2-dieu che tuong tu

Chapter 3Analog Modulation

Contents3.1 Linear Modulation . . . . . . . . . . . . . . . . . . . . 3-3

3.1.1 Double-Sideband Modulation (DSB) . . . . . . 3-3

3.1.2 Amplitude Modulation . . . . . . . . . . . . . . 3-8

3.1.3 Single-Sideband Modulation . . . . . . . . . . . 3-21

3.1.4 Vestigial-Sideband Modulation . . . . . . . . . . 3-35

3.1.5 Frequency Translation and Mixing . . . . . . . . 3-38

3.2 Angle Modulation . . . . . . . . . . . . . . . . . . . . 3-463.2.1 Narrowband Angle Modulation . . . . . . . . . 3-48

3.2.2 Spectrum of an Angle-Modulated Signal . . . . 3-50

3.2.3 Power in an Angle-Modulated Signal . . . . . . 3-56

3.2.4 Bandwidth of Angle-Modulated Signals . . . . . 3-56

3.2.5 Narrowband-to-Wideband Conversion . . . . . . 3-63

3.2.6 Demodulation of Angle-Modulated Signals . . . 3-63

3.3 Interference . . . . . . . . . . . . . . . . . . . . . . . 3-733.3.1 Interference in Linear Modulation . . . . . . . . 3-74

3.3.2 Interference in Angle Modulation . . . . . . . . 3-76

3.4 Feedback Demodulators . . . . . . . . . . . . . . . . . 3-81

3-1

CHAPTER 3. ANALOG MODULATION

3.4.1 Phase-Locked Loops for FM Demodulation . . . 3-81

3.4.2 PLL Frequency Synthesizers . . . . . . . . . . . 3-102

3.4.3 Frequency-Compressive Feedback . . . . . . . . 3-106

3.4.4 Coherent Carrier Recovery for DSB Demodulation 3-108

3.5 Sampling Theory . . . . . . . . . . . . . . . . . . . . . 3-1123.6 Analog Pulse Modulation . . . . . . . . . . . . . . . . 3-117

3.6.1 Pulse-Amplitude Modulation (PAM) . . . . . . . 3-117

3.6.2 Pulse-Width Modulation (PWM) . . . . . . . . . 3-119

3.6.3 Pulse-Position Modulation . . . . . . . . . . . . 3-119

3.7 Delta Modulation and PCM . . . . . . . . . . . . . . . 3-1203.7.1 Delta Modulation (DM) . . . . . . . . . . . . . 3-120

3.7.2 Pulse-Code Modulation (PCM) . . . . . . . . . 3-123

3.8 Multiplexing . . . . . . . . . . . . . . . . . . . . . . . 3-1263.8.1 Frequency-Division Multiplexing (FDM) . . . . 3-127

3.8.2 Quadrature Multiplexing (QM) . . . . . . . . . . 3-130

3.8.3 Time-Division Multiplexing (TDM) . . . . . . . 3-131

3.9 General Performance of Modulation Systems in Noise 3-135

3-2 ECE 5625 Communication Systems I

3.1. LINEAR MODULATION

We are typically interested in locating a message signal to somenew frequency location, where it can be efficiently transmitted

The carrier of the message signal is usually sinusoidal

A modulated carrier can be represented as

xc.t/ D A.t/ cos2fct C .t/

where A.t/ is linear modulation, fc the carrier frequency, and.t/ is phase modulation

3.1 Linear Modulation

For linear modulation schemes, we may set .t/ D 0 withoutloss of generality

xc.t/ D A.t/ cos.2fct /

with A.t/ placed in one-to-one correspondence with the mes-sage signal

3.1.1 Double-Sideband Modulation (DSB)

Let A.t/ / m.t/, the message signal, thus

xc.t/ D Acm.t/ cos.2fct /

From the modulation theorem it follows that

Xc.f / D1

2AcM.f fc/C

1

2AcM.f C fc/

ECE 5625 Communication Systems I 3-3


t t

m(t) xc(t) carrier filled envelope

DSB time domain waveforms

M(0)M(f)

Xc(f)

AcM(0)1

2

fc

-fc

f

f

LSB USB

DSB spectra

Coherent Demodulation

The received signal is multiplied by the signal 2 cos.2fct /,which is synchronous with the transmitter carrier

LPF

m(t) xc(t) x

r(t) d(t) y

D(t)

Accos[2πf

ct] 2cos[2πf

ct]

DemodulatorModulator Channel



For an ideal channel xr.t/ D xc.t/, so

d.t/ DAcm.t/ cos.2fct /

2 cos.2fct /

D Acm.t/C Acm.t/ cos.2.2fc/t/

where we have used the trig identity 2 cos2 x D 1C cos 2x

The waveform and spectra of d.t/ is shown below (assumingm.t/ has a triangular spectrum in D.f /)

D(f)

AcM(0)

2fc

-2fc

f

AcM(0)1

2A

cM(0)1

2

W-W

Lowpass modulation recovery filter

d(t)

t

Lowpass filtering will remove the double frequency carrier term

Waveform and spectrum of d.t/

Typically the carrier frequency is much greater than the mes-sage bandwidth W , so m.t/ can be recovered via lowpass fil-tering

The scale factor Ac can be dealt with in downstream signalprocessing, e.g., an automatic gain control (AGC) amplifier



Assuming an ideal lowpass filter, the only requirement is thatthe cutoff frequency be greater than W and less than 2fc W

The difficulty with this demodulator is the need for a coherentcarrier reference

To see how critical this is to demodulation ofm.t/ suppose thatthe reference signal is of the form

c.t/ D 2 cosŒ2fct C .t/

where .t/ is a time-varying phase error

With the imperfect carrier reference signal

d.t/ D Acm.t/ cos .t/C Acm.t/ cosŒ2fct C .t/yD.t/ D m.t/ cos .t/

Suppose that .t/ is a constant or slowly varying, then thecos .t/ appears as a fixed or time varying attenuation factor

Even a slowly varying attenuation can be very detrimental froma distortion standpoint

– If say .t/ D f t and m.t/ D cos.2fmt /, then

yD.t/ D1

2ŒcosŒ2.fm f /tC cosŒ2.fm Cf /t

which is the sum of two tones

Being able to generate a coherent local reference is also a prac-tical manner



One scheme is to simply square the received DSB signal

x2r .t/ D A2cm

2.t/ cos2.2fct /

D1

2A2cm

2.t/C1

2A2cm

2.t/ cosŒ2.2fc/t

( )2 BPF

LPF

divideby 2

xr(t) y

D(t)

xr(t)2

Acos2πfct

very narrow (tracking) band-pass filter

Carrier recovery concept using signal squaring

Assuming that m2.t/ has a nonzero DC value, then the doublefrequency term will have a spectral line at 2fc which can bedivided by two following filtering by a narrowband bandpassfilter, i.e., Ffm2.t/g D kı.f /C

k

2fc

fSpec

trum

of m

2 (t) Filter this component

for coherent demod

Note that unless m.t/ has a DC component, Xc.f / will notcontain a carrier term (read ı.f ˙ fc), thus DSB is also calleda suppressed carrier scheme



Consider transmitting a small amount of unmodulated carrier

k

m(t) xc(t)

Accos2πf

ct

ffc

-fc

AcM(0)/2

use a narrowband filter (phase-locked loop) to extract the carrier in the demod.

k << 1

3.1.2 Amplitude Modulation

Amplitude modulation (AM) can be created by simply addinga DC bias to the message signal

xc.t/ DACm.t/

A0c cos.2fct /

D Ac1C amn.t/

cos.2fct /

where Ac D AA0c, mn.t/ is the normalized message such thatminmn.t/ D 1,

mn.t/ Dm.t/

jminm.t/j

and a is the modulation index

a Djminm.t/j

A



t

xc(t)

a < 1

m(t)

A

A + m(t)

Accos[2πf

ct]

xc(t)

Bias term

Note that the enve-lope does not cross zero in the case of AM having a < 1

0A

c(1 - a)

A + max m(t) A + min m(t)

Generation of AM and a sample wavefrom

Note that ifm.t/ is symmetrical about zero and we define d1 asthe peak-to-peak value of xc.t/ and d2 as the valley-to-valleyvalue of xc.t/, it follows that

a Dd1 d2

d1 C d2

proof: maxm.t/ D minm.t/ D jminm.t/j, so

d1 d2

d1 C d2D2Œ.AC jminm.t/j/ .A jminm.t/j/2Œ.AC jminm.t/j/C .A jminm.t/j/

Djminm.t/j

AD a



The message signal can be recovered from xc.t/ using a tech-nique known as envelope detection

A diode, resistor, and capacitor is all that is needed to constructand envelope detector

eo(t)x

r(t) C R

t

Recovered envelope with proper RC selection

eo(t)

0

The carrier is removed if 1/fc << RC << 1/W

Envelope detector

The circuit shown above is actually a combination of a nonlin-earity and filter (system with memory)

A detailed analysis of this circuit is more difficult than youmight think

A SPICE circuit simulation is relatively straight forward, but itcan be time consuming if W fc



The simple envelope detector fails if AcŒ1C amn.t/ < 0

– In the circuit shown above, the diode is not ideal andhence there is a turn-on voltage which further limits themaximum value of a

The RC time constant cutoff frequency must lie between bothW and fc, hence good operation also requires that fc W



Digital signal processing based envelope detectors are also pos-sible

Historically the envelope detector has provided a very low-costmeans to recover the message signal on AM carrier

The spectrum of an AM signal is

Xc.f / DAc

2

ı.f fc/C ı.f C fc/

„ ƒ‚ …

pure carrier spectrum

CaAc

2

Mn.f fc/CMn.f C fc/

„ ƒ‚ …

DSB spectrum

AM Power Efficiency

Low-cost and easy to implement demodulators is a plus forAM, but what is the downside?

Adding the bias term to m.t/ means that a fraction of the totaltransmitted power is dedicated to a pure carrier

The total power in xc.t/ is can be written in terms of the timeaverage operator introduced in Chapter 2

hx2c .t/i D hA2cŒ1C amn.t/

2 cos2.2fct /i

DA2c2hŒ1C 2amn.t/C a

2m2n.t/Œ1C cos.2.2fc/t i

If m.t/ is slowly varying with respect to cos.2fct /, i.e.,

hm.t/ cos!cti ' 0;



then

hx2c .t/i DA2c2

1C 2ahmn.t/i C a

2hm2

n.t/i

DA2c2

1C a2hm2.t/i

D

A2c2„ƒ‚…

Pcarrier

Ca2A2c2hm2

n.t/i„ ƒ‚ …Psidebands

where the last line resulted from the assumption hm.t/i D 0

(the DC or average value of m.t/ is zero)

Definition: AM Efficiency

EffD

a2hm2n.t/i

1C a2hm2n.t/i

alsoD

hm2.t/i

A2 C hm2.t/i

Example 3.1: Single Sinusoid AM

An AM signal of the form

xc.t/ D AcŒ1C a cos.2fmt C =3/ cos.2fct /

contains a total power of 1000 W

The modulation index is 0.8

Find the power contained in the carrier and the sidebands, alsofind the efficiency

The total power is

1000 D hx2c .t/i DA2c2Ca2A2c2 hm2

n.t/i



It should be clear that in this problemmn.t/ D cos.2fmt /, sohm2

n.t/i D 1=2 and

1000 D A2c

1

2C1

40:64

D33

50A2c

Thus we see that

A2c D 1000 50

33D 1515:15

and

Pcarrier D1

2A2c D

1515

2D 757:6 W

and thus

Psidebands D 1000 Pc D 242:4 W

The efficiency is

Eff D242:4

1000D 0:242 or 24.2%

The magnitude and phase spectra can be plotted by first ex-panding out xc.t/

xc.t/ D Ac cos.2fct /C aAc cos.2fmt C =3/ cos.2fct /D Ac cos.2fct /

CaAc

2cosŒ2.fc C fm/t C =3

CaAc

2cosŒ2.fc fm/t =3



f

f

0

0

fc+f

mfc-fm

fc

-fc

Ac/2

0.8Ac/4

|Xc(f)|

Xc(f)

π/3

-π/3

Amplitude and phase spectra for one tone AM

Example 3.2: Pulse Train with DC Offset

t

2

-1

m(t)

Tm/3 T

m

Find mn.t/ and the efficiency E

From the definition of mn.t/

mn.t/ Dm.t/

jminm.t/jDm.t/

j 1jD m.t/

The efficiency is

E Da2hm2

n.t/i

1C a2hm2n.t/i



To obtain hm2n.t/i we form the time average

hm2n.t/i D

1

Tm

"Z Tm=3

0

.2/2 dt C

Z Tm

Tm=3

.1/2 dt

#D

1

Tm

Tm

3 4C

2Tm

3 1

D4

3C2

3D7

3

thus

E D.7=3/a2

1C .7=3/a2D

7a2

3C 7a2

The best AM efficiency we can achieve with this waveform iswhen a D 1

Eff

ˇaD1D

7

10D 0:7 or 70%

Suppose that the message signal is m.t/ as given here

Now minm.t/ D 2 and mn.t/ D m.t/=2 and

hm2n.t/i D

1

3 .1/2 C

2

3 .1=2/2 D

1

2

The efficiency in this case is

Eff D.1=2/a2

1C .1=2/a2D

a2

2C a2

Now when a D 1 we have Eff D 1=3 or just 33.3%

Note that for 50% duty cycle squarewave the efficiency maxi-mum is just 50%



Example 3.3: Multiple Sinusoids

Suppose that m.t/ is a sum of multiple sinusoids (multi-toneAM)

m.t/ D

MXkD1

Ak cos.2fkt C k/

where M is the number of sinusoids, fk values might be con-strained over some band of frequencies W , e.g., fk W , andthe phase values k can be any value on Œ0; 2

To find mn.t/ we need to find minm.t/

A lower bound on minm.t/ is PM

kD1Ak; why?

