Digital Communications Tutorial
Cognitive Radio Communications @ Virginia Tech
NSF Research Experiences for Undergraduates (REU) Site
Ratchaneekorn (Kay) Thamvichai
tkay@vt.edu
Outline
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Introduction
Fourier Transform
Sampling
Pulse Amplitude Modulation (PAM)
InterSymbol Interference (ISI)
Digital Bandpass Modulation
Analog vs. Digital

Transmitted bits can be detected and regenerated, so noise does not
propagate additively.

More signal processing techniques are available to improve system
performance: source coding, channel (error-correction) coding,
equalization, encryption, filtering,…

Digital ICs are inexpensive to manufacture

Digital communications permits integration of voice, video, and data on a
single system (ISDN)

Implementation of various algorithms can be done by software instead of
hardware

Security is easier to implement.
Simple Digital Communication System Diagram
Digital communications system block diagram
Fourier Transform

F() is the continuous-time Fourier transform of f(t).

The Fourier transformation F(ω) is the frequency domain
representation of the original function f(t). It describes which
frequencies are present in the original function.
Example 1:
Ex A: Find the Fourier Transform of x(t) = (t)
x(t)
t=0
Ex B: Find the Fourier Transform of x(t) = 0.5cos(500pt)
Ex C: Find the Fourier Transform of f(t) = rect(t/)
t
F     rect


 / 2  jt
  jt
dt   e
dt
e
 / 2

  
  
2 sin 
sin



1
2
2

 

