Digital Passband Transmission
Dr. S. Mohandass
Assistant Professor
Department of ECE
PSG College of Technology
Contents
 Signaling over AWGN channels
 MAP and ML Rule
 Binary Signaling Schemes
 Amplitude-Shift Keying
 Phase-Shift Keying
 Frequency-Shift Keying
 Summary of Three Binary Signaling Schemes
 M-ary Digital Modulation Schemes
 16-QAM Modulation
 Summary and Discussion
 References
Signaling over AWGN channels

The digital source output consists of a sequence of 1s and 0s, with each
binary symbol being emitted every Tb seconds.

The transmitting part of the digital communication system takes the 1s and
0s emitted by the source computer and encodes them into distinct signals
denoted by s1(t) and s2(t), respectively, which are suitable for transmission
over the analog channel.

With the analog channel represented
by an AWGN model, depicted in the
Figure, the received signal is defined by

where w(t) is the channel noise. The receiver has the task of observing the
received signal x(t) for a duration of Tb seconds and then making an estimate of
the transmitted signal si(t), or equivalently the ith symbol, i = 1, 2.

However, owing to the presence of channel noise, the receiver will
inevitably make occasional errors. The requirement, therefore, is to design the
receiver so as to minimize the average probability of symbol error.
3
Geometric Representation of Signals

The received signal vector x is given by,
x = si + w,
i = 1, 2,…, M
where si is the transmitted signal vector and w is additive Gaussian noise
vector. M denotes the possible symbols at the output of the source.
• Let N be the number of basis functions involved in formulating the signal
vector si for all i.
4
MAP Decision Rule
5
MAP Decision Rule
6
Maximum Likelihood Rule
7
Maximum Likelihood Rule
8
Observation space with N = 2 and M = 4
9
Digital Passband modulation

Digital passband modulation techniques
◦ Amplitude-shift keying
◦ Phase-shift keying
◦ Frequency-shift keying

Receivers
◦ Coherent detection
 The receiver is synchronized to the transmitter with respect to carrier phases
◦ Noncoherent detection
 The practical advantage of reduced complexity but at the cost of degraded
performance
At the receiving end, digital demodulation techniques encompass different
forms, depending on whether the receiver is coherent or noncoherent
 Two ways of classifying digital modulation schemes are
(a) by the type of modulation used, and
(b) whether the transmitted data stream is in binary or M-ary form.

10
Binary modulation
Given a binary source, the modulation process involves switching
or keying the amplitude, phase, or frequency of a sinusoidal carrier
wave c(t), between a pair of possible values in accordance with
symbols 0 and 1.
c(t )  Ac cos(2f ct  c )
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.
11
Shift Keying Techniques
12
Digital Band-Pass modulation

Amplitude of the unmodulated carrier is given by,
Ac 

The sinusoidal carrier is given by,
c(t ) 

2
Tb
2
cos(2f ct  c )
Tb
The transmitted signal is given by,
s (t )  b(t )c(t )
s(t ) 
2
b(t ) cos(2f ct )
Tb
where waveform b(t) depends on the modulation used.
• Decreasing the bit duration Tb has the effect of increasing the
transmission bandwidth requirement of a binary modulated wave.

The spectrum of a digital modulated wave is centered on the carrier
frequency fc and the bandwidth depends on Tb.
13
Binary Amplitude-Shift Keying (BASK)
◦ The ON-OFF signaling type of modulation takes following signal waveforms
 E , for binary symbol 1
b(t )   b
for binary symbol 0
0,
 2 Eb
cos( 2f ct ), for symbol 1

s(t )   Tb
0,
for symbol 0

◦ The average transmitted signal energy is ( the two binary symbols must by
equiprobable)
Eb
Eav 
2
◦ There is only one basis function used in BASK given by,
 (t ) 
1
2
cos( 2f ct )
Tb
14
Generation and Detection of ASK Signals

Generation of ASK signal
◦ By using a product modulator with two inputs
 The ON-OFF signal given by
 E , for binary symbol 1
b(t )   b
for binary symbol 0
0,
 The sinusoidal carrier wave given by
c(t ) 

2
cos(2f c t )
Tb
Detection of ASK signal
◦ The simplest way is to use an envelope detector, exploiting the non
constant-envelope property of the BASK signal
◦ The other way is by using a correlation receiver
15
Probability of Error for ASK
• In the on–off keying version of an ASK system,
Symbol 1 is represented by transmitting a sinusoidal carrier o
f
amplitude
, where Eb is the signal energy per bit and
Tb is the bit duration.
Symbol 0 is represented by switching off the carrier.
• Assume that symbols 1 and 0 occur with equal probability
• For an AWGN channel, the average probability of error for
this ASK system under Coherent detection is given by,
16
Complementary Error Function
(erfc)
erfc( x) 
2



