16.543 notes 8: communicating in bandlimited channels

c 2004-2007 Dr. jay Weitzen 1

16.543 Notes 8: Communicating in Bandlimited Channels


Module Objectives

• Different Pulse Shapes and their associated Bandwidths

• Transmission Bandwidth: How much do we need• Reducing Transmission bandwidth

– Intersymbol Interference– Nyquist Criteria for No ISI– Practical Pulse Shaping (root raised cosine, etc)– Duo Binary

• Equalization


Line Coding and Pulse Shaping


Pulse Shaping


Generation of Pulse Shapes


(a) Punched Tape

A

-A0(c) Polar NRZ

A

0(d) Unipolar RZ

A

-A0(e) Bipolar RZ

A

-A

0(f) Manchester NRZ

BINARY DATA1 1 0 1 0 0 1

Mark (hole)

Mark (hole)

Mark (hole)

Mark (hole)

space space space

Binary signaling formats

VoltsA

Time

0(b) Unipolar NRZ

Tb

Line codes


Power Spectra for Binary line codes

; )()( ∑∞

−∞=

==n

sn nTtfatsDigital signal or line code can be represented by:

R(k) is the autocorrelation function given by

Binary signaling :

Multilevel signaling: bs lTT =bs TT =

f(t) - symbol pulse shape;

Ts - duration of one symbol;

{ }na - Set of random data;

; )( ⎟⎟⎠

⎞⎜⎜⎝

⎛Π=

bTttf

= +A V – binary 1na

0 V – binary 0

PSD of a digital signal is given by: ∑∞−

∞=

−=k

kfTj

s

sekRT

π22

)(F(f)Ps(f)

F(f) f(t)where

∑=

+=I

iiiknn PaakR

1)()(

knn aa + and - the levels of the data pulses at the nth and (n+k)th symbol positions

iP - Probability of having the ith product knnaa +

Spectrum of the digital signal depends on: (1) The pulse shape used

(2) Statistical properties used


PSD for line codes

If ‘A’ is chosen so that normalized average power of the polar NRZ signal is unity, then

A=1

Bit rate: R=1/Tb

22 sin

)( ⎟⎟⎠

⎞⎜⎜⎝

⎛∏∏

=b

bbpolar NRZ fT

fTTAf P


1 1 0 1 0 0 1

A

-A0(c) Polar NRZ

Tb

time

Polar NRZ Coding


Polar NRZ signaling

Possible levels for the a’s : +A and -A

22

2

24

1

2

2

1

222

sin)(

0 ,00 ,

)(

04/1)(4/1))((4/1))((4/1)()(

,0 21)(

21)()0(

⎟⎟⎠

⎞⎜⎜⎝

⎛∏∏

=

⎩⎨⎧

≠=

=⇒

=−+−+−+==

≠

=−+==

∑

∑

=+

=

b

bbpolar NRZ

polar

iiknn

iiinn

fTfT

TAf P

kkA

kR

AAAAAAPaakR

kFor

AAAPaaR

( ) ( ) ( )∑∞

−∞=

=

∏∏

=↔=

ks

b

bbb

skfTekRTfF

f

fTfT

TfFTttf

π22

sP

sin)()/()(

C

with alongequation above ngSubstituti

in

gives

∑=

+=I

iiiknn PaakR

1

)()( - the levels of the data pulses at the nth and (n+k)th symbol positions

knn aa + and


Binary-to-multilevel polar NRZ signal conversion

levels 823 ==L

3311 RTT

Dbs

===

lL 2=

lRD =

Baud rate:

Bit rate


PSD of a multilevel polar NRZ waveform

Multilevel signaling is used to reduce the BW of a digital signal

( ) ( )∑=

==8

1

2 210Ri

iin Pa

0kFor =

values.possibleeight theof allfor 81P where i =

( ) .0kR 0,kFor =≠

( ) is for PSD Then the 2 tω

( ) ( ) ( )021P2

w2 +=sTfF

f .3T is width pulse thewhere s bT=

:3T width pulser rectangula For the b

( )2

NRZ multilevelsin

P ⎟⎟⎠

⎞⎜⎜⎝

⎛=

b

b

fTlfTl

Kfππ

constant a isk where

isbandwidth null ThelR

=nullB

∑=

+=I

iiiknn PaakR

1)()(

PSD for a multilevel polar NRZ signal:∑∞−

∞=

−=k

kfTj

s

sekRT

π22

)(F(f)Ps(f)


Spectral Efficiency

( )Hz

sbitBR =η

bandwidth - B rate data - R where

⎟⎠⎞

⎜⎝⎛ +==

NS

BC 1log 2maxη

( )Hz

sbitl =η

If limited BW is desired, then a signaling technique that has high spectral efficiency is desired.

