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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
CHAPTER 3. ANALOG MODULATION
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
3-4 ECE 5625 Communication Systems I
3.1. LINEAR MODULATION
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
ECE 5625 Communication Systems I 3-5
CHAPTER 3. ANALOG MODULATION
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
3-6 ECE 5625 Communication Systems I
3.1. LINEAR MODULATION
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
ECE 5625 Communication Systems I 3-7
CHAPTER 3. ANALOG MODULATION
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
3-8 ECE 5625 Communication Systems I
3.1. LINEAR MODULATION
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
ECE 5625 Communication Systems I 3-9
CHAPTER 3. ANALOG MODULATION
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
3-10 ECE 5625 Communication Systems I
3.1. LINEAR MODULATION
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
ECE 5625 Communication Systems I 3-11
CHAPTER 3. ANALOG MODULATION
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;
3-12 ECE 5625 Communication Systems I
3.1. LINEAR MODULATION
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
ECE 5625 Communication Systems I 3-13
CHAPTER 3. ANALOG MODULATION
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
3-14 ECE 5625 Communication Systems I
3.1. LINEAR MODULATION
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
ECE 5625 Communication Systems I 3-15
CHAPTER 3. ANALOG MODULATION
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%
3-16 ECE 5625 Communication Systems I
3.1. LINEAR MODULATION
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
ECE 5625 Communication Systems I 3-17
CHAPTER 3. ANALOG MODULATION
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
3-18 ECE 5625 Communication Systems I
3.1. LINEAR MODULATION
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%
ECE 5625 Communication Systems I 3-19
CHAPTER 3. ANALOG MODULATION
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
3-20 ECE 5625 Communication Systems I
3.1. LINEAR MODULATION
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
ECE 5625 Communication Systems I 3-21
CHAPTER 3. ANALOG MODULATION
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 /
3-22 ECE 5625 Communication Systems I
3.1. LINEAR MODULATION
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/
ECE 5625 Communication Systems I 3-23
CHAPTER 3. ANALOG MODULATION
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)
3-24 ECE 5625 Communication Systems I
3.1. LINEAR MODULATION
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
ECE 5625 Communication Systems I 3-25
CHAPTER 3. ANALOG MODULATION
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
3-26 ECE 5625 Communication Systems I
3.1. LINEAR MODULATION
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
ECE 5625 Communication Systems I 3-27
CHAPTER 3. ANALOG MODULATION
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/
3-28 ECE 5625 Communication Systems I
3.1. LINEAR MODULATION
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
4AcM.f C fc/CM.f fc/
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
4AcM.f C fc/CM.f fc/
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
ECE 5625 Communication Systems I 3-29
CHAPTER 3. ANALOG MODULATION
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
3-30 ECE 5625 Communication Systems I
3.1. LINEAR MODULATION
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
ECE 5625 Communication Systems I 3-31
CHAPTER 3. ANALOG MODULATION
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)
~
3-32 ECE 5625 Communication Systems I
3.1. LINEAR MODULATION
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
ECE 5625 Communication Systems I 3-33
CHAPTER 3. ANALOG MODULATION
– 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
3-34 ECE 5625 Communication Systems I
3.1. LINEAR MODULATION
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
ECE 5625 Communication Systems I 3-35
CHAPTER 3. ANALOG MODULATION
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
3-36 ECE 5625 Communication Systems I
3.1. LINEAR MODULATION
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
ECE 5625 Communication Systems I 3-37
CHAPTER 3. ANALOG MODULATION
