Digital Communication Systems
Lecture-2, Prof. Dr. Habibullah
Jamal
Under Graduate, Spring 2008
1
Formatting
2
Example 1:
 In ASCII alphabets, numbers, and symbols are encoded using a 7bit code
A total of 27 = 128 different characters can be represented using
a 7-bit unique ASCII code (see ASCII Table, Fig. 2.3)

3
Formatting

Transmit and Receive Formatting
 Transition from information source  digital symbols 
information sink
4

Character Coding (Textual Information)
 A textual information is a sequence of alphanumeric characters
 Alphanumeric and symbolic information are encoded into digital bits
using one of several standard formats, e.g, ASCII, EBCDIC
5
Transmission of Analog Signals


Structure of Digital Communication Transmitter
Analog to Digital Conversion
6
Sampling



Sampling is the processes of converting continuous-time analog
signal, xa(t), into a discrete-time signal by taking the “samples” at
discrete-time intervals
 Sampling analog signals makes them discrete in time but still
continuous valued
 If done properly (Nyquist theorem is satisfied), sampling does not
introduce distortion
Sampled values:
 The value of the function at the sampling points
Sampling interval:
 The time that separates sampling points (interval b/w samples), Ts
 If the signal is slowly varying, then fewer samples per second will
be required than if the waveform is rapidly varying
 So, the optimum sampling rate depends on the maximum
frequency component present in the signal
7

Analog-to-digital conversion is (basically) a 2 step process:
 Sampling
 Convert from continuous-time analog signal xa(t) to discretetime continuous value signal x(n)
 Is obtained by taking the “samples” of xa(t) at discrete-time
intervals, Ts

Quantization
 Convert from discrete-time continuous valued signal to discrete
time discrete valued signal
8
Sampling

Sampling Rate (or sampling frequency fs):
 The rate at which the signal is sampled, expressed as the
number of samples per second (reciprocal of the sampling
interval), 1/Ts = fs

Nyquist Sampling Theorem (or Nyquist Criterion):
 If the sampling is performed at a proper rate, no info is lost about
the original signal and it can be properly reconstructed later on
 Statement:
“If a signal is sampled at a rate at least, but not exactly equal to
twice the max frequency component of the waveform, then the
waveform can be exactly reconstructed from the samples
without any distortion”
f s  2 f max
9
Ideal Sampling ( or Impulse Sampling)
 1   jn s t
x s (t )  x (t )    e
 Ts  n 

Therefore, we have:

Take Fourier Transform (frequency convolution)

1
  jnst  1
jns t
X s ( f )  X ( f )*    e

X
(
f
)*

e


Ts
n 
n
 Ts



s
1
X s ( f )  X ( f )*   ( f  nf s ), f s 
Ts
2
n 
1
Xs ( f ) 
Ts

1
X ( f  nf s ) 

Ts
n 

n
X(f  )

Ts
n 
10
Sampling
If Rs < 2B, aliasing (overlapping of the spectra) results
 If signal is not strictly bandlimited, then it must be passed through
Low Pass Filter (LPF) before sampling
Fundamental Rule of Sampling (Nyquist Criterion)
 The value of the sampling frequency fs must be greater than twice
the highest signal frequency fmax of the signal
Types of sampling
 Ideal Sampling
 Natural Sampling
 Flat-Top Sampling



11
Ideal Sampling ( or Impulse Sampling)



Is accomplished by the multiplication of the signal x(t) by the uniform
train of impulses (comb function)
Consider the instantaneous sampling of the analog signal x(t)
Train of impulse functions select sample values at regular intervals

xs (t )  x(t )   (t  nTs )
n 

Fourier Series representation:

1
 (t  nTs ) 

Ts
n 

e
n 
jns t
,
2
s 
Ts
12
Ideal Sampling ( or Impulse Sampling)
This
shows that the Fourier Transform of the sampled signal is the
Fourier Transform of the original signal at rate of 1/Ts
13
Ideal Sampling ( or Impulse Sampling)


As long as fs> 2fm,no overlap of repeated replicas X(f - n/Ts) will
occur in Xs(f)
fs  fm  fm  f s  2 fm
Minimum Sampling Condition:
Sampling Theorem: A finite energy function x(t) can be completely
reconstructed from its sampled value x(nTs) with