DefinitionDefinition:

Maximum spectral efficiency (which is limited by channel noise) is given by

Shannon’s channel capacity formula

Spectral efficiency of a digital signal is given by

Spectral efficiency for multilevel signaling


Regenerative Repeater

Minimize the effect of channel noise & ISI

Produces a sample value

Generates a clocking signal

Produces a high level o/pif sample value>VTIncreases the amplitude

Regenerate a noise-free digital signal

Amplify and clean-up the signal periodically


Summary: Line Coding


Pulse Shaping for BandlimitedChannels


Transmission Bandwidth: How much do we need?

• The spectrum of a digital signal is very wide.

• Theoretically infinite.• So the answer to our question is: A lot!

Roll Off rate is a function of pulse risetime, for 0 risetime pulsesthe power spectrum rolls off at –20dB/decade

f Logf

-20dB/decade


Pulse Spectrum (baseband)

Sinc(x) -- the envelope of the spectral energy

+ Freq– Freq

Rate (Bit) SymbolInterval Sample

=1

NRZ baseband signal

Sample Interval between symbols (bits shown here)

xxSinxSinc )()( =

0 Hz


The Spectrum Analyzer View

Sinc(x)Magnitude

Sinc(x)MagnitudeOne Sided

Spectrum Analyzer View

All voltages folded over anddoubled, except the DC.


Pulse SpectrumThe Spectral lines

Center lobe width = 2/tMinor lobe widths =1/tSpectral lines every 1/TCenter at 0 Hz

2/t

1/T

freq0 Hz

Spectral Line separation at 1/T where T is the repetition time for pulses or words in wordtransmission.

Word Word Word Word Word Word Word

timeT t = the sample time for each symbol


Base Band To Pulse Modulated RF

t

T

Baseband Pulses

Pulse Modulated RF

fC


Modulation Frequency Shifts the BB Spectrum

0 HzBaseband Signal Fc Hz

Baseband SignalCarrier Modulated

0 Hz

+f–f +f


How much of the signal can be filtered?

By filtering the digital signal we can decrease the amount of Bandwidth necessary to transmit our information!

LPF

Baseband Spectral Display


What Happens When we reduce Bandwidth?

• Filtering in the frequency domain is convolution in the time domain

• As we reduce bandwidth, we introduce intersymbol interference

• In the limit, if we brick wall filter, we create a non-causal infinite time in both directions waveform (sinc in time domain)


Inter Symbol Interference

Zero ISI condition

0 TS

The Challenge: To design a filter so that at sample times the responseof the previous pulse is zero.

Response of two successive impulses

time


Intersymbol Interference

If the rectangular multilevel pulses are filtered improperly as they pass through a communications system, they will spread in time, and the pulse for each symbol may be smeared into adjacent time slots and cause Intersymbol Interference

How can we restrict BW and not introduce ISI?

3 Techniques


( ) ( )sn nTthat −= ∑in ω

( ) ∏ ⎟⎟⎠

⎞⎜⎜⎝

⎛=

sTtth where pulses/s 1D

sT=

( ) ( ) ( )

( ) ( )thnTta

nTtthat

nsn

nsn

*

* in

⎥⎦

⎤⎢⎣

⎡−=

−=⇒

∑

∑

δ

δω

Flat-topped multilevel input signal:

Symbol rate:

( ) ( ) ( )thnTtat en

sn *out ⎥⎦

⎤⎢⎣

⎡−= ∑ δω

( ) ( ) ( ) ( ) ( )tRhtChtThthteh ***=

( ) output at shape pulse theis where the

( ) ( ) ( ) ( ) ( )fHfHfHfHf RCTe =H

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛= ∏ fT

fTT

TtFf

s

ss

s ππsin

H

( ) ( )( ) ( ) ( )fHfHfH

fHf

CT

eR =H

( ) ( )∑ −=n

senout nTthatω

Output signal is given by:

