8.0 Communication Systems

Modulation: embedding an information-bearing
signal into a second signal
e.g.
x(t) : information-bearing signal
c(t) : carrier signal
y(t) = x(t)c(t) : modulated signal
– purposes :



locate the signal on the right band of the spectrum
multiplexing : simultaneous transmission of more than
one signals over the same channel
resistance to noise and disturbance
– demodulation : extracting the information-bearing
signal from the modulated signal
Data Transmission (p.111 of 2.0)
“0”
“1”
…1101 …
Transmitter
1
1 0 1
ℎ𝑐 𝑡
channel distortion
𝑛 𝑡
noise
…1101 …
ℎ2 𝑡
Decision
equalizer
ℎ2 [𝑡] ≈ ℎ𝑐−1 [𝑡]
ℎ1 𝑡
front-end
filter
ℎ1,2 [𝑛]
ℎ1,2 [𝑡]
Communication
𝑐(𝑡)
front-end filter
𝑇𝑥: Transmitter
𝑦(𝑡)
𝑦(𝑡)
𝑦(𝑡)
𝑥(𝑡)
Mod
Amp
ℎ1 (𝑡)
ℎ𝑐 (𝑡)
𝑛(𝑡)
channel
Modulation Property
𝑒
⋅ 𝑥(𝑡)
𝐹
Amp
ℎ2 (𝑡)
equalizer
Modulation
𝑐(𝑡)
𝑗𝜔0 𝑡
𝑐(𝑡)
𝑅𝑥: Receiver
𝑋(𝑗(𝜔 − 𝜔0 ))
ℎ2 (𝑡) ≈ ℎ𝑐−1 (𝑡)
𝑥(𝑡)
Dem
Demodulation
8.1 Amplitude Modulation (AM) and
Frequency-Division Multiplexing (FDM)
Complex Exponential Carrier

Modulation
ct   e j ct c  , c : carrier frequency
xt  : information-bearing signal
y t   xt  e j ct c 
C  j   2   c ,  c  0
Y  j    X  j   j c 

Demodulation
z t   yt  e j ct c   xt 
See Fig. 8.1, p.584 of text
Sinusoidal Carrier

Modulation
ct   cosct   c 
y t   xt  cosct   c 
C  j       c      c ,  c  0
Y  j   1  X  j  jc   X  j  jc 
2
See Fig. 8.4, p.586 of text
Sinusoidal Carrier

Demodulation
– Synchronous demodulation (detection)
z t   y t  cosct   c 
z t   xt  cos 2 ct ,  c  0
 1 xt 1  cos 2ct 
2
See Fig. 8.6, 8.8, p.588, 589 of text
A lowpass filter gives x(t)
Sinusoidal Carrier

Demodulation
– Synchronous demodulation (detection)
If the demodulating carrier is not in phase with the
carrier
z t   xt  cosct  c  cosct  c 
 1 xt cosc  c   cos2ct  c  c 
2
output signal reduced by cos c  c 
synchronization required.
phase-locked loops.
Sinusoidal Carrier

Demodulation
– Asynchronous demodulation (envelope detection)
envelope (the smooth curve connecting the peaks)
carries the information, can be extracted in some other
ways
y t    A  xt  cosct  c 
always positive
See Fig. 8.10, 8.11, p.591, 592 of text
the carrier component consumes energy but carries no
information
See Fig. 8.14, p.593 of text
Sinusoidal Carrier

Double-sideband (DSB)/Single-sideband (SSB)
double-sideband modulation uses twice the bandwidth
DSB/WC (with carrier) , DSB/SC (suppressed carrier),
upper-sideband, lower-sideband
See Figs. 8.19, 8.20, 8.21, 8.22, p.598-601 of text
– a 90°phase-shift network can be used
H  j    j ,   0
j,   0
Vestigial Sideband
0
𝜔
𝜔
Phase Shift Network
90°
𝐼𝑚
𝐼𝑚
𝐼𝑚
−𝜔𝑐
−𝜔𝑐
𝑅𝑒
𝜔𝑐
𝑋𝑝 (𝑗𝜔)
𝑅𝑒
𝑅𝑒
𝜔
90°
𝐹(sin 𝜔𝑐 𝑡) 𝜔
𝜔𝑐
𝑌2 (𝑗𝜔)
𝜔
Frequency-Division Multiplexing (FDM)
each signal allocated with a frequency slot. Many
signals transmitted simultaneously over a single
wideband channel using a single set of transmission
facilities
See Figs. 8.15, 8.16, 8.17, p.594-596 of text
See Fig. 4.27, p.326 of text
Signals mixed in time domain but separated in
frequency domain.
Frequency Division Multiplexing
(p.67 of 4.0)
𝑥1 (𝑡)
𝑥2 (𝑡)
𝑋1 (𝑗𝜔)
𝑋2 (𝑗𝜔)
𝜔
Amp
FDM with SSB
𝜔
0
𝜔
𝜔𝑐1
𝜔𝑐2 𝜔𝑐3 𝜔𝑐4
(p.68 of 4.0)
(p.69 of 4.0)
8.2 Pulse Modulation and
Time-Division Multiplexing