   sinc  

e  j / 2  e j / 2 


j

  
 2 


 2 



sinc function
sinc(x)
1
x
3p
2p
p
0
p
2p
sinc(x) = sin(x)
x
 even function
 zero crossings at x  p ,  2p ,  3p , ...
 Amplitude decreases proportionally to 1/x
3p
Ex D: Pulsed Cosine:
cos(0t)rec(t/T) <=> (T/2) sinc(0 T + sinc(+0T
2
2
Linear Time-Invariant (LTI) system
h(t)
Convolution: y(t) = x(t)*h(t)
Its Fourier Transform:
Y(ω) = X(ω)H(ω)
where H(ω) is a frequency response or a transfer function of a
system h(t).
Ideal filters
 A filter is used to eliminate unwanted parts of the
frequency spectrum of a signal.
 A filter is LTI system with an impulse response h(t).
 The output y(t) of a filter can be founded in time domain
using a convolution.
 However, it is easier to do it in a frequency domain:
Y(ω) = X(ω)H(ω)
Low Pass Filter
with a cutoff frequency c
High Pass Filter
Example 2: Given x(t) = cos(500pt)cos(1000pt), find an
impulse response h(t) of a low-pass filter that passes the
low frequency component of the signal.
x(t)
Low-pass
filter
h(t)
y(t) = low freq.
component of x(t)
H()
Y() = H()X3() = p/2[(-500p) + (+500p)]
=> y(t) = 0.5cos(500pt)
H( ) = rect(/2000p
=> h(t) = 1000sinc(1000t)
Outline
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Introduction
Fourier Transform
Sampling
Pulse Amplitude Modulation (PAM)
InterSymbol Interference (ISI)
Digital Bandpass Modulation
Sampling Continuous-Time signals
Sampling – generating of an ordered number of sequence
by taking values of f(t) as specified instants of time i.e.
f(t1), f(t2), f(t3), … where tm are instants at which sampling
occurs.
Sampling operation is implemented in hardware by an
analog-to-digital converter (ADC) – electronic device used to
sample physical voltage signals.
In most cases, continuous-time signals are sampled at equal
increments of time. The sample increment, called sample
period, is usually denoted as Ts.
Impulse sampling
Define the continuous time impulse train as:
p(t) is an infinite train of continuous time impulse
functions, spaced Ts seconds apart.
 Let x(t) be a continuous time signal we wish to sample. We
will model sampling as multiplying a signal x(t) by p(t).
Sampling Theorem
let
P(ω) be a Fourier Transform of p(t),
X(ω) be a Fourier Transform of x(t),
Xs(ω) be a Fourier Transform of xs (t),
Since
xs(t) = x(t)p(t)
by a multiplication property (Fourier Transform),
where Ck are the Fourier Series coefficients of the periodic signal.
21
 We see that an impulse train in time, p(t), has a Fourier
Transform that is an impulse train in frequency, P().
 The spacing between impulses in time is Ts, and the
spacing between impulses in frequency is ω0 = 2p/Ts.
Note: If we increase the spacing in time between impulses,
this will decrease the spacing between impulses in
frequency, and vice versa.
Spectrum of a sampled signal
replicated scaled versions of X(), spaced every 0 apart in
frequency
Time-domain
Frequency-domain
ω0 = 2p/Ts
If 0c < c ,
ALIASING (overlap area) occurs
If 0c ≥ c ,
Note: if ω0 - ωc ≥ ωc or ω0 ≥ 2ωc, there is no aliasing
Sampling Theorem
Let x(t) be a band-limited signal with X(ω) = 0 for |ω| > ωc.
Then x(t) is uniquely determined by its samples x(nTs), n =
0, ±1, ± 2, … if
ω0 ≥ 2ωc
where ω0 = 2p/Ts.
 This is how to choose a sampling frequency (fs = 1/Ts) or
period (Ts) such that an original continuous-time signal x(t)
can be recovered from a sampled version xs(t).
=> a sampling rate (ω0) MUST be at least twice the
highest frequency (ωc) of a signal to avoid aliasing
problem.
To recover x(t) from its sampled version xs(t), we use a
low pass filter (reconstruction filter) to recover the center
island of Xs():
Ex: Given a signal x(t) with Fourier Transform with cutoff
frequency ωc as shown:
Given three different pulse trains with periods
Draw the sampled spectrum in each case. Which case(s)
experiences aliasing?
Aliasing Phenomenon
 Sampling theorem: the signal is strictly band-limited (c).
However, in practice, no information-bearing signal is
strictly band-limited.
 Aliasing is the phenomenon of a high-frequency
component in the spectrum of the signal seemingly taking
on the identify of a lower frequency in the spectrum of its
sampled version.
 To prevent the effects of aliasing in practice
 Prior to sampling : a low-pass anti-alias filter is
used to attenuate those high-frequency components
of a message signal that are not essential to the
information being conveyed by the signal.
 The filtered signal is sampled at a rate slightly higher
than the Nyquist rate.
Example: Why 44.1 kHz for Audio CDs?
 Sound is audible in 20 Hz to 20 kHz range:
fmax = 20 kHz and the Nyquist rate 2fmax = 40 kHz
 What is the extra 10% of the bandwidth used?
Rolloff from passband to stopband in the magnitude
response of the anti-aliasing filter.
 Okay, 44 kHz makes sense. Why 44.1 kHz?
At the time the choice was made, only recorders capable
of storing such high rates were VCRs.

NTSC: 60-Hz video (30 frames/s) - 490 lines per frame or 245
lines per field, 3 audio samples per line
 the sampling rate is 60 X 245 X 3 = 44.1 KHz
Outline
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Introduction
Fourier Transform
Sampling
Pulse Amplitude Modulation (PAM)
InterSymbol Interference (ISI)
Digital Bandpass Modulation
Pulse-Amplitude Modulation (PAM)
 The amplitude of regularly spaced pulses are varied in
proportion to the corresponding sample values of a
continuous message signal.
 Two operations involved in the generation of the PAM
signal
 Instantaneous sampling of the message signal m(t) every
Ts seconds,
 Lengthening the duration of each sample, so that it
occupies some finite value T.
Sample-and-Hold Filter : Analysis

The PAM signal is

s(t ) 
 m(nT )h(t  nT )
s
n  

s
The h(t) is a standard rectangular pulse of unit amplitude
and duration
 T  1, 0  t  T
t   
2    1 , t  0, t  T
h(t )  rect

 T  2



 
0, otherwise

The instantaneously sampled version of m(t) is
m (t ) 

 m(nT ) (t  nT )
n  
s
s
 To modify mδ(t) so as to assume the same form as the
PAM signal:

s(t )  m (t )  h(t )   m ( )h(t   )d



 m(nT )h(t  nT )
n  
s
s
 The PAM signal s(t) is mathematically equivalent to the
convolution of mδ(t) , the instantaneously sampled version
of m(t), and the pulse h(t).
 Its Fourier Transform:
S( f )  fs

