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Material Type: Notes; Class: Wireless Communications; Subject: Electrical & Computer Engr; University: Villanova University; Term: Unknown 1989;
Typology: Study notes
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Yimin Zhang, Villanova University
ECE 8708 Wireless Communications :Equalization, Diversity, and Channel Coding
Yimin Zhang, Ph.D.
Department of Electrical & Computer Engineering
Villanova University
http://yiminzhang.com/ECE
Yimin Zhang, Villanova University
ECE 8708 Wireless Communications :Equalization, Diversity, and Channel Coding
Outlines
-^
-^
-^
-^
-^
Yimin Zhang, Villanova University
ECE 8708 Wireless Communications : Propagation – Small-Scale Fading
If^
the
channel
possesses
a^
constant-gain
and
linear
phase
response over a bandwidth that is smaller than the bandwidth oftransmitted signal, then the channel creates frequency-selectivefading.
signal spectrum channel response received signal spectrum
f f f
) (^ fS
BC
Yimin Zhang, Villanova University
ECE 8708 Wireless Communications : Propagation – Small-Scale Fading
Frequency-selective
fading
is
due
to
time
dispersion
of
the
transmitted symbols within the channel.– Waveform is distorted by inter-symbol interference (ISI)– Equalization is required (Chapter 7)
-^
Frequency-selective fading channels are much more difficult tomodel than flat fading channels.
-^
For frequency-selective fadingand
C
S^
τ σ
τ σ
More practically
or
τ σ
Yimin Zhang, Villanova University
ECE 8708 Wireless Communications :Equalization, Diversity, and Channel Coding
Equalization Techniques
Equalizer is usually implemented at baseband or at IF in a receiver.
-^
Baseband received signal^ x
( t ):
original information signal f^ (
t ):
combined complex baseband impulse response of thetransmitter, channel, and the RF/IF section of the receiver *^
: convolution (the textbook used
nb
( t ):
baseband noise at the input of the equalizer
) ( ) ( ) ( )
(^
t n t f t x t
y^
b
∗
=
⊗
ECE 8708 Wireless Communications :Equalization, Diversity, and Channel Coding
Equalization Technologies
Denote
h
( eq
t )^
as the impulse response of the equalizer.
The output of the equalizer is
-^
If the impulse of the entire system (including the equalizer) satisfiesor, equivalently,Then, the ISI is cancelled.
) ( ) ( ) ( ) ( ) ( ) ( ) ( )
(^
t h t n t h t f t x t h t y t
d^
eq
b
eq
eq^
∗ + ∗ ∗ = ∗ =
) (
) (
) (^
t
t h t f^
eq
∗
1 ) (
) (^
= f
H f F
eq
ECE 8708 Wireless Communications :Equalization, Diversity, and Channel Coding
Equalization – Time-Domain View
The time-domainimpulse response ofthe channel / equalizercombination should bean impulse.
-^
Symbol-basedprocessing cannotachieve perfectequalization.
Impulse resp.
Τ
Impulse resp.
2Τ 3Τ
time^ time time
Impulse resp.
Channel Equalizer Combined
eq
δ
Yimin Zhang, Villanova University
ECE 8708 Wireless Communications :Equalization, Diversity, and Channel Coding
Equalization Techniques
The
term
equalization
can
be
used
to
describe
any
signal
processing operation that minimizes ISI.
-^
For a time-varying channel, an adaptive equalizer is needed totrack the channel variations.
-^
Two operation modes for an adaptive equalizer: training andtracking.
-^
Three factors affect the time spanning over which an equalizerconverges: equalizer algorithm, equalizer structure and timerate of change of the multipath radio channel.
-^
wireless
systems
are
particularly
well
suited
for
equalizers.
Yimin Zhang, Villanova University
ECE 8708 Wireless Communications :Equalization, Diversity, and Channel Coding
Equalization Techniques
Classical equalization theory : using training sequence (non-blind) tominimize the cost function
-^
Recent techniques for adaptive algorithm : blind algorithms–
Constant Modulus Algorithm (CMA, used for constant envelopemodulation)– Spectral
Coherence
Restoral
Algorithm
exploits
spectral redundancy or cyclostationarity in the Tx signal)
)] ( ) ( [ ] ) (
[^
2
t e t e E t e E k
k
k^
=
Yimin Zhang, Villanova University
ECE 8708 Wireless Communications :Equalization, Diversity, and Channel Coding
Solutions for Optimum Weights
Error signalwhere
-^
Mean square error
[^
T ] N k
k
k
k
k^
y
y
y
y^
−
−
−
=^
.... 2
1
y^
[^
T ] N k
k
k
k
k^
w
w
w
w^
−
−
−
=^
.... 2
1
w
k H k k
k T k k
k H k k H k
k
k^
x
x
x
e^
y w y w w y y w
2
2
k = H k
k
k^
Note: we usedw* as theweights in theformulation.
Yimin Zhang, Villanova University
ECE 8708 Wireless Communications :Equalization, Diversity, and Channel Coding
Optimum weight vector is obtained fromas (provided that
is not singular, or
-^
exists)
Minimum mean square error (MMSE)
-^
Minimizing the MSE tends to reduce the bit error rate.
-^
Note that, the exact statistical expectation cannot be obtained inpractice.
0
2
2
=
−
= ∂ ∂ = ∇
p
Rw
ξ w
p R
w
1
ˆ^
w p
p R p^
ˆ
2
1
2
min
H
k
H
k^
x E
x E
−
=
−
=
−
ξ
Yimin Zhang, Villanova University
ECE 8708 Wireless Communications :Equalization, Diversity, and Channel Coding
Wopt
E(|
(^2) ε| )
Solutions for Optimum Weights
Yimin Zhang, Villanova University
ECE 8708 Wireless Communications :Equalization, Diversity, and Channel Coding
Equalizer Techniques
Linear transversal equalizer (LTE, made up of tapped delaylines)
Basic linear transversal equalizer structure
Yimin Zhang, Villanova University
ECE 8708 Wireless Communications :Equalization, Diversity, and Channel Coding Structure of a Linear Transversal Equalizer
n k
N Nn
k^
y C
ˆ^ d
2 1
−
∑−= = [
]
ω
π
π π^
ω^
d N
e
N
T
e(n)
T T^
o
j
o
∫−^
=
-^
2 t
2
) (F
2
E
) e( F^
ω tj
: frequency response of the channel N^ o
: noise spectral density