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
3-38 ECE 5625 Communication Systems I
3.1. LINEAR MODULATION
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
ECE 5625 Communication Systems I 3-39
CHAPTER 3. ANALOG MODULATION
– 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
3-40 ECE 5625 Communication Systems I
3.1. LINEAR MODULATION
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
ECE 5625 Communication Systems I 3-41
CHAPTER 3. ANALOG MODULATION
– 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
3-42 ECE 5625 Communication Systems I
3.1. LINEAR MODULATION
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
ECE 5625 Communication Systems I 3-43
CHAPTER 3. ANALOG MODULATION
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)
3-44 ECE 5625 Communication Systems I
3.1. LINEAR MODULATION
Example 3.9: Single Diode Mixer
ECE 5625 Communication Systems I 3-45
CHAPTER 3. ANALOG MODULATION
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
3-46 ECE 5625 Communication Systems I
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
ECE 5625 Communication Systems I 3-47
CHAPTER 3. ANALOG MODULATION
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 /
3-48 ECE 5625 Communication Systems I
3.2. ANGLE MODULATION
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
ECE 5625 Communication Systems I 3-49
CHAPTER 3. ANALOG MODULATION
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
3-50 ECE 5625 Communication Systems I
3.2. ANGLE MODULATION
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/
ECE 5625 Communication Systems I 3-51
CHAPTER 3. ANALOG MODULATION
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
3-52 ECE 5625 Communication Systems I
3.2. ANGLE MODULATION
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
ECE 5625 Communication Systems I 3-53
CHAPTER 3. ANALOG MODULATION
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
3-54 ECE 5625 Communication Systems I
3.2. ANGLE MODULATION
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
ECE 5625 Communication Systems I 3-55
CHAPTER 3. ANALOG MODULATION
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
3-56 ECE 5625 Communication Systems I
3.2. ANGLE MODULATION
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 /
ECE 5625 Communication Systems I 3-57
CHAPTER 3. ANALOG MODULATION
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
3-58 ECE 5625 Communication Systems I
3.2. ANGLE MODULATION
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
ECE 5625 Communication Systems I 3-59
CHAPTER 3. ANALOG MODULATION
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
3-60 ECE 5625 Communication Systems I
3.2. ANGLE MODULATION
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
ECE 5625 Communication Systems I 3-61
CHAPTER 3. ANALOG MODULATION
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
3-62 ECE 5625 Communication Systems I
3.2. ANGLE MODULATION
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/
ECE 5625 Communication Systems I 3-63
CHAPTER 3. ANALOG MODULATION
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
3-64 ECE 5625 Communication Systems I
3.2. ANGLE MODULATION
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
ECE 5625 Communication Systems I 3-65
CHAPTER 3. ANALOG MODULATION
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
3-66 ECE 5625 Communication Systems I
3.2. ANGLE MODULATION
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/
ECE 5625 Communication Systems I 3-67
CHAPTER 3. ANALOG MODULATION
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)')
3-68 ECE 5625 Communication Systems I
3.2. ANGLE MODULATION
−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
ECE 5625 Communication Systems I 3-69
CHAPTER 3. ANALOG MODULATION
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
3-70 ECE 5625 Communication Systems I
3.2. ANGLE MODULATION
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)
ECE 5625 Communication Systems I 3-71
CHAPTER 3. ANALOG MODULATION
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/
3-72 ECE 5625 Communication Systems I
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
ECE 5625 Communication Systems I 3-73
CHAPTER 3. ANALOG MODULATION
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
3-74 ECE 5625 Communication Systems I
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!