 M ( f  kf )H ( f )
k  
s
One benefit of PAM
It enables the simultaneous transmission of multiple signals
using time-division multiplexing (TDM).
User 1
User 2
Quantization Process
 Amplitude quantization: The process of transforming the
sample amplitude m(nTs) of a baseband signal m(t) at time
t=nTs into a discrete amplitude v(nTs) taken from a finite
set of possible levels.
It will be represented
by binary number(s)
I k : {mk  m  mk +1}, k  1,2,...,L
39
Outline
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Introduction
Fourier Transform
Sampling
Pulse Amplitude Modulation (PAM)
MATLAB!
InterSymbol Interference (ISI)
Digital Bandpass Modulation
Baseband Transmission of Digital Data
 The transmission of digital data over a physical
communication channel is limited by two
unavoidable factors
1.Intersymbol interference
2.Channel noise
42
 The level-encoded signal and the discrete PAM signal are
+ 1 if t heinput bk is symbol1
ak  
 1 if t heinput bk is symbol0
The transmitted signal is
s(t ) 
The channel output is

 a g (t  kT )
k  
k
b
x(t )  s(t )  h(t )
 The output from the receive-filter is
y(t )  x(t )  q(t )  s(t ) * h(t ) * q(t )
43
The InterSymbol Interference (ISI) Problem
We may express the receive-filter output as the modified

PAM signal
a
y(t ) 
k  
where
After sampling:
k
p(t  kTb )
p(t )  g (t )  h(t )  q(t )
y(iTb ) 

a
k  

a
k  
p[(i  k )Tb ], i  0,1,2,...
pi  p(iTb )
yi  y (iTb )
yi 
k
k
pi k , i  0,1,2,...
44
ISI (cont.)
p0  p(0)  E
Define
where E is the transmitted signal energy / bit (symbol).
What we desire is
However, from
yi 
yi  Eai , for all i

a
k  
yi  E ai +
k
pi k ,

a
k  
k i
k
i  0,1,2,...
pi k , i  0,1,2,...
Residual phenomenon, intersymbol
interference (ISI)
45
Pulse-shaping
 Given the channel transfer function, determine the
transmit-pulse spectrum and receive-filter transfer
function so as to satisfy two basic requirements:
1.Intersymbol interference (ISI) is reduced to zero.
2.Transmission bandwidth is conserved.
The Nyquist Channel
 The optimum solution for zero ISI at the minimum transmission
bandwidth possible in a noise-free environment
 For zero ISI, it is necessary for the overall pulse shape p(t) and
the inverse Fourier transform of the pulse spectrum P(f) to satisfy
the condition
 E , for i  0
pi  p(iTb )  
for all i  0
0,
 i 
 sin c(2 B0t  i)
p(t )   p
i    2 B0 

B0 
1
2Tb
E sin(2pB0t )
Popt (t )  E sin c(2B0t ) 
2pB0t
47
 The overall pulse spectrum is defined by the optimum brickwall function:
 E
, for  B0  f  B0

Popt ( f )   2 B0
0,
ot herwise

The brick-wall spectrum defines B0 as the minimum
transmission bandwidth for zero intersymbol interference.
The optimum pulse shape is the impulse response of an ideal
low-pass channel with an amplitude response Popt(f) in the
passband and a bandwidth B0
48
49
Symbol 1
Symbol 2
Symbol 3
Two difficulties that make its use for a PAM
system impractical:
1. The system requires that the spectrum P(f) be
flat from –B0 to B0, and zero elsewhere
2. The time function p(t) decreases as 1/|t| for
large |t|, resulting in a slow rate of decay
51
Raised-Cosine Pulse Spectrum
 To ensure physical realizability of the overall pulse
spectrum P(f), the modified P(f) decreases toward zero
gradually rather than abruptly
1. Flat portion, which occupies the frequency band 0≤|f| ≤f1 for
some parameter f1 to be defined
2. Roll-off portion, which occupies the frequency band f1 ≤|f|
≤2B0-f1
 E
,
0  f  f1

 2 B0

 p ( f  f1 )  
 E 
p( f )  
1 + cos
 , f1  f  2 B0  f1
 2( B0  f1 )  
 4 B0 
0,
2 B0  f1  f


