ECE 461/561

COMMUNICATIONS I

Dr. Mario Edgardo Magaña

School of EECS

Oregon State University

2

This course deals with the fundamental aspects of analog electronic

telecommunication systems. The first part of the course focuses on learning how

to model, analyze, and design analog amplitude and angle modulated

communication systems (and their variants) and to review the principles of

random processes, in order to model a random noisy communication channel.

The last part of the course is devoted to evaluate analog communication system

performance when a modulated signal is transmitted through a random noisy

channel and to introduce the discretization of analog communication signals to

understand the transition to digital communication systems.

3

4

OBJECTIVE: The objective of a communication system is to efficiently transmit

an information-bearing signal (message) from one location to another through a

communication channel (transmission medium).

WISH LIST FOR EFFICIENT COMMUNICATION:

• High reliability

• Minimum transmitted signal power

• Minimum transmission bandwidth

• Low implementation complexity (low cost)

EFFICIENT TRANSMISSION: Can be achieved by processing the signal through

a technique called modulation.

MODULATION TYPES:

• Analog: Amplitude and angle modulation

• Digital: Amplitude, angle and hybrid (amplitude and angle together) modulation

5

MATHEMATICAL PRELIMINARIES

Let a complex signal in polar form be described by 0( )

j tx t Ae

. Then,

0 0( ) cos sin ( ) ( )r ix t A t jA t x t jx t .

Thus, the real and imaginary parts of ( )x t are

0( ) Re ( ) cosrx t x t A t and 0( ) Im ( ) sinix t x t A t .

Clearly, ( )( ) ( ) j tx t x t e , where

2 2( ) ( ) ( )r ix t x t x t and 1 ( )

( ) tan( )

i

r

x tt

x t

.

Even and Odd signals

( )x t is an even signal iff ( ) ( )x t x t and ( )x t is an odd signal iff ( ) ( )x t x t .

In general,

( ) ( ) ( )e ox t x t x t ,

6

where

( ) ( )( )

2e

x t x tx t

and

( ) ( )( )

2o

x t x tx t

.

Example: Let ( )x t be described by 0( ) cosx t A t . Then

0 0( ) cos cos( ) sin sin( )x t A t A t .

Since 0 0cos( ) cos( )t t and 0 0sin( ) sin( )t t , we get

0

( ) ( )( ) cos cos( )

2e

x t x tx t A t

and 0

( ) ( )( ) sin sin( )

2o

x t x tx t A t

.

Energy and Power Signals

A signal ( )x t is an energy signal iff its energy /2 2

/2lim ( )

T

xTT

E x t dt

.

A signal ( )x t is a power signal iff its power xP is such that /2 2

/2

10 lim ( )

T

xTT

P x t dtT

.

Example: If ( )x t is bounded over a finite time interval and zero outside that interval, then ( )x t

is an energy signal.

7

Example: Let 0( ) cosx t A t . Then we can show that ( )x t is a power signal, i.e.

2

/2 /2 22

0/2 /2

1 1lim ( ) lim cos

2

T T

xT TT T

AP x t dt A t dt

T T

.

Fourier Series and Fourier Transforms:

Let ( )x t be a periodic signal, i.e. ( ) ( )x t x t nT , where T is the period of repetition. Then if

( )x t satisfies the Dirichlet conditions, i.e.

a) ( )x t is absolutely integrable over one period T, i.e.

0

0

( )t T

tx t dt

, for arbitrary 0t .

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

c) The number and size of the discontinuities of ( )x t in each period is finite.

the periodic signal ( )x t has a Fourier series described by

0

0

2( ) ,

jn t

n

n

x t x eT

,

where 0

0

0

1( )

t Tjn t

nt

x x t e dtT

.

8

Let ( )x t be an aperiodic signal. Then if ( )x t satisfies the Dirichlet conditions, i.e.

a) ( )x t is absolutely integrable over the real line, i.e.

( )x t dt

.

b) The number of maxima and minima of ( )x t in any finite interval over the real line is

finite.

c) The number and size of the discontinuities of ( )x t in any finite interval over the real line

is finite.

the signal ( )x t has a Fourier transform in the Hz frequency domain described by

2( ) ( ) j f tX f x t e dt

and the original signal can be recovered from

2( ) ( ) j f tx t X f e df

Parseval’s Theorm:

Let ( )x t be a periodic signal, i.e. ( ) ( )x t x t nT , where n is an integer and T is the period of

repetition. Then the power content of ( )x t is given by

9

0

0

2 21( )

t T

x nt

n

P x t dt xT

.

Let ( )x t be an aperiodic signal which possesses a Fourier transform ( )X f . Then

2 2( ) ( )x t dt X f df

.

