ECE 461 Lectures-1

ECE 461/561COMMUNICATIONS I

Dr. Mario Edgardo Magaña

EECS

Oregon State University

2

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:

• High reliability

• Minimum signal power

• Minimum transmission bandwidth

• Low system complexity (low cost)

EFFICIENT TRANSMISSION: Can be achieved by processing the signal through

a technique called modulation.

MODULATION TYPES:

• Analog modulation

• Digital modulation

3

Amplitude modulation

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

according to the desired 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.

The transmitted AM signal waveform is described by

Requirements:

1.

2. , where Wm is the message bandwidth

)2cos()(1)( tftmkAts cacAM

ttmka ,1)(

mc Wf

4

Taking the Fourier transform of the modulated waveform, we get

Let |M(f)| be described by

Magnitude spectrum of m(t)

)()(2

)()(2

)2cos()()2cos(

)()(

ccca

ccc

caccc

AMAM

ffMffMAk

ffffA

tftmkAtfAF

tsFfS

5

Then the magnitude spectrum of sAM(t) is

Magnitude spectrum of sAM(t)

Observations:

1.

2. Carrier is transmitted explicitly

mAM WB 2

6

Let , then

)2.0cos()( ttm

)2cos()2.0cos(1)( tftkAts cacAM

8

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

9

In the frequency domain,

The modulated waveform now has the following spectral characteristics:

Magnitude spectrum of sDSB(t)

ccc

cc

DSB

ffMffMA

tftmAF

tsFfS

2

2cos)(

)()(

10

Observations:

1. (same as standard AM)

2. Transmitted power is less than standard AM

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

mDSB WB 2

12

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, 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, then it is called non-coherent.

Non-coherent demodulator (receiver) for standard AM

Peak envelope detector (demodulator)

)(ˆ tm)(tS AM R C

+

- -

+D

13

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 = RC is neither too short (rectifier distortion ) nor too long (diagonal

clipping ). Rule of thumb is period of the carrier.

• 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.

1,

2

1/1

(max)

2

am

a kf

k

(max)mf

14

Let the message signal be described by the sinusoidal tone .

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

improperly chosen.

tftm m2cos)(

15Illustration of rectifier distortion and diagonal clipping

16

Coherent demodulator for DSB-SC

At the output of the mixer,

and

cos4cos)(2

ˆ

2cos)(2cosˆ

)(2cosˆ)(

tftmAA

tftmAtfA

tstfAtv

ccc

cccc

DSBcc

cos2

ˆ22

4

ˆ

)(

fMAA

ffMffMAA

tvFfV

cccc

cc

17

Let the message signal have the following magnitude spectrum

Then, if fc>Wm,

18

Suppose is such that

Then, if ,

fH lpf

mcm WfBW 2 cos2

ˆˆ fM

AAfM cc

19

Coherent Costas loop receiver for DSB-SC :

20

I-channel:

After downconverwsion,

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

Q-channel:

cos2cos)(2

coscos)()(

ttmA

tttmAtv

cc

cccI

)(cos2

)( tmA

tm cI

sin2sin)(2

)( ttmA

tv cc

Q

)(sin2

)( tmA

tm cQ

21

Feedback path:

At the output of the multiplier,

At the output of the 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

22

tmA

tmA

tm

c

ce

dthmtm feef )()()(

( ) cos ( )VCO cx t t t

22

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

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

Clearly, if (t) were small, then the instantaneous frequency would be close to

c and the output of the I-path would also be proportional to m(t).

Single-Sideband Modulation

Motivation: Bandwidth occupancy can be minimized by transmitting either the

lower or the upper sideband of the message signal.

,)()(0t

efv dmkt

),(

)(tmk

dt

ttdefvc

c

23

Consider the following transmitter

Suppose we want to transmit the upper sideband, then the following choice of

bandpass filter yields the correct result

24

At the output of the bandpass filter,

Observation: A practical bandpass filter will cause severe distortion.

Suppose the original message signal contains a gap in its magnitude spectrum around the origin, i.e.,

25

Then, SSB demodulation can be obtained by

Synthesis and analysis of SSB:

Let the spectrum of a DSB-SC signal be given by

26

Let this DSB-SC signal be the input of a lowpass filter with magnitude response

Then the resulting (output) signal would have the spectrum

27

The filter can be represented by the algebraic sum

Also,

)( fH L

ccL fffffH sgnsgn2

1)(

ccccc

ccc

ccccc

ccccc

ccc

L

DSBLLSSB

ffffMffffMA

ffMffMA

ffffMffffMA

ffffMffffMA

ffMffMA

fH

fSfHtS

sgnsgn4

4

sgnsgn4

sgnsgn4

2

)(

28

To shed more light on the last equation, let us introduce the concept of the

Hilbert transform.