  2 f (t  nTs )  
 sin 


2
T

s

x(t )   Ts x(nTs )  

 (t  nTs )
n 








T
n 
s
x(nTs ) sin c(2 f s (t  nTs ))
provided that =>
1
1
 Ts 
fs
2 fm
14
Ideal Sampling ( or Impulse Sampling)
This
means that the output is simply the replication of the original
signal at discrete intervals, e.g
15

Ts is called the Nyquist interval: It is the longest time interval that can
be used for sampling a bandlimited signal and still allow
reconstruction of the signal at the receiver without distortion
16
Practical Sampling
In practice we cannot perform ideal sampling


It is practically difficult to create a train of impulses
Thus a non-ideal approach to sampling must be used
We can approximate a train of impulses using a train of very thin
rectangular pulses:


 t  nTs 
x p (t )    




n 

Note:

Fourier Transform of impulse train is another impulse train

Convolution with an impulse train is a shifting operation
17
Natural Sampling
If we multiply x(t) by a train
of rectangular pulses xp(t),
we obtain a gated waveform
that approximates the ideal
sampled waveform, known
as natural sampling or
gating (see Figure 2.8)
xs (t )  x(t ) x p (t )
 x(t )


n 
cn e j 2 nf st
X s ( f )  [ x(t ) x p (t )]




n 

cn [ x(t )e j 2 nf s t ]
c
n 
n
X [ f  nf s ]
18







Each pulse in xp(t) has width Ts and amplitude 1/Ts
The top of each pulse follows the variation of the signal being
sampled
Xs (f) is the replication of X(f) periodically every fs Hz
Xs (f) is weighted by Cn  Fourier Series Coeffiecient
The problem with a natural sampled waveform is that the tops of the
sample pulses are not flat
It is not compatible with a digital system since the amplitude of each
sample has infinite number of possible values
Another technique known as flat top sampling is used to alleviate
this problem
19
Flat-Top Sampling




Here, the pulse is held to a constant height for the whole
sample period
Flat top sampling is obtained by the convolution of the signal
obtained after ideal sampling with a unity amplitude
rectangular pulse, p(t)
This technique is used to realize Sample-and-Hold (S/H)
operation
In S/H, input signal is continuously sampled and then the
value is held for as long as it takes to for the A/D to acquire
its value
20
x '(t )  x(t ) (t )
Flat top sampling (Time Domain)
xs (t )  x '(t )* p(t )



 p(t )* x(t ) (t )  p(t )*  x(t )   (t  nTs ) 
n 


21

Taking the Fourier Transform will result to
X s ( f )  [ xs (t )]



 P( f )   x(t )   (t  nTs ) 
n 



1
 P( f )   X ( f ) *
Ts

1
 P( f )
Ts

 ( f  nf s ) 

n 



 X ( f  nf )
n 
s
where P(f) is a sinc function
22
Flat top sampling (Frequency Domain)
Flat
top sampling becomes identical to ideal sampling as the
width of the pulses become shorter
23
Recovering the Analog Signal

One way of recovering the original signal from sampled signal Xs(f)
is to pass it through a Low Pass Filter (LPF) as shown below

If fs > 2B then we recover x(t) exactly

Else we run into some problems and signal
is not fully recovered
24

Undersampling and Aliasing
 If the waveform is undersampled (i.e. fs < 2B) then there will be
spectral overlap in the sampled signal
The
signal at the output of the filter will be
different from the original signal spectrum
This is the outcome of aliasing!
This
implies that whenever the sampling condition is not met, an
irreversible overlap of the spectral replicas is produced
25

This could be due to:
1. x(t) containing higher frequency than were
expected
2. An error in calculating the sampling rate

Under normal conditions, undersampling of signals causing
aliasing is not recommended
26

Solution 1: Anti-Aliasing Analog Filter




All physically realizable signals are not completely bandlimited
If there is a significant amount of energy in frequencies above
half the sampling frequency (fs/2), aliasing will occur
Aliasing can be prevented by first passing the analog signal
through an anti-aliasing filter (also called a prefilter) before
sampling is performed
The anti-aliasing filter is simply a LPF with cutoff frequency
equal to half the sample rate
27

