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Communications DSP
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Digital Transmission Through Bandlimted Channels
CONTENTS
Characterization of Bandlimited Channels
Characterization of Intersymbol Interference
Signal design for bandlimited Channels
Linear Equalizers
Adaptive Linear Equalizers
Non-Linear Equalizers
Textbook : J. Proakis and M. Salehi: Contemporary communication systems
using MATLAB, 1st
Edit ion. Brooks/Cole, Thomson Learning.
2000. (2nd
Edition available on 2004).
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1. Intersymbol Interference
Many communication channels, including telephone channels and some
radio channels, may be characterized as bandlimited linear fi lters with
frequency response
)()()( f je f A f C θ =
(2-1)
where )( f A and )( f θ are the amplitude and phase responses, respectively.
A channel is non-distort ing or ideal within the bandwidth W if,
c A f A =)( , and f f c ⋅= τ θ )( , W f ∈ , (2-2)
where c A and cτ are constants (i.e. constant amplitude and linear-phase
responses).
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If )( f A is not constant, the distortion is called amplitude distort ion, and
if cτ is not constant, there will be phase distortion.
A measure of the phase linearity or phase distortion of the system is the
envelope delay or group delay
df
f d f
)(
2
1)(
θ
π τ ⋅−= .
(2-3)
Due to the amplitude and phase distortion caused by non-ideal channel, a
succession of pulses transmitted through the channel at a rate
comparable to the bandwidth W are smeared.
Individual pulses might not be distinguishable at the receiver and we
have intersymbol interference (ISI).
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Note the received pulsefor a non-ideal channel
does not have zero
crossings at T ± , T 2± , and
so on.
It is possible to
compensate for the non-
ideal frequency response
characteristic of the
channel by use of a filter
or equalizer at thereceiver.
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2. CHARACTERIZATION OF BANDLIMITED CHANNELS
Telephone Channels
Usable band of the channel :
300Hz to 3200Hz.
Impulse response duration of
an average channel is ~ 10 ms
= L.
If the transmitted symbol rates
is Rs=2500 pulses or symbols
per second, the intersymbol
interference might extend over
20 to 30 symbols
( 310102500 −××=× L Rs ).
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Time-dispersive wireless channels:
e.g. short-wave ionospheric propagation
(HF), tropospheric scatter, and mobile
cellular radio.
Time dispersion, and hence ISI, is
the result of multiple propagation
paths with different path delays.
The number of paths and the
relative time delays can vary with
time. For this reason, they are
called time-variant multipath
channels. The channels can be
characterized by the scattering
function: a 2D representation of
the average received signal power
as a function of relative time delay
and Doppler frequency spread.
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The total time duration (multipath spread) of the channel response is
approximately sμ 7.0 on the average. If transmission occurs at a rate of 710
symbols/sec over the channel, the multipath spread of s7.0 will result in
intersymbol interference that spans about 7 symbols ( 76 10107.0 ⋅× − ).
Exercise: GO THROUGH ILLUSTRATIVE PROBLEM 6.5 (ON MULTIPATH
CHANNEL SIMULATION) AND THE M-FILE.
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2.3 Eye Diagram
The amount of ISI and noise can
be viewed in an oscilloscope:
Display the received signal )(t y on
the vertical input with the
horizontal sweep rate set to 1/T
(the symbol rate).
ISI causes the eye to close:
1) reducing the margin for addit ive
noise (higher detection errors).
2) distorting the position of the
zero-crossings and causes the
system more sensitive to
synchronization errors.
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3 Signal Design for Bandlimited Channels
Necessary and sufficient condition
for a signal )(t x to have zero ISI is
⎩⎨⎧
≠
==
01
00)(
n
nnT x T
T
m f X
m
=+⇔ ∑∞
−∞=
)(
(3-1)
where 1/T is the symbol rate.
One commonly used signals has a
raised-cosine-frequency response
characteristics.
The sampled waveform )(t xδ can be written as
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⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
+>
+≤<
−⎥⎦
⎤⎢⎣
⎡⎟
⎠
⎞⎜⎝
⎛ −−+
−≤≤
=
T f
T f
T T f
T T
f T
f X rc
2
1,0
2
1
2
1,
2
1cos1
2
2
10,
)(
α
α α α
α
π
α
(3-2)
where 10 ≤≤α is called the roll-off factor , range and 1/T is the symbol
rate.