2
exp

z
dz

 x
17
Phase-Shift Keying
Binary Phase-Shift Keying (BPSK)
•
The pair of signals used to represent symbols 1 and 0,
 2 Eb
cos( 2f ct ),
for symbol 1 correspond ing to i  1

T

b
si (t )  
 2 Eb cos( 2f t   )   2 Eb cos( 2f t ), for symbol 0 correspond ing to i  2
c
c
 Tb
Tb
•
These are called antipodal signals:
A pair of sinusoidal wave, which differ only in a relative phase-shift of π radians.
•
The transmitted energy per bit, Eb is constant, equivalently, the average
transmitted power is constant.
•
Demodulation of BPSK cannot be performed using envelope detection,
rather, coherent detection method is used.
•
The orthonormal basis function is given by,
 (t ) 
1
2
cos( 2f ct )
Tb
18
Signal-space diagram of BPSK
19
Generation and Coherent Detection of
BPSK Signals
1. Generation
◦
BPSK modulator consists of two components:
1) Non-return-to-zero level encoder
The input binary data sequence is encoded in polar form with
symbols 1 and 0 represented by the constant-amplitude levels ;
√Eb and -√Eb
2) Product modulator
Multiplies the level-encoded binary wave by the sinusoidal carrier c(t)
of amplitude √2/Tb to produce the BPSK signal
20
2. Detection

To make an optimum decision on the received signal x(t) in favor of
symbol 1 or symbol 0, we assume that the receiver has access to a locally
generated replica of the basis function

In other words, the receiver is synchronized with the transmitter

Two basic components in the binary PSK receiver are as follows:
1. Correlator, which correlates the received signal x(t) with the basis
function
on a bit-by-bit basis.
2. Decision device, which compares the correlator output against a zerothreshold, assuming that binary symbols 1 and 0 are equiprobable. If the
threshold is exceeded, a decision is made in favor of symbol 1; if not, the
decision is made in favor of symbol 0.
21
Probability of Error
• The average probability of symbol error or, equivalently,
the BER for binary PSK using coherent detection and assuming
equiprobable symbols is given by
• As we increase the transmitted signal energy per bit Eb for
a specified noise spectral density N0/2, the message points
corresponding to symbols 1 and 0 move further apart and
the average probability of error Pe is correspondingly reduced
in accordance with the above equation.
22
BER curve for BPSK
23
Quadriphase-Shift Keying (QPSK)
◦ An important goal of digital communication is the efficient
utilization of channel bandwidth
◦ In QPSK (Quadriphase-shift keying)
 The phase of the sinusoidal carrier takes on one of the four equally
spaced values, such as π/4, 3π/4, 5π/4, and 7π/4
 2E



cos 2f c t  (2i  1) , 0  t  T
si (t )   T
4

0,
elsewhere
 where i = 1,2,3,4; E is the transmitted signal energy per symbol and
T is the symbol duration; fc is the carrier frequency.
T  2Tb
 Tb is the bit duration. Each one of the four equally spaced phase
values corresponds to a unique pair of bits called dibit
24
QPSK
•
QPSK transmitted signal si(t) can be redefined as,
si (t ) 
2E

2E



cos (2i  1)  cos( 2f ct ) 
sin (2i  1)  sin( 2f ct )
T
4
T
4


where i = 1,2,3,4.

Based on this representation, we make two observations:
1. There are two orthonormal basis functions, defined by a pair of
quadrature carriers:
2. There are four message points, defined by the two-dimensional signal
vector
25
QPSK

Elements of the signal vectors, namely si1 and si2, have their values
summarized in Table below; the first two columns give the associated dibit
and phase of the QPSK signal.

