Chapter 6: Pass-band Data Transmission
CHAPTER 6
PASS-BAND DATA
TRANSMISSION
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Chapter 6: Pass-band Data Transmission
Outline
• 6.1. Introduction
• 6.2. Pass-band Transmission
• 6.3 Coherent Phase Shift Keying - BPSK
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Chapter 6: Pass-band Data Transmission
6.1 Introduction
• In Ch. 4 we studied digital baseband transmission where the
generated data stream, represented in the form of discrete
pulse-amplitude modulated signal (PAM) is transmitted
directly over a low-pass channel.
• In Ch.6 we will study digital pass-band transmission where
the incoming digital signal is modulated onto a carrier (usually
sinusoidal) with fixed frequency limits imposed by the bandpass channel available
• The communication channel used in pass-band digital
transmission may be microwave radio link, satellite channel
etc.
• Other aspects of study in digital pass-band transmission are
line codes design and orthogonal FDM techniques for
broadcasting.
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Chapter 6: Pass-band Data Transmission
Definitions:
• The modulation of digital signals is a process involving
switching (keying) the amplitude, frequency or phase of a
sinusoidal carrier in some way in accordance with the
incoming digital data.
• Three basic schemes exist:
– amplitude shift keying (ASK)
– frequency shift keying (FSK)
– phase shift keying (PSK)
• REMARKS:
– In continuous wave modulation phase modulated and frequency
modulated signals are difficult to distinguish between, this is not true
for PSK and FSK.
– PSK and FSK both have constant envelope while ASK does not.
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Chapter 6: Pass-band Data Transmission
Figure 6.1
Illustrative waveforms for the three basic forms of signaling binary
information. (a) Amplitude-shift keying. (b) Phase-shift keying. (c)
Frequency-shift keying with continuous phase.
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Chapter 6: Pass-band Data Transmission
Hierarchy of Digital Modulation Techniques
• Depending on whether the receiver does phase-recovery or not
the modulation techniques are divided into:
– Coherent
– Non-coherent
• Phase recovery circuit - ensures synchronization of locally
generated carrier wave (both frequency and phase), with the
incoming data stream from the Tx.
• Binary versus M-ary schemes
– binary – use only two symbol levels;
– M-ary schemes – pure M-ary scheme exists as M-ary ASK, M-ary PSK
and M-ary FSK, using more then one level in the modulation process;
Also hybrid M-ary schemes – quadrature-amplitude modulation
(QAM); preferred over band-pass transmissions when the requirement
is to preserve bandwidth at the expense of increased power
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Chapter 6: Pass-band Data Transmission
Remarks:
• Linearity
– M-ary PSK and M-ary QAM are both linear modulation
schemes; M-ary PSK – constant envelope; M-ary QAM –
no
– M-ary PSK – used over linear channels
– M-ary QAM – used over non-linear channels
• Coherence
– ASK and FSK – used with non-coherent systems; no need
of maintaining carrier phase synchronization
– “noncoherent PSK” means no carrier phase information;
instead pseudo PSK = differential PSK (DPSK);
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Chapter 6: Pass-band Data Transmission
Probability of Error
• Design goal – minimize the average probability of
symbol error in the presence of AWGN.
• Signal-space analysis is a tool for setting decision
