Inter Symbol Interference (ISI) Nyquist Criteria … : Advanced Digital Communications Week 12, 13:...

EE5713 : Advanced Digital Communications

Week 12, 13: � Inter Symbol Interference (ISI)

� Nyquist Criteria for ISI

� Pulse Shaping and Raised-Cosine Filter

20-May-15 Muhammad Ali Jinnah University, Islamabad Advanced Digital Communications (EE5713) 1

� Eye Pattern

� Equalization (On Board)

Baseband Communication System� We have been considering the following baseband system

� The transmitted signal is created by the line coder according to


to

where an is the symbol mapping and g(t) is the pulse shapeProblems with Line Codes

� One big problem with the line codes is that they are not bandlimited

� The absolute bandwidth is infinite

� The power outside the 1st null bandwidth is not negligible. That is, the power in the sidelobes can be quite high

∑∞

−∞=

−=n

bn nTtgats )()(

� If the transmission channel is bandlimited, then high frequency components will be cut off

– Hence, the pulses will spread out

– If the pulse spread out into the adjacent symbol periods, then it is said that intersymbol interference (ISI) has occurred

Intersymbol Interference (ISI)

Intersymbol Interference (ISI)


� Intersymbol interference (ISI) occurs when a pulse spreads out in such a way that it interferes with adjacent pulses at the sample instant

� Causes

– Channel induced distortion which spreads or disperses the pulses

– Multipath effects (echo)

– Due to improper filtering (@ Tx and/or Rx), the received pulses overlap one another thus making detection difficult

Pulse spreading


another thus making detection difficult

� Example of ISI

– Assume polar NRZ line code

– Input data stream and bit superposition

Inter Symbol Interference


� The channel output is the sum of the contributions from each bit

Note:

� ISI can occur whenever a non-bandlimited line code is used over a bandlimited channel

� ISI can occur only at the sampling instants

� Overlapping pulses will not cause ISI if they have zero

ISI


� Overlapping pulses will not cause ISI if they have zero amplitude at the time the signal is sampled

ISI Baseband Communication System Model

channel,theofresponseImpulse)(

r,transmittetheofresponseImpulse)(where

==

th

th

C

T


receivertheofresponseImpulse)(

channel,theofresponseImpulse)(

==

th

th

R

C

∑ −=∞

−∞=nTn nTthats ),()(

∑∞

−∞=

==+−=n

sCTTn fTththtgwheretnnTtgatr /1),(*)()(),()()(

∑∞

−∞=

+−=n

een tnnTthaty )()()( ),(*)(*)()( ththththwhere RCTe =)(*)(*)()( ththtntn RCe =

� Note that he(t) is the equivalent impulse response of the receiving filter

� To recover the information sequence {an}, the output y(t) is sampled at t = kT, k = 0, 1, 2, …

� The sampled sequence is

∑∞

−∞=

+−=n

een kTnnTkThakTy )()()(

AWGN term

ISI Baseband Communication System Model


or equivalently

– h0 is an arbitrary constant

−∞=n

∑ ∑∞

−∞=

∞

≠−∞=−− ++=+=

n knnknknkknknk nhaahnhay

,0

,..2,1,0),(),(where ±±=== kkTnnkThh ekek

AWGN term

Effect of other symbols at the sampling instants t=kT

Desired symbol scaled by gain parameters h0

Signal Design for Bandlimited Channel

Zero ISI

� To remove ISI, it is necessary and sufficient to make the term

∑∞

≠−∞=

+−+=knn

eenk kTnnTkThaahkTy,

0 )()()(

0,0)(0≠≠=− handknfornTkTh

e


� Nyquist Criterion

– Pulse amplitudes can be detected correctly despite pulse spreading or overlapping, if there is no ISI at the decision-making instants

Nyquist Criteria� Nyquist’s three criteria


• 1: At sampling points, no ISI

• 2: At threshold, no ISI

• 3: Areas within symbol period is zero, then no ISI

Muhammad Ali Jinnah University, Islamabad Advanced Digital Communications (EE5713)

1st Nyquist Criterion: Time domain

p(t): impulse response of a transmission system (infinite length)

1p(t)

� shaping function

Equally spaced zeros,

interval Tfn

=2

1

TfN

=2

1

02t0t

t0

-1

no ISI !