Definition

A Hilbert transform filter is a filter that simply phase shifts all frequency

components by π/2 radians, i.e., its transfer function is described by

where the signum function is defined by

Observations:

1.

2.

fjfH sgn

0,1

0,0

0,1

sgn

f

f

f

f

1fH

0,2/

0,2/arg

f

ffH

29

In the time domain, the impulse response of the filter is given by

Therefore, the Hilbert transform of a function g(t) can be interpreted as the

convolution of the impulse response of the filter h(t) and g(t), i.e.,


Therefore,

Using the exponential modulation theorem,

tth

1

)(

d

t

g

tgthtg

)(1

)()()(~

fGfjfG sgn~

fGfjtgF

sgn)(~

cc

Ftfj ffffjGetg c sgn)(~ 2

30

In the time domain, we can synthesize lower SSB mathematically as follows:

where is the Hilbert transform of m(t)

Likewise, the upper SSB signal is described by

tftmAtftmA

etmAj

etmAj

tftmA

ffffMffffMAF

ffMffMAF

fSFts

cccc

tfjc

tfjccc

ccccc

ccc

LSSB

LSSB

cc

2sin)(~2

12cos)(

2

1

)(~4

1)(~

4

12cos)(

2

1

sgnsgn4

1

4

1

)(

22

1

1

1

)(~ tm

tftmAtftmAts ccccUSSB 2sin)(~

2

12cos)(

2

1)(

31

A widely used form of SSB is vestigial sideband, which is used in TV broadcasts.

SSB is a special case of VSB, since the filter that is used to obtain SSB is a

limiting case of the filter used to generate VSB.

Generation of VSB

Instead of the ideal filter used in SSB, let us use the following filter:

can be analytically described by fH LVSB

cQcQL

VSB ffHffHjfH

32

The transfer function has the following characteristics:

Observations about :

• Odd symmetry about the origin

• Linear phase

Mathematically,

and

The ‘+’ sign means that a vestige of the upper sideband is transmitted and a ‘-’ sign means that a vestige of the lower sideband is transmitted.

fH Q

fH Q

tftmAtftmAts ccccLVSB 2sin)(

2

12cos)(

2

1)(

tftmAtftmAts ccccUVSB 2sin)(

2

12cos)(

2

1)(

33

Example: Now, if the magnitude spectrum of the DSB signal is as described

before, then the magnitude spectrum of is

Vestigial sideband magnitude spectrum (upper sideband vestige).

)(ts LVSB

34

To help in the demodulation process, let us insert a pilot signal to the VSB signal

or

where

Therefore, the complex envelope of this pilot aided VSB signal is

tfAtftmAtftmAkts ccccccaPVSB 2cos2sin)(

2

12cos)(

2

1)(

,2sin)(2

12cos)(

2

11)( tftmAktftmkAts ccacac

PVSB

tftmtftmts cQcIPVSB 2sin)(2cos)()(

)(2

1)(

)(2

11)(

tmAktm

tmkAtm

caQ

acI

,)()()()(~ )(tjQI

PVSB etatjmtmts

35

where the real envelope a(t) is described by

and its phase is

If we use an envelope detector to demodulate the pilot-aided VSB, its output would be

2/122

22

22

)(2

1)(

2

11

)()()(

tmkAtmkA

tmtmta

acac

QI

)(

)(tan)( 1

tm

tmt

I

Q

1/ 22 2

1/ 22

1 1( ) 1 ( ) ( )

2 2

1( )1 21 ( ) 1

12 1 ( )2

c a a

a

c a

a

a t A k m t k m t

k m tA k m t

k m t

36

Clearly, if m´(t) is small, then we can extract the message signal m(t) by using

a DC blocking capacitor.

In practice, is minimized by increasing the width of the vestigial

sideband.

Angle Modulation

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

where Ac is the constant amplitude.

The instantaneous frequency is

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

and angle i(t).

)(tm

),(cos)( tAts ic

.)(

)(dt

tdt i

i

37

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 = 0, then

where kp Phase sensitivity of the modulator in rad/volt

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

38

where k Frequency sensitivity of the modulator in rad/s/volt

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.

t

c

t

c

t

ii dmktdmkdt000

)()()()(

t

ccFM dmktAts0

)(cos)(

39

FM modulation

40

PM modulation

41

The figure below shows a comparison between AM, FM and PM modulation of the same message waveform:

42

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

The instantaneous phase angle of the FM signal is

),2cos()( tfAtm mm

).2cos()( tfAkftf mmfci

.mf Akf

)2sin(2

)2sin(2

)(2)(0

tftf

tff

Aktf

dft

mc

mm

mfc

t

ii

43

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:

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

For the general case,

where the complex envelope of the FM signal is described by

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

,/ mmf fAk

( ) 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

,)(Re

Re)(2

)2sin(2

tfje

tftfjcFM

c

mc

ets

eAts

)2sin()( tfjce

meAts

44

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

nne

mects

.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(jmc

f/

f/

tfnj)tfsin(jcm

f/

f/

tfnjem

m

/T

/T

tfnjen

21

21

22

21

21

22

21

21

2

2

2

2 11

45

Let

Then,

and

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

Therefore,

and the FM tone waveform is described by

.tfx m2

dxf

dtm2

1

),(JA

dxeA

c

nc

nxxsinjcn

2

dxe)(J nxxsinj

n 2

1

n

tnfjnce

meJAts 2)()(

.)(2cos)()(

n

mcncFM tnffJAts

46


Average power of the FM waveform:

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

But,

is also the power of the FM waveform.

( ) ( )

( ) cos 2 ( )

( ) cos 2 ( )

( )( ) ( )

2

FM FM

c n c mn

c n c mn

nc c m c m

n

S f s t

A J f nf t

A J F f nf t

JA f f nf f f nf

2/2cAP

n

nc J

AP )(

22

2

48

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

1.

2.

Hence, the average power of an FM tone is

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()( nn

n JJ

1)(2

nnJ

.2/2cA

1)(0 J

2/)(1 J

2,0)( nJ n

49

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


)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

mcmcc

mcmcc

ccc

FM

ffffffA

ffffffA

ffffA

fS

4

42)(

50

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

51

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

52

But,

and

Consequently, the resultant phasor is given by

The magnitude of the resultant may be approximated by

since

)2cos(

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

tf

tftftf

m

mmm

)2sin(

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

tf

tftftf

m

mmm

)2sin( tfjAAS mccFM

,)2(sin2

11

)2(sin

22

2222

tfA

tfAAS

mc

mccFM

1,1)1( nxnxx n

53

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.

FM Transmission Bandwidth:

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

All significant lines are contained in the frequency range

where f is the peak frequency deviation.

)4cos(

441)(

22

tfAtS mcFM

)2sin(tan)( 1 tftS mSFM

.// mmmf fffAkn

,ffff cmc

54

Let be small, i.e., 0 < ≤ 0.3, then

which means that only the first pair of spectral lines is 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

fBFM 2

.2 mFM fB

)/11(2

)/1(2

22

f

fff

ffB

m

mT

55

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

56

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

)/11(2

)/1(2

22

Df

fWf

WfB

m

mT

)(maxˆ tmkft

f

./ˆ mWfD

57

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

58

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

59

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)(

60

with kf (and thus fNB) small

Let us now consider a 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 inn

c

61

cosni(t) can be expanded as follows:

Likewise,

Thus,

Expanding the last term of the previous equality, we get

)(cos)(2cos2

1)(cos

2

1

)(cos)(2cos12

1

)(cos)(cos)(cos

22

2

22

ttt

tt

ttt

in

iin

in

i

in

iin

)(cos)(2cos2

1)(cos

2

1)(cos 442 tttt i

nii

ni

n

)(cos)(2cos4

1)(cos)(2cos

4

1)(cos

2

1)(cos 4242 tttttt i

nii

nii

ni

n

).(cos)(4cos14

1)(cos)(2cos 442 tttt i

nii

ni

62

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

in

i

in

iin

i

in

iin

in

in

iin

in

iin

in

).(cos)(cos2

11

ttk ikn

ik

63

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

So, 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

nc

ii

64

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)(

222 t

aAaAtaAty i

ccic

)(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

ic

ic

iiic

iiic

65

Finally,

Let y(t) be input to an ideal bandpass filter with unity gain, bandwidth wide enough to accommodate spectrum of wide band signal and center frequency nfc, i.e.,

Ideal bandpass filter

Then,

t

fcn

nc

t

fc

t

fc

dmkntfnA

a

dmktfcdmktfccty

01

0

2

0

10

)(22cos2

)(44cos)(22cos)(

t

fcn

nc

WB dmkntfnaA

ts0

1)(22cos

2)(

66

The instantaneous frequency 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

)()( tmnknftf fci

67

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.