Lowpass and Bandpass Signals:

Let ( )x t be described by ( ) ( )cos 2 ( )cx t A t f t t , where ( )A t and ( )t are slowly

varying signals of time (they have small frequency content around zero frequency, i.e. they are

low pass signals). Then,

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

( )cos 2 ( )sin 2

c c

I c Q c

x t A t t f t A t t f t

x t f t x t f t

,

where the in-phase and quadrature components ( )Ix t and ( )Qx t are given by

( ) ( )cos ( )Ix t A t t and ( ) ( )sin ( )Qx t A t t ,

respectively.

10

Hence, ( )Ix t and ( )Qx t are lowpass signals and ( )x t is a bandpass signal with its spectrum

around the center frequency cf . The lowpass complex envelope of ( )x t is given by

( ) ( ) ( )l I Qx t x t jx t

and

2

2

( ) Re ( ) Re ( ) ( ) cos 2 sin 2

Re ( )cos 2 ( )sin 2 ( )cos 2 ( )sin 2

Re ( )cos 2 ( )sin 2 ( )sin 2 ( )cos 2

( )cos 2 ( )sin 2

cj f t

l I Q c c

I c I c Q c Q c

I c Q c I c Q c

I c Q c

x t x t e x t jx t f t j f t

x t f t jx t f t jx t f t j x t f t

x t f t x t f t j x t f t x t f t

x t f t x t f

( ) cos 2 ( )c

t

A t f t t

.

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Standard Amplitude Modulation (AM)

In this type of modulation the amplitude of a sinusoidal carrier is varied

according to the transmitted message signal. Let m(t) be the message signal

we would like to transmit, ka be the amplitude sensitivity (modulation index), and

c(t) = Accos(2πfct) be the sinusoidal carrier signal, where Ac is the amplitude of the

carrier and fc is the carrier frequency. Then the transmitted standard AM signal

waveform is described by

Requirements:

1. , to avoid overmodulation and phase reversals

2. , where Wm is the message bandwidth

( ) 1 ( ) cos(2 )AM c a cs t A k m t f t

ttmka ,1)(

mc Wf

12

Taking the Fourier transform of the modulated waveform, we get

Let be described by

Magnitude spectrum of m(t)

( ) ( ) 1 ( ) cos(2 )

cos(2 ) ( )cos(2 )

( ) ( ) ( ) ( )2 2

AM AM c a c

c c c a c

c a cc c c c

S f s t A k m t f t

A f t A k m t f t

A k Af f f f M f f M f f

( )M f

13

Then the magnitude spectrum of sAM(t) is

Magnitude spectrum of sAM(t)

Observations:

1. The transmission bandwidth is

2. Carrier signal is transmitted explicitly (delta functions are present in

frequency spectrum)

mAM WB 2

14

Let the message signal be the frequency tone , then with )2.0cos()( ttm

( ) 1 cos(2 ) cos(2 ) 10 1 0.5cos 0.2 cos 2AM c a m cs t A k f t f t t t

15

16

To conserve transmitted power, let us suppress the carrier, i.e., let the

transmitted waveform be described by

This is called double side-band suppressed carrier (DSB-SC) modulation.

).2cos()()( tftmAts ccDSB

17

In the frequency domain,

The transmitted modulated waveform now has the following spectral

characteristics:

Magnitude spectrum of sDSB(t)

( ) ( )

( )cos 2

( )) ( ) ( )2 2

DSB

c c

c cc c c c

S f s t

A m t f t

A AM f f f f f M f f M f f

F

F

18

Observations:

1. Transmission bandwidth is (same as standard AM)

2. Transmitted power is less than that used by standard AM

Example: Let a sinc pulse be transmitted using DSB-SC modulation.

mDSB WB 2

19

20

Receivers for AM and DSB

Receivers can be classified into coherent and non-coherent categories.

Definition: If a receiver requires knowledge of the carrier frequency and phase to

extract the message signal from the modulated waveform, then it is called coherent.

Definition: If a receiver does not require knowledge of the phase (only rough

knowledge of the carrier frequency) to extract the message signal from the

modulated waveform, then it is called non-coherent.

Non-coherent demodulator (receiver) for standard AM

Peak envelope detector (Standard AM demodulator)

)(ˆ tm)(tS AMR C

+

- -

+D

21

Observations:

• The net effect of the diode is the multiplication (mixing) of the signals applied

to its input. Therefore, its output will contain the original input frequencies, their

harmonics and their cross products.

• The network is a lowpass filter with a single pole (lossy integrator) that

removes most of the high frequencies. R and C have to be judiciously picked

so that the time constant = RC is neither too short (rectifier distortion ) nor

too long (diagonal clipping ).