Accordingly, a QPSK signal has a two-dimensional signal constellation (i.e.,
N = 2) and four message points (i.e., M = 4) whose phase angles increase in a
counterclockwise direction
26
Signal-space diagram of QPSK
27
Generation of QPSK Signals
◦
The incoming binary data stream is first converted into polar form by a
non-return-to-zero level encoder
◦
The resulting binary wave is next divided by means of a demultiplexer
into two separate binary waves a1(t) and a2(t) consisting of the oddand even- numbered input bits
◦
a1(t) and a2(t) are used to generate two BPSK signals by means of
product modulators
◦
The two BPSK signals are added to produce the desired QPSK signals
28
QPSK Waveform
29
Coherent Detection of QPSK signals
◦
The QPSK receiver consists of an In-phase and quadrature channels with
a common input.
◦
Each channel is made up of a product modulator, integrator, and decisionmaking device.
◦
The I- and Q- channels of the receiver recover the odd- and evennumbered input bits respectively
◦
By applying the outputs of these two channels to a multiplexer, the
receiver recovers the original binary sequence
30
Error Probability of QPSK
In a QPSK system operating on an AWGN channel, the received
signal x(t) is defined by
where w(t) is the sample function of a white Gaussian noise
process of zero mean and power spectral density N0/2.
The average probability of symbol error in terms of the ratio
Eb/N0 is given by
With Gray encoding used for the incoming symbols, we can fin
d that the BER of QPSK is exactly
31
Comparison of BPSK and QPSK
 QPSK system achieves the same average probability of bit error as a
binary PSK system for the same bit rate and the same Eb/N0 , but uses only
half the channel bandwidth.
 For the same Eb/N0 and, therefore, the same average probability of bit
error, a QPSK system transmits information at twice the bit rate of a binary
PSK system for the same channel bandwidth.
 For a prescribed performance, QPSK uses channel bandwidth better than
binary PSK, which explains the preferred use of QPSK over binary PSK in
practice.
32
Binary Frequency-Shift Keying (BFSK)
◦ Each symbols are distinguished from each other by one of two transmitting
sinusoidal waves that differ in frequency by a fixed amount
 2 Eb
cos( 2f1t ), for symbol 1 correspond ing to i  1

 Tb
si (t )  
 2 Eb cos( 2f t ), for symbol 0 correspond ing to i  2
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
◦ It is a continuous-phase signal, in the sense that phase continuity is always
maintained, including the inter-bit switching times.
◦ The most useful form for the set of orthonormal basis functions is described
by
where i=1,2
33
BFSK

The coefficient sij where i = 1, 2 and j = 1, 2 is defined by

Thus, unlike binary PSK, binary FSK is characterized by having a signalspace diagram that is two-dimensional (i.e., N = 2) with two message points
(i.e., M = 2)

The two message points are defined by the vectors
and
• The Euclidean distance ||s1-s2|| is equal to
34
BFSK Constellation diagram
35
BFSK Waveform
36
Generation and Detection of BFSK
37
Probability of Error
• The average probability of bit error or, equivalently, the BER
for binary FSK using coherent detection is given by,
• A binary FSK receiver to maintain the same BER as in a
binary PSK receiver, the bit energy-to-noise density ratio,
Eb/N0, has to be doubled.
• This result is in perfect accord with the signal-space diagra
ms
of BFSK and BPSK, where we see that in a BPSK system the
Euclidean distance between the two message points is equa
l
to 2√Eb, whereas in a BFSK system the corresponding distan
ce
is √(2Eb).
38
M-ary Digital Modulation Schemes
We send any one of M possible signals si(t), where
i=1,2,…,M, 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


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
39
M-ary Phase-Shift Keying

QPSK is a special case of the generic form of PSK commonly referred
to as M-ary PSK

In M-ary PSK, the phase of the carrier takes on one of M possible
values: θi = 2(i – 1)π/M, where i = 1, 2,…,M.

Accordingly, during each signaling interval of duration T, one of the
following M possible signals is transmitted
si (t ) 
2E
2


cos 2f ct 
(i  1) ,
T
M


i  1,2,..., M
0t T


 2
  2
si (t )   E cos
(i  1)  
cos( 2f c t )
M
T







 2
  2
  E sin 
(i  1)  
sin( 2f ct ),
M
  T


i  1,2,..., M
0t T

where E is the signal energy per symbol.

Each si(t) may be expanded in terms of the same two basis functions Φ1(t)
and Φ2(t)

The signal constellation of M-ary PSK is, therefore, two-dimensional.
40
M-ary Phase-Shift Keying

Signal-Space Diagram
◦ Pair of orthogonal functions
1 (t ) 
2
cos( 2f ct ), 0  t  T
T
2 (t ) 
2
sin( 2f ct ), 0  t  T
T
 The average probability of symbol error for coherent
M-ary PSK is given by,
where it is assumed that M ≥ 4. For M = 4, it reduces to
QPSK.
41
Signal-space diagram of 8-PSK
• M-ary PSK is described in
geometric
terms
by
a
constellation of M signal points
distributed uniformly on a circle
of radius √E
• The figure in the Right, shows
the constellation of 8-PSK,
where 8 points are located on
the circle of radius √E
• Each signal point in the figure
corresponds to the signal si(t)
for a particular value of the
index i.
• The squared length from the
origin to each signal point is
equal to the signal energy E.
42
8-PSK
The Euclidean distance for each of the two points m2 and m8
from m1 is (for M = 8)
The average probability of symbol error for coherent M-ary
PSK is given by
where it is assumed that M≥4
43
M-ary Quadrature Amplitude Modulation (QAM)
•
In an M-ary PSK system, the in-phase and quadrature components
of the modulated signal are interrelated in such a way that the
envelope is constrained to remain constant and hence we have
circular constellation of the message points.
•
However, if this constraint is removed so as to permit the in-phase
and quadrature components to be independent, we get a new
modulation scheme called M-ary QAM.
•
The QAM is a hybrid form of modulation, in that the carrier
experiences amplitude as well as phase-modulation.
•
In M-ary PAM, the signal-space diagram is one-dimensional. M-ary
QAM is a two dimensional generalization of M-ary PAM, in that its
formulation involves two orthogonal passband basis functions:
44
M-ary Quadrature Amplitude Modulation (QAM)