Amplitude Modulation with a pulse train carrier
See Figs. 8.23, 8.24, p.602, 603 of text
y t   xt  ct 

2


a



k

,



k
c
c
T
k  
sin kc  2 
ak 
k
C  j   2
Y  j  

 a X  j   k 
k  
k
c
A lowpass filter gives x(t) if sampling theorem is
satisfied, ωc > 2 ωM
Practical Sampling (p.19 of 7.0)
𝑥𝑝 (t)
x(t)
𝑝(𝑡)
(any other pulse shape)
𝜏
𝑥𝑝 (t)
x(t)
t
T
𝜏
𝑃(𝑗𝜔)
T
1
t
T
𝜏
t
𝐹
𝐹
2𝜋
𝑇
𝜔
0
𝑋𝑝 (𝑗𝜔)
0
2𝜋
𝑇
𝜔
1
𝛼
𝜏
𝜔
−𝑇
0
2𝜋
𝑇
𝑇
t
𝐹
𝜔

Amplitude Modulation with a pulse train carrier
– This remains true as long as c(t) is periodic,
represented by a sequence ak. Sinusoidal AM is a
special case here. Impulse train sampling is the case
∆ → 0.

Pulse-Amplitude Modulation
– pulse amplitudes corresponds to the sample values
example : rectangular pulses (sample-and-hold)
See Fig. 8.26, p.606 of text
Sampling theorem applies.
Practical Sampling (p.20 of 7.0)
𝑥(𝑡)
𝑥𝑝 𝑡 =
𝑥 𝑛𝑇 𝑝(𝑡 − 𝑛𝑇)
𝑛
𝑇
𝑥(𝑡)
𝑡
𝜏
𝑋(𝑗𝜔)
𝜏
𝑡
𝑇
𝐹
𝑡
2𝜋
𝑇
𝐹
𝜔
𝜔
𝑋𝑝 (𝑗𝜔)
𝜔
0
2𝜋
𝑇
0 𝜔𝑐
𝜔
(any other
pulse shape)
𝑡
1
𝛼
𝜏
𝐹
𝜔

Time-Division Multiplexing (TDM)
Each signal allocated with a time slot in a period T.
Many signals transmitted simultaneously over a single
channel using a single set of facilities
See Figs. 8.25, 8.27, p.605, 606 of text
Signals mixed in frequency domain but separated in
time domain.
Time Division Multiplexing (TDM)
𝑥1 (𝑡)
𝑥3 (𝑡)
𝑥2 (𝑡)
𝑡

Intersymbol interference
pulses distorted during transmission and causing
interference to adjacent symbols
See Figs. 8.28, 607 of text
Pulses with zero intersymbol interference
pt   0, t  T1 ,  2T1 ,  3T1 ,......
(1)
pt  
T1 sin t T1 
t
See Figs. 8.30, 609 of text
Intersymbol Interference
x[1]
x[2]
x[3]
intersymbol
interference
It is the sample values
rather than pulse shapes
to be transmitted
Distortionless
transmission via
distorted channels

Intersymbol interference
Pulses with zero intersymbol interference
pt   0, t  T1 ,  2T1 ,  3T1 ,......
(2)
P  j   1  P1  j ,
  