Aliasing is prevented by forcing the bandwidth of the sampled
signal to satisfy the requirement of the Sampling Theorem
28

Solution 2: Over Sampling and Filtering in the Digital
Domain
 The signal is passed through a low performance (less costly)
analog low-pass filter to limit the bandwidth.
 Sample the resulting signal at a high sampling frequency.
 The digital samples are then processed by a high
performance digital filter and down sample the resulting
signal.
29
Summary Of Sampling

xs (t ) 
Ideal Sampling
(or Impulse Sampling)

x(t ) x (t )  x(t )   (t  nTs )
n 


 x(nT ) (t  nT )
n 

Natural Sampling
(or Gating)
s

xs (t )  x(t ) x p (t )  x(t )  cn e
s
j 2 nf s t
n 

Flat-Top Sampling

For all sampling techniques
 If fs > 2B then we can recover x(t) exactly
 If fs < 2B) spectral overlapping known as aliasing will occur



xs (t )  x '(t )* p(t )   x(t )   (t  nTs )  * p(t )
n 


30
Example 1:

Consider the analog signal x(t) given by
x(t )  3cos(50 t )  100sin(300 t )  cos(100 t )

What is the Nyquist rate for this signal?
Example 2:

Consider the analog signal xa(t) given by
xa (t )  3cos 2000 t  5sin 6000 t  cos12000 t



What is the Nyquist rate for this signal?
What is the discrete time signal obtained after sampling, if
fs=5000 samples/s.
What is the analog signal x(t) that can be reconstructed from the
sampled values?
31
Practical Sampling Rates



Speech
- Telephone quality speech has a bandwidth of 4 kHz
(actually 300 to 3300Hz)
- Most digital telephone systems are sampled at 8000
samples/sec
Audio:
- The highest frequency the human ear can hear is
approximately 15kHz
- CD quality audio are sampled at rate of 44,000
samples/sec
Video
- The human eye requires samples at a rate of at
least 20 frames/sec to achieve smooth motion
32
Pulse Code Modulation (PCM)


Pulse Code Modulation refers to a digital baseband signal that is
generated directly from the quantizer output
Sometimes the term PCM is used interchangeably with quantization
33
See Figure 2.16 (Page 80)
34
35
Advantages of PCM:
 Relatively inexpensive
 Easily multiplexed: PCM waveforms from different
sources can be transmitted over a common digital
channel (TDM)
 Easily regenerated: useful for long-distance
communication, e.g. telephone
 Better noise performance than analog system
 Signals may be stored and time-scaled efficiently (e.g.,
satellite communication)
 Efficient codes are readily available
Disadvantage:
 Requires wider bandwidth than analog signals
36
2.5 Sources of Corruption in the sampled,
quantized and transmitted pulses


Sampling and Quantization Effects
 Quantization (Granularity) Noise: Results when
quantization levels are not finely spaced apart enough
to accurately approximate input signal resulting in
truncation or rounding error.
 Quantizer Saturation or Overload Noise: Results when
input signal is larger in magnitude than highest
quantization level resulting in clipping of the signal.
 Timing Jitter: Error caused by a shift in the sampler
position. Can be isolated with stable clock reference.
Channel Effects
 Channel Noise
 Intersymbol Interference (ISI)
37
Signal to Quantization Noise Ratio
The level of quantization noise is dependent on how close any
particular sample is to one of the L levels in the converter

For a speech input, this quantization error resembles a noiselike disturbance at the output of a DAC converter

38
Uniform Quantization



A quantizer with equal quantization level is a Uniform Quantizer
Each sample is approximated within a quantile interval
Uniform quantizers are optimal when the input distribution is
uniform



i.e. when all values within the range are equally
likely
q
q
Most ADC’s are implemented using uniform quantizers
 e
2
2
Error of a uniform quantizer is bounded by
39
Signal to Quantization Noise Ratio