• A rule of thumb for choosing is based on the highest modulating signal

(message) frequency that can be demodulated by a peak envelope detector

without attenuation, that is,

where ( ) is the maximum modulating signal (message) frequency in

Hz (rad/s).

2 2

(max) (max)

1/ 1 1/ 1, 1

2

a a

a

m m

k kk

f

(max)mf (max)m

22

Let the message signal be described by the sinusoidal tone .

Then, the following graphs show the types of distortion that can occur when is

properly and improperly chosen.

tftm m2cos)(

Properly chosen

23

Illustration of rectifier distortion and diagonal clipping (with

improperly chosen )

Rectifier distortion

Diagonal clipping

24

Coherent demodulator for DSB-SC: Consider the following demodulator which

assumes fc has been estimated perfectly at the receiver, though is not known.

At the output of the mixer,

and

cos4cos)(2

ˆ

2cos)(2cosˆ

)(2cosˆ)(

tftmAA

tftmAtfA

tstfAtv

ccc

cccc

DSBcc

( )

ˆ ˆ2 2 cos

4 2

j jc c c cc c

V f F v t

A A A AM f f e M f f e M f

25

4

Let the message signal have the following magnitude spectrum

Then, if fc >>Wm,

-2 2

26

Suppose is such that

Then, if ,

fHlpf

mcm WfBW 2 cos2

ˆˆ fM

AAfM cc

27

Coherent Costas loop receiver for DSB-SC :

28

I-channel:

After downconverwsion,

At the output of the lowpass filter, with |H(0)| = 1,

Q-channel:

cos2cos)(2

coscos)()(

ttmA

tttmAtv

c

c

cccI

)(cos2

)( tmA

tm c

I

sin2sin)(2

)( ttmA

tv c

c

Q

)(sin2

)( tmA

tm c

Q

29

Feedback path:

At the output of the multiplier,

At the output of the feedback lowpass filter,

The purpose of hf (t) is to smooth out fast time variations of me(t)

The output of the VCO is described by

2sin)(8

cossin)(4

)(

22

2

2

tmA

tmA

tm

c

ce

dthmtm feef )()()(

( ) cos ( ) ,VCO cx t t t

30

where c is the VCO’s reference frequency and is the

residual phase angle due to the tracking error. The constant kv is the frequency

sensitivity of the VCO in rad/s/volt (it depends on the circuit implementation).

The instantaneous frequency in radians/sec of the VCO’s output is given by

Clearly, if (t) were small and slowly varying, then the instantaneous frequency

would be close to c and the output of the I-path would also be proportional to

the desired message signal m(t), since , for (t) small.

,)()(0t

efv dmkt

0( )( )

( ),

t

c v efc

c v ef

d t k m dd t tk m t

dt dt

cos 1

31

Angle Modulation

Let i(t) be the instantaneous phase angle of a modulated sinusoidal carrier,

i.e.,

where Ac is the constant amplitude.

The instantaneous frequency of the sinusoidal carrier is

Observation: The signal s(t) can be thought of as a rotating phasor of length Ac

and angle i(t), i.e.

),(cos)( tAts ic

.)(

)(dt

tdt i

i

( ) ( )c is t A t

32

If s(t) were an unmodulated carrier signal, then the instantaneous angle would be

where c Constant angular velocity in rad/s

c Constant but arbitrary phase angle in radians

Let i(t) be varied linearly with the message signal m(t) and c = 0, then

where kp Phase sensitivity of the modulator in rad/volt (circuit dependent)

In this case we say that the carrier has been phase modulated.

The phase modulated waveform is given by

Let the instantaneous frequency i(t) be varied linearly with the message signal

m(t), i.e.,

cci tt )(

),()( tmktt pci

))(cos()( tmktAts pccPM

)()( tmkt ci

33

where k Frequency sensitivity of the modulator in rad/s/volt (circuit dependent)

In this case we say that the carrier has been frequency modulated and the

instantaneous angle is obtained by integrating the instantaneous frequency, i.e.,

The modulated waveform is therefore described by

Observation: Both phase and frequency modulation are related to each other and

one can be obtained from the other. Hence, we could deduce the properties of one

of the two modulation schemes once we know the properties of the other. This is

illustrated in the following 2 block diagrams:

t

c

t

c

t

ii dmktdmkdt000

)()()()(

t

ccFM dmktAts0

)(cos)(

34

FM modulation

0( )

t

m d

35

PM modulation

( )dm t

dt

36

The figure below shows a comparison between AM, FM and PM modulation of

the same message waveform:

37

Frequency Modulation

Consider the frequency modulation of a message signal (frequency tone)