When 0=α , )( f X rc reduces to an ideal “ brick wall” physical
nonrealizable response with bandwidth occupancy 1/(2T), called the
Nyquist frequency.
For 0>α , the bandwidth occupied by )( f X rc beyond the Nyquist
frequency is called the excess bandwidth, usually expressed as a
percentage of the Nyquist frequency. For 2/1=α , the excess
bandwidth is 50%, and when 1=
α , the excess bandwidth is 100%.
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The signal pulse )(t xrc having the raised-cosine spectrum is
222 /41
)/cos(
)/(
)/sin()(
T t
T t
T t
T t t xrc
α
πα
π
π
−= (3-3)
In absence of channel distortion, there is no ISI from adjacent symbols.
In presence of channel distortion, a channel equalizer is needed to
minimize its effect on system performance.
Exercise: GO THROUGH ILLUSTRATIVE PROBLEM 6.7 (DESIGN OF
TRANSMIT AND RECEIVE FILTER).
This method is called frequency sampling. Suppose the fi lter length is 2N+1.
The desired analog frequency response is )(Ωd H . The desired frequency
response of the discrete-time fil ter is then )()(~
)(~
Ω== Ωd
T j
d
j
d H e H e H sω . [For
designing the pulse shaping fi lter, to avoid aliasing, we chooseT T s
41= .]
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Sample )(~ ω j
d e H at 2N+1 equally spaced points over the unit circle:
},...,0,...,:)12/(2{ N N k N k k −=+= π ω , we get a set of points ][)( k H e H k jd =
ω . If the
DT-FT of the filter pass through these points then,
][][)( k H enhe H N
N n
jn j k k == ∑−=
− ω ω
Taking the inverse, one gets ∑∑−=
+
−=
== N
N k
N kn j N
N k
jk ek H ek H nh n )12/(2][][][ π ω .
Other methods for designing the transmit and receive fil ters (Nyquist f ilters)
include semidefinite programming (SDP) and eigenfi lter methods.
Normally, )( f X RC will be designed first. It is then factored by a method
called spectral factorization to obtain )( f GT and )( f G R .
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4 Linear Equalizers
The most common type of channel equalizer used in practice to reduce ISI is
a linear FIR fi lter with adjustable coefficients }{ k c .
The ISI is usually negligible
beyond a certain number
of symbols.
The number of terms in the
ISI term is thus fini te.
The linear fi lter is therefore
usually implemented as
finite-duration impulse
response (FIR) fil ter , with
adjustable tap coefficients.
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4.1 Symbol- and Fractionally-spaced Linear Equalizers
The time delay τ between adjacent tap may be selected as large as T, and
the equalizer is called a symbol-spaced equalizer . The input to the
equalizer is )(kT y . However, the excessive bandwidth of the signal causes
those components above the Nyquist frequency 1/(2T) to aliase with those
below. The equalizer only compensates for the aliased channel-distorted
signal.
If τ is shorten to T/2, i.e. the received signal is sampled at 1/(2T) Hz, then
even for 100% excess bandwidth, there will not be any aliasing in the
received signal. The channel equalizer is said to have fractionally spacedtaps, and it is called a fractionally spaced equalizer . However, higher
operating speed is needed.
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For zero ISI, the product of the various transfer functions should equal to
a Nyquist channel, say )( f X rc , the raised cosine spectrum
)()()()()( f X f G f G f C f G rc E RT = . (4-1)
The receive pulse shaping filter )( f G R are usually matched to transmit
pulse shaping f ilter )( f GT with
)()()( f X f G f G rc RT = , (4-2)
i.e. it forms a Nyquist channel with no ISI in absence of channel distortion.
The frequency response of the equalizer should be
)( f G R )(
)(
1
)(
1)(
f j
E ce f C f C f G θ −
== , (4-3)
In this case, the equalizer is said to be the inverse channel filter to the
channel response. It might not be stable and realizable.