The mean-squared value (noise variance) of the quantization error is
given by:

q/2
1

1
2
2   e p(e)de   e
de

e
de

q
q q / 2
q / 2
q / 2
 
q/2
q/2
2
2
2
q
1
e
q

3 q / 2 12
3
q/2
40

The peak power of the analog signal (normalized to 1Ohms )can be
expressed as:
2
 V pp
P

 2
1

V p2

  L2 q 2 



  4 

Therefore the Signal to Quatization Noise Ratio is given by:
L2 q 2 / 4
SNRq  2
 3L2
q /12
41
If q is the step size, then the maximum quantization error that can
occur in the sampled output of an A/D converter is q

q
V pp
L
where L = 2n is the number of quantization levels for the converter.
(n is the number of bits).

Since L = 2n, SNR = 22n or in decibels




S 
2n )  6n dB

10log
(2
10
N dB
42
Nonuniform Quantization


Nonuniform quantizers have unequally spaced levels
 The spacing can be chosen to optimize the Signal-to-Noise Ratio
for a particular type of signal
It is characterized by:
 Variable step size
 Quantizer size depend on signal size
43

Many signals such as speech have a nonuniform distribution

See Figure on next page (Fig. 2.17)
Basic principle is to use more levels at regions with large probability
density function (pdf)


Concentrate quantization levels in areas of largest pdf
Or use fine quantization (small step size) for weak signals and
coarse quantization (large step size) for strong signals

44
Statistics of speech Signal Amplitudes
Figure 2.17: Statistical distribution of single talker speech signal
magnitudes (Page 81)
45
Nonuniform quantization using companding




Companding is a method of reducing the number of bits required in
ADC while achieving an equivalent dynamic range or SQNR
In order to improve the resolution of weak signals within a converter,
and hence enhance the SQNR, the weak signals need to be
enlarged, or the quantization step size decreased, but only for the
weak signals
But strong signals can potentially be reduced without significantly
degrading the SQNR or alternatively increasing quantization step size
The compression process at the transmitter must be matched with an
equivalent expansion process at the receiver
46


The signal below shows the effect of compression, where the
amplitude of one of the signals is compressed
After compression, input to the quantizer will have a more uniform
distribution after sampling
At the receiver, the signal is
expanded by an inverse
operation

The process of COMpressing
and exPANDING the signal is
called companding

Companding is a technique
used to reduce the number of bits
required in ADC or DAC while
achieving comparable SQNR

47

Basically, companding introduces a nonlinearity into the signal
 This maps a nonuniform distribution into something that more
closely resembles a uniform distribution
 A standard ADC with uniform spacing between levels can be used
after the compandor (or compander)
 The companding operation is inverted at the receiver

There are in fact two standard logarithm based companding
techniques
 US standard called µ-law companding
 European standard called A-law companding
48
Input/Output Relationship of Compander


Logarithmic expression Y = log X is the most commonly
used compander
This reduces the dynamic range of Y
49
Types of Companding
 -Law Companding Standard (North & South
America, and Japan)
y  ymax
loge 1   (| x | / xmax 
loge (1   )
sgn( x)
where
 x and y represent the input and output voltages
  is a constant number determined by experiment
 In the U.S., telephone lines uses companding with  = 255




Samples 4 kHz speech waveform at 8,000 sample/sec
Encodes each sample with 8 bits, L = 256 quantizer levels
Hence data rate R = 64 kbit/sec
 = 0 corresponds to uniform quantization
50
A-Law Companding Standard (Europe, China, Russia, Asia,
Africa)


 ymax

y ( x)  


 ymax

| x|
A
xmax
sgn( x),
(1  A)

 |x|
1  log e  A
 xmax

(1  log e A)
| x| 1
0

xmax A



sgn( x),
1 | x|

1
A xmax
where
 x and y represent the input and output voltages
 A = 87.6
 A is a constant number determined by experiment
51
Pulse Modulation



Recall that analog signals can be represented by a sequence of discrete
samples (output of sampler)
Pulse Modulation results when some characteristic of the pulse (amplitude,
width or position) is varied in correspondence with the data signal
Two Types:

Pulse Amplitude Modulation (PAM)


