ECE 6332, Spring, 2014
Wireless Communication
Zhu Han
Department of Electrical and Computer Engineering
Class 15
Mar. 17th, 2014
Outline

Match Filter

Equalizer
– Chapter 11 but just overview this class
– Next class, more in details

Timing
Receiver Structure

Matched filter: match source impulse and maximize SNR
– grx to maximize the SNR at the sampling time/output

Equalizer: remove ISI

Timing
– When to sample. Eye diagram

Decision
– d(i) is 0 or 1
Noise na(t)
d(i)
gTx(t)
i T
gRx(t)
r (iT )  r0 (iT )  n(iT )
?
S
 max
N
Matched Filter

Input signal s(t)+n(t)

Maximize the sampled SNR=s(T0)/n(T0) at time T0
Matched filter example

Received SNR is maximized at time T0
S
Matched Filter: optimal receive filter for maximized
N
example:
gTx (T0  t )  g Rx (t )
gTx (t )
gTx (t )
T0 t
transmit filter
T0
t
T0
receive filter
(matched)
t
Matched Filter
Matched Filter
Define thepeak pulse signal - to - noise ratioas
2
go (T )

E n 2 (t )


(4.3)
2
where go (T ) is theinst antaneous outputsignal power
We have to maximaze to obtainoptimalperformance
Let G ( f ) and H ( f ) denotethe Fourier transformsof g (t ) and h(t )

go (t )   H ( f )G ( f ) exp( j 2 ft)df
(4.4)

g0 (T ) 
2



H ( f )G ( f ) exp( j 2 fT )df
2
Since w(t) is white, the P SD SN ( f ) of n(t )is
N
2
SN ( f )  0 H ( f )
2


(4.5)
(4.6)

 E n (t )   S N ( f )df
2


N0
2



2
H ( f ) df
(4.7)
Matched Filter
 



H ( f )G ( f ) exp( j 2 fT )df
2
(4.8)

N0
2
H ( f ) df



2
Given G ( f ), find H ( f ) t hatmakes a maximum.
Recall t heSchwarz' s inequalit y


-
2


1 ( x )2 ( x )dx   1 ( x ) dx 2 ( x ) dx
2

2

(4.9)
where 1 ( x ) and 2 ( x ) are complexfunct ionsin real variable x.




-
-
1 ( x ) dx  
2
2 ( x ) dx  
2
T heequalit y in (4.9)holds iff 1 ( x )  k2* ( x )
(4.10)
Matched Filter
Since g(t)
Properties of Matched Filters
Consider a knownsignal g (t ) ,
G0 ( f )  H opt ( f )G ( f )
 kG* ( f )G ( f ) exp( j 2 fT )
 k G ( f ) exp( j 2 fT )
2
(4.17)

g0 (T )   G0 ( f ) exp( j 2 fT )df


 k  G ( f ) df
2

siganl energy  E ( by Rayleigh' s energy theorem)
g0 (T )  kE
(4.18)
From(4.7)and (4.14)theaverageoutput noise power is



E n 2 (t )   S N ( f ) df
-
k 2 N0 
2

G
(
f
)
df
2 
 k 2 N0 E 2
(4.19)
( kE ) 2
2E
max 

( kN 0 E 2) N 0
which is independent of waveform( E
(4.20)
N0
 signal energy tonoise PSD ratio)
Equalizer

When the channel is not ideal, or when signaling is not Nyquist,
There is ISI at the receiver side.

In time domain, equalizer removes ISR.

In frequency domain, equalizer flat the overall responses.

In practice, we equalize the channel response using an equalizer
Zero-Forcing Equalizer

The overall response at the detector input must satisfy Nyquist’s
criterion for no ISI:

The noise variance at the output of the equalizer is:
– If the channel has spectral nulls, there may be significant noise
enhancement.
Transversal Transversal Zero-Forcing Equalizer

If Ts<T, we have a fractionally-spaced equalizer

For no ISI, let:
Zero-Forcing Equalizer continue

Zero-forcing equalizer,

Example: Consider a baud-rate sampled equalizer for a system
for which

Design a zero-forcing equalizer having 5 taps.
MMSE Equalizer

In the ISI channel model, we need to estimate data input
sequence xk from the output sequence yk

Minimize the mean square error.
Adaptive Equalizer

Adapt to channel changes; training sequence
Decision Feedback Equalizer

To use data decisions made on the basis of precursors to take
care of postcursors

Consists of feedforward, feedback, and decision sections
(nonlinear)

DFE outperforms the linear equalizer when the channel has
severe amplitude distortion or shape out off.
Different types of equalizers

Zero-forcing equalizers ignore the additive noise and may
significantly amplify noise for channels with spectral nulls

Minimum-mean-square error (MMSE) equalizers minimize the meansquare error between the output of the equalizer and the transmitted
symbol. They require knowledge of some auto and cross-correlation
functions, which in practice can be estimated by transmitting a known
signal over the channel

Adaptive equalizers are needed for channels that are time-varying

Blind equalizers are needed when no preamble/training sequence is
allowed, nonlinear

Decision-feedback equalizers (DFE’s) use tentative symbol decisions
to eliminate ISI, nonlinear

Ultimately, the optimum equalizer is a maximum-likelihood sequence
estimator, nonlinear
Timing Extraction

Received digital signal needs to be sampled at precise instants.
Otherwise, the SNR reduced. The reason, eye diagram

Three general methods
– Derivation from a primary or a secondary standard. GPS, atomic
closk


Tower of base station
Backbone of Internet
– Transmitting a separate synchronizing signal, (pilot clock, beacon)

Satellite
– Self-synchronization, where the timing information is extracted
from the received signal itself


Wireless
Cable, Fiber
Example

Self Clocking, RZ

Contain some clocking information. PLL
Timing/Synchronization Block Diagram

After equalizer, rectifier and clipper

Timing extractor to get the edge and then amplifier

Train the phase shifter which is usually PLL

Limiter gets the square wave of the signal

Pulse generator gets the impulse responses
Timing Jitter

Random forms of jitter: noise, interferences, and mistuning of
the clock circuits.

Pattern-dependent jitter results from clock mistuning and,
amplitude-to-phase conversion in the clock circuit, and ISI,
which alters the position of the peaks of the input signal
according to the pattern.

Pattern-dependent jitter propagates

Jitter reduction
– Anti-jitter circuits
– Jitter buffers
– Dejitterizer