Transcript Modulation, Demodulation and Coding Course

Digital communications I:
Modulation and Coding Course
Spring - 2013
Jeffrey N. Denenberg
Lecture 3d: ISI and Equalization
Last time we talked about:

Signal detection in AWGN channels



Minimum distance detector
Maximum likelihood
Average probability of symbol error


Union bound on error probability
Upper bound on error probability based
on the minimum distance
Lecture 6
2
Today we are going to talk about:

Another source of error:

Inter-symbol interference (ISI)
Nyquist theorem
 The techniques to reduce ISI



Pulse shaping
Equalization
Lecture 6
3
Inter-Symbol Interference (ISI)
ISI in the detection process due to the
filtering effects of the system
 Overall equivalent system transfer function

H
(
f)

H
f)
H
(
f)
H
(
f)
t(
c
r


creates echoes and hence time dispersion
causes ISI at sampling time
z
s
n


k
k
k
is
i
i
k
Lecture 6
4
Inter-symbol interference

xk 
Baseband system model
x1 x2
T

xk 
ht (t )
Channel
hc (t )
Ht ( f )
Hc ( f )
Tx filter
x3
hr (t )
Hr ( f )
zk
t  kT
Detector
xˆk 
n(t )
Equivalent model
x1 x2
T
T
r (t ) Rx. filter
Equivalent system
x3
zk
z (t )
h(t )
H( f )
t  kT
T
H
(
f)

H
f)
H
(
f)
H
(
f)
t(
c
r
Lecture 6
nˆ (t )
filtered noise
5
Detector
xˆk 
Nyquist bandwidth constraint

Nyquist bandwidth constraint:
The theoretical minimum required system bandwidth to
detect Rs [symbols/s] without ISI is Rs/2 [Hz].
 Equivalently, a system with bandwidth W=1/2T=Rs/2
[Hz] can support a maximum transmission rate of
2W=1/T=Rs [symbols/s] without ISI.

R
1R
s
s


W


2
[symbo
Hz]
2
T
2
W

Bandwidth efficiency, R/W [bits/s/Hz] :
An important measure in DCs representing data
throughput per hertz of bandwidth.
 Showing how efficiently the bandwidth resources are
used by signaling techniques.
Lecture 6
6

Ideal Nyquist pulse (filter)
Ideal Nyquist filter
Ideal Nyquist pulse
H( f )
h
(t)sinc(
t/T)
T
1
2T
1
0
W
1
2T
1
2T
f
Lecture 6
 2T  T
0
7
T 2T
t
Nyquist pulses (filters)

Nyquist pulses (filters):


Nyquist filter:


Its transfer function in frequency domain is
obtained by convolving a rectangular function with
any real even-symmetric frequency function
Nyquist pulse:


Pulses (filters) which results in no ISI at the
sampling time.
Its shape can be represented by a sinc(t/T)
function multiply by another time function.
Example of Nyquist filters: Raised-Cosine filter
Lecture 6
8
Pulse shaping to reduce ISI

Goals and trade-off in pulse-shaping



Reduce ISI
Efficient bandwidth utilization
Robustness to timing error (small side
lobes)
Lecture 6
9
The raised cosine filter

Raised-Cosine Filter

A Nyquist pulse (No ISI at the sampling time)
1
for
|f|

2
W

W

0

|f|

W

2
W
2
0
H
(
f
)

cos
2
W

W

|f|

W

0

for
4W

W
0 


0
for
|f|

W



cos[
2
(
W

W
)
t
]
0
h
(
t
)

2
W
(sinc(
2
W
t
))
0
0
2
1

[
4
(
W

W
)
t
]
0
Excess bandwidth: W  W
0
Lecture 6
WW0
r

Roll-off factor
W0
0  r 1
10
The Raised cosine filter – cont’d
h(t)hRC(t)
|H
(f)|
|H
(f)|
RC
r 0
1
1
r  0.5
0.5
1  3 1
T 4T 2T
0
r 1
1 3
2T 4T
1
T
r 1
0.5
 3T  2T  T
0
r  0.5
r 0
T
2T
3T
R
s
Baseband
W

(
1

r
)
Passband
W

(
1

r
)
R
sSB
DSB
s
2
Lecture 6
11
Pulse shaping and equalization to
remove ISI
No ISI at the sampling time
H
(
f
)

H
(
f
)
H
(
f
)
H
(
f
)
H
(
f
)
RC
t
c
r
e

Square-Root Raised Cosine (SRRC) filter and Equalizer
H
(
f
)

H
(
f
)
H
(
f
)
RC
t
r
Taking care of ISI
H
(
f
)

H
(
f
)

H
(
f
)

H
(
f
)caused by tr. filter
r
t
RC SRRC
1
H
e(f)
H
c(f)
Lecture 6
Taking care of ISI
caused by channel
12
Example of pulse shaping
Square-root Raised-Cosine (SRRC) pulse shaping
Amp. [V]

Baseband tr. Waveform
Third pulse
t/T
First pulse
Second pulse
Data symbol
Lecture 6
13
Example of pulse shaping …

