EE5713 : Advanced Digital Communications...Pulse Shaping and Raised-Cosine Filter ... Matched Filter and Correlator Receiver (On Board) Equalization (On Board) 3/27/2013 Muhammad Ali

3/27/2013 Muhammad Ali Jinnah University, Islamabad Advanced Digital Communications (EE5713) 1

EE5713 : Advanced Digital Communications

Week 4, 5:

Inter Symbol Interference (ISI)

Nyquist Criteria for ISI

Pulse Shaping and Raised-Cosine Filter

Eye Pattern

Error Performance Degradation (On Board)

Demodulation and Detection (On Board)

Eb/No and Error Probability (On Board)

Matched Filter and Correlator Receiver (On Board)

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

where an is the symbol mapping and g(t) is the pulse shape

Problems 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) 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)

Intersymbol Interference (ISI)


– Due to improper filtering (@ Tx and/or Rx), the received pulses overlap one

another thus making detection difficult

Example of ISI

– Assume polar NRZ line code

Pulse spreading


– Input data stream and bit superposition

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

Inter Symbol Interference


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

amplitude at the time the signal is sampled

ISI


ISI Baseband Communication System Model

receivertheofresponseImpulse)(

channel,theofresponseImpulse)(

r,transmittetheofresponseImpulse)(where

th

th

th

R

C

T

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

or equivalently

– h0 is an arbitrary constant

n

een kTnnTkThakTy )()()(

n knn

knknkknknk 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

ISI Baseband Communication System Model


Signal Design for Bandlimited Channel

Zero ISI

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

Nyquist Criterion

– Pulse amplitudes can be detected correctly despite pulse

spreading or overlapping, if there is no ISI at the decision-

making instants

knn

eenk kTnnTkThaahkTy,

0 )()()(

0,0)(0

handknfornTkThe


Nyquist Criterion: Time domain

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

Suppose 1/T is the sample rate

The necessary and sufficient condition for p(t) to satisfy Nyquist

Criterion is

0,0

0,1

n

nnTp


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

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

)2(sinsin)(or)( WtcT

tctpthe

WT

2

1

1st Nyquist Criterion: Time domain


1st Nyquist Criterion: Time domain

Equally spaced zeros,

interval Tf s

2

1

Tf s

2

1

02t0t

t

0

1 p(t)

-1

shaping function

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

WfTfP

T

t

t

Tttp

;rect2

rect2

1fTT

W

f

WfP

T

W



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

condition

– Raised Cosine Filter

– Duobinary Signaling (Partial Response Signals)

– 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

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

2

22

2

22 41

cos

sinc4

1

cossin

)(

T

tr

T

tr

T

t

T

tr

T

tr

T

tr

T

tr

tp

T

rf

T

rf

T

r

T

rf

r

TT

T

rfT

fP

2

1||,0

2

1||

2

1,

2

1||cos12/

2

1||0,

)(


Raised cosine shaping

W

W ω

P(ω)

r=0

r = 0.25

r = 0.50

r = 0.75

r = 1.00

W

π

0

t 0

p(t)

W

π

2w

Tradeoff: higher r, higher bandwidth, but smoother in

time.


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

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

Rolloff and bandwidth


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

opening

r = 0 corresponds to Sinc(.) function

Rolloff and bandwidth


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

achieve a signaling rate equal to the Nyquist rate

i.e.

Duobinary and Polibinary Signaling

(Covered in the previous lectures)


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


Concept of the eye pattern

Eye Patterns


Concept of Eye diagram Mask. Waveform must not intrude into the shaded regions.

Eye Patterns


Cosine rolloff filter: Eye pattern

2nd Nyquist

1st Nyquist

2nd Nyquist:

1st Nyquist:

2nd Nyquist:

1st Nyquist:

2nd Nyquist:

1st Nyquist:

2nd Nyquist:

1st Nyquist:


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time (sec)

EYE DIAGRAM

Eye Diagram Examples


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1.5

-1

-0.5

0

0.5

1

1.5

Time (sec)

EYE DIAGRAM WITH NOISE (Variance =0.1)



0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-3

-2

-1

0

1

2

3

Time (sec)

EYE DIAGRAM WITH NOISE (Variance =0.5)


Documents

EE5713 : Advanced Digital Communications...Pulse Shaping and Raised-Cosine Filter ... Matched Filter and Correlator Receiver (On Board) Equalization (On Board) 3/27/2013 Muhammad Ali