The instantaneous frequency (in Hz) of the FM signal is

Define the maximum frequency deviation as

Since

The instantaneous phase angle of the FM signal is

),2cos()( tfAtm mm

( ) cos(2 ), since 2 and 2 .i c f m m c c ff t f k A f t f k k

max ( ) ,i c f mf f t f k A

0 0

( ) 2 ( ) 2 cos 2

2 sin(2 )

2 sin(2 )

t t

i i c f m m

f m

c m

m

c m

t f d f k A f d

k Af t f t

f

f t f t

0

( ) cos cos 2

t

FM c c m ms t A t k A f d

max ( ) max cos(2 ) .i c f m m c f m cf t f k A f t f k A f f

38

where , is known as the FM modulation index (for a tone) or

the maximum phase deviation (in rad) produced by the tone in question

The FM modulated tone is therefore given by

This sFM(t) signal is nonperiodic unless fc = nfm, where n is a positive integer.

For the general case, fc is not necessarily a multiple of fm. Now,

where the complex envelope of the FM signal is described by

Observation: Unlike sFM(t), se(t) is periodic with period 1/fm.

/ /f m m mk A f f f

( ) cos 2 sin(2 )

cos(2 )cos( sin(2 )) sin(2 )sin( sin(2 )) .

FM c c m

c c m c m

s t A f t f t

A f t f t f t f t

2 sin(2 )

sin(2 ) 2 2

( )

( ) ,

c m

m c c

j f t f t

FM c

j f t j f t j f t

c e

s t e A e

e A e e e s t e

sin(2 )( ) mj f t

e cs t A e

39

Since se(t) meets the Dirichlet conditions, we can compute its Fourier series, i.e.,

where the Fourier series coefficients are given by

,)(2 tfnj

n

nemects

.dtefA

dteeAf

dte)t(sf

f/T,dte)t(sT

c

m

m

mm

m

m

mm

m

m

m

m

f/

f/

tfn)tfsin(j

mc

f/

f/

tfnj)tfsin(j

cm

f/

f/

tfnj

em

m

/T

/T

tfnj

en

21

21

22

21

21

22

21

21

2

2

2

21

1

40

Let

Then,

and

where is the Bessel function of the first kind of order n.

Alternatively,

where the Gamma function (t) is defined by

.tfx m2

dxf

dtm2

1

),(JA

dxeA

c

nc

nxxsinjcn

2

dxe)(J nxxsinj

n2

1

2

0

( 1)( ) ,

! ( 1) 2

m nm

n

m

Jm m n

1

0

( ) t xt x e dx

41

Plot of Jn(x), n = 0, 1, 2

Therefore,

and the FM tone waveform is described by

n

tnfj

ncemeJAts

2)()(

2 2( ) ( ) ( )cos 2 ( ) .m cj nf t j f t

FM c n c n c m

n n

s t e A J e e A J f nf t

42

Spectrum of the FM modulated frequency tone:

Average power of the FM waveform:

The average power delivered to a 1 ohm load resistor by the FM waveform is

Now,

is also the average power of the FM waveform delivered to a 1 ohm resistor.

( ) ( )

( ) cos 2 ( )

( ) cos 2 ( )

( )( ) ( )

2

FM FM

c n c m

n

c n c m

n

nc c m c m

n

S f s t

A J f nf t

A J f nf t

JA f f nf f f nf

2/2

cAP

/2 2

/2

1lim ( )

T

FMTT

P s t dtT

43

Examples:

44

It can be shown, in the limit, that

Let us now take a look at the properties of the Bessel function.

1.

2.

Hence, the average power of an FM tone is, as expected,

Suppose is small, i.e., 0 < ≤ 0.3, then

•

•

•

Under the assumption that is small, the Fourier series representation of the FM

waveform can be simplified to three terms.

)()1()( n

n

n JJ

1)(2

n

nJ

.2/2

cA

1)(0 J

2/)(1 J

2,0)( nJ n

2/2 2 2

/2

1lim ( ) ( )

2

Tc

FM nTT

n

As t dt J

T

45

Thus, for small, the FM tone may be described by

In the frequency domain,

)t)ff(cos(A)t)ff(cos(A)tfcos(A

)t)ff(cos()t)ff(cos()tfcos(A)t(s

mccmcccc

mcmcccFM

22

22

2

22

22

2

mcmc

c

mcmc

c

cc

c

FM

ffffffA

ffffffA

ffffA

fS

4

42)(

46

A plot of the magnitude spectrum of the FM tone with small is shown below

The time domain FM waveform can be represented in phasor form as follows:

For arbitrary t = t0, and small , we can illustrate graphically the phasor

representation and arrive at some conclusion.

tfAtfAAS mcmccFM 22

12

2

10

47

The following figure shows an example of the phasor representation

Observation:

The resultant phasor , has magnitude and is out of phase with

respect to the carrier phasor

Analytically,

FMS

,cFM AS

.0cA

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

1 tfjtftfjtfAAS mmmmccFM

48

But,

and

Consequently, the resultant phasor (in rectangular form) is given by

The magnitude of the resultant may be approximated by

since , n = ½ and in our case.

