Probability of Error, Digital Signaling on a Fading Channel
And Equalization Schemes for ISI
Wireless Communications Technologies Spring 2005 Lectures 11&12 R1
Department of Electrical Engineering, Rutgers University, Piscataway, NJ
Instructor: Dr. Narayan Mandayam
Summary by Randy Mitch
Abstract – This document discusses the probability of error for non-coherent FSK and
DPSK. Digital signaling on a frequency selective channel will be reviewed as well as
equalization schemes to eliminate ISI.
1.0 Probability of Error for non-coherent BFSK
Recall from the last lecture that we have Binary FSK orthogonal modulation. When a
“1” is selected then S1(t) is transmitted and when a “0” is selected then S2(t)is
transmitted. S1(t) and S2(t) are orthogonal. The received signal is g1(t) if S1(t) was
transmitted and g2(t) if S2(t) is transmitted. The received signals g1(t) and g2(t) are also
orthogonal. The receiver structure is shown in figure 1.
Filter
Matched to
S1(t)
t=T
Envelope
Detector
l1
l1  l2  1
X(t)
Compare
l2  l1  0
Filter
Matched to
S2(t)
Envelope
Detector
l2
t=T
Figure 1. Non-coherent Receiver for BFSK
Let S1(t) be transmitted, then an error occurs if l2 > l1.
l2  xI 2  xQ2
Where XI2 and XQ2 are both Gaussian with zero mean and psd = N0/2.
 2l2
 l2 2 
 exp  

fl2 (l2 )   N 0
 N0 

0




, otherwise 
, l2  0
1

Pr obability of error Pe1  Pr l2  l1   Pr l2  1 l1 fl1  l1  dl1
0

 l2 
Pr l2  1 l1   fl2  l2  dl2  exp   1 
 N0 
l1
N 

where l1  xI1  xQ1 , xI1 N  E , 0  and xQ1
2 

 x  E 2
I1
1


f xI xI1 
exp 

N0
 N0




2
1
 xQ 
f xQ xQ1 
exp  1 
 N0
 N 0 

 

 N 
N  0, 0 
 2 
 


Pr obability of error Pe1   p error xI1 xQ1 f xI xQ dxI1 dxQ1
1
1

  exp  x



 xQ1 2  xI1 2 
 xI1  E
Pe1    exp 
 exp 
N0
N0
 




 

Rewriting xQ1 2  xI1 2  xI1  E
Pe1 

2
 xQ1  2 xI1  E
2



 E 
 2
1
exp  
xI1  E
  exp  
 N0
2
N
N
0  
0



2

2


 2 xQ1 2 

 dxI dxQ
N 0  1 1
2
Q1
E
2
2



 xQ1 

dx
exp


dxQ1
 I1 
N


0 
 

 E 
1
Pe1  exp  

2
 2 N0 
2.0 Probability of error for non-coherent M-ary FSK
In general for M-ary FSK with non-coherent detection:
Pe 
M
   i  1 kEb 
1
i
 1 exp 


2  m  1 i 2
iN0


Where k  log 2 M
A non-coherent receiver for M-ary FSK is shown in Figure 2. As with the binary FSK
receiver a comparison of the output of the envelope detectors selects the appropriate
branch.
Figure 3 illustrates the probability of bit error for various values of M. As M increases
the bandwidth efficiency decreases but the power efficiency increases. More
constellations require more chunks of orthogonal spaces.
2
Filter
Matched to
S1(t)
t=T
Envelope
Detector
t=T
X(t)
Filter
Matched to
S2(t)
Envelope
Detector
Compare
Filter
Matched to
Sm(t)
t=T
Envelope
Detector
Figure 2. Non-coherent Receiver for M-ary FSK
Figure 3. Non-coherent M-ary FSK BER
3
3.0 Differential Phase Shift Keying
If information is keyed onto the phase, PSK, it cannot be detected by non-coherent
methods. Use phase difference between two consecutive waveforms to carry the
information.
Assumption: It works provided the unknown phase introduced by the channel varies
slowly (i.e. slow enough to be considered constant over 2 bit intervals). Consider input
sequence {mk}. Generate differentially encoded sequence {dk} from {mk} as follows:
1. Sum dk-1 and mk Modulo 2.
2. Set dk to be the complement of the result from 1 above.
3. Use dk to phase shift a carrier as follows:
dk = 1
 =0
dk = 0
 = 
Example: if
m k   10010011
dk 1  11011011
dk   10110111
   0 00 000
Observe: Symbol d k is unchanged from previous symbol if incoming symbol is 1.
Symbol d k is toggled from previous symbol if incoming symbol is 0.
4.0

- DQPSK
4
Exploit the above observation to derive the probability of error. Let the DPSK signal in
Eb
0  t  Tb be
cos  2 f ct  . If the next bit (in the interval Tb  t  2 Tb) is 1, then the
2Tb
phase is unchanged. If the next bit is 0, then the phase is shifted by  .
4



