Chapter 10 Intersymbol Interference In this chapter we examine optimum demodulation when the transmitted signal is filtered by the channel and there is additive white Gaussian noise. The optimum demodulator chooses the possible transmitted vector that would result in the received vector (in the absence of noise) to be as close as possible (In Euclidean distance) to what was received. This we show can be implemented by a filter matched to the received signal for a given data symbol followed by a nonlinear processing via the Viterbi algorithm. The filter is sampled at the data rate. We also analyze the performance of such a system. The analysis is very similar to that of convolutional codes. Because the received signal is filtered and sampled, the output of the filter consists of two components. One due to the transmitted signal and one due to the noise. The output due to noise is, however, not white. However, in the next section we show that the output of the matched filter can be whitened. With a whitened matched filter the optimum receiver (Viterbi algorithm) becomes clear. Finally, in the last section we show how to design a system to eliminate intersymbol interference. 1. Optimum Demodulation Consider transmitting data at rate 1 T through a channel with bandwidth W or through a distorting channel. We would like to find the optimum (minimum sequence error probability) receiver. Assume the modulator is a filter acting on a infinite sequence of impulses (at rate 1 T with impulse response ft . The channel is characterized by an impulse response of gt and the receiver is a filter sampled at rate 1 T with impulse response ht . ft Modulator s T t gt Channel zt rt nt Data Rate 1 T The transmitted signal is of the form s T t u N ∑ m N u m ft mT where u m is the data symbol transmitted during the m-th signaling interval assumed to be in the alphabet A and ft is the waveform used for transmission. We assume a transmission of 2N 1 data symbols (think of N as being very large). The output of the channel filter is then zt ∞ ∞ gt τ s τ d τ 10-1
22
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Chapter 10
Intersymbol Interference
In this chapter we examine optimum demodulation when the transmitted signal is filtered by the channel and thereis additive white Gaussian noise. The optimum demodulator chooses the possible transmitted vector that wouldresult in the received vector (in the absence of noise) to be as close as possible (In Euclidean distance) to whatwas received. This we show can be implemented by a filter matched to the received signal for a given data symbolfollowed by a nonlinear processing via the Viterbi algorithm. The filter is sampled at the data rate. We also analyzethe performance of such a system. The analysis is very similar to that of convolutional codes. Because the receivedsignal is filtered and sampled, the output of the filter consists of two components. One due to the transmitted signaland one due to the noise. The output due to noise is, however, not white. However, in the next section we showthat the output of the matched filter can be whitened. With a whitened matched filter the optimum receiver (Viterbialgorithm) becomes clear. Finally, in the last section we show how to design a system to eliminate intersymbolinterference.
1. Optimum Demodulation
Consider transmitting data at rate 1T through a channel with bandwidth W or through a distorting channel. We
would like to find the optimum (minimum sequence error probability) receiver. Assume the modulator is a filteracting on a infinite sequence of impulses (at rate 1
T with impulse response f t . The channel is characterized by
an impulse response of g t and the receiver is a filter sampled at rate 1T with impulse response h t .
f t Modulator
sT t g t Channel
z t r t n t
Data
Rate 1T
The transmitted signal is of the form
sT t u N
∑m N
um f t mT where um is the data symbol transmitted during the m-th signaling interval assumed to be in the alphabet A andf t is the waveform used for transmission. We assume a transmission of 2N 1 data symbols (think of N as beingvery large). The output of the channel filter is then
z t ∞ ∞g t τ s τ dτ
10-1
10-2 CHAPTER 10. INTERSYMBOL INTERFERENCE
∞ ∞g t τ N
∑m N
um f τ mT dτ
N
∑m N
um ∞ ∞g t τ f τ mT dτ
N
∑m N
umh t mT where
h t ∞ ∞g t τ f τ dτ
The received signal consist of two terms. One due to signal and one due to noise.
r t N
∑m N
umh t mT n t su t n t
where
su t N
∑m N
umh t mT Since n t is white Gaussian noise the optimum receiver computes for each data sequence v
ΛN v 2 r t sv t sv
2
2 ∞ ∞r t sv t dt ∞ ∞
s2v t dt
2 ∞ ∞r t N
∑k N
vkh t kT dt ∞ ∞∑m
∑k
vmvkh t kT h t mT dt
2N
∑k N
vk ∞ ∞r t h t kT dt ∑
m∑k
vmvk ∞ ∞h t kT h t mT dt
2N
∑k N
vkyk N
∑k N
N
∑m N
vkvmxm k
where
yk ∞ ∞r t h t kT dt
xm k ∞ ∞h t kT h t mT dt
Thus the optimum decision rule isChose v if ΛN v max
uΛN u
Since the decision statistic depends on the received signal only through yk it is clear that yk is a sufficient statisticto implement optimal receiver.