)2cos(

)2sin(sincos)2cos()2cos(0

tf

tftftf

m

mmm

)2sin(

)2cos(sincos)2sin()2sin(0

tf

tftftf

m

mmm

)2sin( tfjAAS mccFM

1/22 2 2 2 2 2

2 2

sin (2 ) 1 sin (2 )

11 sin (2 ) ,

2

FM c c m c m

c m

S A A f t A f t

A f t

(1 ) 1 , 1nx nx nx 2 2sin 2 mx f t

49

Finally, the magnitude and phase of the resultant are found to be

Observation:

• For an FM tone, the spectral lines sufficiently away from the carrier may be

ignored because their contribution (amplitude) is very small when β is small.

FM Transmission Bandwidth:

For an FM tone, as becomes large Jn() has significant components only for

All significant lines are contained in the frequency range

where f is the peak frequency deviation.

)4cos(

441)(

22

tfAtS mcFM

1 1sin(2 )( ) tan tan sin(2 )c m

FM mS

c

A f tS t f t

A

.// mmmf fffAkn

,ffff cmc

50

Let be small, i.e., 0 < ≤ 0.3, then

and only the first pair of spectral lines are significant, i.e., the significant lines are

contained in the range fc± fm

Observation: The previous analysis of an FM tone suggests that

1. For large the FM bandwidth is

2. For small the FM bandwidth is

In general, the FM transmission bandwidth may be approximated by

This is known as the Carson’s rule.

Observation: Carson’s rule underestimates the transmission bandwidth by about

10%.

0),()(0 nJJ n

2FMB f

.2 mFM fB

)/11(2

)/1(2

22

f

fff

ffB

m

mT

51

Alternative definition of FM tone transmission bandwidth:

A band of frequencies that keeps all spectral lines whose magnitudes are greater

than 1% of the unmodulated carrier amplitude Ac, i.e.,

where

,2 max mT fnB

.01.0)(:maxmax nJnn

52

General Case:

Let an arbitrary message signal m(t) have bandwidth Wm.

Define the peak frequency deviation and the deviation ratio by

and

Then Carson’s rule can be used to define the transmission bandwidth of an

arbitrary FM signal, i.e., when m(t) is arbitrary.

Specifically, the FM transmission bandwidth can be defined by

)(maxˆ tmkft

f

./ˆ mWfD

2 2

12 1 2 1

2 1 2 1

m

m

m m

m

BT f W

Wf f

f D

fW W D

W

53

Example: In commercial FM in the US, f = 75 kHz, Wm = 15 kHz.

Therefore, the deviation ratio is D = 75 kHz/15 kHz = 5.

Using Carson’s rule, the transmission bandwidth is

Using the Universal curve, the transmission bandwidth is

In practice, FM radio in the US uses a transmission bandwidth of BT = 200 kHz.

Generation of FM

The frequency of the carrier can be varied by the modulating signal m(t) directly or

indirectly.

,180)/11(2 kHzDfBT

.kHzf.BT 24023

54

Direct generation of FM

If a very high degree of stability of the carrier frequency is not a concern, then we

can generate FM directly using circuits without external crystal oscillators. Examples

of this method are VCO’s, varactor diode modulators, reactance modulators, Crosby

modulators (modulators that use automatic frequency control), etc..

Reactance FM modulator

m(t)

+

-

R2

R1

R3

C1

RFC1

C2

R2 C3

R6

R5RFC2

R7 C4

C7

C6

C5

L1 L2

+

-

+VCC

)(tsFM

55

Indirect generation of FM

Commercial applications of FM (as established by the FCC and other spectrum

governing bodies) require a high degree of stability of the carrier frequency. Such

restrictions can be satisfied by using external crystal oscillators, a narrowband

phase modulator, several stages of frequency multiplication and mixers.

Let us begin with the synthesis of narrow-band FM.

Narrowband frequency modulator

The narrow band FM signal is given by

t

fccNB dmktfAts0

)(22cos)(

56

with kf (and thus fNB) small, since m(t) is in the mV range.

Let us now consider a nonlinear technique to increase the FM signal bandwidth.