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4.2 Zero-forcing equalizers (ZF equalizers)
The impulse response of the FIR equalizer is
∑−=
−=K
K n
n E nt ct g )()( τ δ , (4-4)
and the corresponding frequency response is
∑−=
−=K
K n
fn j
n E ec f G τ π 2)( , (4-5)
where }{ nc are the 2K+1 equalizer coefficients, and K is chosen sufficiently
large so that the equalizer spans the length of the ISI.
Zero-forcing assumes the received pulse shape (or channel) is known and finds an equalizer to minimize the ISI at t ime instants, nT, n=1,…..
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Let )()()()( f X f G f C f G RT = and )(t x the time waveform corresponding to
)( f X . The equalized output signal pulse q(t) is
∑−=
−==K
K n
n E nt xct qt gt x )()()(*)( τ , (4-6)
The samples of )(t q taken at times mT t = , are given by
∑−=
−=K
K n
n nmT xcmT q )()( τ , K m ±±= ,....,1,0 . (4-7)
For zero ISI, )(mT q should ideally be zero except at m=0.
Since there are 2K+1 equalizer coefficients, we can control only 2K+1
sampled values of )(t q . We therefore impose the zero-forcing
condit ions to determine the ZF-equalizer coefficient }{ nc
⎩⎨⎧
±±±=
==
K m
mmT q
,...,2,1,0
0,1)( . (4-8)
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(4.7) is a system of linear equation in unknown }{ nc which can be written
as
q Xc = , (4-9)
where X is an )12()12( +×+ K K matrix with elements )( τ nmT x − , c is the
)12( +K coefficient vector, and q is the )12( +K column vector with one
nonzero element.
The equalizer coefficient is given by
q X c1−= , (4-10)
FIR zero-forcing equalizer does not completely eliminate the ISI
because it has a finite length. As K is increased, the residual ISI can be
reduced, and as ∞→K , ISI is completely eliminated under noise free
condition.
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A drawback of zero-forcing equalizer is that it ignores the presence of
additive noise. It might lead to significant noise enhancement. In a
frequency range where )( f C is small, the channel equalizer
)(/1)( f C f G E = compensates by placing large gain in that frequency
range. The noise in that frequency range is greatly enhanced.
Example: Suppose that the received output is equal to
2)/2(1
1
)( T t t x +=
,(4-11)
where T /1 is the symbol rate. The pulse is sampled at the rate T /2 and is
equalized by a zero-forcing equalizer. Determine the coefficients of a
five-tap zero-forcing equalizer.
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Solution
Since the pulse is sampled at the rate T /2 , it is a fractionally-spaced equalizer
with 2/T =τ . Also, as the equalizer is of 5 taps, 22/)15( =−=K .
The zero-forcing condition in (4.8) becomes
∑−=
−=K
K n
n
nT mT xcmT q )
2()( , 2,....,1,0 ±±=m . (4-12)
From (4.7), the matrix X and vector q are given by
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
=
5/110/117/126/137/1
12/15/110/117/1
5/12/112/15/1
17/110/15/12/11
37/126/117/110/15/1
X , and
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
=
0
0
1
0
0
q . (4-13)
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Solving for l inear equation q Xc = yields
⎥⎥
⎥⎥⎥⎥
⎦
⎤
⎢⎢
⎢⎢⎢⎢
⎣
⎡
−
−
−
==
⎥⎥
⎥⎥⎥⎥
⎦
⎤
⎢⎢
⎢⎢⎢⎢
⎣
⎡
= −
−
−
2.2
9.4
3
9.4
2.2
1
2
1
0
1
2
q X c
c
c
c
c
c
opt . (4-14)
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4.3 Minimum Mean-Square Error (MMSE) Equalizers
Let )(t z be the noise-corrupted output of the FIR equalizer
∑−=
−=K
K n
n nt yct z )()( τ , (4-14)
where )(t y is the input to the equalizer given by (2.3).
The equalizer is sampled at t imes mT t = , and we have
∑−=
−=K
K n
n nmT ycmT z )()( τ . (4-15)
The desired response at the output of the equalizer at mT t = is the
transmitted symbol ma (assume to be known during training of the
equalizer).