Consider a filter hr t h t . Then if the received sequence is passed through this filter the output would be
y s ∞ ∞r t hr s t dt
∞ ∞r t h t s dt
1. 10-3
The sampled output would be
y kT ∞ ∞r t h t kT dt
Since this is yk defined earlier the received signal should be filtered and sampled as shown below before doingsome processing.
f t Modulator
s t g t Channel
z t r t hr t
Demodulator
y t t kT
n t Data
Rate 1T yk
yk ∞ ∞r t h t kT dt ∞ ∞
N
∑k N
umh t mT h t kT dt ηk
ηk ∞ ∞n t h t kT dt
Now the original continuous time detection problem can be replaced with a discrete time problem.
yk N
∑m N
umxm k ηk
Note that ηk is Gaussian.
E ηk 0
Var ηk N0
2 ∞ ∞
h2 t dt
E ηkηm E
∞ ∞n t h t kT dt ∞ ∞
n s h s mT dt ∞ ∞
∞ ∞h t kT h s mT E n t n s dtds
N0
2 ∞ ∞
h t kT h t mT dt N0
2xk m
However, ηk is not an i.i.d. sequence.
ΛN v 2N
∑k N
vkyk N
∑k N
N
∑m N
vkvmxm k
10-4 CHAPTER 10. INTERSYMBOL INTERFERENCE
2N
∑k N
vkyk N
∑k N
x0 vk 2 2N
∑k N
vk
k 1
∑m N
vmxm k
2N
∑k N
vkyk N
∑k N
x0v2k 2
N
∑k N
vk
k N
∑j 1
vk jx j
N
∑k N
2vkyk N
∑k N
x0v2k 2
N
∑k N
vk
k N
∑j 1
vk jx j
(Note that v j 0 j N). Assume xm 0 m L (finite intersymbol interference)
ΛN v 2N
∑k N
vkyk N
∑k N
x0v2k 2
N
∑k N
vk
min L k N ∑j 1
vk jx j
N
∑k N
2vkyk x0v2
k 2vk
min L k N ∑j 1
vk jx j The decision rule can be implemented in a Viterbi algorithm like structure. Define the state at time k to be the lastL data symbols. (These are the only symbols that affect the output at time k).
σk vk L 1 vk Let
λ σk σk 1 2vk 1yk 1 x0v2k 1 2vk 1
L
∑m 1
vk 1 mxm
Then
ΛN v N 1
∑k N
λ σk σk 1 2v Ny N x0v2 N
Thus we can apply the Viterbi algorithm.Let Γ σm be the length (optimization criteria) of the shortest (optimum) path to state σm at time m. Let σm
be the shortest path to state σm at time m. Let Γ σm 1 σm be the length of the path to state σm 1 at time m thatgoes through state σm at time m. The algorithm works as follows.
Storage:
k, time index,Ω σm σm AL
Γ σm σm AL
Initialization
k N,Ω σ N σ N σ N ΦL 1 A (Φ is the empty set).Γ σ N 2v Ny N x0v2 N
Now consider the performance of the above maximum likelihood sequence detector (MLSD). We will evaluate theunion upper bound to the error probability. To do this we need to determine the pairwise error probability betweentwo sequences. This is the probability that sequence v is demodulated given sequence u is transmitted for a systemwith only two possible transmitted sequences (u v). Let P u v denote the conditional error probability given
2. 10-7
u transmitted. Then
P u v P ΛN v ΛN u u Q
sv t su t
2N0 e sv t su t 2 4N0
As expected the pairwise error probability depends only on the square Euclidean distance between the signals suand sv.
sv t su t 2 ∞ ∞
sv t su t 2 dt
∞ ∞ N
∑k N
vk uk h t kT 2
dt
∞ ∞
N
∑k N
N
∑m N
vk uk vm um h t kT h t mT dt
N
∑k N
N
∑m N
vk uk vm um ∞ ∞h t kT h t mT dt
N
∑k N
N
∑m N
vk uk vm um xk m
Let εk 12 vk uk 1 vk 1 uk 1
0 vk uk 1 vk 1 uk 1
sv t su t 2 4
N
∑k N
N
∑m N
εkεmxk m
4N
∑k N
ε2kx0 4
N
∑k N 1
2k 1
∑m N
εkεmxk m
4
N
∑k N ε2
kx0 2k N
∑j 1
εkεk jx j 4
N
∑k N ε2
kx0 2L
∑j 1
εkεk jx j P u v
exp 1N0
N
∑k N ε2
kx0 2L
∑m 1
εkεk mxm P e
Thus the incremental Euclidean distance between two paths for a given time index k is depends on the past Lerrors. The error state is defined to be the last L errors εk εk L 1 . The all zero error state corresponds to thepast L symbols being correctly demodulated.