Let sNB(t) be input to a nonlinear device with transfer characteristic y(t) = axn(t),

where x(t) is its input, namely,

Nonlinear device.

Let , then at the output of the nonlinear device, we

observe

Let us expand this last equation to infer the effect of this nonlinear device.

t

fci dmktft0

)(22)(

)(cos)( taAty i

nn

c

n

a( )NBs t ( )y t

57

cosni(t) can be expanded as follows:

Likewise,

Thus,

Expanding the last term of the last equation, we get

)(cos)(2cos2

1)(cos

2

1

)(cos)(2cos12

1

)(cos)(cos)(cos

22

2

22

ttt

tt

ttt

i

n

ii

n

i

n

i

i

n

ii

n

)(cos)(2cos2

1)(cos

2

1)(cos 442 tttt i

n

ii

n

i

n

)(cos)(2cos4

1)(cos)(2cos

4

1)(cos

2

1)(cos 4242 tttttt i

n

ii

n

ii

n

i

n

2 4 41cos 2 ( )cos ( ) 1 cos 4 ( ) cos ( ).

2

n n

i i i it t t t

58

Rewriting the equation before the last one, we get

The last term in the expansion of cosni(t) is given by

)(cos)(6cos32

1

)(cos)(2cos32

1)(cos)(4cos

16

1

)(cos)(2cos4

1)(cos

8

1)(cos

2

1

)(cos)(4cos8

1)(cos

8

1

)(cos)(2cos4

1)(cos

2

1)(cos

6

66

442

44

42

tt

tttt

tttt

ttt

tttt

i

n

i

i

n

ii

n

i

i

n

ii

n

i

n

i

n

ii

n

i

n

ii

n

i

n

).(cos)(cos2

11

ttk i

kn

ik

59

Let n be an even number, then, when k = n, the last term is

If, on the other hand, n is an odd number, then when k = n-1, the last term is

Therefore, the last term in the expansion of cosni(t) is always

So, y(t) can be expanded as

)(cos2

11

tn in

)(cos2

1)()2cos(

2

1)(cos)()1cos(

2

1112

tntnttn ininiin

)(cos2

11

tn in

)(cos2

)(2cos)(cos)(1210 tn

Aatctccty in

n

c

ii

60

Example: Consider the cases when n = 2 and n = 3.

Let n = 2, then

or

Let n = 3, then

)(cos)( 22 taAty ic

)(2cos222

)(2cos1)(

22

2 taAaAt

aAty i

cci

c

)(3cos4

)(cos4

3

)(cos2

1)(3cos

2

1

2

1)(cos

2

1

)(cos)(2cos2

1)(2cos

2

1)(

33

3

3

taA

taA

tttaA

tttaAty

i

c

i

c

iiic

iiic

61

Finally,

Let y(t) be input to an ideal bandpass filter with unity gain, bandwidth wide enough

to accommodate spectrum of a wide band signal with center frequency nfc, i.e.,

Ideal bandpass filter

Then,

t

fcn

n

c

t

fc

t

fc

dmkntfnA

a

dmktfcdmktfccty

0

1

0

2

0

10

)(22cos2

)(44cos)(22cos)(

t

fcn

n

cWB dmkntfn

aAts

0

1)(22cos

2)(

( )BP

H f ( )WB

s t( )y t

Ideal filter

62

The instantaneous frequency in Hz of sWB(t) is

Observations about sWB(t) :

1. The carrier frequency is nfc

2. The peak frequency deviation is nfNB

These are the desired properties of the WB FM signal.

The overall frequency multiplier device is shown below:

Complete frequency multiplier

( ) ( )i c ff t nf nk m t

( )BP

H f 1

cos2

n

cin

aAn t

Ideal filter

center cf nf

( )na cosc iA t

63

Example: Noncommercial FM broadcast in the US uses the 88-90 MHz band and

commercial FM broadcast uses the 90-108 MHz band (divided into 200 kHz

channels). In either case f = 75 kHz. Suppose we target a station with fc = 90.1

MHz. Then the indirect FM generation method suggested by Armstrong enables

us to achieve our goals.

Let us start with a 400 kHz crystal oscillator and a narrow band phase modulator

with f = 14.47 Hz.

64

Armstrong indirect method of FM generation

33.808

fc = 1.408

fc = 1.408

65

Demodulation of FM signals

Consider the following receiver architecture

Frequency discriminator implementation of an FM demodulator

The slope circuit is characterized by purely imaginary transfer functions

.2,1),( isH i

66

Let H1(f ) be described by

Graphically,

elsewhere,0

2/2/),2/(2

2/2/),2/(2

)(1 TcTcTc

TcTcTc

BffBfBffaj

BffBfBffaj

fH

2

Tc

Bf

2

Tc

Bf cf

2

Tc

Bf

cf2

Tc

Bf

1( )H f

2 Tj aB

2 Tj aB

f

67

where a > 0 is a constant that determines the slope of H1(s ).