The mathematical description of the QAM signal is given by
si (t ) 
i  0,1,..., M  1
2 E0
2 E0
ai cos( 2f ct ) 
bi sin( 2f ct ),
0t T
T
T
where E0 is the energy of the message signal with the lowest
amplitude
 The signal si(t) involves two phase-quadrature carriers, each one of
which is modulated by a set of discrete amplitudes; hence the
terminology “quadrature amplitude modulation.”
 The amplitude parameters ai and bi for in-phase and quadrature
components are independent of each other for all i
 M-ary QAM is a hybrid form of M-ary modulation that combines
M-ary ASK and M-ary PSK.
 In M-ary QAM, the constellation of message points depends on the
number of possible symbols, M.
45
16-QAM Constellation Diagram
In this example, the case of square constellation is considered
,
for which the number of bits per symbol is even (M=16).
The message points in each quadrant are identified with
Gray-encoded quadbits.
46
16-QAM Modulation

The signal-space representation of M-ary QAM for M=16, shows that the
message points form a square constellation

Unlike M-ary PSK, the different signal points of M-ary QAM are
characterized by different energy levels

With an even number of bits per symbol L, we can write L=√M

Therefore L=4 for 16-QAM modulation

Each signal point in the constellation corresponds to a specific quadbit

Under this condition, an M-ary QAM square constellation can always be
viewed as the Cartesian product of a one-dimensional L-ary PAM
constellation with itself.

By definition, the Cartesian product of two sets of coordinates
(representing a pair of one-dimensional constellations) is made up of the
set of all possible ordered pairs of coordinates with the first coordinate
in each such pair being taken from the first set involved in the product
and the second coordinate taken from the second set in the product.
47
16-QAM Modulation

The ordered pairs of coordinates naturally form a square
matrix, as shown by

To calculate the probability of symbol error for this M-ary
QAM, we exploit the property: A QAM square constellation
can be factored into the product of the corresponding L-ary
PAM constellation with itself.
48
16-QAM Modulation
• Two signal constellations for the 4-ary PAM, one vertically oriented
along the Φ2-axis in part (a) of the figure, and the other horizontally
oriented along the Φ1-axis in part (b) of the figure.
• These two parts are spatially orthogonal to each other, accounting
for the two-dimensional structure of the M-ary QAM
49
Probability of Error
The probability of symbol error for M-ary QAM is approximately given by
The transmitted energy in M-ary QAM is variable, in that its instantaneous
value naturally depends on the particular symbol transmitted.
Therefore, it is more logical to express Pe in terms of the average value of
the transmitted energy rather than E0. Assuming that the L amplitude level
s of the in-phase or quadrature component of the M-ary QAM signal are
equally likely, we have
The probability of symbol error for M-ary QAM in terms of Eav is given by
50
Summary and Discussion







BASK, BPSK, and BFSK are the digital counterparts of amplitude
modulation, phase modulation, and frequency modulation in Analog
Communication
Both BASK and BPSK exhibit discontinuity. It is possible to configure BFSK
in such a way that phase continuity is maintained across the entire input
binary data stream.
Both BASK and BPSK are examples of linear modulation, with increasing
complexity in going from BASK and BPSK.
BFSK is in general an example of nonlinear modulation
In coherent detection, the receiver must be synchronized with the
transmitter in two respects – carrier phase and bit timing
In noncoherent detection, the receiver ignores knowledge of the carrier
phase between its own operation and that of the transmitter
M-ary signaling schemes are preferred over binary modulation schemes
when bandwidth is of profound importance
51
References
Haykin S, “Digital Communication Systems”, John Wiley & Sons,
2014.
 Lathi B P, “Modern Digital and Analog communication Systems”,
Oxford University Press, 2010.
 Proakis J.G and Salehi
M, “Fundamentals of Communication
Systems” Pearson, 2011.
 Bernard Sklar, “Digital Communications”, Pearson Education Asia,
Sixth reprint, 2005.

52