1st Nyquist Criterion: Time domain

Suppose 1/T is the sample rateThe necessary and sufficient condition for p(t) to satisfy

( ) ( )( )

≠=

=0,0

0,1

n

nnTp

Is that its Fourier transform P(f) satisfy

( ) TTmfPm

=+∑∞

−∞=


1st Nyquist Criterion: Frequency domain

( ) TTmfPm

=+∑∞

−∞=

2a Nf f= 4 Nff

0(limited bandwidth)


Proof

( ) ( ) ( )( )

( )∑ ∫

∞

−∞=

+

−=

m

Tm

TmdffnTjfPnTp

212

2122exp π

( ) ( ) ( )∫∞

∞−= dfftjfPtp π2exp

( ) ( ) ( )∫∞

∞−= dffnTjfPnTp π2expAt t=T

Fourier Transform

( ) ( )

( ) ( )

( ) ( )∫

∫ ∑

∑ ∫

−

−

∞

−∞=

∞

−∞=−

−∞=

=

+=

+=

T

T

T

Tm

m

T

T

m

dffnTjfB

dffnTjTmfP

dffnTjTmfP

21

21

21

21

21

21

2exp

2exp

2exp

π

π

π

( ) ( )∑∞

−∞=

+=m

TmfPfB


Proof

( ) ( )∑∞

−∞=

=n

n nfTjbfB π2exp

( ) ( )∫− −=T

Tn nfTjfBTb21

212exp π

( ) ( )∑∞

−∞=

+=m

TmfPfB

( )nTTpbn −= ( )( )

==

0nTb ( )

≠=

00 nbn

( ) TfB = ( ) TTmfPm

=+∑∞

−∞=


� Pulse shape that satisfy this criteria is Sinc(.) function, e.g.,

� The smallest value of T for which transmission with zero ISI is possible is

)2(sinsin)(or)( WtcT

tctpthe =

=

T1=

Nyquist Criterion: Time domain


� Problems with Sinc(.) function

– It is not possible to create Sinc pulses due to

– Infinite time duration– Sharp transition band in the frequency domain

– Sinc(.) pulse shape can cause ISI in the presence of timing errors

• If the received signal is not sampled at exactly the bit instant, then ISI will occur

WT

21=

Nyquist Criterion: Time domain

1p(t)

� shaping function


Equally spaced zeros,

interval Tf s

=2

1

Tf s

=2

1

02t0t

t0

-1

no ISI !

Sample rate vs. bandwidth

� W is the channel bandwidth for P(f)

� When 1/T > 2W, there is no way, we can design a system with no ISI

P(f)



� When 1/T = 2W (The Nyquist Rate), rectangular function satisfy Nyquist condition

( ) ( ) ( )( )

,otherwise,0

,;sinc

sin

<

=

==WfT

fPT

t

t

Tttp

πππ

( ) ( );rect2

rect21

fTTW

f

WfP =

=


( ) ( );rect2

rect2

fTTWW

fP =

=

T

W


� When 1/T < 2W, numbers of choices to satisfy Nyquist condition

– Raised Cosine Filter

– Duobinary Signaling (Partial Response Signals)