S1  t   




Eb
cos  2 f ct  , 0  t  Tb 
2Tb


Eb
cos  2 f ct  , Tb  t  2Tb 
2Tb




S2  t   



Eb
cos  2 f ct  , 0  t  Tb
2Tb




Eb
cos  2 f ct    , Tb  t  2Tb 
2Tb

It is possible to consider S1  t  and S2  t  as similar to non-coherent orthogonal
modulation over a 2 bit interval, 0  t  2Tb.
 E 
1
Pe  exp  

2
 N0 
Figure 4 compares the BER for DPSK and FSK. Observe that DPSK BER is
approximately 3 dB better than non-coherent FSK.
Figure 4 BER Comparisons of DPSK and FSK
5
5.0 Digital Signaling on Frequency Selective Fading Channels
Modeling: Recall any modulated signal is given as v(t )  A b  t  kT , x k  . We restrict
k
ourselves to linear modulation.
b  t , x k   xk ha  t 
xk   complex symbol sequence
amplitude sampling pulse
The above signal is transmitted through a channel c(t) that results in the received signal
w(t).

w(t )   xk h  t  kT   z (t )
k 0
zero-mean AWGN

h(t ) 
 h ( )c(t   )d .
a


For causal channels  h(t )   ha ( )c(t   )d , t  0 .
0
We also assume h(t) = 0 for t  0 and h(t) = 0 for t  LT. L is some real positive integer.
 x  (t  nT )
n
k
ha (t )
c (t )
u (t )

h* (t )
y (n )
t T
z (t )
h (t )
Figure 5. Matched Filter Receiver in AWGN Channel
If we know h(t), we can implement a matched filter as above.
y (t ) 

x
k 
k
f (t  kT )  v(t )
Filter noise
6

f (t ) 
 h  h(  t )d .
*

f (t ) is the overall pulse response and it accounts for transient filter, channel and receive
filters.
yn  y (nt ) 

x
k 
k
f (nT  kT )  v(nT )
  xk f n k  vn
k
 xn f 0 

x
k 
k n
k
f n  2  vn
called intersymbol interference(ISI)
ISI is caused by the channel so if we can eliminate it then we can treat the channel as
AWGN channel. Therefore to achieve the same performance as on a AWGN channel, we

require
x
k 
k n
k
f n  2  0 . In order to do this we must make f n2 = 0. Alternately,
0 if i  j 
f k   k0 f 0 where  ij  
.
1 if i  j 
Equally in the frequency domain
1 
n

F  f    f0

T n  
T
, F ( f )  F  f (t )
It can be shown that f(t) can be any function that has equally spaced zero crossings.
What is the optimum receiver? We must express w(t )  lim  wn n (t ) using the
n
Karhunen-Loeve expansion.
T
hnk   h(t  kT ) n* (t )dt
0
T
zn   z (t ) n* (t )dt
0
w   w1 , w2 , ... wn  is a multivariate Gaussian distribution because z(t) is Gaussian.
7


1
 1
p ( w x, H )  
exp 
wn   xk hnk
k 
n 1  N 0

 N0
where H   h1 , h 2 , h3 , ... ,hn  and hn  (...., hn3 , hn2 , hn1, h0 )
N
2





Therefore the optimum receiver is the Maximum Likelihood receiver, for the case of
AWGN it reduces to
N



arg max u ( x)   wn   xk hnk
x
n 1
k 


2





The bottom line is:
1. We need knowledge of  f n  to understand the channel response and knowledge of
cn  in order to perfectly eliminate ISI.
Therefore we need to estimate the
channel in order to equalize it.
2. An additional problem results as well by observing the following:

x
y (t ) 
k 
k
f (t  kT )  v(t )
where the noise function is

 h ( ) z (t   )d
v(t ) 
*

is still Gaussian but is not white. Therefore the noise samples at the output are
correlated. In this case we can obtain a discrete time white noise model as
follows.
 x  (t  nT )
n
k
ha (t )
c (t )

h* (t )
Whitening
Filter
vk
t T
z (t )
h (t )
Figure 6. Whitening Filter with Matched Filter Receiver
The output of the whitening filter vk is given by
L
vk   g n xk  n  k
n 0
8
where g n is the overall filter for the channel and the whitening filter and  k is the
white noise process.
3. Another important consideration is the design of ISI equalizing filters is
extremely sensitive to timing information. We can solve this sensitivity in two
ways.
a. Use pulse shaping. Raised cosine pulse shaping allows us to derive the
length of the pulse by sampling at even points.
b. Use fractional sampling. Sample the output y(t) at a rate higher than 2/T.
Although we will still have correlated noise and will need a whitening
filter the overall pulse shape will be less sensitive to timing errors.
6.0 Equalization Schemes
There are two types of equalization schemes.
1. Symbol by symbol equalizers that can be linear or non-linear.
2. Sequence estimation equalizers that are non-linear.
6.1 Symbol-by-Symbol Equalizers
We consider the discrete time white noise model shown in figure 7. Where an  is the
input signal, L is the memory of the channel,  n is the additive noise and
g  ( g0 , g1, .....,gl )T is the channel vector that describes the overall channel impulse
response.
an 
T
g0
……...
T
X
g1
T
X
X
gL


