ECE 6640
Digital Communications
Dr. Bradley J. Bazuin
Assistant Professor
Department of Electrical and Computer Engineering
College of Engineering and Applied Sciences
Chapter 4
Chapter 4:
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
ECE 6640
Optimum Receivers for AWGN Channels
Waveform and Vector Channel Models
Waveform and Vector AWGN Channels
Optimal Detection and Error Probability for
Band-Limited Signaling
Optimal Detection and Error Probability for
Power-Limited Signaling
Optimal Detection in Presence of Uncertainty:
Noncoherent Detection
A Comparison of Digital Signaling Methods
Lattices and Constellations Based on Lattices
Detection of Signaling Schemes with Memory
Optimum Receiver for CPM Signals
Performance Analysis for Wireline and Radio
Communication Systems
Bibliographical Notes and References
Problems
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
160
160
167
188
203
210
226
230
242
246
259
265
266
2
4.3 Optimal Detection and Error
Probability for Band-Limited Signaling
• These are for “lower bandwidth”, low dimensionality
signaling types.
• This section is an excellent reference for some of the
primary signal types discussed.
• Explicit BER vs. Eb/No equations are derived based on the
previous material presented.
– An assumption of equally likely symbols is made for each
derivation.
– 4.3-1 Derives ASK or PAM
– 4.3-2 Derives MPSK
– 4.3-3 Derives QAM
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
3
MASK Summary
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
4
MASK
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
5
MASK Performance
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
6
MPSK
•
The marginal probability density
function for a symbol can be defined as
•
The pdf is a function of the average
symbol energy
•
The higher the number of symbols, the
tighter the symbol decision regions must
become and more errors can be
expected.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
7
MPSK
• In general, the integral of p(θ) does not reduce to a simple
form and must be evaluated numerically, except for M = 2
and M = 4.
• For M=2
• For M=4
• For other M (M large and SNR large)
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
8
MPSK
M=2
M=4
M other
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
9
QAM
QAM is dependent upon the symbol
constellation selected.
• Default to square constellations
of 4, 16, 64, & 256
• Numerous others are possible
with potentially better system
performance
• The optimal detector uses 2 basis
symbols to resolve the in-phase
and quadrature components
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
10
Square Constellation QAM
• This case appears as two dimensional ASK/PAM
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
11
Square Constellation QAM
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
12
Comparing QAM and MPSK
• Looking at the ratio of the Q(x) arguments
– At M=4 the systems are equivalent, but for higher M QAM has
better performance.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
13
Demodulation and Detection of BandLimited Signals
• Matched filter involve the basis form of the signals.
Note: Filter are
matched to basis,
not matched to
symbols!
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
14
4.4 Optimal Detection and Error
Probability for Power-Limited Signaling
• These are for “wider bandwidth”, higher dimensionality
signaling types.
• BER vs. Eb/No equations are derived based on the
previous material presented.
– An assumption of equally likely symbols is made for each
derivation.
– 4.4-1 Orthogonal FSK
– 4.4-2 Biorthogonal
– 4.4-3 Simplex
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Orthogonal Signals - MFSK
• For equiprobable, equal-energy orthogonal signals, the
optimum detector selects the signal resulting in the largest
cross-correlation between the received vector r and each of
the M possible transmitted signal vectors {sm}, i.e.,
d min  2  E
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Orthogonal Signals (cont)
•
The probability of correct symbol detection can be described as
•
assuming Gaussian noise elements where the elements are independent and
identically distributed (IID)
•
with an individual dimension represented as
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Orthogonal Signals (cont)
•
The integral becomes
•
The error is the complement, therefore
•
In general, Equation 4.4–10 cannot be made simpler, and the error probability
can be found numerically for different values of the SNR.
To determine bit errors, let us assume that s1 corresponds to a data sequence of
length k with a 0 at the first component. The probability of an error at this
component is the probability of detecting an sm corresponding to a sequence
with a 1 at the first component. Since there are 2k−1 such sequences, we have
•
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FSK Signaling
Another Union Bound
•
The orthogonal case is easier than the previous derivation as all symbols have
the minimum distance. Taking the result
•
For orthogonal signaling
•
Using M = 2k and Eb = E/k, we have
•
Note that if
•
Then Pe  0 as k  ∞ (not Pe  infinite like the text says!!)