areas for signal detection over AWGN (i.e. based on
maximum likelihood signal detection) (Ch.5!)
• Based on these decisions probability of symbol error
Pe is calculated
– for simple binary coherent methods as coherent binary PSK
and coherent binary FSK, there are exact formulas for Pe
– for coherent M-ary PSK and coherent M-ary FSK
approximate solutions are sought.
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Chapter 6: Pass-band Data Transmission
Power Spectra
• power spectra of resulting modulated signals is
important for:
– comparison of virtues and limitations of different
schemes
– study of occupancy of channel bandwidth
– study of co-channel interference
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Chapter 6: Pass-band Data Transmission
A modulated signal is described in terms of in-phase and
quadrature component as follows:
s(t )  sI (t ) cos(2 f cT )  sQ (t )sin(2 f ct )
~
 Re[ s(t ) exp( j 2 f ct )]
complex
envelope
(6.1)
~
s(t )  sI (t )  jsQ (t )
exp( j 2 fcT )  cos(2 fct )  j sin(2 fct )
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Chapter 6: Pass-band Data Transmission
• The complex envelope is actually the baseband
version of the modulated (bandpass) signal.
~
• sI(t) and sQ(t) as components of s (t ) are low-pass
signals.
Let SB(f) denote
the power spectral density of the complex
~
envelope s (t ), known as baseband power spectral density.
The power spectral density Ss(f) of the original band-pass
signal s(t) is a frequency shifted version of SB(f) except for a
scaling factor:
1
S s (t )  [ S B ( f  f c )  S B ( f  f c )]
4
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Chapter 6: Pass-band Data Transmission
So,
• as far as the power spectrum is concerned it is
sufficient to evaluate the baseband
power spectral
~
density SB(f) and since s (t ) is a low-pass
signal, the calculation of SB(f) should be simpler
than the calculation of Ss(f).
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Chapter 6: Pass-band Data Transmission
Bandwidth efficiency
• Main goal of communication engineering –
spectrally efficient schemes
– maximize bandwidth efficiency = ratio of the data
rate in bits per seconds to the effectively utilized
channel bandwidth.
– achieve bandwidth at minimum practical
expenditure of average SNR
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Chapter 6: Pass-band Data Transmission
The effectiveness of a channel with bandwidth B can be
expressed as:
Rb
  bits / s / Hz
B
data rate
(6.5)
bandwidth
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Chapter 6: Pass-band Data Transmission
• Before (Ch.4) we discussed that the bandwidth
efficiency is the product of two independent
factors:
– multilevel encoding – use of blocks of bits instead
of single bits.
– spectral shaping – bandwidth requirements on the
channel are reduced by the use of suitable pulseshaping filters
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Chapter 6: Pass-band Data Transmission
Outline
• 6.1. Introduction
• 6.2. Pass-band Transmission
• 6.3 Coherent Phase Shift Keying
– Binary Phase shift Keying (BPSK)
– Quadriphase-Shift Keying (QPSK)
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Chapter 6: Pass-band Data Transmission
6.2 Pass-band transmission model
• Functional blocks of the model
• Transmitter side
– message source, emitting a symbol every T seconds; a
symbol belongs to an alphabet of M symbols, denoted by
m1, m2, ….mM; the a priori probabilities P(m1),
P(m2),…P(mM) specify the message source output; when
symbols are equally likely we can express the
probability pi as:
pi  P(mi )
1