52
 The roll-off factor:
f1
  1
B0
 Time-domain of the overall channel
 cos(2pB0t ) 

p(t )  E sin c(2 B0t )
2 2 2 
 1  16 B0 t 
 The amount of intersymbol interference resulting from
a timing error ∆t decreases as the roll-off factor is
increased form zero to unity.
53
Frequency domain P(f)
Time domain p(t)
54
 Transmission-Bandwidth Requirement
 The transmission bandwidth required by using the
raised-cosine pulse spectrum is
BT  2B0  f1
BT  B0 (1 +  )
 Excess channel
 The transmission bandwidth requirement of the
raised-cosine spectrum exceeds that of the
optimum Nyquist channel by the amount
f v  B0
1. When the roll-off factor is zero, the excess BW is
reduced to zero
2. When the roll-off factor is unity, the excess BW is
increased to B0.
55
Summary (ISI)

The intersymbol interference problem, which arises due to
imperfections in the frequency response of the channel

ISI refers to the effect on that pulse due to cross-talk or
spillover from all other signal pulses in the data stream applied
to the channel input

A corrective measure widely used in practice is to shape the
overall pulse spectrum of the baseband system, starting from
the source of the message signal all the way to the receiver.

ISI is a signal-dependent phenomenon, it therefore disappears
when the information-bearing signal is switched off.

Noise is always there, regardless of whether there is data
transmission or not.

Another corrective measure for dealing with the ISI: channel
equalization.
56
Outline

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


Introduction
Fourier Transform
Sampling
Pulse Amplitude Modulation (PAM)
InterSymbol Interference (ISI)
Digital Bandpass Modulation Techniques
Digital band-pass modulation techniques

Baseband Communication: Signals are transmitted without any shift
in the range of frequency of the signal.

Band-pass Communication: Uses modulation to shift the frequency
spectrum of a (carrier) sinusoidal signal. Usually, one of the basic
parameters (amplitude, frequency, or phase) of the carrier signal is
varied in proportion to the baseband signal (information-bearing
data stream).

Why modulate signals?
 Convert signals to a form that is suitable for transmission
 Sharing the frequency band with other stations

Three



basic modulation schemes:
Amplitude-shift keying (ASK)
Phase-shift keying (PSK)
Frequency-shift keying (FSK)
58
Given a binary source
 The modulation process involves switching or keying the
amplitude, phase, or frequency of a sinusoidal carrier
wave between a pair of possible values in accordance
with symbol (bit) 0 and 1.
c(t )  Ac cos(2pf ct + c )
 Examples of a band-pass process
1. Binary amplitude shift-keying (BASK)
 The carrier amplitude is keyed between the two
possible values used to represent symbols 0 and 1
2. Binary phase-shift keying (BPSK)
 The carrier phase is keyed between the two possible
values used to represent symbols 0 and 1.
3. Binary frequency-shift keying (BFSK)
 The carrier frequency is keyed between the two
possible values used to represent symbols 0 and 1.
60
 In digital comm., the usual practice is to assume that
the carrier c(t) has unit energy measured over one
symbol (bit) duration (Tb).
2
c(t ) 
cos(2pf ct + c )
Tb
where
Ac 
2
Tb
 Decreasing the bit duration Tb has the effect of
increasing the transmission bandwidth requirement of
a binary modulated wave. (Fourier Transform
property).
61
 Band-Pass Assumption
 The spectrum of a digital modulated wave s(t) is centered on
the carrier frequency fc
s(t )  b(t )c(t )
where b(t) is an incoming binary stream with bandwidth W.
2
s(t ) 
b(t ) cos(2pf ct )
Tb
 Assumption: fc>> BW,
There will be no spectral overlap in the generation of s(t)
 The transmitted signal energy per bit can be approximated as:
Tb
Eb   s(t ) dt
2
0

2
Tb
Tb

0
b(t ) cos2 (2pf ct )dt 
2
1
Tb
Tb

0
2
b(t ) dt
62
Binary Amplitude-Shift Keying (BASK)