ECE 6640
d min  2  E
Note: a necessary and sufficient
condition for reliable comm. that is
slightly lower is derived in Chap. 6.
It is called the Shannon limit.
19
Orthogonal Signaling
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4.4-2 Biorthogonal Signaling
d min  2  E
d other  2  E
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Biorthogonal Signaling
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COMMENT ON SIMULATIONS
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Correlation Receiver Design
•
Maximizing Design (maximum symbol correlation)
– Create a matched filter based on every one of the symbols being
transmitted.
*
sm L  T    , for 1  m  M
– Determine the correct symbol sampling time. (Proportional to LT)
– The maximum of the sampled correlates is selected as the output symbol.
max s t   nt   sm L  T  t 
*
1 m  M
•
@ t  k  T  filter _ delay
Minimizing Design (minimum distance from signal to symbol)
– Create a matched filter based om every one of the basis symbol set.
*
rn t   s t   nt   n t  ,
for 1  n  N
– Determine the correct symbol sampling time.
– Find the symbol vector, sm, that is a minimum distance.
ECE 6640
min rn k   sm
1 m  M
for t  k  T  filter _ delay
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Design Considerations
• If the dimensionality of the symbols is small, the
minimization approach using the basis set for the symbols
will require less hardware/processing.
– For ASK N=1
– For PSK N=2
• If the dimensionality of the symbols is large, the
maximization approach using the matched filter of the
symbols will be a likely choice
– For FSK N=M
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Matlab Minimization for MPSK
% RECEIVER ---------------------------------------------% Process the array of symbol plus noise values
% Detect the correlator output using an optimal threshold
corrc= (ccarrier/sqrt(NSampSym))'*symbol_noise;
corrs=-(scarrier/sqrt(NSampSym))'*symbol_noise;
csym = corrc + sqrt(-1)*corrs;
Note: Correlator
perfectly matched to
symbol time
% map to the closest symbol for the symbol detection/decision
% The first column is noise only
[ee ind]=min(abs(sym_map-csym(1)));
sd(:,1)= [real(sym_map(ind));imag(sym_map(ind))];
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Matlab Maximization for MFSK
% RECEIVER ---------------------------------------------Note: Correlator
% Process the array of symbol plus noise values
perfectly matched to
% Detect the correlator output using an optimal threshold
symbol time
for nnn=1:M
FcorrM=Fstart+(2*(nnn-1)*Fstep);
corrsym=sqrt(2)*(-1)^(n-1)*sin(2*pi*(FcorrM)*time_symbol)/sqrt(NSampSym);
corr(nnn,:) = corrsym'*symbol_noise;
end
[val received_sym]=max(corr);
bits_rcvd=Sym_bits(received_sym',:)';
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Matlab Matched Filter for
MASK, QAM & MPSK
%%
% Symbol plus scaled filter and SNR noise
RX_samples = TX_samples + sqrt(1/SNR)*input_noise;
%%
% Received signal baseband processing
%
% firrcos matched filter receiver
pred_samples =filter(hsqnyq,1,RX_samples);
…
% optimal symbol time sampling
pred_sym = pred_samples(Tdelay-1:expand:end);
num_pred_sym = length(pred_sym);
…
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%%
% Symbol and Bit Error Analysis
%
% map to the closest symbol for the symbol detection/decision
eepower = 0;
sd = zeros(num_pred_sym,2);
ygsym = zeros(num_pred_sym,1);
% minimum distance computations
for jj=1:num_pred_sym
[ee,ind]=min(abs(complex_const_map/sigma_s-pred_sym(jj)));
sd(jj,:)= [real(complex_const_map(ind)) imag(complex_const_map(ind))];
ygsym(jj) = ind-1;
eepower = eepower + ee^2;
end
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Useful Matlab Study
• Compare BER performance based on using different
filters.
– Using different forms of matched filters.
• rectangular, square-root Nyquist, truncated sinc functions, etc.