An error event is defined to be an error sequence that diverges once from the all zero state and then remergeslater. Since a necessary condition for an error of a particular type (first event error or symbol error) is that anerror event occurs that causes the demodulator/decoder to follow a path that diverges and then at some later timeremerges we can calculate the error probability for a particular node by counting the number of paths (and theirdistance) that diverge and remerge. We can use the state diagram to determine the number of error sequences witha particular distance.
10-8 CHAPTER 10. INTERSYMBOL INTERFERENCE
Let e ε N εN wH e Hamming weight of e (number of nonzero terms)
PE m First event error probability P at time m decoder is not at correct state for the first time Pb Bit error probability P bit error occurs for symbol m
The union bound on the probability of error at time 0 is
PE m ∑u vε 0
P u v P u ∑
eε 0
∑u v
e 12v u P u v P u
where the sum is over all sequences that diverge from the all zero state and then remerge later. Each of the22N 1 u sequences are equally likely. In each position where εk 0 the components of the sequences u and v aredetermined. If εk 1 then vk 1 and uk 1. Similarly if εk 1 then vk 1 and uk 1. The components ewhere εk 0 there are two choices for uk and vk (uk vk 1 or uk vk 1). Since there are 2N 1 wH e places where εk 0 there are 22N 1 wH e such sequences u and v. Hence
PE f ∑e
ε0 0
P e 2 2N 1 wH e 2 2N 1 ∑
eε0 0
2 wH e P e The bit error probability is bounded by
Pb ∑
e
w e 2w e P e
P e 2w e N
∏k N
1
2wH εk exp 1N0 ε2
kx0 2L
∑m 1
εkεk mxm For L 1
P e 2wh e N
∏k N
1
2wH εk exp
1N0 ε2
kx0 2εkεk 1x1 We calculate these union bounds by enumerating the sequences that diverge from the all zero error state and
remerge (error events) that correspond to a given Euclidean distance between two data sequences and has a givennumber of nonzero terms (or is a given length). To do this we draw a state diagram (similar to that for convolutionalcodes) and label each path with DxNlMl where x is the incremental Euclidean distance squared (divided by 4) ingoing from one state to another and l is 1 if the error path is nonzero and is zero if the error is zero. (This redundantuse of l will be explained when we determine the bit error probability).
Error State Diagram L 1
2. 10-9
MNDx0
MNDx0
1
1
MNDx0 2x1MNDx0 2x1
MNDx0 2x1
MNDx0 2x1
0 0
1
-1
a
d
b
c
The transfer function is calculated by solving the following equations for TdTa.
Td Tc Tb
Tb MNDx0 2x1Tb MNDx0 2x1Tc MNDx0Ta
Tc MNDx0 2x1Tc MNDx0 2x1Tb MNDx0Ta
Adding the last two equations and solving for Tb Tc and substituting the result into the first equation yields
Thus there are two paths with 1 error and Euclidean distance squared of 4x0. There are two paths with two errorsand Euclidean distance squared of 8x0 8x1 and so on.
PE
T D N M D e 1 N0 N 1 M 1 2 e x0 N0
1 12 e x0
N0 e2x1 N0 e 2x1
N0 Pb
∂T D N M ∂N D e 1 N0 N 1 M 1 2 Dx0 1 N Dx0 2x1 Dx0 2x1 2 D e 1 N0 N 1 M 1 2
e x0 N0
1 12 e x0
N0 e2x1 N0 e 2x1
N0 2
10-10 CHAPTER 10. INTERSYMBOL INTERFERENCE
For large SNR this is the same as no ISI! Just as with convolutional codes this Union-Bhattacharyya bound can beimproved by using the exact error probability for the first few terms and then upper bounding the error probabilityfor higher order terms with the Bhattacharyya bound.
For L 2 the error state diagram is shown below.