M
for all i
(6.6)
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Chapter 6: Pass-band Data Transmission
– signal transmission encoder , producing a vector si made up
of N real elements, one such set for each of the M symbols
of the source alphabet; dimension- wise N ≤ M;
– si is fed to a modulator that constructs a distinct signal si(t)
of duration T seconds as the representation of symbol mi
generated by the message source; the signal si is an energy
signal (what does this mean?); si is real valued
• Channel:
– linear channel wide enough to accommodate the
transmission of the modulated signal with negligible or no
distortion
– the channel white noise is a sample function of AWGN with
zero mean and N0/2 power spectral density
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Chapter 6: Pass-band Data Transmission
• Receiver side (blocks described in detail p.326-327)
– detector
– signal transmission decoder; reverses the operations
performed in the transmitter;
Figure 6.2
Functional model of pass-band data transmission system.
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Chapter 6: Pass-band Data Transmission
Outline
• 6.1. Introduction
• 6.2. Pass-band Transmission
• 6.3 Coherent Phase Shift Keying
– Binary Phase shift Keying (BPSK)
– Quadriphase-Shift Keying (QPSK)
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Chapter 6: Pass-band Data Transmission
6.3 Coherent Phase Shift Keying
- Binary Phase Shift Keying (BPSK)
• In a coherent binary PSK the pair of signals used to
represent binary 0 and 1 are defined as:
2 Eb
s1 (t ) 
cos(2 f ct )
Tb
(6.8)
duration of one bit
fc=nc/Tb
2 Eb
2 Eb
s2 (t ) 
cos(2 f ct   )  
cos(2 f ct )
Tb
Tb
(6.9)
transmitted energy
per bit
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Chapter 6: Pass-band Data Transmission
• The equations (6.8) and (6.9) represent antipodal signals –
sinusoidal signals that differ only in a relative phase shift of
180 degrees.
• In BPSK there is only one basis function of unit energy
expressed as:
2
1 (t ) 
cos(2 f ct ),
Tb
0  t  Tb
(6.10)
• So the transmitted signals can be expressed as:
s1 (t )  Eb 1 (t ),
s2 (t )   Eb 1 (t ),
0  t  Tb
(6.11)
0  t  Tb
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Chapter 6: Pass-band Data Transmission
• A coherent BPSK system can be characterized by
having a signal space that is one dimensional (N= 1),
with signal constellation consisting of two message
points (M = 2)
• The coordinates of the message points are:
Tb
Tb
s11   s1 (t )1 (t )dt
s21   s2 (t )1 (t )dt
0
  Eb
0
(6.13)
  Eb
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Chapter 6: Pass-band Data Transmission
message point
corresponding to s2
message point
corresponding to s1
nc is an integer such that
Tsymbol = nc/Tbit
Figure 6.3
Signal-space diagram for coherent binary PSK system. The waveforms depicting the
transmitted signals s1(t) and s2(t), displayed in the inserts, assume nc  2.
Note that the frequency fc is chosen to ensure that each transmitted bit contains an integer
number of cycles..
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Chapter 6: Pass-band Data Transmission
Error Probability of Binary PSK
• Decision rule: based on the maximum likelihood
decision algorithm/rule which in this case means
that we have to choose the message point closest to
the received signal point
observation vector x lies in region Zi if
the Euclidean distance ||x-sk|| is minimum for k = i
• For BPSK: N= 1, space is divided into two areas (fig.6.3)
– the set of points closest to message point 1 at +E1/2
– the set of points closest to message point 2 at – E1/2
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Chapter 6: Pass-band Data Transmission
• The decision rule is simply to decide that signal s1(t) (i.e.
binary 1) was transmitted if the received signal point falls in
region Z1, and decide that signal s2(t) (i.e. binary symbol 0)
was transmitted if the received signal falls in region Z2.
• Two kinds of errors are possible due to noise:
– sent s1(t), received signal point falls in Z2
– sent s2(t), received signal point falls in Z1
• This can be expressed as:
Zi: 0 < x1 < æ
• and the observed element is expressed as a function of the
received signal x(t) as:
Tb
x1   x(t )1 (t )dt
(6.15)
0
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Chapter 6: Pass-band Data Transmission
So,
• In Ch.5 it was deduced that memory-less AWGN channels,
the observation elements Xi are Gaussian RV with mean sij
and variance N0/2.
• The conditional probability density function that xj (signal sj
was received providing mi was sent) is given by:
1
1
2
f x j ( x j / mi ) 
exp[
( x1  sij ) ]
N0
 N0
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Chapter 6: Pass-band Data Transmission
• When we substitute for the case of BPSK
1
1
f x1 ( x1 / 0) 
exp[ ( x1  s21 ) 2 ]
N0
 N0
1
1