• must have appropriate nulls to minimize ISI
– Using “unmatched” filters.
• rect TX with Nyquist filter receive
• Section 4.5 is on Non-coherent Detection
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4.5 Noncoherent Detection
• There are times when the coherence assumption is invalid.
– The channel can introduces random changes to the signal as either
a random attenuation or a random phase shift. Chapter 10 deals
with equalization and Chapter 13 deals with fading channels.
– Alternately, imperfect knowledge of the signals at the receiver
arises when the transmitter and the receiver are not perfectly
synchronized. It can use only signals in the form of {sm(t − td )},
where td represents the time difference and is model as a random
variable or as a random received signal phase.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
31
Noncoherent Basis
•
By the Karhunen-Loeve expansion theorem discussed in Section 2.8–
2, we can find an orthonormal basis for expansion of the random
process sm(t; θ) and by Example 2.8–1, the same orthonormal basis can
be used for expansion of the white Gaussian noise process n(t).
•
By using this basis, the waveform channel given in Equation 4.5–1
becomes equivalent to the vector channel for which the optimal
detection rule is given by
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
32
Noncoherent Detection
• Equation 4.5–3 represents the optimal decision rule and the
resulting decision regions. The minimum error probability,
when the optimal detection rule of Equation 4.5–3 is
employed, is given by
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
33
Example 4.5-1
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
34
Example 4.5-1 cont
Ebavg
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2
 5
1
1   1 
  Eb      Eb    Eb
 8
2
2   2 

Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
35
Random Phase
• The random time can be translated into a random phase
element based on the following
• The time/phase does not modify the noise term, but the
signal term now has a random rotation.
– This directly effects the detection process.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
36
Detection
• To design the optimal detector under these circumstances a
generalized form is needed
– note that for a simple MPSK system, the correct symbol is not
likely to be detected.
– what can be computed is
– leading to …
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
37
Detection (cont)
• A summation in absolute value and phase
– where θ denotes the phase of rl · sml . Note that the integrand in
Equation 4.5–19 is a periodic function of φ with period 2π, and we
are integrating over a complete period; therefore θ has no effect on
the result.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
38
Detection (cont)
• The integral of an cosine exponential term involves the
modified Bessel function so that
• As the zeroth order modified Bessel function is
monotonically increasing in x we can also use
– this is an envelope/magnitude detector.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
39
Noncoherent Detection
• Envelope Detection
– As long as correlator /filter magnitude provide a valid measure, the
system can be processed.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
40
Practical Noncoherent Signals
• Phase blind signal based on filter/correlator magnitudes
readily work for
– MFSK
– Differential signaling methods (relative phase changes, not
absolute phase changes)
• They will be limited or not function properly with
– MASK – phase dependent for +/– MPSK – phase driven
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4.5-3 Error Prob. Noncoherent FSK
• Each independent symbol processor result in
• moreover, the symbol statistics now differ with matching
symbols having a Ricean pdf and noise symbols having 2D Gaussian or Rayleigh pdfs.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
42
Error Prob (cont)
• For the resulting outputs
• The prob. of correct detection and symbol error is now
dependent on an exponential …
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
43
Error Prob (cont)
• For the resulting outputs
• The prob. of correct detection and symbol error is now
dependent on an exponential …
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
44
Error Prob (cont)
• For binary orthogonal signaling, including binary
orthogonal FSK with noncoherent detection, Equation 4.5–
44 simplifies to
– as compared to coherent systems with
– Therefore, non coherent detection does not perform as well as
coherent, but high Eb/No they may be relatively close.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
45
Bit Error for Noncoherent FSK
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
46
AB CARLSON TEXT ANALYSIS
ECE 6640
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Noncoherent Binary Systems
• Synchronous coherent receiver can be very difficult to
design.
• Can noncoherent systems be more easily designed without
giving up significant BER performance?
– For a 1-2 dB Eb/No performance loss, YES!
Notes and figures are based on the textbook: A. Bruce Carlson, P.B. Crilly,
“Communication Systems, 5th ed.”, McGraw-Hill, 2010.