The worst case value may not occur in practice depending uponthe phase and frequency values, so we may have to resort to anumerical search or a plot of the waveform

Suppose that M D 3 with fk D f65; 100; 35g Hz, Ak Df2; 3:5; 4:2g, and k D f0; =3;=4g rad.

>> [m,t] = M_sinusoids(1000,[65 100 35],[2 3.5 4.2],...[0 pi/3 -pi/4], 20000);>> plot(t,m)

>> min(m)

ans = -7.2462e+00

>> -sum([2 3.5 4.2]) % worst case minimum value

ans = -9.7000e+00

>> subplot(311)>> plot(t,(1 + 0.25*m/abs(min(m))).*cos(2*pi*1000*t))>> hold



Current plot held>> plot(t,1 + 0.25*m/abs(min(m)),'r')>> subplot(312)>> plot(t,(1 + 0.5*m/abs(min(m))).*cos(2*pi*1000*t))>> holdCurrent plot held>> plot(t,1 + 0.5*m/abs(min(m)),'r')>> subplot(313)>> plot(t,(1 + 1.0*m/abs(min(m))).*cos(2*pi*1000*t))>> holdCurrent plot held>> plot(t,1 + 1.0*m/abs(min(m)),'r')

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05−8

−6

−4

−2

0

2

4

6

8

Time (s)

m(t

) A

mpl

itude

min m(t)

Finding minm.t/ graphically

The normalization factor is approximately given by 7.246, thatis

mn.t/ Dm.t/

7:246

Shown below are plots of xc.t/ for a D 0:25; 0:5 and 1 usingfc D 1000 Hz



0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05−2

0

2

x c(t),

a =

0.2

5

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05−2

0

2

x c(t),

a =

0.5

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05−2

0

2

x c(t),

a =

1.0

Time (s)

Modulation index comparison (fc D 1000 Hz)

To obtain the efficiency of multi-tone AM we first calculatehm2

n.t/i assuming unique frequencies

hm2n.t/i D

MXkD1

A2k2jminm.t/j2

D22 C 3:52 C 4:22

2 7:2462D 0:3227

The maximum efficiency is just

Eff

ˇaD1D

0:3227

1C 0:3227D 0:244 or 24.4%



A remaining interest is the spectrum of xc.t/

Xc.f / DAc

2

ı.f fc/C ı.f C fc/

CaAc

4

MXkD1

Ak

hejkı.f .fc C fk//

C ejkı.f C .fc C fk//i

(USB terms)

CaAc

4

MXkD1

Ak

hejkı.f .fc fk//

C ejkı.f C .fc fk//i

(LSB terms)

ï1000 ï800 ï600 ï400 ï200 0 200 400 600 800 10000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Frequency (Hz)

Ampl

itude

Spe

ctra

(|X c(f)

|)

SymmetricalSidebands for

a = 0.5

Carrier with Ac = 1

Amplitude spectra



3.1.3 Single-Sideband Modulation

In the study of DSB it was observed that the USB and LSBspectra are related, that is the magnitude spectra about fc haseven symmetry and phase spectra about fc has odd symmetry

The information is redundant, meaning thatm.t/ can be recon-structed one or the other sidebands

Transmitting just the USB or LSB results in single-sideband(SSB)

For m.t/ having lowpass bandwidth of W the bandwidth re-quired for DSB, centered on fc is 2W

Since SSB operates by transmitting just one sideband, the trans-mission bandwidth is reduced to just W

M(f)

f

fc - W

fc - W f

c+W

fc+Wf

c

fc

fc

XDSB

(f)

f

XSSB

(f)XSSB

(f)

f

LSB USB

W

USBremoved

LSBremoved

DSB to two forms of SSB: USSB and LSSB

The filtering required to obtain an SSB is best explained withthe aid of the Hilbert transform, so we divert from text Chapter



3 back to Chapter 2 to briefly study the basic properties of thistransform

Hilbert Transform

The Hilbert transform is nothing more than a filter that shiftsthe phase of all frequency components by =2, i.e.,

H.f / D j sgn.f /

where

sgn.f / D

8<:1; f > 0

0; f D 0

1; f < 0

The Hilbert transform of signal x.t/ can be written in terms ofthe Fourier transform and inverse Fourier transform

Ox.t/ D F1 j sgn.f /X.f /

D h.t/ x.t/

where h.t/ D F1fH.f /g

We can find the impulse response h.t/ using the duality theo-rem and the differentiation theorem

d

dfH.f /

F ! .j 2t/h.t /

where here H.f / D j sgn.f /, so

d

dfH.f / D 2jı.f /



Clearly,F1f2jı.f /g D 2j

so

h.t/ D2j

j 2tD

1

t

and1

t

F ! j sgn.f /

In the time domain the Hilbert transform is the convolutionintegral

Ox.t/ D

Z1

1

x./

.t /d D

Z1

1

x.t /

d

Note that since the Hilbert transform of x.t/ is a =2 phaseshift, the Hilbert transform of Ox.t/ is

OOx.t/ D x.t/

why? observe that .j sgn.f //2 D 1

Example 3.4: x.t/ D cos!0t

By definition

OX.f / D j sgn.f / 1

2

ı.f f0/C ı.f C f0/

D j

1

2ı.f fc/C j

1

2ı.f C f0/



so from ej!0tF) ı.f f0/

Ox.t/ D j1

2ej!0t C j

1

2ej!0t

Dej!0t ej!0t

2jD sin!0t

or2cos!0t D sin!0t

It also follows that

2sin!0t D22cos!0t D cos!0t

since OOx.t/ D x.t/

Hilbert Transform Properties

1. The energy (power) in x.t/ and Ox.t/ are equal

The proof follows from the fact that jY.f /j2 D jH.f /j2jX.f /j2

and jj sgn.f /j2 D 1

2. x.t/ and Ox.t/ are orthogonal, that isZ1

1

x.t/ Ox.t/ dt D 0 (energy signal)

limT!1

1

2T

Z T

T

x.t/ Ox.t/ dt D 0 (power signal)



The proof follows for the case of energy signals by generaliz-ing Parseval’s theoremZ

1

1

x.t/ Ox.t/ dt D

Z1

1

X.f / OX.f / df

D

Z1

1

.j sgn.f //„ ƒ‚ …odd

jX.f /j2„ ƒ‚ …even

df D 0

3. Given signals m.t/ and c.t/ such that the corresponding spec-tra are

M.f / D 0 for jf j > W (a lowpass signal)C.f / D 0 for jf j < W (c.t/ a highpass signal)

then3m.t/c.t/ D m.t/ Oc.t/

Example 3.5: c.t/ D cos!0t

Suppose that M.f / D 0 for jf j > W and f0 > W then

5m.t/ cos!0t D m.t/2cos!0tD m.t/ sin!0t

Analytic Signals

Define analytic signal z.t/ as

z.t/ D x.t/C j Ox.t/

where x.t/ is a real signal



The envelope of z.t/ is jz.t/j and is related to the envelopediscussed with DSB and AM signals

The spectrum of an analytic signal has single-sideband charac-teristics

In particular for zp.t/ D x.t/C j Ox.t/

Zp.f / D X.f /C j˚ j sgn.f /X.f /

D X.f /

1C sgn.f /

D

(2X.f /; f > 0

0; f < 0

Note: Only positive frequencies present

Similarly for zn.t/ D x.t/ j Ox.t/

Zn.f / D X.f /1 sgn.f /

D

(0; f > 0

2X.f /; f < 0



W-W

W-W

W-W

f

f

f

X(f)

Zp(f)

Zn(f)

2

2

1

The spectra of analytic signals can suppress positive or negativefrequencies

Return to SSB Development

SidebandFilter

Accosω

ct

m(t)

xDSB

(t)x

SSB(t)

LSB or USB

Basic SSB signal generation

In simple terms, we create an SSB signal from a DSB signalusing a sideband filter

The mathematical representation of LSSB and USSB signalsmakes use of Hilbert transform concepts and analytic signals



DSB Signal Starting Point

Formation of HL(f)

-fc

fc

-fc

fc

f

f

f

f

sgn(f + fc)/2

-sgn(f - fc)/2

+1/2

+1/2

-1/2

-1/2

HL(f) = [sgn(f + f

c) - sgn(f - f

c)]/21

An ideal LSSB filter

From the frequency domain expression for the LSSB, we canultimately obtain an expression for the LSSB signal, xcLSSB.t/,in the time domain

Start with XDSB.f / and the filter HL.f /

XcLSSB.f / D1

2AcM.f C fc/CM.f fc/

1

2

sgn.f C fc/ sgn.f fc/



XcLSSB.f / D1

4AcM.f C fc/sgn.f C fc/

CM.f fc/sgn.f fc/

1

4AcM.f C fc/sgn.f fc/

CM.f fc/sgn.f C fc/

D1


C1

4AcM.f C fc/sgn.f C fc/

M.f fc/sgn.f fc/

The inverse Fourier transform of the first term is DSB, i.e.,

1

2Acm.t/ cos!ct

F !

1


The second term can be inverse transformed using

Om.t/F ! j sgn.f / M.f /

soF1

˚M.f C fc/sgn.f C fc/

D j Om.t/ej!ct

since m.t/e˙j!ctF !M.f ˙ fc/

Thus

1

4AcF1

˚M.f Cfc/sgn.f Cfc/M.f fc/sgn.f fc/

D

1

4Acj Om.t/ej!ct j Om.t/ej!ct

D

1

2Om.t/ sin!ct



Finally,

xcLSSB.t/ D1

2Acm.t/ cos!ct C

1

2Ac Om.t/ sin!ct

Similarly for USSB it can be shown that

xcUSSB.t/ D1

2Acm.t/ cos!ct

1

2Ac Om.t/ sin!ct

The direct implementation of SSB is very difficult due to therequirements of the filter

By moving the phase shift frequency from fc down to DC (0Hz) the implementation is much more reasonable (this appliesto a DSP implementation as well)

The phase shift is not perfect at low frequencies, so the modu-lation must not contain critical information at these frequencies

H(f) = -jsgn(f)

0o

-90o

m(t)

xc(t)

sinωct

cosωct

cosωct

Carrier Osc.+

+-LSBUSB

Phase shift modulator for SSB



Demodulation

The coherent demodulator first discussed for DSB, also worksfor SSB

LPFxr(t)

d(t) yD(t)

4cos[2πfct + θ(t)]

1/Ac scale factor

included

Coherent demod for SSB

Carrying out the analysis to d.t/, first we have

d.t/ D1

2Acm.t/ cos!ct ˙ Om.t/ sin!ct

4 cos.!ct C .t//

D Acm.t/ cos .t/C Acm.t/ cosŒ2!ct C .t/ Ac Om.t/ sin .t/˙ Ac Om.t/ sinŒ2!ct C .t/

so

yD.t/ D m.t/ cos .t/ Om.t/ sin .t/.t/ small' m.t/ Om.t/.t/

– The Om.t/ sin .t/ term represents crosstalk

Another approach to demodulation is to use carrier reinsertionand envelope detection

EnvelopeDetector

xr(t)

e(t)yD(t)

Kcosωct



e.t/ D xr.t/CK cos!ct

D

1

2Acm.t/CK

cos!ct ˙

1

2Ac Om.t/ sin!ct

To proceed with the analysis we must find the envelope of e.t/,which will be the final output yD.t/

Finding the envelope is a more general problem which will beuseful in future problem solving, so first consider the envelopeof

x.t/ D a.t/„ƒ‚…inphase

cos!ct b.t/„ƒ‚…quadrature

sin!ct

D Re˚a.t/ej!ct C jb.t/ej!ct

D Re

˚Œa.t/C jb.t/„ ƒ‚ …QR.t/Dcomplex envelope

ej!ct

In a phasor diagram x.t/ consists of an inphase or direct com-ponent and a quadrature component

R(t)

θ(t)

a(t)

b(t)

In-phase - I

Quadrature - Q

Note: R(t) =

R(t)ejθ(t) = a(t) + jb(t)

~



where the resultant R.t/ is such that

a.t/ D R.t/ cos .t/b.t/ D R.t/ sin .t/

which implies that

x.t/ D R.t/

cos .t/ cos!ct sin .t/ sin!ct

D R.t/ cos!ct C .t/

where .t/ D tan1Œb.t/=a.t/

The signal envelope is thus given by

R.t/ Dpa2.t/C b2.t/

The output of an envelope detector will be R.t/ if a.t/ andb.t/ are slowly varying with respect to cos!ct

In the SSB demodulator

yD.t/ D

s1

2Acm.t/CK

2C

1

2Ac Om.t/

2 If we choose K such that .Acm.t/=2CK/2 .Ac Om.t/=2/

2,then

yD.t/ '1

2Acm.t/CK

Note:

– The above analysis assumed a phase coherent reference

– In speech systems the frequency and phase can be ad-justed to obtain intelligibility, but not so in data systems



– The approximation relies on the binomial expansion

.1C x/1=2 ' 1C1

2x for jxj 1

Example 3.6: Noncoherent Carrier Reinsertion

Let m.t/ D cos!mt , !m !c and the reinserted carrier beK cosŒ.!c C!/t

Following carrier reinsertion we have

e.t/ D1

2Ac cos!mt cos!ct

1

2Ac sin!ct sin!ct CK cos

.!c C!/t

D1

2Ac cos

.!c ˙ !m/t

CK cos

.!c C!/t

We can write e.t/ as the real part of a complex envelope times

a carrier at either !c or !c C!