The error is defined as the difference between ma and )(mT z .
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The mean-square error (MSE) between the actual output sample )(mT z and
the desired values ma is
])([2
mamT z E MSE −=⎥⎥⎦
⎤
⎢⎢⎣
⎡−−= ∑
−=
2
)( m
K
K n
n anmT yc E τ
[ ]∑ ∑−= −=
−−=K
K n
K
K k
k n k mT ynmT y E cc )()(* τ τ
[ ]∑−=
−⋅−K
K k
mk k mT ya E c )(2 * τ )|(| 2ma E + .
∑ ∑−= −=
−=K
K n
K
K n
yk n k n Rcc )( ∑−=
−K
K k
ayk k Rc )(2 )|(| 2ma E + .
(4-16)
where )]()([)( * τ τ k mT ynmT y E k n R y −−=− , )]([)( * τ k mT ya E k R may −⋅= ,
The expectation [ ]⋅ E is taken with respect to the random information
sequence }{ ma and the additive noise.
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The minimum MSE solution is obtained by differentiating (4.15) with
respect to the equalizer coefficients }{ nc .
The condition for minimum MSE is
∑−=
=−K
K nay yn
k Rk n Rc )()( , K k ±±±= ,...,2,1,0 , (4-17)
which is a system of linear equation with )12( +K equations in )12( +K
unknown }{ nc .
In practice, the autocorrelation sequence )(n R y and the cross-correlation
sequence )(n Ray are unknown a priori. They have to be estimated using
the time-average estimates
∑=
−=K
k
y kT ynkT yK
n R1
* )()(1
)(ˆ τ , ∑=
−=K
k
k ay ankT yK
n R1
** )(1
)(ˆ τ (4-18)
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The symbols k a are assumed known, which are transmitted to the
receivers during the so-called training mode.
In contrast to the zero-forcing solution, these equations depend on the
statistical properties (the autocorrelation) of the noise as well as the ISI
through the autocorrelation )(n R y .
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5 Adaptive Linear Equalizers
On channels whose frequency response characteristics are unknown
but time-invariant (does not change with time, such as telephone lines),
we may measure the channel characteristics by sending knownsymbols (training symbols) to the receiver and adjust the parameters of
the equalizer. Once adjusted, the parameters remain fixed during the
transmission of data. Such equalizers are called preset equalizers.
Adaptive equalizers update their parameters by sending periodically
training symbols to the receivers during the transmission of data, so
they are capable of tracking a slowly time-varying channel response.
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Both the zero-forcing and MMSE equalizers require the solution of a systemof linear equation of the form
d Bc = (5-1)
where B is a )12()12( +×+ K K matrix, c is a column vector representing the
)12( +K equalizer coefficients, and d is an )12( +K column vector.
The solution is
d B c1−=opt (5-2)
Solving (5.2) directly will require very high arithmetic complexity:
))12(( 3+K O . In practical implementation, it is solved using iterative
methods such as the Least mean squares (LMS) or recursive least squares
(RLS) algori thms.
C i i DSP
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5.1 LMS algorithm
The LMS algori thm is based on the method of steepest descent:
1. One begins with an arbitrarily chosen coefficient vector say 0 c .
2. Each tap coefficient is changed in the direction opposite to its
corresponding gradient component in the gradient vector g , which is the
derivative of the MSE with respect to the )12( +K filter coefficients:
k k k g c c ⋅Δ−=
+1 (5-3)
where Δ is the step-size parameter.
C i ti DSP
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In the LMS algorithm, g is estimated continuously. A commonly used
method is to approximate the MSE by the instantaneous error2*2 )()( k
T
k k k ae y c−= . Thus, the estimated gradient is
*2 22)(ˆk k k k k k yeeee
k k −=∇⋅=∇= c c g (5-4)
The LMS or stochastic gradient algorithm is given by
*1 k k k k e y c c ⋅Δ+=+ ,
*k
T
k k k ae y c−= .(5-5)
Step-size selection:
One commonly used step-size parameter Δ in order to ensure
convergence and good tracking capabilities in slowly varying channels is
RPK )12(5
1
+=Δ (5-6)
RP denotes the received signal-plus-
noise power, which can be estimated
from the received signal.