Raised Cosine pulse at the output of matched filter
Amp. [V]
Baseband received waveform at
the matched filter output
(zero ISI)
t/T
Lecture 6
14
Eye pattern

Eye pattern:Display on an oscilloscope which
sweeps the system response to a baseband signal at
the rate 1/T (T symbol duration)
Distortion
due to ISI
amplitude scale
Noise margin
Sensitivity to
timing error
Timing jitter
Lecture 6
time scale
15
Example of eye pattern:
Binary-PAM, SRRQ pulse

Perfect channel (no noise and no ISI)
Lecture 6
16
Example of eye pattern:
Binary-PAM, SRRQ pulse …

AWGN (Eb/N0=20 dB) and no ISI
Lecture 6
17
Example of eye pattern:
Binary-PAM, SRRQ pulse …

AWGN (Eb/N0=10 dB) and no ISI
Lecture 6
18
Equalization – cont’d
Step 1 – waveform to sample transformation
Step 2 – decision making
Demodulate & Sample
Detect
z (T )
r (t )
Frequency
down-conversion
For bandpass signals
Received waveform
Receiving
filter
Equalizing
filter
Threshold
comparison
Compensation for
channel induced ISI
Baseband pulse
(possibly distored)
Lecture 6
Baseband pulse
19
Sample
(test statistic)
mˆ i
Equalization

ISI due to filtering effect of the
communications channel (e.g. wireless
channels)

Channels behave like band-limited filters
j

(f)
c
H
(f)
H
(f)e
c
c
Non-constant amplitude
Non-linear phase
Amplitude distortion
Phase distortion
Lecture 6
20
Equalization: Channel examples

Example of a frequency selective, slowly changing (slow fading)
channel for a user at 35 km/h
Lecture 6
21
Equalization: Channel examples …

Example of a frequency selective, fast changing (fast fading)
channel for a user at 35 km/h
Lecture 6
22
Example of eye pattern with ISI:
Binary-PAM, SRRQ pulse

Non-ideal channel and no noise
h
(
t
)

(
t
)

0
.
7

(
t
T
)
c
Lecture 6
23
Example of eye pattern with ISI:
Binary-PAM, SRRQ pulse …

AWGN (Eb/N0=20 dB) and ISI
h
(
t
)

(
t
)

0
.
7

(
t
T
)
c
Lecture 6
24
Example of eye pattern with ISI:
Binary-PAM, SRRQ pulse …

AWGN (Eb/N0=10 dB) and ISI
h
(
t
)

(
t
)

0
.
7

(
t
T
)
c
Lecture 6
25
Equalizing filters …
Baseband system model

a1
a(tkT
) Tx filter

ht (t )
Channel
hc (t )
Ht ( f )
Hc ( f )
k
k
Ta a
2
3

r (t ) Equalizer
he (t )
He ( f )
aˆk 
Rx. filter z (t ) z k
hr (t )
Detector
t  kT
Hr ( f )
n(t )
Equivalent model
H
(
f)

H
f)
H
(
f)
H
(
f)
t(
c
r
a1
a(tkT
) Equivalent system

k
k
Ta a
2
3
h(t )
H( f )
z (t )
x(t )
Equalizer z (t )
he (t )
He ( f )
nˆ (t )
filtered noise
ˆ(t)n
n
(t)h
t)
r(
Lecture 6
26
aˆk 
zk
t  kT
Detector
Equalization – cont’d

Equalization using


MLSE (Maximum likelihood sequence
estimation)
Filtering – See notes on
z-Transform and Digital Filters

Transversal filtering



Decision feedback


Zero-forcing equalizer
Minimum mean square error (MSE) equalizer
Using the past decisions to remove the ISI contributed
by them
Adaptive equalizer
Lecture 6
27
Equalization by transversal filtering

Transversal filter:

A weighted tap delayed line that reduces the effect
of ISI by
proper adjustment of the filter taps.
N
z
(
t
)

c
x
(
t

n
)
n


N
,...,
N
k


2
N
,...,
2
N

n
n


N

x(t )
c N



c N 1

c N 1
cN

Coeff.
adjustment
Lecture 6
28
z (t )
Transversal equalizing filter …

Zero-forcing equalizer:

The filter taps are adjusted such that the equalizer output
is forced to be zero at N sample points on each side:
Adjust
cn nN N

1 k

0

z
(
k
)

0k


1
,...,

N

Mean Square Error (MSE) equalizer:

The filter taps are adjusted such that the MSE of ISI and
noise power at the equalizer output is minimized.


2
min
E
(z
(
kT
)
a
)
k
Adjust
c 
N
n n N
Lecture 6
29
Example of equalizer
2-PAM with SRRQ
 Non-ideal channel
h
(
t
)

(
t
)

0
.
3

(
t
T
)
c


Matched filter outputs at the sampling time
One-tap DFE
ISI-no noise,
No equalizer
ISI-no noise,
DFE equalizer
ISI- noise
No equalizer
ISI- noise
DFE equalizer
Lecture 6
30