exp[ ( x1  Eb ) 2 ]
N0
 N0
(6.16)
• Then the conditional probability of the receiver in
favor of 1 provided 0 was transmitted is:

1
10   f x1 (x1 / 0)dx1 
 N0
0


0
1
exp[
( x1  Eb )2 ]dx1 (6.17)
N0
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Chapter 6: Pass-band Data Transmission
• if we substitute and change the integration variable:
1
z
( x1  Eb )]
N0
10 
1

(6.18)


exp( z 2 )dz
Eb / N0
Eb
1
 erfc(
2
N0
(6.19)
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Chapter 6: Pass-band Data Transmission
• Considering an error of the second kind:
– signal space is symmetric about the origin
– p01 is the same as p10
• Average probability of symbol error or the bit error
rate for coherent BPSK is:
Eb
1
Pe  erfc(
)
2
N0
(6.20)
• So increasing the signal energy per bit makes the
points and
move farther apart which
correspond to reducing the error probability.
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Chapter 6: Pass-band Data Transmission
Generation and Detection of Coherent
BPSK Signals
• Transmitter side:
– Need to represent the binary sequence 0 and 1 in polar form
with constant amplitudes, respectively –
and +
(polar non-return-to-zero – NRZ - encoding).
– Carrier wave is with frequency fc=(nc/Tb)
– Required BPSK modulated signal is at the output of the
product modulator.
• Receiver side
– noisy PSK is fed to a correlator with locally generated
reference signal
– correlator output is compared to a threshold of 0 volts in
the decision device
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Chapter 6: Pass-band Data Transmission
Figure 6.4
Block diagrams for (a) binary PSK transmitter and (b)
coherent binary PSK receiver.
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Chapter 6: Pass-band Data Transmission
Power Spectra of BPSK
• From the modulator – the complex envelope of the
BPSK has only in-phase component
• Depending on whether we have a symbol 1 or 0
during the signaling interval 0 ≤ t ≤ Tb the in-phase
component is +g(t) or – g(t).
 2 Eb

, 0  t  Tb 

g (t )   Tb



otherwise 
 0,
(6.21)
symbol
shaping function
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Chapter 6: Pass-band Data Transmission
• We assume that the input binary wave is random, with symbols
1 or 0 equally likely and that symbols transmitted during the
different time slots are statistically independent.
• So, (Ch.1) the power spectra of such a random binary wave is
given by the energy spectral density of the symbol shaping
function divided by the symbol duration.(See Ex.1.3 and 1.6)
• g(t) is an energy signal – FT
• Finally, the energy spectral density is equal to the squared
magnitude of the signals FT.
2 Eb sin 2 ( Tb f )
SB ( f ) 
2
( Tb f )
 2 Eb sin c 2 (Tb f )
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Chapter 6: Pass-band Data Transmission
Outline
• 6.1. Introduction
• 6.2. Pass-band Transmission
• 6.3 Coherent Phase Shift Keying
– Binary Phase shift Keying (BPSK)
– Quadriphase-Shift Keying (QPSK)
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Chapter 6: Pass-band Data Transmission
6.3 Coherent Phase Shift Keying - QPSK
• Reliable performance
– Very low probability of error
• Efficient utilization of channel bandwidth
– Sending more then one bit in a symbol
• Quadriphase-shift keying (QPSK) - example of quadraturecarrier multiplexing
–
–
–
–
Information is carried in the phase
Phase can take one of four equally spaced values – π/4, 3π/4, 5π/4, 7π/4
We assume gray encoding (10, 00, 01, 11)
Transmitted signal is defined as:
 2E

cos[2 f ct  (2i  1) ], 0  t  T

si (t )   T
4

0,
elsewhere

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Chapter 6: Pass-band Data Transmission
Signal-Space Diagram of QPSK
• From 6.23 we can redefine the transmitted signal
using a trigonometric identity:
2E

2E

si (t ) 
cos[(2i 1) ]cos(2 fct ) 
sin[(2i 1) ]sin(2 fct ) (6.24)
T
4
T
4
• From this representation we can use Gram-Schmidt
Orthogonal Procedure to create the signal-space
diagram for this signal.
• It allows us to find the orthogonal basis functions
used for the signal-space representation.
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Chapter 6: Pass-band Data Transmission
• In our case there exist two orthogonal basis functions
in the expansion of si(t). These are φ1(t) and φ2(t),
defined by a pair of quadrature carriers:
2
1 (t ) 
cos(2 f ct ),
T
2
2 (t ) 
sin(2 fct ),
T
0  t  T (6.25)
0  t  T (6.26)
• Based on these representations we can make the
following two important observations:
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Chapter 6: Pass-band Data Transmission
• There are 4 message points and the associated vectors
are defined by:
 

 E cos[(2i  1) 4 ]
si  
 , i  1, 2,3, 4
 E sin[(2i  1)  ] 