L
rn   g k an  k   n
k 0
n
Figure 7. Discrete Time White Noise Model
9
Linear equalizers estimate the nth transmitted symbol aˆ n as
aˆn 
 
M
cr
j n j
j  M
Where c j
M
M
is the equalizer filter of (2M+1) taps. M is a
design choice and C is the chosen number of taps to estimate aˆ n .
6.2 Zero Forcing Equalizer
The Zero Forcing Equalizer combines the channel and equalizer impulse response to
force to zero at all but one of the taps in the TDL filter. The tap coefficients, c, are chosen
to “zero-out” the ISI. Given the channel vector, g , we can select the tap coefficients, c,
to get the desired response q  (0, 0, 0,......, qn , 0,...) . The Zero Forcing Equalizer zeros
out all but the desired result qn . The tap coefficients are calculated using the
relationship qn  c g n . Zero forcing equalizers can perfectly eliminate ISI as M   but
T
it also enhances the noise. In practice, the receiver does not know the channel vector and
finite length training sequences are used to choose the tap coefficients.
6.3 Minimum Mean Square Equalizer (MMSE)
The MMSE is another symbol-by-symbol equalizer but is superior to the zero forcing
equalizer in performance. The MSME utilizes the mean square error criterion to adjust
the tap coefficients. We define an estimation error:
 n  an  aˆn
Where an is the symbol sent and aˆ n is the estimated symbol. The function to be
minimized is:
J  min E  n 2   min E (an  aˆn ) 2 
c
c
2
M


 min E (an   c j rn  j ) 
c
j  M


The error is minimized by choosing c j  so as to make the error sequence orthogonal to
the signal sequence rn l , for l  M , i.e. E  n rnl   0 , l  M . The optimum c is
obtained by solving the following set of equations.
M
 c E r
j  M
j
r   E  an rn l  ,
n  j n l
l M
10
Define the following functions.
and
E  rn j rnl  , j  l ,  M ,....,  M
b   E  an rn  m  E  an rn  m 1 ........E  an rn  m 
then
copt  1 b

To implement the MSME we need to us adaptive algorithms because the channel
response is not readily known. Typically, we use the steepest descent algorithm.
2
1 E   n  
c j (n  1)  c j (n)  
, j   M .......  M
2
c j
Where  is a positive number.
Note:
E  2  n 

 (n) 
 2 E  (n)
  2 E  (n)rn j   2 Rer ( j )
c j
c j 

The term Rer is the cross correlation between the input and the error. The equalizer taps
can be obtained by implementing a stochastic gradient algorithm.
1
c j (n  1)  c j (n)   (n)rn  j
2
To evaluate  (n) in this algorithm we use a training sequence.
6.4 Decision Feedback Equalizer (DFE)
A DFE is a nonlinear equalizer that consists of a feed forward section and a feedback
section. DFEs are very effective in frequency selective channels because they mitigate
the effects of noise enhancements that degrade the performance of linear equalizers. The
DFE uses previous decisions to eliminate ISI caused by previously detected symbols on
current symbols. The output of the equalizer can be expressed as:
aˆn 
0

j  M1
c j rn  j
feed forwad filter M1 1 taps

M2
c j an j

j 1
feedback filter M 2 taps
Where an 1 , an  2 , ....., an  M 2 are earlier detected symbols. Both the feed forward loop
and the feedback loop need adaptation to adjust the coefficients.
11
6.5 Maximum Likelihood Sequence Estimation (MLSE)
Recall the discrete-time white noise channel model.
L
rn   g m an  m   n
m 0
Assume k symbols are transmitted over the channel. Then after receiving the sequence
k
k
rn n 1 , the ML receiver decides in favor of the sequence an n 1 that maximizes the
likelihood function

log rk , rk 1, ..., r1 ak , ak 1 ,..., a1

Since the noise samples are independent and rn depends only on the L most recent
transmitted symbols.





log rk , rk 1, ..., r1 ak , ak 1 ,..., a1  log rk ak , ak 1 ,., ak l  log rk 1 , rk 2, .., r1 ak 1, ak 2 ,.., a1

This has already been calculated
where ak  L  0 for k  L  0
Since the 2nd term has been calculated at the previous time (k-1), only the first term needs
to be computed at time k for each incoming signal rk . The Veterbi algorithm can be used
to implement the MLSE equalizer. Adaptive MLSE is used in GSM.
References:
[1]
[2]
[3]
[4]
[5]
[6]
N. Mandayam, Wireless Communication Technologies, course notes.
T. Rappaport, Wireless Communications. Principles and Practice. 2nd Edition,
Prentice-Hall, Englewood Cliffs, NJ: 1996.
J. Proakis, Digital Communications, 3rd Edition, McGraw-Hill, NY: 1995.
G. Stuber, Principles of Mobile Communication, 2nd Edition, Kluwer Academic
Publishers, Norwell, MA: 2001.
B. Sklar, Digital Communications. Fundamentals and Applications, Prentice-Hall,
Englewood Cliffs, NJ: 1988.
D. Wu, Wireless Communication Technologies, Rutgers University, Piscataway,
NJ: 2002
12