The amplitude of the periodic pulse train is varied in proportion to the
sample values of the analog signal
Pulse Time Modulation



Encodes the sample values into the time axis of the digital signal
Pulse Width Modulation (PWM)
 Constant amplitude, width varied in proportion to the signal
Pulse Duration Modulation (PDM)
 sample values of the analog waveform are used in determining the
width of the pulse signal
52
53
PCM Waveform Types







The output of the A/D converter is a set of binary bits
But binary bits are just abstract entities that have no physical definition
We use pulses to convey a bit of information, e.g.,
In order to transmit the bits over a physical channel they must be
transformed into a physical waveform
A line coder or baseband binary transmitter transforms a stream of bits
into a physical waveform suitable for transmission over a channel
Line coders use the terminology mark for “1” and space to mean “0”
In baseband systems, binary data can be transmitted using many kinds of
pulses
54



There are many types of waveforms. Why?  performance criteria!
Each line code type have merits and demerits
The choice of waveform depends on operating characteristics of a
system such as:
 Modulation-demodulation requirements
 Bandwidth requirement
 Synchronization requirement
 Receiver complexity, etc.,
55
Goals of Line Coding (qualities to look for)
 A line code is designed to meet one or more of the following goals:
 Self-synchronization
 The ability to recover timing from the signal itself
 That is, self-clocking (self-synchronization) - ease of clock lock
or signal recovery for symbol synchronization
 Long series of ones and zeros could cause a problem
 Low probability of bit error
 Receiver needs to be able to distinguish the waveform associated
with a mark from the waveform associated with a space
 BER performance
 relative immunity to noise
 Error detection capability
 enhances low probability of error
56




Spectrum Suitable for the channel
 Spectrum matching of the channel
 e.g. presence or absence of DC level
 In some cases DC components should be avoided
 The transmission bandwidth should be minimized
Power Spectral Density
 Particularly its value at zero
 PSD of code should be negligible at the frequency near zero
Transmission Bandwidth
 Should be as small as possible
Transparency
 The property that any arbitrary symbol or bit pattern can be
transmitted and received, i.e., all possible data sequence should
be faithfully reproducible
57
Line Coder


The input to the line encoder is
the output of the A/D converter
or a sequence of values an that
is a function of the data bit
The output of the line encoder
is a waveform:
s(t ) 

a
n 
n
f (t  nTb )
where f(t) is the pulse shape and Tb is the bit period (Tb=Ts/n for n
bit quantizer)

This means that each line code is described by a symbol mapping
function an and pulse shape f(t)

Details of this operation are set by the type of line code that is
being used
58
Summary of Major Line Codes


Categories of Line Codes
 Polar - Send pulse or negative of pulse
 Unipolar - Send pulse or a 0
 Bipolar (a.k.a. alternate mark inversion, pseudoternary)
 Represent 1 by alternating signed pulses
Generalized Pulse Shapes
 NRZ -Pulse lasts entire bit period
 Polar NRZ
 Bipolar NRZ
 RZ - Return to Zero - pulse lasts just half of bit period
 Polar RZ
 Bipolar RZ
 Manchester Line Code
 Send a 2-  pulse for either 1 (high low) or 0 (low high)
 Includes rising and falling edge in each pulse

No DC component
59

When the category and the generalized shapes are combined, we have
the following:

Polar NRZ:
 Wireless, radio, and satellite applications primarily use Polar
NRZ because bandwidth is precious
Unipolar NRZ
 Turn the pulse ON for a ‘1’, leave the pulse OFF for a ‘0’
 Useful for noncoherent communication where receiver can’t
decide the sign of a pulse
 fiber optic communication often use this signaling format
Unipolar RZ
 RZ signaling has both a rising and falling edge of the pulse
 This can be useful for timing and synchronization purposes


60
Bipolar RZ
 A unipolar line code, except now we alternate
between positive and negative pulses to send a ‘1’
 Alternating like this eliminates the DC component
 This is desirable for many channels that cannot
transmit the DC components
Generalized Grouping
 Non-Return-to-Zero: NRZ-L, NRZ-M NRZ-S
 Return-to-Zero: Unipolar, Bipolar, AMI
 Phase-Coded: bi-f-L, bi-f-M, bi-f-S, Miller, Delay
Modulation
 Multilevel Binary: dicode, doubinary