T1
P1  j 
,     2
T1
T1
0
, else
P1(jw) with odd symmetry about j 
T1






P1   j  j    P1  j  j , 0    
T1 
T1 
T1


See Figs. 8.31, 610 of text

Pulse coded modulation (PCM)
– binary representation of pulse amplitude (sample
values) and binary transmission of signals
– much more easier to distinguish between 1’s and 0’s
Date Transmission (p.111 of 2.0)
“0”
“1”
…1101 …
Transmitter
1
1 0 1
ℎ𝑐 𝑡
channel distortion
𝑛 𝑡
noise
…1101 …
ℎ2 𝑡
Decision
equalizer
ℎ2 [𝑡] ≈ ℎ𝑐−1 [𝑡]
ℎ1 𝑡
front-end
filter
ℎ1,2 [𝑛]
ℎ1,2 [𝑡]
8.3 Angle/Frequency Modulation
Angle Modulation
– y t   cosct   c t   cos t 
 c t    0  k p xt 
phase modulation (PM)
d t 
 c  k f xt  frequency modulation (FM)
dt
phase modulation with x(t)
dxt  corresponds to frequency
d t 
 c  k p
dt modulation with dx(t)/dt
dt
Angle Modulation
𝑦(𝑡)
𝑦 𝑡 = 𝐴 cos(𝜔0 𝑡 + 𝜃)
𝑡
𝑦(𝑡)
𝑃𝑀
𝑥(𝑡)
𝑑
𝑑𝑡
𝑑
𝑥(𝑡)
𝑑𝑡
𝑦(𝑡)
𝐹𝑀
𝜃(𝑡)
Angle Modulation
– instantaneous frequency
i t   d  t 
dt
phase modulation
i t   c  k p d x t 
dt
frequency modulation i t   c  k f x t 
Highly nonlinear process
See Figs. 8.32, p.612 of text
– features
constant envelope: transmitter always operates at peak
power
information not carried by amplitudes : amplitude
disturbances eliminated to a large extent
Fig. 8.32
(b)
(a)
𝜔𝑐
𝜔𝑐
𝑡
(c)
𝑡
Instantaneous frequency
𝜔𝑐
𝑡
𝑑
𝜔𝑖 𝑡 = 𝜃(𝑡)
𝑑𝑡
Spectrum of FM Signals

Consider the easiest case
x t   A cos m t
i t   c  k f A cos mt  c   cos mt ,   k f A
y t   cosc t  (  / m ) sin m t 
 / m  m, modulaton index

Narrowband FM, m << π/2
cosm sin mt   1
sin m sin mt   m sin mt
y t   cos ct  msin mt sin ct  narrowband FM
y2 t   cos ct  mcos mt cos ct  DSB/WC
See Figs. 8.33, 8.34, p.615 of text
Narrowband FM vs. DSB/WC
𝑦2 𝑡 = [1 + 𝑚 cos 𝜔𝑚 𝑡] cos 𝜔𝑐 𝑡
= [1 + 𝑥(𝑡)] cos 𝜔𝑐 𝑡
𝜔
−𝜔𝑚 0
𝜔𝑚
𝑌2 (𝑗𝜔)
𝜔
−𝜔𝑐
0
−𝜔𝑐 + 𝜔𝑚
𝜔𝑐
𝜔𝑐 + 𝜔𝑚
Narrowband FM
𝐼𝑚
−𝜔𝑐
𝑅𝑒
𝐼𝑚
−𝜔𝑐
𝐼𝑚
𝑅𝑒
−𝜔𝑚
𝜔𝑚
𝑅𝑒
0
𝐼𝑚
𝜔𝑐
𝜔
𝐹(sin 𝜔𝑚 𝑡)
𝜔𝑐
𝑅𝑒
𝜔
𝐹(sin 𝜔𝑐 𝑡)
𝜔
𝐹[ sin 𝜔𝑚 𝑡 sin 𝜔𝑐 𝑡 ]
Spectrum of FM Signals

Wideband FM , m not small
yt   cos ct cosm sin mt   sin ct sin m sin mt 
– cos(msinωmt), sin(msinωmt)
periodic with fundamental frequency ωm
with spectrum of impulses at multiples of ωm
See Figs. 8.35, p.616 of text
n-th harmonics considered negligible, | n | > m
B  2mm  2  2k f A
Spectrum of FM Signals

Example : periodic square wave signal
xt  : periodic square - wave, k f  1,   A
i t   c   or c  
y t   r t  cosc    t   r  t  T  cosc    t 
2

See Figs. 8.36-8.39, p.617, 618 of text
Y  j   1 R  j  jc  j   R j  jc  j 
2
 1 R j  jc  j   R j  jc  j 
2
R j   e  jT 2 R  j 
8.4 Discrete-time Modulation

Complex jexponential
carrier
n
cn   e
C e
j
c

   2   
k  
c
 k 2 
y n   xn  cn 
Y e j   1  X e j  C e j     d
2 2
See Figs. 8.41, p.620 of text