48
Noncoherent OOK receiver
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Figure 14.3-2
• Using an envelope detector, the noise pdf for a zero
symbol becomes Rician and is non-longer Gaussian.
• The noise pdf for a one symbol remains Gaussian
Notes and figures are based on the textbook: A. Bruce Carlson, P.B. Crilly,
“Communication Systems, 5th ed.”, McGraw-Hill, 2010.
49
Conditional PDFs for
noncoherent OOK
Figure 14.3-3
Pe 0 Vopt   Pe1 Vopt 

E 
Pe 0  exp  b 
 2  N0 
Pe 
Vopt 
Ac

E 
1
1
1
 Pe 0  Pe1    Pe 0    exp  b 
2
2
2
 2  N0 
2
Notes and figures are based on the textbook: A. Bruce Carlson, P.B. Crilly,
“Communication Systems, 5th ed.”, McGraw-Hill, 2010.
50
Noncoherent detection of binary FSK
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Figure 14.3-5
Notes and figures are based on the textbook: A. Bruce Carlson, P.B. Crilly,
“Communication Systems, 5th ed.”, McGraw-Hill, 2010.
51
Noncoherent FSK
• Qualitative comments
– Using envelope detectors on each symbol output, the Rician error
distribution effects the z detection statistic.
Pe 

E 
1
 exp  b 
2
 2  N0 
Notes and figures are based on the textbook: A. Bruce Carlson, P.B. Crilly,
“Communication Systems, 5th ed.”, McGraw-Hill, 2010.
52
Binary error probability curves
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
(a) coherent BPSK (b) DPSK (c) coherent OOK or FSK (d) noncoherent
FSK (e) noncoherent OOK: Figure 14.3-4
10
10
BER
10
10
10
10
10
BER Simulation for BPSK and BFSK
0
-1
-2
-3
-4
BPSK
BPSK
BFSK
BFSK
-5
-6
0
2
simulation
(theoretical)
simulation
(theoretical)
4
6
8
E b/No (dB)
10
Notes and figures are based on the textbook: A. Bruce Carlson, P.B. Crilly,
“Communication Systems, 5th ed.”, McGraw-Hill, 2010.
12
14
16
53
Binary error probability curves
(a) coherent BPSK (b) DPSK (c) coherent OOK or FSK
(d) noncoherent FSK (e) noncoherent OOK
Figure 14.3-4
Notes and figures are based on the textbook: A. Bruce Carlson, P.B. Crilly,
“Communication Systems, 5th ed.”, McGraw-Hill, 2010.
4.6 Comparison Of Methods
• One can compare the methods on the basis of the SNR
required to achieve a specified probability of error.
• It would not address data rate of transmission or
bandwidth.
• To measure the bandwidth efficiency, we define a
parameter r , called the spectral bit rate, or the bandwidth
efficiency, as the ratio of bit rate of the signaling scheme to
the bandwidth
R
r
W
bits / sec/ Hz
– larger r is a more bandwidth-efficient
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
55
4.6–1 Bandwidth and Dimensionality
• The sampling theorem states that in order to reconstruct a
signal with bandwidth W, we need to sample this signal at
a rate of at least 2W samples per second. In other words,
this signal has 2W degrees of freedom (dimensions) per
second. Therefore, the dimensionality of signals with
bandwidth W and duration T is
N  2  TS  W
r
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R R
R
log 2 M 2  log 2 M
  2  TS   2 

W N
N
R
N
bit / sec/ Hz
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
56
Common Modulation b/s/Hz
• The bandwidth efficiency can be stated in terms of the
degrees of freedom and the the M array allowed.
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– MASK: N=1
r  2  log 2 M bit / sec/ Hz
– MPSK: N=2
r  log 2 M
– MFSK: N=N
r  2  log 2 M
bit / sec/ Hz
N
bit / sec/ Hz
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
57
Comparing Eb/No to r
• Using a predetermined BER, a
family of curves for each
modulation type can be generated
and plotted as shown.
– Pe=10^-5
– Two regions develop, those with
bandwidths less than symbol data
rates R>W and those with greater
bandwidths than symbol data rates
W>R.
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