Digital Communications I: Modulation and Coding Course

Digital communications I:
Modulation and Coding Course
Period 3 - 2007
Catharina Logothetis
Lecture 6
Lecture 6 2
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 3
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 4
Inter-Symbol Interference (ISI)

ISI in the detection process due to the
filtering effects of the system

Overall equivalent system transfer function

creates echoes and hence time dispersion

causes ISI at sampling time
)()()()( fHfHfHfH
rct
=
i
ki
ikkk
snsz
∑
≠
++=
α
Lecture 6 5
Inter-symbol interference

Baseband system model


Equivalent model
Tx filter
Channel
)(tn
)(tr
Rx. filter
Detector
k
z
kTt =
{ }
k
x
ˆ
{}
k
x
1
x
2
x
3
x
T
T
)(
)(
fH
th
t

t
)(
)(
fH
th
r
r
)(
)(
fH
th
c
c
Equivalent system
)(
ˆ
tn
)(
tz
Detector
k
z
kTt =
{ }
k
x
ˆ
{}
k
x

1
x
2
x
3
x
T
T
)(
)(
fH
th
filtered noise
)()()()( fHfHfHfH
rct
=
Lecture 6 6
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.

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.
Hz][symbol/s/ 2
22
1
≥⇒≤=
W
R
W
R
T
ss
Lecture 6 7
Ideal Nyquist pulse (filter)
T2
1
T2
1−
T
)( fH
f
t
)/sinc()( Ttth =
1
0
T
T2

T−
T2−
0
T
W
2
1
=
Ideal Nyquist filter Ideal Nyquist pulse
Lecture 6 8
Nyquist pulses (filters)

Nyquist pulses (filters):

Pulses (filters) which results in no ISI at the
sampling time
.

Nyquist filter:

Its transfer function in frequency domain is
obtained by convolving a rectangular function with
any real even-symmetric frequency function

Nyquist pulse:

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 9
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 10
The raised cosine filter

Raised-Cosine Filter

A Nyquist pulse (No ISI at the sampling time)
⎪
⎪
⎩
⎪
⎪
⎨
⎧
>
<<−
⎥
⎦
⎤
⎢

⎣
⎡
−
−+
−<
=
Wf
WfWW
WW
WWf
WWf
fH
||for 0
||2for
2||
4
cos
2||for 1
)(
0
0
0
2
0
π
Excess bandwidth:
0
WW −
Roll-off factor
0

0
W
WW
r
−
=
10 ≤≤ r
2
0
0
00
])(4[1
])(2cos[
))2(sinc(2)(
tWW
tWW
tWWth
−−
−
=
π
Lecture 6 11
The Raised cosine filter – cont’d
2
)1( Baseband
sSB
s
R
rW +=
|)(||)(| fHfH

RC
=
0=r
5.0
=r
1=r
1=r
5.0=r
0=r
)()( thth
RC
=
T2
1
T4
3
T
1
T4
3−
T2
1−
T
1−
1
0.5
0
1
0.5
0

T
T2
T3
T−
T2−
T3−
s
RrW )1( Passband
DSB
+=
Lecture 6 12
Pulse shaping and equalization to
remove ISI

Square-Root Raised Cosine (SRRC) filter and Equalizer
)()()()()(
RC
fHfHfHfHfH
erct
=
No ISI at the sampling time
)()()()(
)()()(
SRRCRC
RC
fHfHfHfH
fHfHfH
tr
rt
===

=
Taking care of ISI
caused by tr. filter
)(
1
)(
fH
fH
c
e
=
Taking care of ISI
caused by channel