In this case, since K will be large compared to Ac=2, we write

e.t/ D1

2AcRe

ne˙j!mtej!ct

oCKRe

n1 ej.!cC!/t

oD Re

n 12Ace

j.˙!m!/t CK

„ ƒ‚ …complex envelope QR.t/

ej.!cC!/to



Finally expanding the complex envelope into the real and imag-inary parts we can find the real envelope R.t/

yD.t/ Dhn12Ac cosŒ˙!m C!/tCK

o2C

n12Ac sinŒ.˙!m C!/t

o2i1=2'1

2Ac cosŒ.!m !/tCK

where the last line follows for K Ac

Note that the frequency error ! causes the recovered mes-sage signal to shift up or down in frequency by !, but notboth at the same time as in DSB, thus the recovered speechsignal is more intelligible

3.1.4 Vestigial-Sideband Modulation

Vestigial sideband (VSB) is derived by filtering DSB such thatone sideband is passed completely while only a vestige remainsof the other

Why VSB?

1. Simplifies the filter design

2. Improves the low-frequency response and allows DC topass undistorted

3. Has bandwidth efficiency advantages over DSB or AM,similar to that of SSB



A primary application of VSB is the video portion of analogtelevision (note HDTV has replaced this in the US)

The generation of VSB starts with DSB followed by a filterthat has a 2ˇ transition band, e.g.,

jH.f /j D

8<:0; f < Fc ˇf .fcˇ/

2ˇ; fc ˇ f fc C ˇ

1; f > fc C ˇ

ffcf

c - β f

c + β

|H(f)|

1

Ideal VSB transmitter filter amplitude response

VSB can be demodulated using a coherent demod or using car-rier reinsertion and envelope detection

ff - f

1f + f

1f - f

2f + f

2fc

B/2A(1 - ε)/2

Aε/2

Transmitted Two-Tone Spectrum(only single-sided shown)

0

Two-tone VSB signal



Suppose the message signal consists of two tones

m.t/ D A cos!1t C B cos!2t

Following the DSB modulation and VSB shaping,

xc.t/ D1

2A cos.!c !1/t

C1

2A.1 / cos.!c C !1/t C

1

2B cos.!c C !2/t

A coherent demod multiplies the received signal by 4 cos!ctto produce

e.t/ D A cos!1t C A.1 / cos!1t C B cos!2tD A cos!1t C B cos!2t

which is the original message signal

The symmetry of the VSB shaping filter has made this possible

In the case of broadcast TV the carrier in included at the trans-mitter to insure phase coherency and easy demodulation at theTV receiver (VSB + Carrier)

– Very large video carrier power is required for typical TVstation, i.e., greater than 100,000 W

– To make matters easier still, the precise VSB filtering isnot performed at the transmitter due to the high powerrequirements, instead the TV receiver does this



0

0

-0.75

-0.75 0.75

-1.75 4.0

4.0

4.5

4.75

4.75(f - f

cv) MHz

(f - fcv

) MHz

Video CarrierAudio Carrier

TransmitterOutput

ReceiverShapingFilter

1

2β interval

Broadcast TV transmitter spectrum and receiver shaping filter

3.1.5 Frequency Translation and Mixing

Used to translate baseband or bandpass signals to some newcenter frequency

BPFatf2 ff

f1 f

2

Local oscillator of the form

e(t)

m(t)cos1t

Frequency translation system

Assuming the input signal is DSB of bandwidth 2W the mixer(multiplier) output is

e.t/ D m.t/ cos.!1t /

local osc (LO)‚ …„ ƒ2 cos.!1 ˙ !2/t

D m.t/ cos.!2t /Cm.t/ cosŒ.2!1 ˙ !2/t



The bandpass filter bandwidth needs to be at least 2W Hz wide

Note that if an input of the form k.t/ cosŒ.!1˙2!2/t is presentit will be converted to !2 also, i.e.,

e.t/ D k.t/ cos.!2t /C k.t/ cosŒ.2!1 ˙ 3!2/t ;

and the bandpass filter output is k.t/ cos.!2t /

The frequencies !1˙2!2 are the image frequencies of !1 withrespect to !LO D !1 ˙ !2

Example 3.7: AM Broadcast Superheterodyne Receiver

TunableRF-Amp

IF Filt/Amp

LocalOsc.

EnvDet

AudioAmp

Joint tuning

Automatic gaincontrol

fIF

AM Broadcast Specs: fc = 540 to 1600 kHz on 10 kHz spacings

carrier stability Modulated audio flat 100 Hz to 5 kHz Typical f

IF = 455 kHz

For AM BT = 2W

AM Superheterodyne receiver

We have two choices for the local oscillator, high-side or low-side tuning



– Low-side: 540455 fLO 1600455 or 85 fLO

1145, all frequencies in kHz

– High-side: 540 C 455 fLO 1600 C 455 or 995 fLO 2055, all frequencies in kHz

The high-side option is advantageous since the tunable oscil-lator or frequency synthesizer has the smallest frequency ratiofLO,max=fLO,min D 2055=995 D 2:15

Suppose the desired station is at 560 kHz, then with high-sidetuning we have fLO D 560C 455 D 1015 kHz

The image frequency is at fimage D fc C 2fIF D 560 C 2

455 D 1470 kHz (note this is another AM radio station centerfrequency

f (kHz)

f (kHz)

f (kHz)

f (kHz)0

Input

fLO

MixerOutput

ImageOut ofmixer

455 560 1470

1470

1575(560+1015)

2485(1470+1015)

1015(560+455)

455

Desired Potential Image

This is removedwith RF BPF

IF BPF

1470-1015

1015-560

fIF

fIF

BRF

BIF

Receiver frequency plan including images



Example 3.8: A Double-Conversion Receiver

TunableRF-Amp

10.7 MHzIF BPF

455 kHzIF BPF

1stLO

2ndLO

FMDemod

fc = 162.475 MHz

(WX #4)

fLO1

= 173.175 MHz fLO2

= 11.155 MHz

10.7 MHz &335.65 MHz

455 kHz &21.855 MHz

Double-conversion superheterodyne receiver

Consider a frequency modulation (FM) receiver that uses double-conversion to receive a signal con carrier frequency 162.475MHz (weather channel #4)

– Frequency modulation will be discussed in the next sec-tion

The dual-conversion allows good image rejection by using a10.7 MHz first IF and then can provide good selectivity byusing a second IF at 455 kHz; why?

– The ratio of bandwidth to center frequency can only be sosmall in a low loss RF filter

– The second IF filter can thus have a much narrower band-width by virtue of the center frequency being much lower

A higher first IF center frequency moves the image signal fur-ther away from the desired signal



– For high-side tuning we have fimage D fc C 2fIF D fc C

21:4 MHz

Double-conversion receivers are more complex to implement

Mixers

The multiplier that is used to implement frequency translationis often referred to as a mixer

In the world of RF circuit design the term mixer is more ap-propriate, as an ideal multiplier is rarely available

Instead active and passive circuits that approximate signal mul-tiplication are utilized

The notion of mixing comes about from passing the sum of twosignals through a nonlinearity, e.g.,

y.t/ D Œa1x1.t/C a2x2.t/2C other terms

D a21x21.t/C 2a1a2x1.t/x2.t/C a

22x

22.t/

In this mixing application we are most interested in the centerterm

ydesired.t/ D 2a1a2x1.t/ x2.t/

Clearly this mixer produces unwanted terms (first and third),

and in general many other terms, since the nonlinearity willhave more than just a square-law input/output characteristic



A diode or active device can be used to form mixing productsas described above, consider the dual-gate MEtal Semiconduc-tor FET (MESFET) mixer shown below

+5V

R2

C4

L3

47pF

L4

C8C7Q1NE25139

IF270nH270nH

82pF

C3

42pF

0.01uF

10Ω

C8R1 C5

47pF0.01uF

270Ω

L15 turns, 28 AWG.050 I.D.

C10.5pFLO

5 turns, 28 AWG.050 I.D.

C2

L2

0.5pF

RF

G1

G2D

S

Dual-Gate MESFET Active Mixer

VRF

VOUT

VIN

VLO

zL

Nonlinear Device

Mixer concept

The double-balanced mixer (DBM), which can be constructedusing a diode ring, provides better isolation between the RF,LO, and IF ports

When properly balanced the DBM also allows even harmonicsto be suppressed in the mixing operation



A basic transformer coupled DBM, employing a diode ring, isshown below, followed by an active version

The DBM is suitable for use as a phase detector in phase-locked loop applications

02 91 81 71

DNG

+FI

-FI

DNG

61DNG

13

12

11

14

15

GND

GND

GND

LO2LO2

LO1LO1

4

3

2

1

RFBIAS

TAP

RFRFIN

5GND

LO SELECT

GND

6 7 8 9

VCC VCCD NG

D NG

LE SOL

01

MAX9982

IF OUT

4:1 (200:50)TRANSFORMER

41

25V

3C9

C10

C7

C6

C11C8

R3L1

R4L2

T1 6

5V5V

C5C4

R1

C2

C1

C3

825 MHz to 915 MHz SiGe High-Linearity Active DBM

vo(t)

mixer

D3 D4

D1D2inputLO

R G

LO source

vp(t)

inputRF

R G

RF source

R L

IF load

vi (t)

IFout

Passive Double-Balanced Mixer (DBM)



Example 3.9: Single Diode Mixer



3.2 Angle Modulation

A general angle modulated signal is of the form

xc.t/ D Ac cosŒ!ct C .t/

Definition: Instantaneous phase of xc.t/ is

i.t/ D !ct C .t/

where .t/ is the phase deviation

Define: Instantaneous frequency of xc.t/ is

!i.t/ Ddi.t/

dtD !c C

d.t/

dt

where d.t/=dt is the frequency deviation

There are two basic types of angle modulation

1. Phase modulation (PM)

.t/ D kp„ƒ‚…phase dev. const.

m.t/

which implies

xc.t/ D Ac cosŒ!ct C kpm.t/

– Note: the units of kp is radians per unit of m.t/

– If m.t/ is a voltage, kp has units of radians/volt


3.2. ANGLE MODULATION

2. Frequency modulation (FM)

d.t/

dtD kf„ƒ‚…

freq. dev. const.

m.t/

or

.t/ D kf

Z t

t0

m.˛/ d˛ C 0

– Note: the units of kf is radians/sec per unit of m.t/

– If m.t/ is a voltage, kf has units of radians/sec/volt

– An alternative expression for kf is

kf D 2fd

where fd is the frequency-deviation constant in Hz/unitof m.t/

Example 3.10: Phase and Frequency Step Modulation

Consider m.t/ D u.t/ v

We form the PM signal

xPM.t/ D Ac cos!ct C kpu.t/

; kp D =3 rad/v

We form the FM signal

xFM.t/ D Ac cosh!ct C 2fd

Z t

m.˛/ d˛i; fd D 3 Hz/v



1 1

1

1

1 1

1

1

t t

π/3 phase step at t = 0 3 Hz frequency step at t = 0

Phase Modulation Frequency Modulation

fc

fc

fc + 3 Hzf

c

Phase and frequency step modulation

3.2.1 Narrowband Angle Modulation

Begin by writing an angle modulated signal in complex form

xc.t/ D ReAce

j!ctej.t/

Expand ej.t/ in a power series

xc.t/ D ReAce

j!ct

1C j.t/

2.t/

2Š

The narrowband approximation is j.t/j 1, then

xc.t/ ' ReAce

j!ct C jAc.t/ej!ct

D Ac cos.!ct / Ac.t/ sin.!ct /



Under the narrowband approximation we see that the signal issimilar to AM except it is carrier plus modulated quadraturecarrier

90o

φ(t)

Ac sin(ω

ct)

NBFMx

c(t)

+

−

NBFM modulator block diagram

Example 3.11: Single tone narrowband FM

Consider NBFM with m.t/ D cos!mt

.t/ D 2fd

Z t

cos!mt d˛

D2fd

2fmsin!mt D

fd

fmsin!mt

Now,

xc.t/ D cos!ct C

fd

fmsin!mt

' Ac

cos!ct

fd

fmsin!mt sin!ct

D Ac cos!ct C

fd

2fmsin.fc C fm/t

fd

2fmsin.fc fm/t

This looks very much like AM



fc

fc + f

m

fc - f

m f

0

Single tone NBFM spectra

3.2.2 Spectrum of an Angle-Modulated Signal

The development in this obtains the exact spectrum of an anglemodulated carrier for the case of

.t/ D ˇ sin!mt

where ˇ is the modulation index for sinusoidal angle modula-tion

The transmitted signal is of the form

xc.t/ D Ac cos!ct C ˇ sin!mt

D AcRe

˚ej!ct ejˇ sin!mt

Note that ejˇ sin!mt is periodic with period T D 2=!m, thus

we can obtain a Fourier series expansion of this signal, i.e.,

ejˇ sin!mt D

1XnD1

Ynejn!mt



The coefficients are

Yn D!m

2

Z =!m

=!m

ejˇ sin!mtejn!mt dt

D!m

2

Z =!m

=!m

ej.n!mtˇ sin!mt / dt

Change variables in the integral by letting x D !mt , then dx D!mdt , t D =!m! x D , and t D =!m! x D

With the above substitutions, we have

Yn D1

2

Z

ej.nxˇ sin x/ dx

D1

Z

0

cos.nx ˇ sin x/ dx D Jn.ˇ/

which is a Bessel function of the first kind order n with argu-ment ˇ

Jn.ˇ/ Properties

Recurrence equation:

JnC1.ˇ/ D2n

ˇJn.ˇ/ Jn1.ˇ/

n – even:Jn.ˇ/ D Jn.ˇ/

n – odd:Jn.ˇ/ D Jn.ˇ/



J0(β)

J1(β)

J2(β) J

3(β)

β2 4 6 8 10

0.4

0.2

0.2

0.4

0.6

0.8

1

Bessel function of order 0–3 plotted

The zeros of the Bessel functions are important in spectralanalysis

First five Bessel function zeros for order 0 – 5

J0(β) = 0

2.40483, 5.52008, 8.65373, 11.7915, 14.9309

J1(β) = 0

3.83171, 7.01559, 10.1735, 13.3237, 16.4706

J2(β) = 0

5.13562, 8.41724, 11.6198, 14.796, 17.9598

J3(β) = 0

6.38016, 9.76102, 13.0152, 16.2235, 19.4094

J4(β) = 0

7.58834, 11.0647, 14.3725, 17.616, 20.8269

J5(β) = 0

8.77148, 12.3386, 15.7002, 18.9801, 22.2178



Spectrum cont.