Note:There are many other variations of line codes (see Fig. 2.22,
page 80 for more)
61
Commonly Used Line Codes

Polar line codes use the antipodal mapping


  A, when X n  1
an  
  A, when X n  0
Polar NRZ uses NRZ pulse shape
Polar RZ uses RZ pulse shape
62
Unipolar NRZ Line Code
 Unipolar non-return-to-zero (NRZ) line code is defined by
unipolar mapping
when X n  1 Where X is the nth data bit
 A,
n
a 
n


0,
when X n  0
In addition, the pulse shape for unipolar NRZ is:
where Tb is the bit period f (t )    t  , NRZ Pulse Shape
 
 Tb 
63
Bipolar Line Codes
 With bipolar line codes a space is mapped to zero and
a mark is alternately mapped to -A and +A
 A, when X n  1 and last mark   A

an   A, when X n  1 and last mark   A
0,
when X n  0

It
is also called pseudoternary signaling or alternate mark inversion
(AMI)
Either
RZ or NRZ pulse shape can be used
64
Manchester Line Codes
 Manchester line codes use the antipodal mapping
and the following split-phase pulse shape:
 Tb
t 4
f (t )   
T
 b
 2

 Tb

t 4
 T

 b

 2





65
Summary of Line Codes
66
67
Comparison of Line Codes



Self-synchronization
 Manchester codes have built in timing information because they
always have a zero crossing in the center of the pulse
 Polar RZ codes tend to be good because the signal level always
goes to zero for the second half of the pulse
 NRZ signals are not good for self-synchronization
Error probability
 Polar codes perform better (are more energy efficient) than
Unipolar or Bipolar codes
Channel characteristics
 We need to find the power spectral density (PSD) of the line
codes to compare the line codes in terms of the channel
characteristics
68
Comparisons of Line Codes
Different pulse shapes are used
 to control the spectrum of the transmitted signal (no DC value,
bandwidth, etc.)
 guarantee transitions every symbol interval to assist in symbol timing
recovery
1. Power Spectral Density of Line Codes (see Fig. 2.23, Page 90)
 After line coding, the pulses may be filtered or shaped to further
improve there properties such as
 Spectral efficiency
 Immunity to Intersymbol Interference
 Distinction between Line Coding and Pulse Shaping is not easy
2. DC Component and Bandwidth
 DC Components
 Unipolar NRZ, polar NRZ, and unipolar RZ all have DC components
 Bipolar RZ and Manchester NRZ do not have DC components

69

First Null Bandwidth



Unipolar NRZ, polar NRZ, and bipolar all have 1st null bandwidths
of Rb = 1/Tb
Unipolar RZ has 1st null BW of 2Rb
Manchester NRZ also has 1st null BW of 2Rb, although the
spectrum becomes very low at 1.6Rb
70
Generation of Line Codes


The FIR filter realizes the different pulse shapes
Baseband modulation with arbitrary pulse shapes can be
detected by
 correlation detector
 matched filter detector (this is the most common detector)
71
Bits per PCM word and M-ary Modulation

Section 2.8.4: Bits per PCM Word and Bits per Symbol


Section 2.8.5: M-ary Pulse Modulation Waveforms


L=2l
M = 2k
Problem 2.14: The information in an analog waveform, whose
maximum frequency fm=4000Hz, is to be transmitted using a 16-level
PAM system. The quantization must not exceed ±1% of the peak-topeak analog signal.
(a) What is the minimum number of bits per sample or bits per PCM
word that should be used in this system?
(b) What is the minimum required sampling rate, and what is the
resulting bit rate?
(c) What is the 16-ary PAM symbol Transmission rate?
72
Solution to Problem 2.14
| e | pV pp
V pp  Lq
 1 
l  log 2 

2
p


fs  8000
q
| e |max 
2
V pp
q
L
1
2 L
2p
l
 l  log 2 (50)  6
Rs  48000
M  16
R
48000
R2 

 12000 symbols / sec
log 2 ( M )
4
73