The ON-OFF signaling variety
 Eb , for binary symbol1
b(t )  
for binary symbol0
0,
 2 Eb
cos(2pf ct ), for symbol1

s(t )   Tb
0,
for symbol0


The average transmitted signal energy is (the two binary
symbols must be equi-probable)
Eb
Eav 
2
64
fc = 8 Hz, Tb = 1s
65
fc = 8 Hz, Tb = 0.5 s
66
From figures:
 The spectrum of the BASK signal contains a line
component at f=fc
 When the carrier is fixed and the bit duration is halved,
the width of the main lobe of the sinc function defining
the envelope of the BASK spectrum is doubled, which,
in turn, means that the transmission bandwidth of the
BASK signal is doubled.
Tb halved <= => W is doubled
 The transmission bandwidth of BASK, measured in
terms of the width of the main lobe of its spectrum, is
equal to 2/Tb, where Tb is the bit duration.
67
Phase-Shift Keying
 Binary Phase-Shift Keying (BPSK)
 The pair of signals used to represent symbols 1 and 0,
 2 Eb
cos(2pf c t ),
for symbol1

 Tb
si (t )  
 2 Eb cos(2pf t + p )   2 Eb cos(2pf t ), for symbol0
c
c
 Tb
Tb
 An antipodal signals

A pair of sinusoidal wave, which differ only in a relative
phase-shift of p radians.
Note: The transmitted energy per bit, Eb, is constant.
Equivalently, the average transmitted power is constant.
68
Signal Space diagram of BPSK
70
fc = 8 Hz, Tb = 1s
71
fc = 8 Hz, Tb = 0.5 s
72
From figures:
 BASK and BPSK signals occupy the same transmission
bandwidth (2/Tb), which defines the width of the main
lobe of the sinc-shaped power spectra.
 The BASK spectrum includes a carrier component,
whereas this component is absent from the BPSK
spectrum.
73
 Quadriphase-Shift Keying (QPSK)
 An important goal of digital communication is the
efficient utilization of channel bandwidth.
 In QPSK, the phase of the sinusoidal carrier takes on
one of the four equally spaced values, such as p/4,
3p/4, 5p/4, and 7p/4
 2E
p


cos2pf ct + (2i  1) , 0  t  T
si (t )   T
4


elsewhere
0,
 Each one of the four equally spaced phase values
corresponds to a unique symbol which is a pair of bits
(00, 01, 10, 11).
Symbol duration
T  2Tb
74
p
2E
p
2E


sin (2i  1)  sin(2pf ct )
cos(2i  1)  cos(2pf ct ) 
si (t ) 
4
T
4
T


1. In reality, the QPSK signal consists of the sum of two BPSK signals.
2. One BPSK signal, represented by the first term defined the product
of modulating a binary wave by the sinusoidal carrier
p

2 E / T cos(2i  1)  cos(2pf ct ),
4

p   E / 2 for i  1, 4

E cos(2i  1)   
4   E / 2 for i  2 , 3

3. The second binary wave
p

 2 E / T sin (2i  1)  sin( 2pf c t ),
4

p    E / 2 for i  1, 2

 E sin (2i  1)   
4  E/2
for i  3, 4

75
76
Signal Space diagram of QPSK
78
QPSK Transmitter
QPSK Receiver
80
fc = 8 Hz, Tb = 1s
BW = 1/Tb
81
fc = 8 Hz, Tb = 0.5 s
BW = 1/Tb
82
Frequency-Shift Keying
 Binary Frequency-Shift Keying (BFSK)

Each symbols are distinguished from each other by
transmitting one of two sinusoidal waves that differ in
frequency by a fixed amount:
 2 Eb
cos(2pf1t ), for symbol1