We obtain the spectrum of xc.t/ by inserting the series repre-sentation for ejˇ sin!mt

xc.t/ D AcRe

"ej!ct

1XnD1

Jn.ˇ/ejn!mt

#

D Ac

1XnD1

Jn.ˇ/ cos.!c C n!m/t

We see that the amplitude spectrum is symmetrical about fcdue to the symmetry properties of the Bessel functions

ffc

f c +

2f m

f c +

3f m

f c +

4f m

f c +

5f m

f c +

f m

f c - f m

f c -

2fm

f c -

3fm

f c -

5fm

f c -

4fm

|AcJ

0(β)|

|AcJ

1(β)|

|AcJ

2(β)||A

cJ

-2(β)|

|AcJ

3(β)||A

cJ

-3(β)|

|AcJ

4(β)||A

cJ

-4(β)|

|AcJ

5(β)||A

cJ

-5(β)|

|AcJ

-1(β)|

Am

plitu

de S

pect

rum

(one

-sid

ed)

For PMˇ sin!mt D kp .A sin!mt /„ ƒ‚ …

m.t/

) ˇ D kpA

For FM

ˇ sin!mt D kf

Z t

A cos!m˛ d˛ Dfd

fmA sin!mt

) ˇ D .fd=fm/A



When ˇ is small we have the narrowband case and as ˇ getslarger the spectrum spreads over wider bandwidth

-5 0 5 10

0.2

0.4

0.6

0.8

1

-5 0 5 10

0.2

0.4

0.6

0.8

1

-5 0 5 10

0.2

0.4

0.6

0.8

1

-5 0 5 10

0.2

0.4

0.6

0.8

1

β = 0.2, Ac = 1

(narrowband case)

β = 1, Ac = 1

β = 2.4048, Ac = 1

(carrier null)

β = 3.8317, Ac = 1

(1st sideband null)

β = 8, Ac = 1

(spectrum becoming wide)

-5 0 5 10

0.2

0.4

0.6

0.8

1

(f - fc)/f

m

(f - fc)/f

m

(f - fc)/f

m

(f - fc)/f

m

(f - fc)/f

m

Am

plitu

deSp

ectr

umA

mpl

itude

Spec

trum

Am

plitu

deSp

ectr

umA

mpl

itude

Spec

trum

Am

plitu

deSp

ectr

um

The amplitude spectrum relative to fc as ˇ increases



Example 3.12: VCO FM Modulator

Consider again single-tone FM, that is m.t/ D A cos.2fmt /

We assume that we know fm and the modulator deviation con-stant fd

FindA such that the spectrum of xc.t/ contains no carrier com-ponent

An FM modulator can be implemented using a voltage con-trolled oscillator (VCO)

VCO

CenterFreq = f

c

m(t)

sensitivity fd MHz/v

A VCO used as an FM modulator

The carrier term is AcJ0.ˇ/ cos!ct

We know that J0.ˇ/ D 0 for ˇ D 2:4048; 5:5201; : : :

The smallest ˇ that will make the carrier component zero is

ˇ D 2:4048 Dfd

fmA

which implies that we need to set

A D 2:4048 f m

fd



Suppose that fm D 1 kHz and fd D 2:5 MHz/v, then wewould need to set

A D 2:4048 1 103

2:5 106D 9:6192 104

3.2.3 Power in an Angle-Modulated Signal

The average power in an angle modulated signal is

hx2c .t/i D A2chcos2

!ct C .t/

i

D1

2A2c C

1

2A2chcos

˚2!ct C .t/

i

For large fc the second term is approximately zero (why?),thus

Pangle mod D hx2c .t/i D

1

2A2c

which makes the power independent of the modulation m.t/(the assumptions must remain valid however)

3.2.4 Bandwidth of Angle-Modulated Signals

With sinusoidal angle modulation we know that the occupiedbandwidth gets larger as ˇ increases

There are an infinite number of sidebands, but

limn!1

Jn.ˇ/ limn!1

ˇn

2nnŠD 0;

so consider the fractional power bandwidth



Define the power ratio

Pr DPcarrier C P˙k sidebands

PtotalD

12A2cPk

nDk J2n .ˇ/

12A2c

D J 20 .ˇ/C 2

kXnD1

J 2n .ˇ/

Given an acceptable Pr implies a fractional bandwidth of

B D 2kfm (Hz)

In the text values of Pr 0:7 and Pr 0:98 are single anddouble underlined respectively

It turns out that for Pr 0:98 the value of k is IPŒ1C ˇ, thus

B D B98 ' 2.ˇ C 1/fm sinusoidal mod only

For arbitrary modulation m.t/, define the deviation ratio

D Dpeak freq. deviationbandwidth of m.t/

Dfd

W

max jm.t/j

In the sinusoidal modulation bandwidth definition let ˇ ! D

and fm! W , then we obtain what is known as Carson’s rule

B D 2.D C 1/W

– Another view of Carson’s rule is to consider the maxi-mum frequency deviationf D max jm.t/jfd , thenB D2.W Cf /



Two extremes in angle modulation are

1. Narrowband: D 1 ) B D 2W

2. Wideband: D 1 ) B D 2DW D 2f

Example 3.13: Single Tone FM

Consider an FM modulator for broadcasting with

xc.t/ D 100 cos2.101:1 106/t C .t/

where fd D 75 kHz/v and

m.t/ D cos2.1000/t

v

The ˇ value for the transmitter is

ˇ Dfd

fmA D

75 103

103D 75

Note that the carrier frquency is 101.1 MHz and the peak de-viation is f D 75 kHz

The bandwidth of the signal is thus

B ' 2.1C 75/1000 D 152 kHz

-50 0 50 100

2.55

7.510

12.515

17.5

(f - 101.1 MHz)1 kHz

Am

plitu

deSp

ectr

um

B = 2(β + 1)fm

76-76

101.1 MHz 101.1 MHz + 76 kHz101.1 MHz - 76 kHz



Suppose that this signal is passed through an ideal bandpassfilter of bandwidth 11 kHz centered on fc D 101:1 MHz, i.e.,

H.f / D …

f fc

11000

C…

f C fc

11000

The carrier term and five sidebands either side of the carrier

pass through this filter, resulting an output power of

Pout DA2c2

"J 20 .75/C 2

5XnD1

J 2n .75/

#D 241:93 W

Note the input power is A2c=2 D 5000 W

Example 3.14: Two Tone FM

Finding the exact spectrum of an angle modulated carrier is notalways possible

The single-tone sinusoid case can be extended to multiple tonewith increasing complexity

Suppose that

m.t/ D A cos!1t C B cos!2t

The phase deviation is given by 2fd times the integral of thefrequency modulation, i.e.,

.t/ D ˇ1 sin!1t C ˇ2 sin!2t

where ˇ1 D Afd=f1 and ˇ2 D Afd=f2



The transmitted signal is of the form

xc.t/ D Ac cos!ct C ˇ1 sin!1t C ˇ2 sin!2t

D AcRe

ej!ctejˇ1 sin!1tejˇ2 sinˇ2t

We have previously seen that via Fourier series expansion

ejˇ1 sin!1t D

1XnD1

Jn.ˇ1/ejn!1t

ejˇ2 sin!1t D

1XnD1

Jn.ˇ2/ejn!2t

Inserting the above Fourier series expansions into xc.t/, wehave

xc.t/ D AcRe

(ej!ct

1XnD1

Jn.ˇ1/ejn!1t

mXmD1

Jm.ˇ2/ejm!2t

)

D Ac

1XnD1

1XmD1

Jn.ˇ1/Jm.ˇ2/ cos.!c C n!1 Cm!2/t

The nonlinear nature of angle modulation is clear, since we see

not only components at !c C n!1 and !c C m!2, but also atall combinations of !c C n!1 Cm!2

To find the bandwidth of this signal we can use Carson’s rule(the sinusoidal formula only works for one tone)

Recall that B D 2.f CW /, wheref is the peak frequencydeviation



The frequency deviation is

fi.t/ D1

2

d

dt

ˇ1 sin!1t C ˇ2 sin!2t

D ˇ1f1 cos.2f1t /C ˇ2f2 cos.2f2t / Hz

The maximum of fi.t/, in this case, is ˇ1f1 C ˇ2f2

Suppose ˇ1 D ˇ2 D 2 and f2 D 10f1, then we see that W Df2 D 10f1 and

B D 2.WCf / D 210f1C2.f1C10f1/

D 2.32f1/ D 64f1

Am

plitu

deSp

ectr

um

(f - fc)

f1

β1 = β

2 = 2, f

2 = 10f

1

B = 2(W + ∆f) = 2(10f1 + 2(11)f

1) = 64f

1

-40 -20 0 20 40

0.050.10.150.2

0.250.3

0.35

Example 3.15: Bandlimited Noise PM and FM

This example will utilize simulation to obtain the spectrum ofan angle modulated carrier

The message signal in this case will be bandlimited noise hav-ing lowpass bandwidth of W Hz



In MATLAB we can generate Gaussian amplitude distributedwhite noise using randn() and then filter this noise using ahigh-order lowpass filter (implemented as a digital filter in thiscase)

We can then use this signal to phase or frequency modulatea carrier in terms of the peak phase deviation, derived fromknowledge of maxŒj.t/j



3.2.5 Narrowband-to-Wideband Conversion

Narrowband FM Modulator(similar to AM)

x nFreq.

Multiplier

LO

BPF

Frequencytranslate

Narrowband FMCarrier = f

c1

Peak deviation = fd1

Deviation ratio = D1

Wideband FMCarrier = nf

c1

Peak deviation = nfd1

Deviation ratio = nD1

xc(t)

m(t)

Ac1

cos[ωct + φ(t)]

Ac2

cos[nωct + nφ(t)]

narrowband-to-wideband conversion

Narrowband FM can be generated using an AM-type modula-tor as discussed earlier (a VCO is not required, so the carriersource can be very stable)

A frequency multiplier, using say a nonlinearity, can be usedto make the signal wideband FM, i.e.,

Ac1 cosŒ!ct C .t/n! Ac2 cosŒn!ct C n.t/

so the modulator deviation constant of fd1 becomes nfd1

3.2.6 Demodulation of Angle-Modulated Signals

To demodulate FM we require a discriminator circuit, whichgives an output which is proportional to the input frequencydeviation

For an ideal discriminator with input

xr.t/ D Ac cosŒ!ct C .t/



the output is

yD.t/ D1

2KD

d.t/

dt

IdealDiscriminatorx

c(t) y

D(t)

Ideal FM discriminator

For FM

.t/ D 2fd

Z t

m.˛/ d˛

soyD.t/ D KDfdm.t/

slope = KD

InputFrequency

OutputSignal (voltage)

fc

Ideal discriminator I/O characteristic

For PM signals we follow the discriminator with an integrator

yD(t)x

r(t)

IdealDiscrim.

Ideal discriminator with integrator for PM demod



For PM .t/ D kpm.t/ so

yD.t/ D KDkpm.t/

We now consider approximating an ideal discriminator with:

EnvelopeDetectorx

r(t) y

D(t)

e(t)

Ideal discriminator approximation

If xr.t/ D Ac cosŒ!ct C .t/

e.t/ Ddxr.t/

dtD Ac

!c C

d

dt

sin!ct C .t/

This looks like AM provided

d.t/

dt< !c

which is only reasonable

Thus

yD.t/ D Acd.t/

dtD 2Acfdm.t/ (for FM)

– Relative to an ideal discriminator, the gain constant isKD D 2Ac

To eliminate any amplitude variations onAc pass xc.t/ througha bandpass limiter



LimiterBPF

EnvelopeDetector

Bandpass Limiter

xr(t) y

D(t)

e(t)

FM discriminator with bandpass limiter

We can approximate the differentiator with a delay and subtractoperation

e.t/ D xr.t/ xr.t /

since

lim!0

e.t/

D lim

!0

xr.t/ xr.t /

Ddxr.t/

dt;

thuse.t/ '

dxr.t/

dt

In a discrete-time implementation (DSP), we can perform asimilar operation, e.g.

eŒn D xŒn xŒn 1

Example 3.16: Complex Baseband Discriminator

A DSP implementation in MATLAB that works with complexbaseband signals (complex envelope) is the following:

function disdata = discrim(x)% function disdata = discrimf(x)% x is the received signal in complex baseband form%% Mark Wickert



xI=real(x); % xI is the real part of the received signalxQ=imag(x); % xQ is the imaginary part of the received signalN=length(x); % N is the length of xI and xQb=[1 -1]; % filter coefficientsa=[1 0]; % for discrete derivativeder_xI=filter(b,a,xI); % derivative of xI,der_xQ=filter(b,a,xQ); % derivative of xQ% normalize by the squared envelope acts as a limiterdisdata=(xI.*der_xQ-xQ.*der_xI)./(xI.^2+xQ.^2);

To understand the operation of discrim() start with a generalangle modulated signal and obtain the complex envelope

xc.t/ D Ac cos.!ct C .t//

D Re˚Ace

j.t/ej!ct

D AcRe˚Œcos.t/C j sin.t/ej!ct

The complex envelope is

Qxc.t/ D cos.t/C j sin.t/ D xI .t/C jxQ.t/

where xI and xQ are the in-phase and quadrature signals re-spectively

A frequency discriminator obtains d.t/=dt

In terms of the I and Q signals,

.t/ D tan1xQ.t/

xI .t/

The derivative of .t/ is

d.t/

dtD

1

1C .xQ.t/=xI .t//2d

dt

xQ.t/

xI .t/

DxI .t/x

0

Q.t/ x0

I .t/xQ.t/

x2I .t/C x2Q.t/



In the DSP implementation xI Œn D xI .nT / and xQŒn DxQ.nT /, where T is the sample period