68

Armstrong indirect method of FM generation

69

Demodulation of FM signals

Consider the following receiver architecture

Frequency discriminator implementation of an FM demodulator

The slope circuit is characterized by a purely imaginary transfer function

Let H1(f ) be described by

.2,1),( isH i

elsewhere,0

2/2/),2/(2

2/2/),2/(2

)(1 TcTcTc

TcTcTc

BffBfBffaj

BffBfBffaj

fH

70

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

Define G1(f ) H1(f )/j, then g1(t ) 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

)2cos()()2cos()()( ,1,11 tftgtftgtg cQcI

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

tfj cetgtg 211 )(~Re)(

)(2

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

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

1

,1,1

2,1,1

2,1,1

21

21

tg

tftgtftg

etjgtgetjgtgetgetg

cQcI

tfjQI

tfjQI

tfjtfj cccc

71

But,

or

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

tfjtfj cc etgetgFtgF 21

211 )(~)(~)(2

))((~

)(~

)(2 111 cc ffGffGfG

)(~

1 fG 2/TBf

.0),()(~

11 ffGffG c

)(~

1 fG

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

1 fG

./)(~

1 jfH

72

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

~1 fH

73

From the previous derivation,

But,

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

elsewhere,0

2/),2/(4)(

~1

TT BfBfajfH

.)(~Re

Re

)(cos)(

)(

tfjFM

tfjdmkjc

t

fccFM

c

ct

f

ets

eeA

dmktfAts

2

22

0

0

22

)(~)(~

2

1)(~

11 tsthts FM

74


But,

which implies that

elsewhere

BffSBfaj

fSfHfS

TFMT

FM

,0

2/),(~

)2/(2

)(~

)(~

2

1)(

~11

)(2)(

ffXjdt

tdxF

tf

tf

tf

dmkj

T

fTc

dmkjTc

dmkjfc

FMTFM

etmB

kBaAj

eBaAetmkaAj

tsBjadt

tsdats

0

00

)(2

)(2)(2

1

)(2

1

2)(2

)(~)(~)(~

75

Therefore,

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

However, if

2)(22cos)(

21

)(22sin)(2

1

)(2

1Re

)(~Re)(

0

0

)(22

211

0

t

fcT

fTc

t

fcT

fTc

dmkjtfj

T

fTc

tfj

dmktftmB

kBaA

dmktftmB

kBaA

etmB

kBaAj

etsts

tfc

c

ttmB

k

T

f ,1)(2

76

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

fTce

)(~

)(~

12 fHfH

.)(2

1)(2

tm

B

kBaAts

T

fTce

)(4

)()()(ˆ 21

tmaA

tststm

c

ee

77

Frequency discriminator

FM demodulation via phase-locked loops

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

Phase-locked loop FM detector

78

From the previous block diagram,

Let the phase detector be described by

then,

))(sin()(

))(cos()(

0 ttAte

ttAtx

cv

ccr

)()(sin)()(2sin2

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

tttttAAk

kttAttAte

cvcd

dcvcc

79

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 vcdd

dt

tdf

t

)(max

)(ˆ)(

tmkdt

tdvco

.)(ˆ)(

0t

vco dmkt

80

Let the PLL operate in the 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

)()(

)()(2

)(ˆ

ttk

ttAAk

tm

t

vcd

81

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 is therefore given by

t

vcot

t

vco dkkdmkt00

)()()(ˆ)(

)()()(

0 ttkdt

td

),()()(

00 tktkdt

td

),()()( 00 sksks

0

0

)(

)()(

ks

k

s

ssH cl

82

The corresponding impulse response is

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

Clearly,

or , for large k0.

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

In this case,

In the s-domain, and

)()( 00 tuekth tk

cl

1lim)(lim0

0

00

ks

ksH

kcl

k

)()( ss

)()( tt

t

ccFM dAuktAtx0

)(cos)(

tAkduAktt

0

)()(

2)(

s

Aks )(

)(0

20

kss

kAks

83

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

.)(

)()()(ˆ

0000

0

11111

kssk

Ak

ksskkAk

kss

kAkssksM

vco

tt

)()()()()(ˆ 001111 tueAktmktueAktuAktm tktk

).()(ˆ 1 tmktm

0,/)( asassH LF

)()()(

)( 0 sss

sHks LF

84

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

Define then

where

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

)()( tt

aksks

ask

sHks

sHk

s

ssH

LF

LFcl

002

0

0

0

)(

)(

)(

)()(

),()(ˆ)( sss

)(2

)()()(

)()()(

22

2

002

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

tAkdAkdAukttt

00

)()(

85

In the complex frequency domain,

Let kA, then (s) = /s2

and

If 0 < < 1, then

and

Hence the steady-state phase error is zero.

A typical value of is 0.707.

2)(

s

Aks

22 2)(

nn sss

tet nt

n

n 2

21sin

1)(

.0)(lim

tt

86

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

87

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

spectrum located at 11.95 ≤ f ≤ 12.05 MHz. If no filter is used in the RF section,

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

Spectra when no RF filter is used

ECE 461 Lectures-1

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