Define G1(f ) H1(f )/j, then is the impulse response of a real

bandpass system described by G1(f ).

In the time domain,

where g1,I(t) and g1,Q(t) are the in-phase and quadrature components of g1(t).

Therefore, the complex envelope of g1(t) is described by

which implies that

Using this information, we get

1 1, 1,( ) ( )cos(2 ) ( )sin(2 )I c Q cg t g t f t g t f t

)()()(~,1,11 tjgtgtg QI

2

1 1( ) ( ) cj f tg t e g t e

)(2

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

)()()()()(~)(~

1

,1,1

2

,1,1

2

,1,1

2

1

2

1

tg

tftgtftg

etjgtgetjgtgetgetg

cQcI

tfj

QI

tfj

QI

tfjtfj cccc

1

1 1( ) ( )g t G f

68

But,

or

since

which implies that has a lowpass frequency response limited to

and that

Observations: can be obtained by taking the part of G1( f ) that corresponds

to positive frequencies, shifting it to the origin and then scaling it by a factor of 2.

In the next figure is replaced by and replaced by

2 2

1 1 12 ( ) ( ) ( )c cj f t j f tg t F g t e g t e

))((~

)(~

)(2 111 cc ffGffGfG

)(~

1 fG 2/TBf

1 1( ) 2 ( ), 0.cG f f G f f

)(~

1 fG

)(1 fG jfH /)(1 )(~

1 fG

./)(~

1 jfH

* *

1 1( ) ( )g t G f

69

Frequency responses of H1(f ), and H2(f )),(~

1 fH

70

From the previous derivation,

But,

which implies that

If s1(t) is the output of the slope filter H1(f ) when the input is sFM(t) then the complex

envelope of the output is

1

4 ( / 2), / 2( )

0, elsewhere

T Tj a f B f BH f

cos ( )

( )

tf 0 c

c

tj2πk m(τ)dτ j2π f t

FM c c f c

0

j2π f t

FM

s (t)= A 2π f t+2πk m τ dτ = e A e ×e

= e s t ×e .

)(~)(~

2

1)(~

11 tsthts FM

( )t

f 0j2πk m(τ)dτ

FM cs t A e

71

In the frequency domain,

But,

which implies that in the time domain we get

1 1

1( ) ( ) ( )

2

2 ( / 2) ( ) 2 ( ) ( ), / 2

0,

FM

T FM FM T FM T

S f H f S f

j a f B S f a j f S f j aB S f f B

elsewhere

( )2 ( )

dx tj f X f

dt

F

0 0

0

1

2 ( ) 2 ( )

2 ( )

( )( ) ( )

2 ( )

21 ( )

t tf f

tf

FMT FM

j k m d j k m d

c f c T

j k m df

c T

T

ds ts t a ja B s t

dt

j aA k m t e j aA B e

kj aA B m t e

B

72

Therefore,

Observation: s1(t ) contains both AM and FM.

However, if

0

2

1 1

2 2 ( )

0

0

( ) ( )

21 ( )

21 ( ) sin 2 2 ( )

21 ( ) cos 2 2 ( )

2

c

tc f

j f t

j f t j k m df

c T

T

tf

c T c f

T

tf

c T c f

T

s t e s t e

ke j aA B m t e

B

kaA B m t f t k m d

B

kaA B m t f t k m d

B

ttmB

k

T

f ,1)(2

73

Then a distortionless envelope detector can extract m(t) plus a bias, i.e.,

Finally, if

then

Moreover,

The cascade of a slope circuit and an envelope detector is known as a frequency

discriminator.

.)(2

1)(1

tm

B

kBaAts

T

f

Tce

)(~

)(~

12 fHfH

.)(2

1)(2

tm

B

kBaAts

T

f

Tce

1 2ˆ ( ) ( ) ( )

4 ( )

e e

c f

m t s t s t

aA k m t

74

Frequency discriminator

FM demodulation via phase-locked loops

We consider the phase-locked loop (PLL) FM detector shown below

Phase-locked loop FM detector

75

From the previous block diagram, assuming an ideal propagation channel, and

The phase detector is modeled by

then,

kd depends on the multiplier implementation.