The derivatives, x0I .t/ and x0Q.t/ are approximated by the back-wards difference xI Œn xI Œn 1 and xQŒn xQŒn 1 re-spectively

To put this code into action, consider a single tone message at1 kHz with ˇ D 2:4048

.t/ D 2:4048 cos.2.1000/t/

The complex baseband (envelope) signal is

Qxc.t/ D ej.t/D ej 2:4048 cos.2.1000/t/

A MATLAB simulation that utilizes the function Discrim() is:

>> n = 0:5000-1;>> m = cos(2*pi*n*1000/50000); % sampling rate = 50 kHz>> xc = exp(j*2.4048*m);>> y = Discrim(xc);>> % baseband spectrum plotting tool using psd()>> bb_spec_plot(xc,2^11,50);>> axis([-10 10 -30 30])>> grid>> xlabel('Frequency (kHz)')>> ylabel('Spectral Density (dB)')>> t = n/50;>> plot(t(1:200),y(1:200))>> axis([0 4 -.4 .4])>> grid>> xlabel('Time (ms)')>> ylabel('Amplitude of y(t)')



−10 −8 −6 −4 −2 0 2 4 6 8 10−30

−20

−10

0

10

20

30

Frequency (kHz)

Spe

ctra

l Den

sity

(dB

)

0 0.5 1 1.5 2 2.5 3 3.5 4−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Time (ms)

Am

plitu

de o

f y(t

)Note: no carrier term since β = 2.4048

Baseband FM spectrum and demodulator output wavefrom



Analog Circuit Implementations

A simple analog circuit implementation is an RC highpass fil-ter followed by an envelope detector

Highpass Envelope Detector

ReR C

e

C

C R

|H(f)|

1

0.707

fc

f1

2πRC

Linear operating region converts FM to AM

Highpass

RC highpass filter + envelope detector discriminator (slope detector)

For the RC highpass filter to be practical the cutoff frequencymust be reasonable

Broadcast FM radio typically uses a 10.7 MHz IF frequency,which means the highpass filter must have cutoff above thisfrequency

A more practical discriminator is the balanced discriminator,which offers a wider linear operating range



R

R

L1

C1

C2

L2

Re

Re

Ce

Ce

yD(t)x

c(t)

Bandpass Envelope Detectors

f

f

Filte

r A

mpl

itude

Res

pons

eFi

lter

Am

plitu

deR

espo

nse

|H1(f)|

|H1(f)| - |H

2(f)|

|H2(f)|

f2

f2

f1

f1

Linear region

Balanced discriminator operation (top) and a passive implementation(bottom)



FM Quadrature Detectors

C1

Cp

Lp

xc(t)

xquad

(t)

xout

(t)

Tank circuit tuned to f

c

Usually a lowpass filter is added here

Quadrature detector schematic

In analog integrated circuits used for FM radio receivers andthe like, an FM demodulator known as a quadrature detectoror quadrature discriminator, is quite popular

The input FM signal connects to one port of a multiplier (prod-uct device)

A quadrature signal is formed by passing the input to a capaci-tor series connected to the other multiplier input and a paralleltank circuit resonant at the input carrier frequency

The quadrature circuit receives a phase shift from the capacitorand additional phase shift from the tank circuit

The phase shift produced by the tank circuit is time varying inproportion to the input frequency deviation

A mathematical model for the circuit begins with the FM inputsignal

xc.t/ D AcŒ!ct C .t/


3.3. INTERFERENCE

The quadrature signal is

xquad.t/ D K1Ac sin!ct C .t/CK2

.t/

dt

where the constants K1 and K2 are determined by circuit pa-rameters

The multiplier output, assuming a lowpass filter removes thesum terms, is

xout.t/ D1

2K1A

2c sin

K2

d.t/

dt

By proper choice of K2 the argument of the sin function issmall, and a small angle approximation yields

xout.t/ '1

2K1K2A

2c

d.t/

dtD1

2K1K2A

2cKDm.t/

3.3 Interference

Interference is a fact of life in communication systems. A throughunderstanding of interference requires a background in random sig-nals analysis (Chapter 6 of the text), but some basic concepts canbe obtained by considering a single interference at fc C fi that liesclose to the carrier fc



3.3.1 Interference in Linear Modulation

ffc - f

mfc + f

mfc + f

ifc

Xr(f)

Sing

le-S

ided

Spe

ctru

m12A

m

12A

m

Ac

Ai

AM carrier with single tone interference

If a single tone carrier falls within the IF passband of the re-ceiver what problems does it cause?

Coherent Demodulator

xr.t/ DAc cos!ct C Am cos!mt cos!ct

C Ai cos.!c C !i/t

– We multiply xr.t/ by 2 cos!ct and lowpass filter

yD.t/ D Am cos!mt C Ai cos!i t„ ƒ‚ …interference

Envelope Detection: Here we need to find the received enve-lope relative to the strongest signal present

– Case Ac Ai

– We will expand xr.t/ in complex envelope form by firstnoting that

Ai cos.!cC!i/t D Ai cos!i t cos!ctAi sin!i t sin!ct


3.3. INTERFERENCE

now,

xr.t/ D Re˚Ac C Am cos!mt C Ai cos!i t

jAi sin!i tej!ct

D Re

˚QR.t/ej!ct

so

R.t/ D j QR.t/j

D

h.Ac C Am cos!mt C Ai cos!i t /2

C .Ai sin!i t /2i1=2

' Ac C Am cos!mt C Ai cos!i t

assuming that Ac Ai

– Finally,

yD.t/ ' Am cos!mt C Ai cos!i t„ ƒ‚ …interference

– Case Ai >> Ac– Now the interfering term looks like the carrier and the re-

maining terms look like sidebands, LSSB sidebands rela-tive to fc C fi to be specific

– From SSB envelope detector analysis we expect

yD.t/ '1

2Am cos.!i C !m/t C Ac cos!i t

C1

2Am cos.!i !m/t

and we conclude that the message signal is lost!



3.3.2 Interference in Angle Modulation

Initially assume that the carrier is unmodulated

xr.t/ D Ac cos!ct C Ai cos.!c C !i/t

In complex envelope form we have

xr.t/ D Re˚.Ac C Ai cos!i t jAi sin!i t /ej!ct

with QR.t/ D Ac C Ai cos!i t jAi sin!i t

The real envelope or envelope magnitude is, R.t/ D j QR.t/j,

R.t/ Dp.Ac C Ai cos!i t /2 C .Ai sin!i t /2

and the envelope phase is

.t/ D tan1

Ai sin!i tAc C Ai cos!i t

For future reference note that:

tan1 x D x x3

3Cx5

5x7

7C

jxj1' x

We can thus write that

xr.t/ D R.t/ cos!ct C .t/

If Ac Ai

xr.t/ ' .Ac C Ai cos!i t /„ ƒ‚ …R.t/

cos!ct C

Ai

Acsin!i t„ ƒ‚ ….t/


3.3. INTERFERENCE

Case of PM Demodulator: The discriminator recovers d.t/=dt ,so the output is followed by an integrator

yD.t/ D KD

Ai

Acsin!i t

Case of FM Demodulator: The discriminator output is used di-rectly to obtain d.t/=dt

yD.t/ D1

2KD

Ai

Ac

d

dtsin!i t D KD

Ai

Acfi cos!i t

We thus see that the interfering tone appears directly in theoutput for both PM and FM

For the case of FM the amplitude of the tone is proportional tothe offset frequency fi

For fi > W , recall W is the bandwidth of the message m.t/,a lowpass filter following the discriminator will remove theinterference

When Ai is similar to Ac and larger, the above analysis nolonger holds

In complex envelope form

xr.t/ D Re˚Ac C Aie

j!i tej!ct

The phase of the complex envelope is

.t/ D †Ac C Aie

j!i tD tan1

Ai sin!i t

Ac C Ai cos!i t



We now consider Ai Ac and look at plots of .t/ and thederivative

1 0.5 0.5 1

0.1

0.05

0.05

0.1

1 0.5 0.5 1

1

0.5

0.5

1

1 0.5 0.5 1

3

2

1

1

2

3

1 0.5 0.5 1

0.6

0.4

0.2

0.2

0.4

0.6

1 0.5 0.5 1

50

40

30

20

10

1 0.5 0.5 110

20

30

40

50

60

70

φ(t)

φ(t)

φ(t)

Ai = 0.1A

c

fi = 1

Ai = 0.9A

c

fi = 1

Ai = 1.1A

c

fi = 1

dφ(t)/dt

dφ(t)/dt

dφ(t)/dt

t t

t

t

t

t

Phase deviation and discriminator outputs when Ai Ac

We see that clicks (positive or negative spikes) occur in thediscriminator output when the interference levels is near thesignal level

When Ai Ac the message signal is entirely lost and thediscriminator is said to be operating below threshold


3.3. INTERFERENCE

To better see what happens when we approach threshold, applysingle tone FM to the carrier

.t/ D †Ace

jAm cos.!mt / C Aiej!i t

Plot the discriminator output d.t/=dt with Am D 5, fm D 1,fi D 3, and various values of Ai

1 0.5 0.5 1

30

20

10

10

20

30

1 0.5 0.5 1

30

20

10

10

20

30

1 0.5 0.5 1

80

60

40

20

201 0.5 0.5 1

400

300

200

100

Ai = 0.005,

fi = 3Am = β = 5,

fm = 1

Ai = 0.5,

fi = 3Am = β = 5,

fm = 1

Ai = 0.9,

fi = 3Am = β = 5,

fm = 1

Ai = 0.1,

fi = 3Am = β = 5,

fm = 1

t

t

t

t

dφ(t)/dt dφ(t)/dt

dφ(t)/dt dφ(t)/dt

Discriminator outputs as Ai approaches Ac with single tone FM ˇ D 5

The Use of Preemphasis in FM

We have seen that when Ai is small compared to Ac the inter-ference level in the case of FM demodulation is proportionalto fi

The generalization from a single tone interferer to backgroundnoise (text Chapter 6), shows a similar behavior, that is wide



bandwidth noise entering the receiver along with the desiredFM signal creates noise in the discriminator output that hasamplitude proportional with frequency (noise power propor-tional to the square of the frequency)

In FM radio broadcasting a preemphasis boosts the high fre-quency content of the message signal to overcome the increasednoise background level at higher frequencies, with a deem-phasis filter used at the discriminator output to gain equal-ize/flatten the end-to-end transfer function for the modulationm.t/

FMMod Discrim

C

Rr C

r

HP(f) H

d(f)

ff1

f2

|Hp(f)|

ff1

|Hd(f)|

Wf1

f0

Dis

crim

inat

or O

utpu

tw

ith I

nter

fere

nce/

Noi

se No preemphasis

With preemphasis

Message Bandwidth

FM broadcast preemphasis and deemphasis filtering

The time constant for these filters is RC D 75 s (f1 D1=.2RC/ D 2:1 kHz ), with a high end cutoff of aboutf2 D 30 kHz


3.4. FEEDBACK DEMODULATORS

3.4 Feedback Demodulators

The discriminator as described earlier first converts and FMsignal to and AM signal and then demodulates the AM

The phase-locked loop (PLL) offer a direct way to demodulateFM and is considered a basic building block by communicationsystem engineers

3.4.1 Phase-Locked Loops for FM Demodula-tion

The PLL has many uses and many different configurations,both analog and DSP based

We will start with a basic configuration for demodulation ofFM

PhaseDetector

VCO

LoopFilter

LoopAmplifier

µxr(t)

xr(t)

ed(t)

ev(t)

eo(t)

-eo(t)

Kd

Sinusoidal phase detectorwith invertinginput

Basic PLL block diagram



Let

xr.t/ D Ac cos!ct C .t/

eo.t/ D Av sin

!ct C .t/

– Note: Frequency error may be included in .t/ .t/

Assume a sinusoidal phase detector with an inverting operationis included, then we can further write

ed .t/ D1

2AcAvKd sin

.t/ .t/

– In the above we have assumed that the double frequency

term is removed (e.g., by the loop filter eventually)

Note that for the voltage controlled oscillator (VCO) we havethe following relationship

VCOK

v

ev(t) ω

o + dθ

dt

but

d.t/

dtD Kvev.t/ rad/s

) .t/ D Kv

Z t

ev.˛/ d˛

In its present form the PLL is a nonlinear feedback controlsystem



f(t)sin( )

µ

φ(t)

θ(t)

ev(t)

ed(t)+

- Loop filterimpulse response

Loopnonlinearity

ψ(t)

Nonlinear feedback control model

To shown tracking we first consider the loop filter to have im-pulse response ı.t/ (a straight through connection or unity gainamplifier)

The loop gain is now defined as

KtD1

2AcAvKdKv rad/s

The VCO output is

.t/ D Kt

Z t

sinŒ.˛/ .˛/ d˛

ord.t/

dtD Kt sinŒ.t/ .t/

Let .t/ D .t/.t/ and apply an input frequency step!,i.e.,

d.t/

dtD ! u.t/

Now,

d.t/

dtD !

d .t/

dtD Kt sin .t/; t 0



We can now plot d =dt versus , which is known as a phaseplane plot

Stablelock point

dψ(t)/dt

ψ(t)

∆ω > 0Β

Α

∆ω

∆ω + Kt

∆ω - Kt

ψss

Phase plane plot (1st-order PLL)

d .t/

dtCKt sin .t/ D !u.t/

At t D 0 the operating point is at B

Since dt is positive ifd

dt> 0 ! d is positive

Since dt is positive ifd

dt< 0 ! d is negative

therefore the steady-state operating point is at A

The frequency error is always zero in steady-state

The steady-state phase error is ss

– Note that for locking to take place, the phase plane curvemust cross the d =dt D 0 axis



The maximum steady-state value of! the loop can handle isthus Kt

The total lock range is then

!c Kt ! !c CKt ) 2Kt

– For a first-order loop the lock range and the hold-rangeare identical

For a given! the value of ss can be made small by increas-ing the loop gain, i.e.,

ss D sin1!