4 
• Values are summarized in Table 6.1
• Conclusion:
(6.27)
– QPSK has a two-dimensional signal constellation (N = 2)
and four message points (M = 4).
– As binary PSK, QPSK has minimum average energy
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Chapter 6: Pass-band Data Transmission
Figure 6.6
Signal-space diagram of coherent QPSK system.
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Chapter 6: Pass-band Data Transmission
Example 6.1
• Generate a QPSK signal for the given
binary input.
Input binary sequence is: 01101000
Divided into odd- even- input bits sequences
Two waveforms are created: si1φ1(t) and si2
φ2(t) – individually viewed as binary PSK
signals.
By adding them we get the QPSK signal
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Chapter 6: Pass-band Data Transmission
Example 6.1 – cont’d
 To define the decision rule for the detection of the
transmitted data sequence the signal space is
partitioned into four regions in accordance with:
observation vector x lies in region Zi if
the Euclidean distance ||x-sk|| is minimum for k = i
 Result: Four regions – quadrants – are defined,
whose vertices coincide with the origin.
 Marked in fig. 6.6 (previous pages)
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Figure 6.7
Chapter 6: Pass-band Data Transmission
(a) Input binary sequence. (b) Odd-numbered bits of input sequence and
associated binary PSK wave. (c) Even-numbered bits of input sequence
and associated binary PSK wave. (d) QPSK waveform defined as s(t) 
si11(t)  si22(t).
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Chapter 6: Pass-band Data Transmission
Error probability of QPSK
• In a coherent system the received signal is defined
as:
x(t )  si (t )  w(t ),
 0t T

i  1, 2,3, 4
(6.28)
• w(t) is the sample function of a white Gaussian noise
process of zero mean and N0/2.
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Chapter 6: Pass-band Data Transmission
The observation vector has two elements, x1 and x2,
defined by:
T
x1   x (t )1 (t )dt
0



E cos  2i  1   w1
4

(6.29)
E

 w1
2
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Chapter 6: Pass-band Data Transmission
The observation vector has two elements, x1 and x2,
defined by:
T
x1   x (t )1 (t )dt
0



E cos  2i  1   w1
4

E

 w1
2
(6.29)
i=1 and 3 so
cos(π/4)
= 1/2
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Chapter 6: Pass-band Data Transmission
T
x2   x(t )2 (t )dt
0


  E sin  2i  1   w2
4

(6.30)
E

 w2
2
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Chapter 6: Pass-band Data Transmission
T
x2   x(t )2 (t )dt
0


  E sin  2i  1   w2
4

E

 w2
2
(6.30)
i=2 and 4 so
sin(3π/4)
= 1/2
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Chapter 6: Pass-band Data Transmission
So,
• The observable elements x1 and x2 are
sample values of independent Gaussian RV
with mean equal to +/-√E/2 and -/+√E/2
and variance equal to N0/2.
• The decision rule is to find whether the
received signal si is in the expected zone Zi
or not.
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Chapter 6: Pass-band Data Transmission
Calculation of the error
probability:
• QPSK is actually equivalent to two BPSK systems
working in parallel and using carriers that are
quadrature in phase.
• According to 6.29 and 6.30 these two BPSK are
characterized as follows:
– The signal energy per bit is √E/2
– The noise spectral density is N0/2.
• Calculate the average probability of bit error for
each channel as:
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Chapter 6: Pass-band Data Transmission
• In one of the previous
classes we derived the
formula for the bit error
rate for coherent binary
PSK as:
• Using 6.20 we can find
the average probability
for bit error in each
channel of the coherent
QPSK as:
Eb
1
Pe  erfc(
)
2
N0
(6.20)
 E/2 
1
P '  erfc 