Complex exponential carrier
cn  cos c n
See Figs. 8.42, 8.43, p.621, 612 of text

Example : Software Defined Radio
Discrete-time Realization of
Continuous-time Modulation
𝑦 𝑡 = 𝐴 cos[𝜃 𝑡 ]
𝑦(𝑡)
𝑦(𝑡)
𝑦[𝑛]
𝑥(𝑡)
𝑡
FM
Mod
FM
Dem
𝑥(𝑡)
𝑦[𝑛] = 𝐴 cos(𝜃 𝑛 )
𝑥(𝑡)
𝑥[𝑛]
Discrete-time
Mod
FM Mod
Dem
𝑦(𝑡)
𝑦(𝑡) 𝑦[𝑛]
𝑥[𝑛]
FM Dem
𝑥(𝑡)
Problems
•
Problem 8.25, p.633 of text
− Frequency inverter as a speech scrambler for secure
speech communication
𝑥(𝑡)
𝑦(𝑡)
−𝜔𝑀
𝜔𝑀
𝜔
cos 𝜔𝑀 𝑡
𝜔𝑘
−𝜔𝑘
−𝜔𝑀 0 𝜔𝑀
𝜔
−𝜔𝑀 0
𝜔𝑀 − 𝜔𝑘
𝜔𝑀
− Inverse system is itself
𝜔
Problems
•
Problem 8.34, p.640 of text
− Implementing AM with a nonlinear element
(multiplier is difficult to implement)
removed by filtering
Problems
•
Problem 8.39, p.645 of text
− Frequency Shift Keying (FSK) for digital transmission
Data Transmission (p.111 of 2.0) (p.2 of 8.0)
“0”
“1”
…1101 …
Transmitter
1
1 0 1
ℎ𝑐 𝑡
channel distortion
𝑛 𝑡
noise
…1101 …
ℎ2 𝑡
Decision
equalizer
ℎ2 [𝑡] ≈ ℎ𝑐−1 [𝑡]
ℎ1 𝑡
front-end
filter
ℎ1,2 [𝑛]
ℎ1,2 [𝑡]
Problems
•
Problem 8.39, p.645 of text
(a)
“orthogonal” but evaluated in a period of T
(b)
when T is a common multiple of the periods of
both
and
Problems
•
Problem 8.40, p.646 of text
− Quadrature multiplexing
Problems
•
Problem 8.40, p.646 of text
Sinusoidals
cos 𝜔0 𝑡
sin 𝜔0 𝑡
𝐹
(p.25 of 4.0)
𝜋 𝛿 𝜔 − 𝜔0 + 𝛿 𝜔 + 𝜔0 ,
𝐹 𝜋
𝑗
𝛿 𝜔 − 𝜔0 − 𝛿 𝜔 + 𝜔0 ,
[𝑒 𝑗𝜔0 𝑡 + 𝑒 −𝑗𝜔0 𝑡 ]
1
2
1
2𝑗
[𝑒 𝑗𝜔0 𝑡 − 𝑒 −𝑗𝜔0 𝑡 ]
cos 𝜔0 (𝑡 − 𝑡0 )
𝐼𝑚
−𝜔0
sin 𝜔0 𝑡
𝑅𝑒
𝜔0
cos 𝜔0 𝑡
𝜔
𝑥 𝑡 − 𝑡0
𝑒 −𝑗𝜔0 𝑡 ⋅ 𝑋(𝑗𝜔)
Problems
•
Problem 8.40, p.646 of text
𝐹[cos 𝜔𝑐 𝑡]
𝐼𝑚
𝐹[𝑟(𝑡) cos 𝜔𝑐 𝑡]
𝐼𝑚
𝑋1 (𝑗𝜔)
𝐼𝑚
𝑅𝑒
𝑅𝑒
𝑅𝑒
𝑋2 (𝑗𝜔)
𝜔
𝐹[sin 𝜔𝑐 𝑡]
𝜔
𝜔
𝐹[𝑟(𝑡) cos 𝜔𝑐 𝑡]
Problems
•
Problem 8.44, p.649 of text
− Zero-forcing equalizer for pulse transmission
𝑥(𝑡)
𝑝(𝑡)
𝑦(𝑡)
𝑥(𝑡)
𝑇1
𝑇1
𝑎−𝑁
𝑇1
𝑎𝑁
𝑦(𝑡)
Intersymbol Interference
(p.43 of 8.0)
x[1]
x[2]
x[3]
intersymbol
interference
It is the sample values
rather than pulse shapes
to be transmitted
Distortionless
transmission via
distorted channels
Problems
•
Problem 8.44, p.649 of text
Requirement:
Example:
…