Kt

Thus for largeKt the in-lock operation of the loop can be mod-

eled with a fully linear model since .t/ .t/ is small, i.e.,

sinŒ.t/ .t/ ' .t/ .t/

The s-domain linear PLL model is the following

Kv/s

AcAvKd/2 F(s) µ

Ev(s)Φ(s)

Θ(s)

+−

Ψ(s)

Linear PLL model

Solving for ‚.s/ we have

‚.s/ DKt

s

ˆ.s/ ‚.s/

F.s/

or ‚.s/

1C

Kt

sF.s/

DKt

sˆ.s/F.s/



Finally, the closed-loop transfer function is

H.s/D‚.s/

ˆ.s/D

KtsF.s/

1C KtsF.s/

DKtF.s/

s CKtF.s/

First-Order PLL

Let F.s/ D 1, then we have

H.s/ DKt

Kt C s

Consider the loop response to a frequency step, that is for FM,we assume m.t/ D Au.t/, then

.t/ D Akf

Z t

u.˛/ d˛

so ˆ.s/ DAkf

s2

The VCO phase output is

‚.s/ DAkfKt

s2.Kt C s/

The VCO control voltage should be closely related to the ap-plied FM message

To see this write

Ev.s/ Ds

Kv

‚.s/ DAkf

Kv

Kt

s.s CKt/



Partial fraction expanding yields,

Ev.s/ DAkf

Kv

1

s

1

s CKt

thus

ev.t/ DAkf

Kv

h1 eKt t

iu.t/

t

Am(t)

0

1st-Order PLL frequency step response at VCO input Kvev.t/=kf

In general,

ˆ.s/ DkfM.s/

sso

Ev.s/ DkfM.s/

ss

Kv

Kt

s CKt

Dkf

Kv

Kt

s CKt

M.s/

Now if the bandwidth of m.t/ is W Kt=.2/, then

Ev.s/ kf

Kv

M.s/ ) ev.t/ kf

Kv

m.t/

The first-order PLL has limited lock range and always has anonzero steady-state phase error when the input frequency isoffset from the quiescent VCO frequency



Increasing the loop gain appears to help, but the loop band-width becomes large as well, which allows more noise to enterthe loop

Spurious time constants which are always present, but not aproblem with low loop gains, are also a problem with highgain first-order PLLs

Example 3.17: First-Order PLL Simulation Example

Tool such as MATLAB, MATLAB with Simulink, VisSim/Comm,ADS, and others provide an ideal environment for simulatingPLLs at the system level

Circuit level simulation of PLLs is very challenging due to theneed to simulate every cycle of the VCO

The most realistic simulation method is to use the actual band-pass signals, but since the carrier frequency must be kept lowto minimize the simulation time, we have difficulties removingthe double frequency term from the phase detector output

By simulating at baseband, using the nonlinear loop model,many PLL aspects can be modeled without worrying abouthow to remove the double frequency term

– A complex baseband simulation allows further capability,but will not be discussed at his time

The most challenging aspect of the simulation is dealing withthe integrator found in the VCO block (Kv=s)



We consider a discrete-time simulation where all continuous-time waveforms are replaced by their discrete-time counter-parts, i.e., xŒn D x.nT / D x.n=f s/, where fs is the samplefrequency and T D 1=fs is the sampling period

The input/output relationship of an integration block can beapproximated via the trapezoidal rule

yŒn D yŒn 1CT

2

xŒnC xŒn 1

function [theta,ev,phi_error] = PLL1(phi,fs,loop_type,Kv,fn,zeta)% [theta, ev, error, t] = PLL1(phi,fs,loop_type,Kv,fn,zeta)%%% Mark Wickert, April 2007

T = 1/fs;Kv = 2*pi*Kv; % convert Kv in Hz/v to rad/s/v

if loop_type == 1% First-order loop parametersKt = 2*pi*fn; % loop natural frequency in rad/s

elseif loop_type == 2% Second-order loop parametersKt = 4*pi*zeta*fn; % loop natural frequency in rad/sa = pi*fn/zeta;

elseerror('Loop type must be 1 or 2');

end

% Initialize integration approximation filtersfilt_in_last = 0; filt_out_last = 0;vco_in_last = 0; vco_out = 0; vco_out_last = 0;

% Initialize working and final output vectorsn = 0:length(phi)-1;theta = zeros(size(phi));ev = zeros(size(phi));phi_error = zeros(size(phi));

% Begin the simulation loop



for k = 1:length(n)phi_error(k) = phi(k) - vco_out;% sinusoidal phase detectorpd_out = sin(phi_error(k));% Loop gaingain_out = Kt/Kv*pd_out; % apply VCO gain at VCO% Loop filterif loop_type == 2

filt_in = a*gain_out;filt_out = filt_out_last + T/2*(filt_in + filt_in_last);filt_in_last = filt_in;filt_out_last = filt_out;filt_out = filt_out + gain_out;

elsefilt_out = gain_out;

end% VCOvco_in = filt_out;vco_out = vco_out_last + T/2*(vco_in + vco_in_last);vco_in_last = vco_in;vco_out_last = vco_out;vco_out = Kv*vco_out; % apply Kv% Measured loop signalsev(k) = vco_in;theta(k) = vco_out;

end

To simulate a frequency step we input a phase ramp

Consider an 8 Hz frequency step turning on at 0.5 s and a -12Hz frequency step turning on at 1.5 s

.t/ D 28.t 0:5/u.t 0:5/ 12.t 1:5/u.t 1:5/

>> t = 0:1/1000:2.5;>> idx1 = find(t>= 0.5);>> idx2 = find(t>= 1.5);>> phi1(idx1) =2*pi* 8*(t(idx1)-0.5).*ones(size(idx1));>> phi2(idx2) = 2*pi*12*(t(idx2)-1.5).*ones(size(idx2));>> phi = phi1 - phi2;>> [theta, ev, phi_error] = PLL1(phi,1000,1,1,10,0.707);>> plot(t,phi_error); % phase error in radians



0 0.5 1 1.5 2 2.5−0.5

0

0.5

1

Time (s)

Pha

se E

rror

,φ(

t) −

θ(t

), (

rad)

With Kt = 2π(10) and

Kv = 2π(1) rad/s/v, we

know that with the 8 Hz step e

v(t) = 8, so

working backwards, sin(φ - θ) = 8/10 = 0.8 and φ - θ = 0.927 rad.

0.927

-0.412

Phase error for input within lock range

In the above plot we see the finite rise-time due to the loop gainbeing 2.10/

This is a first-order lowpass step response

The loop stays in lock since the frequency swing either side ofzero is within the˙10 Hz lock range

Suppose now that a single positive frequency step of 12 Hz isapplied, the loop unlocks and cycle slips indefinitely; why?

>> phi = 12/8*phi1; % scale frequency step from 8 Hz to 12 Hz>> [theta, ev, phi_error] = PLL1(phi,1000,1,1,10,0.707);>> subplot(211)>> plot(t,phi_error)>> subplot(212)>> plot(t,sin(phi_error))



0 0.5 1 1.5 2 2.50

50

100

Time (s)

Pha

se E

rror

(φ(

t) −

θ(t

))

0 0.5 1 1.5 2 2.5−1

−0.5

0

0.5

1

Time (s)

Pha

se E

rror

sin

(φ(t

) −

θ(t

))

Cycle slips

Phase error for input exceeding lock range by 2 Hz

By plotting the true phase detector output, sinŒ.t/ .t/, wesee that the error voltage is simply not large enough to pull theVCO frequency to match the input which is offset by 12 Hz

In the phase plane plot shown earlier, this scenario correspondsto the trajectory never crossing zero



Second-Order Type II PLL

To mitigate some of the problems of the first-order PLL, wecan include a second integrator in the open-loop transfer func-tion

A common loop filter for building a second-order PLL is anintegrator with lead compensation

F.s/ Ds C a

s

The resulting PLL is sometimes called a perfect second-orderPLL since two integrators are now in the transfer function

In text Problem 3.52 you analyze the lead-lag loop filter

F.s/ Ds C a

s C a

which creates an imperfect, or finite gain integrator, second-order PLL

Returning to the integrator with phase lead loop filter, the closed-loop transfer function is

H.s/ DKtF.s/

s CKtF.s/D

Kt.s C a/

s2 CKts CKta

The transfer function from the input phase to the phase error .t/ is

G.s/D‰.s/

‚.s/D‚.s/ ˆ.s/

‚.s/

or G.s/ D 1 H.s/ Ds2

s2 CKts CKta



In standard second-order system notation we can write the de-nominator of G.s/ D 1 H.s/ (and also H.s/) as

s2 CKts CKta D s2C 2!ns C !

2n

where

!n DpKta D natural frequency in rad/s

D1

2

rKt

aD damping factor

For an input frequency step the steady-state phase error is zero

– Note the hold-in range is infinite, in theory, since the in-tegrator contained in the loop filter has infinite DC gain

To verify this we can use the final value theorem

ss D lims!0

s

!

s2

s2

s2 CKts CKta

D lim

s!0

!s

s2 CKts CKtaD 0

In exact terms we can find .t/ by inverse Laplace transform-ing

‰.s/ D!

s2 C 2!ns C !2n

The result for < 1 is

.t/ D!

!np1 2

e!nt sin!np1 2 t

u.t/



Example 3.18: Second-Order PLL Simulation Example

As a simulation example consider a loop designed with fn D10 Hz and D 0:707

Kt D 2!n D 2 0:707 2 10 D 88:84

a D!n

2D

2 10

2 0:707D 44:43

The simulation code of Example 3.17 includes the needed loopfilter via a software switch

The integrator that is part of the loop filter is implemented us-ing the same trapezoidal formula as used in the VCO

We input a 40 Hz frequency step and observe the VCO controlvoltage (ev.t/) as the loop first slips cycles, gradually pulls in,then tracks the input signal that is offset by 40 Hz

The VCO gain Kv D 1 v/Hz or 2 rad/s, so ev.t/ effectivelycorresponds to the VCO frequency deviation in Hz

>> t = 0:1/1000:2.5;>> idx1 = find(t>= 0.5);>> phi(idx1) = 2*pi*40*(t(idx1)-0.5).*ones(size(idx1));>> [theta, ev, phi_error] = PLL1(phi,1000,2,1,10,0.707);>> plot(t,ev)>> axis([0.4 0.8 -10 50])



0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8−10

0

10

20

30

40

50

Time (s)

VC

O C

ontr

ol V

olta

ge e

v(t)

(Kv =

1 H

z/v)

Cycle slipping, butpulling in to match40 Hz frequency step Cycle slipping stops,

and the loop settles

VCO control voltage for a 40 Hz frequency step

Example 3.19: Bandpass Simulation of FM Demodulation

Baseband PLL simulations are very useful and easy to imple-ment, but sometimes a full bandpass level simulation is re-quired

The MATLAB simulation file PLL1.m is modified to allow pass-band simulation via the function file PLL2.m

The phase detector is a multiplier followed by a lowpass filterto remove the double frequency term

function [theta, ev, phi_error] = PLL2(xr,fs,loop_type,Kv,fn,zeta)% [theta, ev, error, t] = PLL2(xr,fs,loop_type,Kv,fn,zeta)%



%% Mark Wickert, April 2007

T = 1/fs;% Set the VCO quiescent frequency in Hzfc = fs/4;% Design a lowpass filter to remove the double freq term[b,a] = butter(5,2*1/8);fstate = zeros(1,5); % LPF state vector

Kv = 2*pi*Kv; % convert Kv in Hz/v to rad/s/v

if loop_type == 1% First-order loop parametersKt = 2*pi*fn; % loop natural frequency in rad/s

elseif loop_type == 2% Second-order loop parametersKt = 4*pi*zeta*fn; % loop natural frequency in rad/sa = pi*fn/zeta;

elseerror('Loop type musy be 1 or 2');

end

% Initialize integration approximation filtersfilt_in_last = 0; filt_out_last = 0;vco_in_last = 0; vco_out = 0; vco_out_last = 0;

% Initialize working and final output vectorsn = 0:length(xr)-1;theta = zeros(size(xr));ev = zeros(size(xr));phi_error = zeros(size(xr));

% Begin the simulation loopfor k = 1:length(n)

% Sinusoidal phase detector (simple multiplier)phi_error(k) = 2*xr(k)*vco_out;% LPF to remove double frequency term[phi_error(k),fstate] = filter(b,a,phi_error(k),fstate);pd_out = phi_error(k);% Loop gaingain_out = Kt/Kv*pd_out; % apply VCO gain at VCO% Loop filterif loop_type == 2

filt_in = a*gain_out;filt_out = filt_out_last + T/2*(filt_in + filt_in_last);



filt_in_last = filt_in;filt_out_last = filt_out;filt_out = filt_out + gain_out;

elsefilt_out = gain_out;

end% VCOvco_in = filt_out + fc/(Kv/(2*pi)); % bias to quiescent freq.vco_out = vco_out_last + T/2*(vco_in + vco_in_last);vco_in_last = vco_in;vco_out_last = vco_out;vco_out = Kv*vco_out; % apply Kv;vco_out = sin(vco_out); % sin() for bandpass signal% Measured loop signalsev(k) = filt_out;theta(k) = vco_out;

end

Note that the carrier frequency is fixed at fs=4 and the lowpassfilter cutoff frequency is fixed at fs=8

The double frequency components out of the phase detectorare removed with a fifth-order Butterworth lowpass filter

The VCO is modified to include a bias that shifts the quiescentfrequency to fc D fs=4

