “anti ping-pong” characteristic.

2

Hold-in range calculation.

( ) ( ) sin ( ) ( )

(0)sin ( ) (assuming the input frequency changes slow enough.)

1(0)

2

Pull-in range is determined by the physical

ov v d e

v d e

h v d

d tK v t K K t f t

dtw K K F t

f K K F

characteristics of

components in the PLL (e.g., PD, VCO).

Note: is always true for PLL.h pf f

2

Let ( ) sin ( )

( ) = ( ) and ( ) ( )2

Assume that the PLL is in lock (i.e., ).

2( )

( )

PLL as FM Detector

t

in i c f

t fi f i

c o

v

v t A w t D m d

Dt D m d f M f

j f

f f

fj F f

KV f

2

2

( )

( ) ( )2 2

( ) ( )

( ) ( ) (Assuming ( ) 1 for , and .)2

( ) ( )

f

vi i

v d v d

f v d

v

DF f

Kf f

f fF f j F f j

K K K K

D K KV f M f F f f B B

K

v t Cm t

A coherent detector uses an internally generated sinusoidal signal that has

the same frequency and phase with the incoming signal to detect th

PLL as Coherent AM Detector

o o

e messages.

The - 90 phase shifter is needed to eliminate the +90 phase shift between

( ) and ( ) in the basic PLL shown earlier.o inv t v t

( ) has the frequency, = where is the frequency of the input

signal, ( ), which is generated by a local oscillator.

PLL as Frequency Synthesizer

o out x x

x

Nv t f f f

M

v t

Direct Digital Synthesis (DDS)One way to generate any waveform is to store the desired waveform in the

memory (e.g., RAM or ROM), and play-back according to the desired timing.

- For arbitrary waveforms, signal is sampled/recorded for a desired length.

- For periodic waveforms, signals can be stored for one period only.

Playback repeats the same pattern over and over.

- For mathematical functions, signal values can be calculated and stored.

With DDS devices, one can eliminate unreliable analog devices

(e.g., oscillator). DDS devices are also useful for sound and music synthesis.

Generalized Transmitter (Type 1)

Bandpass signal (i.e., signal to be transmitted in the channel)

( ) Re ( ) ( ) ( ) cos ( )

Thus the messsage, ( ), is modulated in either AM or PM signal.

cjw tcv t g t e v t R t w t t

m t

Generalized Transmitter (Type 2)

( ) Re ( ) ( ) ( ) cos ( )sin

Thus the messsage, ( ), is modulated in quadrature signal.

cjw tc cv t g t e v t x t w t y t w t

m t

Generalized ReceiverNearly all receivers are superheterodyne type. Superheterodyne implies

there are two different local oscillator frequencies that the incoming

signals are manipulated. (Heterodyne receivers have only one local operating

frequencies.) Two frequencies are: intermediate frequency (IF) and baseband.

Superheterodyne receivers need two local oscillators: one to bring

the incoming signals to the IF frequency, and another one to bring down

the IF frequency signals to the baseband to recover the messages.

The reasons for using the IF filter:

- Making a high quality filter is expensive. Design one good IF filter and use

it to achieve high gain and rejection of unwanted signals.

- Selection of IF filter frequency can be made for better signal processing

capability such as rejection of image frequency, reduction of non-linear

effects, reduction of noise, etc.

Image frequency is those happened to fall in the bandpass region of the filter

after the initial frequency translation.

2 if For down converters:

2 if

For up converters: 2

c IF LO cimage

c IF LO c

image c IF

f f f ff

f f f f

f f f

Example of Image Frequency

Zero IF ReceiverZero IF receivers has a center frequency of its bandpass region at 0.

Thus it is a lowpass filter. Sometimes it is desired to use a zero IF receiver

to reduce the cost and design a high quality fil

IFf

ter. For zero IF receivers, there

is no image frequency. When using a zero IF receiver, the system is a

heterodyne system.

At the sending transmitter: impurity in sinusoidal carrier

At the receiver: signal overload, non-linearity in RF and IF filters,

Possible Sources of Interference

cross modulation, adjacent signals

In the channel: non-linearity in the transmission medium, signals from adjacent

broadcasting region.

Note. If the receivers were made in digital circuit, the incoming signal must be sampled at the bandpass frequency. It is not easy to do so.

2sin cw t