Equivalent impulse response:

Equivalent transfer function:where

Receiving filter is given by:

Output signal can be rewritten as:

He(f) – overall filtering characteristic (chosen to minimize ISI)

Intersymbol Interference


Using Eye Patterns to Visualize ISI


How to Interpret Eye patterns


Distorted polar NRZ waveform and corresponding eye pattern

Resemble human eye

Effect of channel filtering & channel noise

Timing error eye opening

Sensitivity slope of the open eye

noise margin height of the eye opening

Information from the eye pattern:

Received line code

Oscilloscopepresentations

Normal- Eye openNoise - Eye close


Example of eye pattern:Binary-PAM, SRRQ pulse

• Perfect channel (no noise and no ISI)


Example of eye pattern:Binary-PAM, SRRQ pulse …

• AWGN (Eb/N0=20 dB) and no ISI


Example of eye pattern:Binary-PAM, SRRQ pulse …

• AWGN (Eb/N0=10 dB) and no ISI


Examples of Eye Diagrams


Avoiding ISI


Avoiding ISI (2)


BaseBand ISI Model


Baseband ISI Model (2)


( )⎩⎨⎧

≠=

=+0k ,00k ,

he

CkTs τ

( )tf

tft

s

s

ππsin

h e =

( ) ∏ ⎟⎟⎠

⎞⎜⎜⎝

⎛=

ss ff

ff 1He

)(hfor function x

sinx a choose Now

constant nonzero a is Csymbolsinput the

of esclock tim e with thcompared esclock tim samplingreceiver in theoffset theis period clocking (sample) symbol theis T

integeran isk where

e

s

t

τ

sT1f s =where

:function transfer thisof bandwidth Absolute2f

B s=

Nyquist’s first method (Zero ISI)ISI can be eliminated by using an equivalent transfer function, He(f),

such that the impulse response satisfies the condition:

and 0let =τ


The Nyquist Bandwidth

fn= Nyquist Frequency = Symbol Rate/2

This condition gives zero ISI (Inter Symbol Interference)

Ideal “brick-wall” filter at the minimum bandwidth

frequency

*Remember: In a radio transmitter the filtering is done at baseband.

Envelope of digital baseband spectrum.


0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

α = 0.3

α = 0.5

α = 0

α= 1.0

Fs : Symbol Rate

Alpha describes the "sharpness" of the filterOccupied bandwidth is approximately: Symbol rate X (1 + α)

Filter Bandwidth Parameter "α“Practical filter shapes

brick wall


Nyquist Zero ISI Condition


Nyquist ISI (2)


Nyquist Sampling (3)


Raised Cosine-Rolloff Nyquist Filtering

( ) ( )

⎪⎪

⎩

⎪⎪

⎨

⎧

>

<<⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎦

⎤⎢⎣

⎡ −+

<

=∆

Bf

Bff

ff

Bf

fH e

,0

f ,2

cos121

,1

11π

0 fBf −=∆

∆−≡ fff 01

0

ffr ∆=

( ) ( )[ ]( ) ⎥

⎥⎦

⎤

⎢⎢⎣

⎡

−⎟⎟⎠

⎞⎜⎜⎝

⎛==

∆

∆−2

0

00

1

412cos

22sin

2h t

ee ftf

tftf

ffHFtπ

ππ

filter theofbandwidth dB-6 theis f where o

Transfer Function: B- Absolute BW

Rolloff factor:

Impulse response is given by:

Definition:


The raised cosine filter

• Raised-Cosine Filter– A Nyquist pulse (No ISI at the sampling time)

⎪⎪⎩

⎪⎪⎨

⎧

>

<<−⎥⎦

⎤⎢⎣

⎡−−+

−<

=

Wf

WfWWWW

WWfWWf

fH

||for 0

||2for 2||4

cos

2||for 1

)( 00

02

0

π

Excess bandwidth:0WW − Roll-off factor

0

0

WWWr −

=10 ≤≤ r

20

000 ])(4[1

])(2cos[))2(sinc(2)(tWW

tWWtWWth−−−

=π


The Raised cosine filter – cont’d

2)1( Baseband sSB

sRrW +=

|)(||)(| fHfH RC=

0=r5.0=r

1=r1=r

5.0=r

0=r

)()( thth RC=

T21

T43

T1

T43−

T21−

T1−

1

0.5

0

1

0.5

0 T T2 T3T−T2−T3−

sRrW )1( Passband DSB +=


Raised Cosine (III)