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The optimal solution can be approached after a few hundred iterations.
As the equalizer is updated at the symbol rate, it corresponds to afraction of a second.
Initially, the adaptive equalizer is trained by the transmission of a
known pseudorandom sequence }{ ma over the channel. At the
demodulator, the equalizer employs the known sequence to adjust i ts
coefficients.
Upon initial adjustment, the adaptive equalizer switches from a
training mode to a decision-directed mode, in which case the
decisions at the output of the detector are sufficiently reliable so that
the error signal is formed by computing the difference between thedetector output and the equalizer output
k k k zae −= ˆ (5.7) k a is the output of the detector.
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In general, decision errors at the output of the detector occur infrequently.
Such errors have litt le effect on the performance of the tracking algorithm.
EXERCISE: GO THROUGH ILLUSTRATIVE PROBLEM 6.12 AND RUN THE M-
FILE.
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EXAMPLE
The LMS algori thm is used to identify the following channel
]088.0,038.0,126.0,0,25.0,9047.0,25.0,126.0,088.0,063.0,05.0[ −−−= x
with 11)12( =+K .
Smaller stepsize
leads to faster
convergence but
higher errors.
RLS has faster
convergence but a
high complexity.
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6 Nonlinear Equalizers
The linear filter equalizers are very effective on channels, such as wire line
telephone channels, where ISI is not severe.
The severity of the ISI is directly related to the spectral characteristics of the
channel and not necessarily to the time span of the ISI.
There is a spectral nul l in channel B at f=1/2T (more severe ISI). Channel A
does not have a channel null and has a large span of ISI.
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The energy of thetotal response is
normalized to unity
for both channels.
The time span of the ISI in channel A is 5 symbol intervals on each side of
the desired signal component, which has a value of 0.72.
The time span for the ISI in channel B is one symbol interval on each side
of the desired signal component, which has a value of 0.815.
In spite of the shorter ISI, channel B results in more severe ISI. A linear
equalizer will introduce a large gain in its frequency response to
compensate for the channel null in channel B at f=1/2T, noise is enhanced.
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6.1 Decision Feedback Equalizers (DFE)
A DFE is a nonlinear equalizer that employs previous decisions to
eliminate the ISI caused by previously detected symbols on the current
symbol to be detected.
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It consists of two filters. The first fil ter is a feedfoward filter , which is
generally a fractionally-spaced FIR filter with adjustable tap coefficients.The second one is called a feedback filter , which is an FIR filter with
symbol-spaced taps having adjustable coefficients. Its input is the set
of previously detected symbols.
The output of the feedback filter is subtracted from the output of the
feedforward fi lter to form the input to the detector.
∑∑=
−=
−−=21
11
~)( N
n
nmn
N
n
nm abnmT yc z τ (6.1)
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where }{ nc and }{ nb are the adjustable coefficients of the feedforward
and feedback filters, respectively; nma −~ , 2,..,2,1 N n = , are the previously
detected symbols; 1 N and 2 N are the length of the feedforward fil ter
and feedback f ilters, respectively.
The tap coefficients are usually selected to minimize the MSE criterion
using the stochastic gradient (LMS) algorithm or RLS algorithm.
Decision errors from the detector that are fed to the feedback filterhave a small effect on the performance of the DFE. A small loss in
performance of 1 to 2dB is possible at error rates below 210 , but the
decision errors in the feedback fil ters are not catastrophic.
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In a digital communication system that transmit information over a
channel that causes ISI, the optimum detector is a maximum-likelihoodsymbol detector (MLSD) that produces at its output the most probable symbol
sequence }~{ k a for the given received sampled sequence }{ k y .
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That is, the detector finds the sequence }~{ k a that maximizes the
likelihood function.
}){|}({ln})({ k k k a y pa =Λ (6.2)
where }){|}({ k k a y p is the joint probabili ty of the received sequence}{ k y conditioned on }{ k a . The Viterbi algorithm can be used to
implement the MLSD, but its complexity grows exponentially with the
span of the ISI. They are suitable for short channel with severe ISI,
such as mobile channels.