))(sin()(

))(cos()(

0 ttAte

ttAtx

cv

ccr

)()(sin)()(2sin2

))(sin())(cos()(1

tttttAAk

kttAttAte

c

vcd

dcvcc

76

with the proper choice of lowpass filter, the output of the phase detector is

A VCO is an FM modulator with peak frequency deviation

where implies that

An equivalent nonlinear model is now shown

Nonlinear model of PLL FM demodulator

)()(sin2

)( ttAAk

te vcd

d

( )max

t

d t

dt

)(ˆ)(

tmkdt

tdvco

.)(ˆ)(

0

t

vco dmkt

77

Assuming the PLL is operating in the near lock condition, i.e., , or that

is small. Then,

and the linear approximation of the PLL is given by

Linear model of the phase-locked loop

Let the loop filter have the transfer function HLF(s) = 1, then

)()( tt

)()( tt

)()()()(sin tttt

ˆ ( ) ( ) ( )2

( ) ( ) ,2

d c v

d c vt t

k A Am t t t

k A Ak t t k

78

Thus, the output of the VCO is given by

Let k0 = ktkvco, then

or

In the s-domain, assuming zero initial conditions,

The closed-loop transfer function from input (t) to output (t) is therefore given by

0 0

ˆ( ) ( ) ( ) ( )

t t

vco t vcot k m d k k d

)()()(

0 ttkdt

td

),()()(

00 tktkdt

td

0 0( ) ( ) ( ),s k s k s

0 0

0 0

( )( ) ( ) ( )

( )cl

k ksH s s s

s s k s k

79

The corresponding impulse response is

, where u(t) is the unit step function.

Let us now find out what happens when the loop gain k0 is increased, i.e.,

Clearly,

or , faster for large .

Example: Let the message signal be a step function, i.e., m(t) = Au(t), then

In this case,

In the s-domain, and

1lim)(lim0

0

00

ks

ksH

kcl

k

)()( ss

)()( tt

t

ccFM dAuktAtx0

)(cos)(

0

( ) ( )

t

t k A u d k At

2)(

s

Aks

)()(

0

2

0

kss

kAks

0 0k

01

0( ) ( ) ( )k t

cl clh t H s k e u t L

80

The Laplace transform of is then given by

Let k1 = kω/kvco, then in the time domain,

Clearly, as t , the estimate

Observation: This result is valid when the initial phase error is small.

Remark: A large loop gain k0 results in practical difficulties, hence, a different loop

filter has to be used.

Consider the loop filter described by

Then the output of the VCO is given by

)(ˆ tm

0 0 0 0

1 1 1 1 1ˆ ( ) ( ) ( ) .( )

tt t

vco

Ak k AkM s k s s Ak k

s s k k s s k k s s k

)()()()()(ˆ 00

1111 tueAktmktueAktuAktmtktk

).()(ˆ 1 tmktm

0,/)( asassH LF

0ˆ ( ) ( )( )

( ) ( ) ( ) ( ) ( )t LF LFvco vco

k H s k H sM ss k k s s s s

s s s

81

Let be small, then the closed-loop transfer function is

Define the phase angle error by then

where

Consider again the step function message m(t) = Au(t). Then

)()( tt

aksks

ask

sHks

sHk

s

ssH

LF

LF

cl

00

2

0

0

0

)(

)(

)(

)()(

),()(ˆ)( sss

)(2

)()()(

)()()(

22

2

00

2

2

0

0 sss

ss

aksks

ss

sHks

sHkss

nnLF

LF

akn 0

02 kn

a

k

ak

kk

n

0

0

00

2

1

22

tAkdAkdAukt

tt

00

)()(

82

In the complex frequency domain,

Let kA, then (s) = /s2

and

If 0 < < 1, then

and

Hence the steady-state phase difference error is zero and (t) (t) as t.

A typical value of is 0.707.

2)(

s

Aks

22 2)(

nnsss

tet n

t

n

n 2

21sin

1)(

.0)(lim

tt

83

Superheterodyne Receiver

Definition: To heterodyne means to combine a radio frequency wave with a locally

generated wave of different frequency, in order to produce a new frequency equal

to the sum or difference of the two.

Specifically, a superheterodyne receiver is one that performs the operations of

carrier frequency tuning of the desired signal, filtering it to separate it from

unwanted signals, in most instances, amplifying it to compensate for signal power

loss due to propagation medium.

Generic superheterodyne receiver

84

Example: Let m(t) modulate a sinusoidal carrier with frequency fc = 10 MHz. Let the

bandwidth of the modulated carrier be BT = 200 kHz and let fIF = 1 MHz. Let the

local oscillator frequency be fLO = 11 MHz and let an interferer have its spectrum

located at 11.95 ≤ f ≤ 12.05 MHz. Then, if no bandpass filter is used in the RF

section, at the output of the RF block and of the mixer we have (for f 0 ).

Spectra when no RF filter is used

output