– Gaussian Filter Approximation


– Gaussian Filter Approximation

� The most typical one is the raised cosine function

Raised Cosine Pulse

� The following pulse shape satisfies Nyquist’s method for zero ISI

� The Fourier Transform of this pulse shape is

2

22

2

22 41

cossinc

41

cossin)(

T

trT

tr

T

t

T

trT

tr

T

trT

tr

tp

−

=−

=

ππ

π

π


� The Fourier Transform of this pulse shape is

� where r is the roll-off factor that determines the bandwidth

+≥

+≤≤−

−−+

−≤≤

=

T

rf

T

rf

T

r

T

rf

r

TT

T

rfT

fP

21

||,0

21

||2

1,

21

||cos12/

21

||0,

)(π

Responses for different roll-off factors (a) Frequency response. (b) Time response

� Bandwidth occupied beyond 1/2T is called the excess bandwidth (EB)

� EB is usually expressed as a %tage of the Nyquist frequency, e.g.,

– Rolloff factor, r= 1/2 ===> excess bandwidth is 50 %

– Rolloff factor, r= 1 ===> excess bandwidth is 100 %

� RC filter is used to realized Nyquist filter since the transition band can be changed using the roll-off factor

Rolloff and bandwidth


changed using the roll-off factor

� The sharpness of the filter is controlled by the parameter r

� When r = 0 this corresponds to an ideal rectangular function

� Bandwidth B occupied by a RC filtered signal is increased from its minimum value

� So the bandwidth becomes:sT

B2

1min =

( )rBB += 1min

� Benefits of large roll off factor

– Simpler filter – fewer stages (taps) hence easier to implement with less processing delay

– Less signal overshoot, resulting in lower peak to mean excursions of the transmitted signal

– Less sensitivity to symbol timing accuracy – wider eye

Rolloff and bandwidth


– Less sensitivity to symbol timing accuracy – wider eye opening

� r = 0 corresponds to Sinc(.) function

Partial Response Signals

� To improve the bandwidth efficiency

– Widen the pulse, the smaller the bandwidth.

– But there is ISI. For binary case with two symbols, there is only few possible interference patterns.

– By adding ISI in a controlled manner, it is possible to


– By adding ISI in a controlled manner, it is possible to achieve a signaling rate equal to the Nyquist rate

i.e.

Duobinary and Polibinary Signaling

(Covered in the previous lectures)

2nd Nyquist Criterion

� Values at the pulse edge are distortion-less

Nyquist Three Criteria� Nyquist’s Three Criteria


• 1: At sampling points, no ISI

• 2: At threshold, no ISI• 2: At threshold, no ISI

• 3: Areas within symbol period from other symbols is zero, then no ISI


Example (Two Criteria)

3rd Nyquist Criterion

� Within each symbol period, the integration of signal (area) is proportional to the integration of the transmit signal (area)

=0,1 n

≠=

==∫+

− 0,0

0,1)(2

12

212 n

ndttpA

T

T

n

n

Eye Patterns� An eye pattern is obtained by superimposing the actual waveforms for large

numbers of transmitted or received symbols

– Perfect eye pattern for noise-free, bandwidth-limited transmission of an alphabet of two digital waveforms encoding a binary signal (1’s and 0’s)


– Actual eye patterns are used to estimate the bit error rate and the signal to- noise ratio

Eye Patterns


Concept of the eye pattern

Eye Diagram� The eye diagram is created by taking the time domain

signal and overlapping the traces for a certain number of symbols.

� The open part of the signal represents the time that we can safely sample the signal with fidelity

Vertical and Horizontal Eye Openings

� The vertical eye opening or noise margin is related to the SNR, and thus the BER

– A large eye opening corresponds to a low BER

� The horizontal eye opening relates the jitter and the sensitivity of the the jitter and the sensitivity of the sampling instant to jitter

– The red brace indicates the range of sample instants with good eye opening

– At other sample instants, the eye opening is greatly reduced, as governed by the indicated slope

Interpretation of Eye Diagram

Cosine rolloff filter: Eye pattern

2nd Nyquist

1st Nyquist

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Inter Symbol Interference (ISI) Nyquist Criteria … : Advanced Digital Communications Week 12, 13:...

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