2
 No 
 E
1
 erfc 
2
 2 No
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

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Chapter 6: Pass-band Data Transmission
• The bit errors for the in-phase and
quadrature channels of the coherent QPSK
are statistically independent
• The in-phase channel makes a decision on
one of the two dibits constituting a symbol;
the quadrature channel – for the other one.
• Then the average probability of a correct
decision is product of two statistically
independent events p1 and p2.
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Chapter 6: Pass-band Data Transmission
• The average probability for a correct decision resulting from
the combined action of the two channels can be expressed as
(p1 * p2):
Pc  (1  P ') 2
1
E 2
 [1  erfc(
)]
2
2 N0
(6.32)
E
1
E
2
 1  erfc(
)  erfc (
)
2 N0
4
2 N0
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Chapter 6: Pass-band Data Transmission
• Thus the average probability for a symbol error for
coherent QPSK can be written as:
Pe  1  Pc
E
1
E
2
 erfc(
)  erfc (
)
2 N0
4
2 N0
(6.33)
• The term erfc2(√E/2N0)<< 1 so it can be ignored,
then:
Pe
E
erfc(
)
2 N0
(6.34)
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Chapter 6: Pass-band Data Transmission
• Since there are two bits per symbol in the QPSK
system, the energy per symbol is related to the
energy per bit in the following way:
E  2Eb
(6.36)
• So, using the ratio Eb/N0 we can express the symbol
error (6.37):
Pe
Eb
erfc(
)
No
(6.37)
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Chapter 6: Pass-band Data Transmission
• Finally we can express the bit error rate (BER) for QPSK
as:
Eb
1
BER  erfc(
)
2
No
(6.38)
Conclusions:
• A coherent QPSK system achieves the same average probability of bit error
as a coherent PSK system for the same bit error rate and the same Eb/N0 but
uses half of the channel bandwidth.
or
• At the same channel bandwidth the QPSK systems transmits information at
twice the bit rate and the same average probability of error.
• Better usage of channel bandwidth!
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Chapter 6: Pass-band Data Transmission
Generation and Detection of Coherent
QPSK Signals
• Algorithm (transmitter)
– input binary data sequence transformed into polar form (nonreturn-to-zero encoder) – symbols 1 and 0 are represented by
+√E/2 and -√E/2
– divided into two streams by a demultiplexer (odd and even
numbered bits) – a1(t) and a2(t)
– in any signaling interval the amplitudes of a1(t) and a2(t)
equal si1 and si2 depending on the particular bit that is sent
– a1(t) and a2(t) modulate a pair of quadrature carriers
(orthogonal basis functions φ1(t) = √2/Tcos(2πfct) and φ2(t)=
√2/Tsin(2πfct) )
– results in a pair of binary PSK which can be detected
independently due to the orthogonallity of the basis functions.
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Chapter 6: Pass-band Data Transmission
• Algorithm (receiver)
– pair of correlators with common input
– locally generated pair of coherent reference signals
φ1(t) and φ2(t).
– correlator outputs – x1 and x2 produced in response
to the input signal x(t)
– threshold comparison for decision
• in-phase – x1>0 decision for 1; x1<0 decision of 0
• quadrature – x2>0 decision for 1; x2<0 decision of 0
– combined in a multiplexer
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Chapter 6: Pass-band Data Transmission
Figure 6.8
Block diagrams of (a)
QPSK transmitter and
(b) coherent QPSK
receiver.
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Chapter 6: Pass-band Data Transmission
Power Spectra of QPSK Signals
•
Assumptions;
–
–
–
•
binary wave is random;
1 and 0 symbols are equally likely;
symbols transmitted in adjacent intervals are statistically
independent
Then:
1. depending on the dibit sent during the signaling
interval Tb ≤ t ≤ Tb
- the in-phase component equals +g(t) or – g(t)
- similar situation exists for the quadrature component
Note: the g(t) denotes the symbol shaping function
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Chapter 6: Pass-band Data Transmission
 E
,
0t T

g (t )   T
 0,
otherwise

(6.39)
So,
it follows that the in-phase and quadrature
components have a common power spectral
density E sinc2(Tf).
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Chapter 6: Pass-band Data Transmission
•
The in-phase and quadrature components
are statistically independent.
•
the baseband power spectral density of QPSK equals
the sum of the individual power spectral densities of
the in-phase and quadrature components
S B ( f )  2 E sin c (Tf )
2
 4 Eb sin c 2 (2Tb f )
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(6.40)
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