The VCO output is not simply a phase deviation, but rather asinusoid with argument the VCO output phase

We will test the PLL using a single tone FM signal

>> t = 0:1/4000:5;>> xr = cos(2*pi*1000*t+2*sin(2*pi*10*t));>> psd(xr,2^14,4000)>> axis([900 1100 -40 30])>> % Process signal through PLL>> [theta, ev, phi_error] = PLL2(xr,4000,1,1,50,0.707);>> plot(t,ev)>> axis([0 1 -25 25])



900 920 940 960 980 1000 1020 1040 1060 1080 1100−40

−30

−20

−10

0

10

20

30

Frequency (Hz)

Pow

er S

pect

rum

of x

r(t)

(dB

)

Single tone FM input spectrum having fm D 10 Hz and f D 20 Hz

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−25

−20

−15

−10

−5

0

5

10

15

20

25

Time (s)

VC

O C

ontr

ol V

olta

ge e

v(t)

(Kv =

1 H

z/v)

Recovered modulation at VCO input, ev.t/



General Loop Transfer Function and Steady-State Errors

We have see that for arbitrary loop filter F.s/ the closed-looptransfer function H.s/ is

H.s/ DKtF.s/

s CKtF.s/

and the loop error function G.s/ D 1 H.s/ is

G.s/ Ds

s CKtF.s/

In tracking receiver applications of the PLL we need to con-sider platform dynamics which give rise to a phase deviationof the received signal of the form

.t/ DRt2 C 2ft C 0

u.t/

which is a superposition of a phase step, frequency step, and afrequency ramp

In the s-domain we have

ˆ.s/ D2R

s3C2f

s2C0

s

From the final value theorem the loop steady-state phase erroris

ss D lims!0

s

2R

s3C2f

s2C0

s

G.s/

If we generalize the loop filters we have been considering tothe form

F.s/ D1

s2

s2 C as C b

D 1C

a

sCb

s2



we have for G.s/

G.s/ Ds3

s3 CKts2 CKtas CKtb

Depending upon the values chosen for a and b, we can createa 1st, 2nd, or 3rd-order PLL using this F.s/

The steady-state phase error when using this loop filter is

ss D lims!0

s0s

2 C 2fs C 2R

s3 CKts2 CKtas CKtb

What are some possible outcomes for ss?



3.4.2 PLL Frequency Synthesizers

A frequency synthesizer is used to generate a stable, yet pro-grammable frequency source

A frequency synthesizer is often used to allow digital tuning ofthe local oscillator in a communications receiver

One common frequency synthesis type is known as indirectsynthesis

With indirect synthesis a PLL is used to create a stable fre-quency source

The basic block diagram of an indirect frequency synthesizeris the following

PhaseDetector

LoopFilter VCO

1M

1N

fref

fout

fref

M

fout

N

Freq Div

Freq Div

Indirect frequency synthesis using a PLL

When locked the frequency error is zero, thus

fout DN

M fref



Example 3.20: A PLL Synthesizer for Broadcast FM

In this example the synthesizer will provide the local oscillatorsignal for frequency converting the FM broadcast band from88.1 to 107.9 MHz down to an IF of 10.7 MHz

The minimum channel spacing should be 200 kHz

We will choose high-side tuning for the LO, thus

88:1C 10:7 fLO 107:9C 10:7 MHz98:8 fLO 118:6 MHz

– The step size must be 200 kHz so the frequency must beno larger than 200 kHz

To reduce the maximum frequency into the divide by countera frequency offset scheme will be employed

The synthesizer with offset oscillator is the following

PhaseDetector

LoopFilter VCO

1M

1N

fref

fout

fref

M

fmix

N

Freq Div

Freq Div

foffset

fmix

DifferenceFrequency

= 200 kHz

FM broadcast band synthesizer producing fLO for fIF D 10:7 MHz



Choose foffset < fout then fmix D fout foffset, and for locking

fref

MDfmix

N) fout D

Nfref

MC foffset

– Note that Fmix D Nfref=M and fout D fmix C foffset, byvirtue of the low side tuning assumption for the offset os-cillator

Let fref=M D 200 kHz and foffset D 98:0 MHz, then

Nmax D118:6 98:0

0:2D 103

andNmin D

98:8 98:0

0:2D 4

To program the LO such that the receiver tunes all FM stationsstep N from 4; 5; 6; : : : ; 102; 103



Example 3.21: Simple PLL Frequency Multiplication

A scheme for multiplication by three is shown below:

PhaseDetector

Loop Filt.& Ampl.

VCOCentered at 3f

c

xVCO

= Acos[2π(3fc)t]

Input at fc

t

f

Input Spectrum

fc

3fc

0

Hard limit sinusoidal input if needed

PLL as a Frequency Multiplier



Example 3.22: Simple PLL Frequency Division

A scheme for divide by two is shown below:

PhaseDetector

LowpassFilter

Loop Filt.& Ampl.

VCOCentered at f

c/2

xLPF

= Acos[2π(fc/2)t]

Keep theFundamental

Input at fc

t

VCO Output

VCO Output Spectrum

f

fc/2

0

0 2T0

PLL as a Frequency Divider

3.4.3 Frequency-Compressive Feedback

BPF Discrim

VCO

xr(t) e

v(t)

Demod.Outpute

o(t)

ed(t) x(t)

Frequency compressive feedback PLL



If we place a discriminator inside the PLL loop a compressingaction occurs

Assume that

xr.t/ D Ac cosŒ!ct C .t/

and

ev.t/ D Av sin.!c !o/t CKv

Z t

ev.˛/ d˛

Then,

ed .t/ D1

2AcAv

blocked by BPF‚ …„ ƒsin.2!c !o/t C other terms

sinŒ!ot C .t/ Kv

Z t

ev.˛/ d˛„ ƒ‚ …passed by BPF

so

x.t/ D 1

2AcAv sin

h!ot C .t/ Kv

Z t

ev.˛/ d˛i

Assuming an ideal discriminator

ev.t/ D1

2KD

d.t/

dtKvev.t/

or

ev.t/

1C

KvKD

2

DKD

2d.t/

dt



For FM d.t/=dt D 2fdm.t/, so

ev.t/ DKDfd

1CKvKD=.2/m.t/

which is the original modulation scaled by a constant

The discriminator input must be

x.t/ D 1

2AcAd sin

!ot C

1

1CKvKD=.2/.t/

Assuming that KvKD=.2/ 1 we conclude that the dis-criminator input has been converted to a narrowband FM sig-nal, which is justifies the name ’frequency compressive feed-back’

3.4.4 Coherent Carrier Recovery for DSB De-modulation

Recall that a DSB signal is of the form

xr.t/ D m.t/ cos!ct

A PLL can be used to obtain a coherent carrier reference di-rectly from xr.t/

Here we will consider the squaring loop and the Costas loop



( )2 LoopFilter VCO

x 2

LPF

-90o

0o

Am2(t)cos(2ωct + 2φ)

xr(t)

m(t)cos(ψ)

Bsin(2ωct + 2θ)

cos(ωct + θ)

m(t)cos(ωct + φ)

static phase error

Squaring Loop

LoopFilterVCO

LPF

LPF

0o

-90o

xr(t)

sin(ωct + θ)

cos(ωct + θ)

ksin(2ψ)

m(t)sin(ψ)

m(t)cos(ψ)

m2(t)sin2ψ12

Costas Loop

Note: For both of the above loops m2.t/ must contain a DCcomponent

The Costas loop or a variation of it, is often used for carrierrecovery in digital modulation

Binary phase-shift keying (BPSK), for example, can be viewedas DSB where

m.t/ D

1XnD1

dnp.t nT /



where dn D ˙1 represents random data bits and p.t/ is a pulseshaping function, say

p.t/ D

(1; 0 t T

0; otherwise

Note that in this case m2.t/ D 1, so there is a strong DC valuepresent

2 4 6 8 10

1

0.5

0.5

1

2 4 6 8 10

1

0.5

0.5

1

t/T

t/T

m(t)

m(t)cos(ωct)

BPSK modulation

Digital signal processing techniques are particularly useful forbuilding PLLs

In the discrete-time domain, digital communication waveformsare usually processed at complex baseband following someform of I-Q demodulation



LPF A/D

A/DLPF

0o

-90o

xIF

(t)cos[2πf

cLt + φ

L]

fs

fs

Sampling clock

Discrete-Time

r[n] = rI[n] + jr

Q[n]

rI(t)

rQ(t)

IF to discrete-time complex baseband conversion

To Symbol Synch

[ ]y n

[ ]x n

FromMatched

Filter

NCO

LUT −1z

−1z

[ ]v n Error Generation

−2 1() 2M M Im()

pk

akLoop Filter

[ ]e n

e− jθ[n]

θ[n]

M th-power digital PLL (DPLL) carrier phase tracking loop

ρ[n]

φ[n]

FromMatched

FilterTo Symbol Synch

( )je

( )je

M

Rect.to

Polar

[ ]x n [ ]y n

1arg()

ML-Tap

MA FIR

L-TapDelay

()F θ[n]

Non-Data Aided (NDA) feedforward carrier phase tracking



3.5 Sampling Theory

We now return to text Chapter 2, Section 8, for an introduc-tion/review of sampling theory

Consider the representation of continuous-time signal x.t/ bythe sampled waveform

xı.t/ D x.t/

"1X

nD1

ı.t nTs/

#D

1XnD1

x.nTs/ı.t nTs/

x(t) xδ(t)

tt0 0 T

s-T

s2T

s3T

s4T

s5T

s

Sampling

How is Ts selected so that x.t/ can be recovered from xı.t/?

Uniform Sampling Theorem for Lowpass Signals

GivenFfx.t/g D X.f / D 0; for f > W

then choose

Ts <1

2Wor fs > 2W .fs D 1=Ts/

to reconstruct x.t/ from xı.t/ and pass xı.t/ through an idealLPF with cutoff frequency W < B < fs W

2W D Nyquist frequencyfs=2 D folding frequency


3.5. SAMPLING THEORY

proof:

Xı.f / D X.f /

"fs

1XnD1

ı.f nfs/

#but X.f / ı.f nfs/ D X.f nfs/, so

Xı.f / D fs

1XnD1

X.f nfs/

-W W

-W W

0

0

f

f

X(f)X0

Xδ(f)

-fs

fs

X0fs

fs-W

Guard band= f

s - 2W

Lowpass reconstruction filter

. . . . . .

f-2f

sfs

-fs

2fs

X0fs

. . . . . .

Aliasingfs < 2W

fs > 2W

Spectra before and after sampling at rate fs

As long as fs W > W or fs > 2W there is no aliasing(spectral overlap)



To recover x.t/ from xı.t/ all we need to do is lowpass filterthe sampled signal with an ideal lowpass filter having cutofffrequency W < fcutoff < fs W

In simple terms we set the lowpass bandwidth to the foldingfrequency, fs=2

Suppose the reconstruction filter is of the form

H.f / D H0…

f

2B

ej 2f t0

we then choose W < B < fs W

For input Xı.f /, the output spectrum is

Y.f / D fsH0X.f /ej 2f t0

and in the time domain

y.t/ D fsH0x.t t0/

If the reconstruction filter is not ideal we then have to designthe filter in such a way that minimal desired signal energy is re-moved, yet also minimizing the contributions from the spectraltranslates either side of the n D 0 translate

The reconstruction operation can also be viewed as interpolat-ing signal values between the available sample values

Suppose that the reconstruction filter has impulse response h.t/,


3.5. SAMPLING THEORY

then

y.t/ D

1XnD1

x.nTs/h.t nTs/

D 2BH0

1XnD1

x.nTs/sincŒ2B.t t0 nTs/

where in the last lines we invoked the ideal filter describedearlier

Uniform Sampling Theorem for Bandpass Signals

If x.t/ has a single-sided bandwidth of W Hz and

Ffx.t/g D 0 for f > fu

then we may choose

fs D2fu

m

where

m D

fu

W

;

which is the greatest integer less than or equal to fu=W

Example 3.23: Bandpass signal sampling

4 2 2 4

0.20.40.60.81

f

X(f)

Input signal spectrum



In the above signal spectrum we see that

W D 2; fu D 4 fu=W D 2 ) m D 2

sofs D

2.4/

2D 4

will work

The sampled signal spectrum is

Xı.f / D 4

1XnD1

X.f nfs/

15 10 5 5 10 15

1234

f

Xδ(f)

Recover withbandpass filter

fs

2fs

3fs

-fs

-2fs

-3fs

Spectrum after sampling


3.6. ANALOG PULSE MODULATION

3.6 Analog Pulse Modulation

The message signal m.t/ is sampled at rate fs D 1=Ts

A characteristic of the transmitted pulse is made to vary in aone-to-one correspondence with samples of the message signal

A digital variation is to allow the pulse attribute to take onvalues from a finite set of allowable values

3.6.1 Pulse-Amplitude Modulation (PAM)

PAM produces a sequence of flat-topped pulses whose ampli-tude varies in proportion to samples of the message signal

Start with a message signal, m.t/, that has been uniformlysampled

mı.t/ D

1XnD1

m.nTs/ı.t nTs/

The PAM signal is

mc.t/ D

1XnD1

m.nTs/…

t .nTs C =2/

m(t)

mc(t)

τ

0 τ Ts

2Ts

3Ts

4Ts

t

PAM waveform



It is possible to create mc.t/ directly from mı.t/ using a zero-order hold filter, which has impulse response

h.t/ D …

t =2

and frequency response

H.f / D sinc.f /ejf

h(t)mδ(t) mc(t)

How does h.t/ change the recovery operation from the case ofideal sampling?