 Tb
si (t )  
 2 Eb cos(2pf t ), for symbol0
2
 Tb

Sunde’s BFSK

When the frequencies f1 and f2 are chosen in such a way that
they differ from each other by an amount equal to the
reciprocal of the bit duration Tb
83
84
fc = 8 Hz, Tb = 1s
f = fc ± 1/(2Tb)
BW = 3/Tb
85
Bandwidth
2/Tb
2/Tb
3/Tb for
f1,2 =
fc ± 1/(2Tb)
QPSK
1/Tb
86
M-ary Digital Modulation Schemes
 We send any one of M possible signals during each
signaling interval of duration T.
 The requirement is to conserve bandwidth at the
expense of both increased power and increased
system complexity.
 When the bandwidth of the channel is less than the
required value, we resort to an M-ary modulation
scheme for maximum bandwidth conservation
87
M-ary Phase-Shift Keying
 If we take blocks of m bits to produce a symbol and
use an M-ary PSK scheme with M=2m and symbol
duration T=mTb
 The bandwidth required is proportional to 1/(mTb).
 The use of M-ary PSK provides a reduction in
transmission bandwidth by a factor m=log2M over
BPSK.
88
89
 M-ary Quadrature Amplitude Modulation (QAM)
 The mathematical description of the new modulated
signal
i  0,1,...,M  1
2E0
2E0
si (t ) 
ai cos(2pf ct ) 
bi sin(2pf ct ),
0t T
T
T
 The level parameter for in-phase component and
quadrature component are independent of each other
for all i.
 M-ary QAM is a hybrid form of M-ary modulation.
 M-ary amplitude-shift keying (M-ary ASK)
si (t ) 
2 E0
ai cos(2pf c t ) i  0,1,...,M  1
T
90
 Signal-Space Diagram
 Figure 7.21 is the signal-space representation of M-ary
QAM for M=16
 Unlike M-ary PSK, the different signal points of M-ary
QAM are characterized by different energy levels
 Each signal point in the constellation corresponds to a
specific quadbit
92
Bit Error Rate
 Average bit error rate (BER)
 Let n denote the number of bit errors observed in a
sequence of bits of length N; then the relative frequency
definition of BER is
n
BER  lim  
N  N
 
 BER goal:
 For data transmission over wireless channels, a bit
error rate of 10-5 to 10-6
 For video transmission, a BER of 10-7 to 10-12
depending upon the quality desired and the encoding
method.
93
Signal to Noise Ratio (SNR)
The ratio of the modulated energy per information bit to the
one-sided noise spectral density; namely,
Modulated energy per bit
Noise spectral density
Eb

N0
SNR digital

ref
The reference SNR is independent of transmission rate. Since it
is a ratio of energies, it has essentially been normalized by the
bit rate.
where
Q x  
1
2p


x
exp( s 2 / 2)ds
95
BASK
Pe = 0.5P(0 decided| 1 is trans.) + 0.5P(1 decided|0 is trans.)
   
 
  
Pe  0.5Q
 + 0.5Q   Q
 for   /2
  
 
 2 

  

Pe  Q

Q


 2 

Eb
N0

 where  


   
Q

  
N 0T / 2 and   AT 
1
2p 



exp(( y   ) 2 / 2 2 )dy
2 EbT
97
Real-world use (Tidbits)
 The wireless LAN standard, IEEE 802.11b-1999, uses a
variety of different PSKs depending on the data-rate
required.
- Basic-rate of 1 Mbit/s,
DBPSK
- Extended-rate of 2 Mbit/s, DQPSK
- 5.5 Mbit/s and the full-rate of 11 Mbit/s, QPSK
is used with complementary code keying.
 The higher-speed wireless LAN standard, IEEE 802.11g2003[1][3] has eight data rates: 6, 9, 12, 18, 24, 36, 48
and 54 Mbit/s.
- The 6 and 9 Mbit/s modes, OFDM modulation
where each sub-carrier is BPSK modulated.
- The 12 and 18 Mbit/s modes use OFDM with QPSK.
- The fastest four modes use OFDM with QAM
 BPSK is appropriate for low-cost passive transmitters,
and is used in RFID standards.
 Bluetooth uses
p/4-DQPSK for the rate 2 Mbit/s and
8-DPSK at its higher rate (3 Mbit/s)
 IEEE 802.15.4 (the wireless standard used by ZigBee)
also relies on PSK. It has two frequency bands:
- 868–915 MHz using BPSK and
- 2.4 GHz using OQPSK
References:
 Simon Haykin and Michael Moher, “Introduction to
Analog and Digital Communications,” 2nd ed., John Wiley
& Sons, Inc., 2007.
 Charles L. Phillips, John M. Parr, Eve A. Riskin, “Signals,
Systems, and Transforms,” 4th ed., Pearson/Prentice
Hall, 2008.