Eye Diagrams for Raised Cosine


Root Raised RC


Practical Pulse Shaping


Example of pulse shaping

• Square-root Raised-Cosine (SRRC) pulse shaping

t/T

Amp. [V]

Baseband tr. Waveform

Data symbol

First pulseSecond pulse

Third pulse


Example of pulse shaping …

• Raised Cosine pulse at the output of matched filter

t/T

Amp. [V]

Baseband received waveform at the matched filter output(zero ISI)


Controlled ISI


Introduction to Equalization


Equalization• ISI due to filtering effect of the communications channel (e.g. wireless

channels)– Channels behave like band-limited filters

)()()( fjcc

cefHfH θ=

Non-constant amplitude

Amplitude distortion

Non-linear phase

Phase distortion


Example of eye pattern with ISI:Binary-PAM

• Non-ideal channel and no noise)(7.0)()( Tttthc −+= δδ


Example of eye pattern with ISI:Binary-PAM…

• AWGN (Eb/N0=20 dB) and ISI)(7.0)()( Tttthc −+= δδ


Example of eye pattern with ISI:Binary-PAM, SRRQ pulse …

• AWGN (Eb/N0=10 dB) and ISI)(7.0)()( Tttthc −+= δδ


What Can You Do to Combat ISI

• When the Multipath in a channel approaches 10% or more of a bit period, significant distortion and ISI result. What Can you do to combat it– Remove it (equalizers) – Make it Not a Factor (OFDM)– Combine it (Spread Spectrum/Rake Receiver)


Equalizing filters …• Baseband system model

• Equivalent model

Tx filter Channel

)(tn

)(tr Rx. filterDetector

kz

kTt =

{ }ka1a

2a 3aT )()(fHth

t

t

)()(fHth

r

r

)()(fHth

c

c

Equivalent system

)(ˆ tn

)(tzDetector

kz

kTt =)(

)(fHth

filtered noise

)()()()( fHfHfHfH rct=

∑ −k

k kTta )(δ Equalizer

)()(fHth

e

e

1a

2a 3aT

∑ −k

k kTta )(δ )(tx Equalizer

)()(fHth

e

e

)()()(ˆ thtntn r∗=

{ }ka)(tz

)(tz


What is Equalization


Equalization process


Combat ISI with Equalization• Equalization is required because channel frequency response is not flat• Zero-forcing equalizer

– Inverts channel, Eliminates ISI– Flattens freq. response– Amplifies noise

• MMSE equalizer– Optimizes trade-off

between noiseamplification and ISI

• Decision-feedbackequalizer– Use Previous Decisions

to remove ISIIncreases complexity

– Propagates error

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

frequency (× fs Hz)

Magnitude

Channel frequency response

Zero-forcing equalizer frequency response

MMSEequalizer frequency response


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)(

fHfH

ce = Taking care of ISI

caused by channel


Equalizing filters …• Baseband system model

• Equivalent model

Tx filter Channel

)(tn

)(tr Rx. filterDetector

kz

kTt =

{ }ka1a

2a 3aT )()(fHth

t

t

)()(fHth

r

r

)()(fHth

c

c

Equivalent system

)(ˆ tn

)(tzDetector

kz

kTt =)(

)(fHth

filtered noise

)()()()( fHfHfHfH rct=

∑ −k

k kTta )(δ Equalizer

)()(fHth

e

e

1a

2a 3aT

∑ −k

k kTta )(δ )(tx Equalizer

)()(fHth

e

e

)()()(ˆ thtntn r∗=

{ }ka)(tz

)(tz


Transversal Filter Model For Channel


Generalized Transversal Filter


Generalized ZFE


Zero Forcing Equalizer


ZFE (2)


Noise Enhancement in ZFE


MMSE Equalization


Adaptive DFE


Example of equalizer• 2-PAM with SRRQ• Non-ideal channel

• One-tap DFE)(3.0)()( Tttthc −+= δδ

Matched filter outputs at the sampling time

ISI-no noise,No equalizer

ISI-no noise,DFE equalizer

ISI- noiseNo equalizer

ISI- noiseDFE equalizer


End of Module 8

16.543 notes 8: communicating in bandlimited channels

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