– If Ts we can get by with just a lowpass reconstruc-tion filter having cutoff frequency at fs=2 D 2=Ts

– In general, there may be a need for equalization if tau ison the order of Ts=4 to Ts=2

f-W W f

s-fs

sinc() function envelope

Lowpass reconstruction filter

Lowpassmc(t) m(t)

Recovery of m.t/ from mc.t/


3.6. ANALOG PULSE MODULATION

3.6.2 Pulse-Width Modulation (PWM)

A PWM waveform consists of pulses with width proportionalto the sampled analog waveform

For bipolar m.t/ signals we may choose a pulse width of Ts=2to correspond to m.t/ D 0

The biggest application for PWM is in motor control

It is also used in class D audio power amplifiers

A lowpass filter applied to a PWM waveform recovers themodulation m.t/

20 10 10 20

1

0.5

0.5

1PWM Signal

Analog input m(t)

t

Example PWM signal

3.6.3 Pulse-Position Modulation

With PPM the displacement in time of each pulse, with re-spect to a reference time, is proportional to the sampled analogwaveform

The time axis may be slotted into a discrete number of pulsepositions, then m.t/ would be quantized

Digital modulation that employs M slots, using nonoverlap-ping pulses, is a form of M -ary orthogonal communications



– PPM of this type is finding application in ultra-widebandcommunications

20 10 10 20

1

0.5

0.5

1PPMSignal

Analog input m(t)

t

Example PPM signal

3.7 Delta Modulation and PCM

This section considers two pure digital pulse modulation schemes

Pure digital means that the output of the modulator is a binarywaveform taking on only discrete values

3.7.1 Delta Modulation (DM)

The message signal m.t/ is encoded into a binary sequencewhich corresponds to changes in m.t/ relative to referencewaveform ms.t/

DM gets its name from the fact that only the difference fromsample-to-sample is encoded

The sampling rate in combination with the step size are the twoprimary controlling modulator design parameters


3.7. DELTA MODULATION AND PCM

1

-1

+

−

m(t)

ms(t) =

xc(t)∆(t)

Pulse Modulator

d(t)

δ0

Control the step size

Delta modulator with step size parameter ı0

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

−1

−0.5

0

0.5

1

Time (s)

m(t

) an

d m

s(t)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

−1

−0.5

0

0.5

1

Time (s)

x c(t)

m(t) (blue)m

s(t) (red)

δ0 = 0.15Slope

overload

Start-up transient

Delta modulator waveforms

The maximum slope that can be followed is ı0=Ts



A MATLAB DM simulation function is given below

function [t_o,x,ms] = DeltaMod(m,fs,delta_0,L)% [t,x,ms] = DeltaMod(m,fs,delta_0,L)%% Mark Wickert, April 2006

n = 0:(L*length(m))-1;t_o = n/(L*fs);ms = zeros(size(m));x = zeros(size(m));ms_old = 0; % zero initial conditionfor k=1:length(m)

x(k) = sign(m(k) - ms_old);ms(k) = ms_old + x(k)*delta_0;ms_old = ms(k);

end

x = [x; zeros(L-1,length(m))];x = reshape(x,1,L*length(m));

The message m.t/ can be recovered from xc.t/ by integratingand then lowpass filtering to remove the stair step edges (low-pass filtering directly is a simplification)

Slope overload can be dealt with through an adaptive scheme

– If m.t/ is nearly constant keep the step size ı0 small

– If m.t/ has large variations, a larger step size is needed

With adaptive DM the step size is controlled via a variable gainamplifier, where the gain is controlled by square-law detectingthe output of a lowpass filter acting on xc.t/



1

-1

+

−

m(t)

ms(t)

xc(t)∆(t)

Pulse Modulator

d(t)

LPF( )2

VGA

Means to obtain a variable step size DM

3.7.2 Pulse-Code Modulation (PCM)

Each sample of m.t/ is mapped to a binary word by

1. Sampling

2. Quantizing

3. Encoding

Sampler Quantizer Encoder

Sample&

Hold

Analog toDigital

Converter

Parallelto Serial

Converter

PCM Output

Serial Data

m(t)

m(t)

EquivalentViews

n



EncodedOutput

111110101100011010001000

Quant.Level

76543210 t

0 Ts

2Ts

3Ts

4Ts

5Ts

6Ts

7Ts

Quantizer Bits: n = 3, q = 2n = 8

Encoded Serial PCM Data: 001 100 110 111 110 100 010 010 ...

m(t)

3-Bit PCM encoding

Assume that m.t/ has bandwidth W Hz, then

– Choose fs > 2W

– Choose n bits per sample (q D 2n quantization levels)

– ) 2nW binary digits per second must be transmitted

Each pulse has width no more than

./max D1

2nW;

so using the fact that the lowpass bandwidth of a single pulseis about 1=.2/ Hz, we have that the lowpass transmissionbandwidth for PCM is approximately

B ' kW n;

where k is a proportionality constant

When located on a carrier the required bandwidth is doubled



Binary phase-shift keying (BPSK), mentioned earlier, is a pop-ular scheme for transmitting PCM using an RF carrier

Many other digital modulation schemes are possible

The number of quantization levels, q D log2 n, controls thequantization error, assumingm.t/ lies within the full-scale rangeof the quantizer

Increasing q reduces the quantization error, but also increasesthe transmission bandwidth

The error betweenm.kTs/ and the quantized valueQŒm.kTs/,denoted e.n/, is the quantization error

If n D 16, for example, the ratio of signal power in the samplesofm.t/, to noise power in e.n/, is about 95 dB (assumingm.t/stays within the quantizer dynamic range)

Example 3.24: Compact Disk Digital Audio

CD audio quality audio is obtained by sampling a stereo sourceat 44.1 kHz

PCM digitizing produces 16 bits per sample per L/R audiochannel



Synch(27 bits)

SubCode Data (96 bits)Data (96 bits)

Parity(32 bits)

Parity(32 bits)

0.163 mm

One Frame of 12 Audio Samples

(8 bits)

CD recoding frame format

The source bit rate is thus 2 16 44:1ksps D 1:4112 Msps

Data framing and error protection bits are added to bring thetotal bit count per frame to 588 bits and a serial bit rate of4.3218 Mbps

3.8 Multiplexing

It is quite common to have multiple information sources lo-cated at the same point within a communication system

To simultaneously transmit these signals we need to use someform of multiplexing


3.8. MULTIPLEXING

There is more than one form of multiplexing available to thecommunications engineer

3.8.1 Frequency-Division Multiplexing (FDM)

With FDM the idea is to locate a group of messages on dif-ferent subcarriers and then sum then together to form a newbaseband signal which can then be modulated onto the carrier

Mod#1

RFMod

fsc1

fc

Mod#2

fsc2

Mod#N

fscN

. . .

m1(t)

m2(t)

mN(t)

xc(t)

Lower bound on the composite signal band-width

Compositebaseband

FDM transmitter

At the receiver we first demodulate the composite signal, thenseparate into subcarrier channels using bandpass filters, thendemodulate the messages from each subcarrier



BPFfsc1

BPFfsc2

BPFfscN

RFDemod

Sub Car.Demod #1

Sub Car.Demod #2

Sub Car.Demod #N

yD1

(t)

yD2

(t)

yDN

(t)

... ... ...

FDM receiver/demodulator

The best spectral efficiency is obtained with SSB subcarriermodulation and no guard bands

At one time this was the dominant means of routing calls in thepublic switched telephone network (PSTN)

In some applications the subcarrier modulation may be combi-nations both analog and digital schemes

The analog schemes may be combinations of amplitude mod-ulation (AM/DSM/SSB) and angle modulation (FM/PM)


3.8. MULTIPLEXING

Example 3.25: FM Stereo

x2Freq. Mult

19 kHzPilot

FMMod

l(t)

r(t)

l(t) + r(t)

l(t) - r(t)

+

++

+

+

−

+

38 kHz 19 kHzpilot

xb(t)

xc(t)

fc

f (kHz)19150 3823 53

Xb(f)

Other subcarrier services can occupy this region

PilotCarrier

FM stereo transmitter

FMDiscrim

LPFW = 15

kHz

LPFW = 15

kHzBPF fc = 19 kHz

x 2 FreqMult

Mono output

l(t) + r(t)

l(t) - r(t)

l(t)

r(t)−

xb(t)x

r(t)

Coherent demod of DSB on 38 kHz subcarrier

FM stereo receiver



3.8.2 Quadrature Multiplexing (QM)

LPF

LPFChannel

m1(t)

m2(t)

d1(t)

yD1(t)

yD2(t)

d2(t)

2sinωct

2cosωctA

ccosω

ct

Acsinω

ct

xc(t) x

r(t)

QM modulation and demodulation

With QM quadrature (sin/cos) carrier are used to send inde-pendent message sources

The transmitted signal is

xc.t/ D Acm1.t/ cos!ct Cm2.t/ sin!ct

If we assume an imperfect reference at the receiver, i.e., 2 cos.!ctC/, we have

d1.t/ D Acm1.t/ cos m2.t/ sin

Cm1.t/ cos.2!ct C /Cm2.t/ sin.2!ct C /„ ƒ‚ …LPF removes these terms

yD1.t/ D Ac

m1.t/ cos Cm2.t/ sin

The second term in yD1.t/ is termed crosstalk, and is due to

the static phase error


3.8. MULTIPLEXING

Similarly

yD2.t/ D Acm2.t/ cos m1.t/ sin

Note that QM acheives a bandwidth efficiency similar to that

of SSB using adjacent two subcarriers or USSB and LSSB to-gether on the same subcarrier

3.8.3 Time-Division Multiplexing (TDM)

Time division multiplexing can be applied to sampled analogsignals directly or accomplished at the bit level

We assume that all sources are sample at or above the Nyquistrate

Both schemes are similar in that the bandwidth or data rate ofthe sources being combined needs to be taken into account toproperly maintain real-time information flow from the sourceto user

For message sources with harmonically related bandwidths wecan interleave samples such that the wideband sources are sam-pled more often

To begin with consider equal bandwidth sources



Info. Source 1

Info. Source 2

Info. Source N

...

Channel

Info.User 1

Info. User 2

Info. User N

...

Commutators

For equal bandwidth: s1s

2s

3 s

1s

2s

3 s

1s

2s

3 s

1s

2s

3 s

1s

2s

3 s

1s

2s

3 s

1s

2s

3 ....

SynchronizationRequired

Analog TDM (equal bandwidth sources)

Suppose thatm1.t/ has bandwidth 3W and sourcesm2.t/,m3.t/,andm4.t/ each have bandwidthW , we could send the samplesas

s1s2s1s3s1s4s1s2s1 : : :

with the commutator rate being fs > 2W Hz

The equivalent transmission bandwidth for multiplexed signalscan be obtained as follows

– Each channel requires greater than 2Wi samples/s

– The total number of samples, ns, over N channels in T sis thus

ns D

NXiD1

2WiT

– An equivalent signal channel of bandwidth B would pro-duce 2BT D ns samples in T s, thus the equivalent base-


3.8. MULTIPLEXING

band signal bandwidth is

B D

NXiD1

Wi Hz

which is the same minimum bandwidth required for FDMusing SSB

– Pure digital multiplexing behaves similarly to analog mul-tiplexing, except now the number of bits per sample, whichtakes into account the sample precision, must be included

– In the earlier PCM example for CD audio this was takeninto account when we said that left and right audio chan-nels each sampled at 44.1 ksps with 16-bit quantizers,multiplex up to

2 16 44; 100 D 1:4112 Msps



Example 3.26: Digital Telephone System

The North American digital TDM hierarchy is based on a sin-gle voice signal sampled at 8000 samples per second using a7-bit quantizer plus one signaling bit

The serial bit-rate per voice channel is 64 kbps

North American Digital TDM Hierarchy)

Digital No. of 64 kbpsSignal Bit Rate PCM VF Transmission

Sys. Number R (Mb/s) Channels Media UsedDS-0 0.064 1 Wire pairs

T1 DS-1 1.544 24 Wire pairsT1C DS-1C 3.152 48 Wire pairsT2 DS-2 6.312 96 Wire pairsT3 DS-3 44.736 672 Coax, radio, fiber

DS-3C 90.254 1344 Radio, fiberDS-4E 139.264 2016 Radio, fiber, coax

T4 DS-4 274.176 4032 Coax, fiberDS-432 432.00 6048 Fiber

T5 DS-5 560.160 8064 Coax, fiber

Consider the T1 channel which contains 24 voice signals

Eight total bits are sent per voice channel at a sampling rate of8000 Hz

The 24 channels are multiplexed into a T1 frame with an extrabit for frame synchronization, thus there are 24 8C 1 D 193bits per frame


3.9. GENERAL PERFORMANCE OF MODULATION SYSTEMS IN NOISE

Frame period is 1=8000 D 0:125 ms, so the serial bit rate is193 8000 D 1:544 Mbps

Four T1 channels are multiplexed into a T2 channel (96 voicechannels)

Seven T2 channels are multiplexed into a T3 channel (672voice channels)

Six T3 channels are multiplexed into a T4 channel (4032 voicechannels)

3.9 General Performance of ModulationSystems in Noise

Regardless of the modulation scheme, the received signal xr.t/,is generally perturbed by additive noise of some sort, i.e.,

xr.t/ D xc.t/C n.t/

where n.t/ is a noise process

The pre- and post-detection signal-to-noise ratio (SNR) is usedas a figure or merit



Pre-Det.Filter

Demod/Detector

xr(t) y

D(t)

Common to all systems

(SNR)T =

PT

<n2(t)>(SNR)

D =

<m2(t)>

<n2(t)>

DSB, SSB, QDSB

Coherent Demod

(SNR)T

(SNR)D

FM, D = 10

FM, D = 5

FM, D = 2

PCMq = 256

PCMq = 64

Nonlinear modu-lation systems have a distinct threshold in noise

General modulation performance in noise


2-dieu che tuong tu

Documents

t d acm

delta modulation

modulation theorem

t d xc

narrowband angle modulation

analog pulse modulation

linear modulation schemes

pulseamplitude modulation