ECE 5625 Communication Systems I 3-75
CHAPTER 3. ANALOG MODULATION
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/
3-76 ECE 5625 Communication Systems I
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
ECE 5625 Communication Systems I 3-77
CHAPTER 3. ANALOG MODULATION
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
3-78 ECE 5625 Communication Systems I
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
ECE 5625 Communication Systems I 3-79
CHAPTER 3. ANALOG MODULATION
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
3-80 ECE 5625 Communication Systems I
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
ECE 5625 Communication Systems I 3-81
CHAPTER 3. ANALOG MODULATION
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
3-82 ECE 5625 Communication Systems I
3.4. FEEDBACK DEMODULATORS
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
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CHAPTER 3. ANALOG MODULATION
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
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3.4. FEEDBACK DEMODULATORS
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/
ECE 5625 Communication Systems I 3-85
CHAPTER 3. ANALOG MODULATION
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/
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3.4. FEEDBACK DEMODULATORS
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
ECE 5625 Communication Systems I 3-87
CHAPTER 3. ANALOG MODULATION
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)
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3.4. FEEDBACK DEMODULATORS
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
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CHAPTER 3. ANALOG MODULATION
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
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3.4. FEEDBACK DEMODULATORS
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))
ECE 5625 Communication Systems I 3-91
CHAPTER 3. ANALOG MODULATION
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
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3.4. FEEDBACK DEMODULATORS
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
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CHAPTER 3. ANALOG MODULATION
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/
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3.4. FEEDBACK DEMODULATORS
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])
ECE 5625 Communication Systems I 3-95
CHAPTER 3. ANALOG MODULATION
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)%
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3.4. FEEDBACK DEMODULATORS
%% 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);
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CHAPTER 3. ANALOG MODULATION
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])
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3.4. FEEDBACK DEMODULATORS
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/
ECE 5625 Communication Systems I 3-99
CHAPTER 3. ANALOG MODULATION
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
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3.4. FEEDBACK DEMODULATORS
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?
ECE 5625 Communication Systems I 3-101
CHAPTER 3. ANALOG MODULATION
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
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3.4. FEEDBACK DEMODULATORS
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
ECE 5625 Communication Systems I 3-103
CHAPTER 3. ANALOG MODULATION
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
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3.4. FEEDBACK DEMODULATORS
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
ECE 5625 Communication Systems I 3-105
CHAPTER 3. ANALOG MODULATION
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
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3.4. FEEDBACK DEMODULATORS
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
ECE 5625 Communication Systems I 3-107
CHAPTER 3. ANALOG MODULATION
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
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3.4. FEEDBACK DEMODULATORS
( )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 /
ECE 5625 Communication Systems I 3-109
CHAPTER 3. ANALOG MODULATION
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
3-110 ECE 5625 Communication Systems I
3.4. FEEDBACK DEMODULATORS
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
ECE 5625 Communication Systems I 3-111
CHAPTER 3. ANALOG MODULATION
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
3-112 ECE 5625 Communication Systems I
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)
ECE 5625 Communication Systems I 3-113
CHAPTER 3. ANALOG MODULATION
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/,
3-114 ECE 5625 Communication Systems I
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
ECE 5625 Communication Systems I 3-115
CHAPTER 3. ANALOG MODULATION
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
3-116 ECE 5625 Communication Systems I
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
ECE 5625 Communication Systems I 3-117
CHAPTER 3. ANALOG MODULATION
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/
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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
ECE 5625 Communication Systems I 3-119
CHAPTER 3. ANALOG MODULATION
– 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
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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
ECE 5625 Communication Systems I 3-121
CHAPTER 3. ANALOG MODULATION
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/
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3.7. DELTA MODULATION AND PCM
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
ECE 5625 Communication Systems I 3-123
CHAPTER 3. ANALOG MODULATION
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
3-124 ECE 5625 Communication Systems I
3.7. DELTA MODULATION AND PCM
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
ECE 5625 Communication Systems I 3-125
CHAPTER 3. ANALOG MODULATION
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
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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
ECE 5625 Communication Systems I 3-127
CHAPTER 3. ANALOG MODULATION
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)
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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
ECE 5625 Communication Systems I 3-129
CHAPTER 3. ANALOG MODULATION
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
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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
ECE 5625 Communication Systems I 3-131
CHAPTER 3. ANALOG MODULATION
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-
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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
ECE 5625 Communication Systems I 3-133
CHAPTER 3. ANALOG MODULATION
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
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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
ECE 5625 Communication Systems I 3-135
CHAPTER 3. ANALOG MODULATION
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
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