3. Signal Design for Filtered Channels
Because the complexity of the Viterbi algorithm grows as the A L where A is the alphabet size and L is the memoryof the channel it is desirable to design a system with zero intersymbol interference. So consider transmitting dataat rate 1
T through a channel with bandwidth W . At what rate is this possible without creating intersymbol
interference? Assume the modulator is a filter acting on a infinite sequence of impulses (at rate 1T with impulse
response f t . The channel is characterized by an impulse response of g t and the receiver is a filter sampled atrate 1
T with impulse response h t .
f t Modulator
s(t) g t Channel
z t r t h t
Demodulator
y t t kT
n t Data
Rate 1T yk
The transmitted signal is of the form
s t ∞
∑m ∞
um f t mT The output of the received filter is then
yk N
∑m N
umxk m ηk
In order that there be no intersymbol interference we require that
xm 1 m 00 m 0
let
x t ∞ ∞h τ h τ t dt
∞ ∞h τ h t τ dt
3. 10-11
where h t h t . Then x t is the convolution of h t with h t and x nT xn. Thus X f H f H f H f 2. If the (absolute) bandwidth of the channel is W then X f also has bandwidth W . That is
X f 0 f W
Thus by the sampling theorem
x t ∞
∑n ∞
x n2W sin 2πW t n
2W 2πW t n
2W ∞
∑n ∞
x nT φn W t where
φn W t sin 2πW t n2W T
2πW t n2W
If W 12T then
x t ∞
∑n ∞
x nT sin π t nT T π t nT T
For no intersymbol interference we require that x nT 0 for n 0. Let x 0 1 then
x t sin π t nT T π t nT T
which implies that
X f T f 1
2T0 f 1
2T
Thus H f 2 T f 1
2T0 f 1
2T
Thus 2W pulses per second can yield zero intersymbol interference. It is easy to see that by signaling faster thanrate 2W we can not guarantee that there is no intersymbol interference.
Problems with this pulse shape are: (1) It is hard to generate and (2) a slight timing error results in infiniteseries decaying as 1
t for intersymbol interference.
Solutions (a) signal slower or (b) allowed intersymbol interference in a controlled fashion.
1. Intersymbol-Interference Free Pulse Shapes
Consider slower signaling first. Consider 1T 2W , W 1
2T . This implies aliasing at the receiver. Since
W 12T we can divide the interval W W into segments of length 1
T . Let N 2WT 1 2 be the
number of such segments. When the signal is sampled these segments get moved to the origin and cause aliasing.
x kT W WX f e j2π f kT d f
N
∑n N
2n 1 2T
f 2n 1 2TX f e j2π f kT d f
N
∑n N
1 2T
f 1 2TX f n
T e j2π f n
T kT d f
1 2T
f 1 2T N
∑n N
X f nT e j2π f kT d f
10-12 CHAPTER 10. INTERSYMBOL INTERFERENCE
4 6 8 10 12 14 16−1
−0.5
0
0.5
1
time
x1(t
)
Pulse Shape
0 0.5 1 1.5 2 2.5
x 104
0
1
2
3
4
5
6x 10
−5
frequency
X1(
f)
Spectrum
Figure 10.1: Nyquist Pulse Shape and Spectrum
0 5 10 15 20 25 30 35 40−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
time
x1(t
)
Data Waveform
Figure 10.2: Nyquist Waveform
3. 10-13
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−3
−2
−1
0
1
2
3
Figure 10.3: Nyquist Eye Diagram
Let
Xeq f N
∑n N
X f nT
Then
x kT 1 2T
f 1 T Xeq f e j2πukT d f
Clearly we can have zero intersymbol interference if
t for the Nyquist pulse. The parameter α is called the rolloff
factor.
H f T 0
f 1 α2T
T2 1 sin πt f 1
2T α 1 α2T
f 1 α2T
h t sin π 1 α t 4αt cos π 1 α t π 1 4αt 2 t
Because we do not have any intersymbol interference the performance of this method for avoiding intersymbolinterference has the same performance as BPSK. The difference is that this modulation scheme requires absolutebandwidth of W 1 α
2T . However, since this does not result in a constant envelope signal for applications requiringconstant envelope transmission this modulation scheme is not acceptable.
The second method is to allow some intersymbol interference. This intersymbol interference is allowed in acontrolled fashion. We still signal at rate 1
T 2W but do not require zero intersymbol interference.
This is called Duobinary Transmission (also called partial response class I).
X f e jπ f 2W
W cos π f2W f W
0 f W
Example (2):
x nT 1 n 1 10 n 0 1
This is called Modified Duobinary Transmission (also called Partial Response Class IV).
X f j
W sin π fW f W
0 f W
This is used in magnetic recording (with maximum likelihood decoding).For these systems with controlled intersymbol interference we still need a way to detect the data. One method
is decision feedback. The other method is precoding.