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Square Root Raised Cosine Filter

Digital Communication, 4th Edition

Chapter 9: Signal Design for Band-Limited Channels

John G. Proakis


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Introduction

We consider the problem of signal design when the channel isband-limited to some specified bandwidth ofWHz.

The channel may be modeled as a linear filter having an equivalentlow-pass frequency response C(f) that is zero for |f| >W.

Our purpose is to design a signal pulse g(t) in a linearly modulatedsignal, represented as

that efficiently utilizes the total available channel bandwidth W.

When the channel is ideal for |f|

W, a signal pulse can be designed thatallows us to transmit at symbol rates comparable to or exceeding thechannel bandwidth W.

When the channel is not ideal, signal transmission at a symbol rate equal toor exceeding Wresults in inter-symbol interference (ISI) among a number ofadjacent symbols.

( ) ( )nn

v t I g t nT =


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For our purposes, a band-limited channel such as a telephone

channel will be characterized as a linear filter having an

equivalent low-pass frequency-response characteristic C(f),

and its equivalent low-pass impulse response c(t).

Then, if a signal of the form

is transmitted over a band-pass telephone channel, the

equivalent low-pass received signal is

wherez(t) denotes the additive noise.

( ) ( ) ( ) ( )

( ) ( ) ( )tztctv

tzdtcvtrl

+=

+=

( ) ( )tfj c

etvts2

Re=

Characterization of Band-LimitedChannels


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Alternatively, the signal term can be represented in thefrequency domain as V(f)C(f), where V(f) = F[v(t)].

If the channel is band-limited to WHz, then C(f) = 0 for |f| >W.

As a consequence, any frequency components in V(f) above

|f| = Wwill not be passed by the channel, so we limit thebandwidth of the transmitted signal to WHz.

Within the bandwidth of the channel, we may express C(f) as

where |C(f)|: amplitude response.

(f): phase response.

The envelope delay characteristic:

( ) ( ) ( )fjefCfC =

( )( )

df

fdf

2

1=



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A channel is said to be nondistorting orideal if the amplituderesponse |C(f)| is constant for all |f| Wand(f) is a linearfunction of frequency, i.e.,(f) is a constant for all |f| W.

If |C(f)| is not constant for all |f| W, we say that the channeldistorts the transmitted signal V(f) in amplitude.

If(f) is not constant for all |f| W, we say that the channel

distorts the signal V(f) in delay.As a result of the amplitude and delay distortion caused by thenonideal channel frequency-response C(f), a succession of

pulses transmitted through the channel at rates comparable tothe bandwidth Ware smeared to the point that they are nolonger distinguishable as well-defined pulses at the receivingterminal. Instead, they overlap, and thus, we have inter-symbol

interference (ISI).



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Fig. (a) is a band-limited pulse having zeros periodically spaced

in time at T, 2T, etc.

If information is conveyed by the pulse amplitude, as in PAM, for

example, then one can transmit a sequence of pulses, each of

which has a peak at the periodic zeros of the other pulses.



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However, transmission of the pulse through a channel modeled as

having a linear envelope delay(f) [quadratic phase(f)]

results in the received pulse shown in Fig. (b), where the zero-

crossings that are no longer periodically spaced.



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A sequence of successive pulses would no longer be

distinguishable. Thus, the channel delay distortion results in ISI.

It is possible to compensate for the nonideal frequency-response

of the channel by use of afilterorequalizerat the demodulator.

Fig. (c) illustrates the output of a linear equalizer that

compensates for the linear distortion in the channel.



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The equivalent low-pass transmitted signal for several different

types of digital modulation techniques has the common form

where {In}: discrete information-bearing sequence of symbols.

g(t): a pulse with band-limited frequency-

response G(f), i.e., G(f) = 0 for |f| > W.This signal is transmitted over a channel having a frequency

response C(f), also limited to |f| W.

The received signal can be represented as

where andz(t) is the AWGN.

( ) ( ) ( )tznTthItrn

nl +=

=0

( ) ( )

=

=0n

n nTtgItv

( ) ( ) ( )

= dtcgth

Signal Design for Band-LimitedChannels


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Suppose that the received signal is passed first through a filter and

then sampled at a rate 1/Tsamples/s, the optimum filter from the

point of view of signal detection is one matched to the received

pulse. That is, the frequency response of the receiving filter is

H*(f).

We denote the output of the receiving filter as

where

x(t): the pulse representing the response of the receiving

filter to the input pulse h(t).

v(t): response of the receiving filter to the noisez(t).

( ) ( ) ( )

=

+=0n

n tvnTtxIty



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Ify(t) is sampled at times t= kT+0, k = 0, 1,, we have

or, equivalently,

where0: transmission delay through the channel.

The sample values can be expressed as

( ) ( ) ( )00

00 +++=+

=

kTvnTkTxIykTyn

nk

0

, 0,1,...k n k n k n

y I x v k

=

= + =

0

00

1, 0,1,...

k k n k n k

n

n k

y x I I x v k x

=

= + + =



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We regardx0 as an arbitrary scale factor, which we arbitrarily set

equal to unity for convenience, then

where

Ik: the desired information symbol at the k-thsampling instant.

: ISI

vk: additive Gaussian noise variable at the k-th

sampling instant.

k

knn

nknkk vxIIy ++=

=

0

=

knn

nknxI

0


l D f B d L d


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The amount of ISI and noise in a digital communication system

can be viewed on an oscilloscope.

For PAM signals, we can display the received signaly(t) on the

vertical input with the horizontal sweep rate set at 1/T.

The resulting oscilloscope display is called an eye pattern.

Eye patterns for binary and quaternary amplitude-shift keying

(or PAM):

The effect of ISI is to cause the eye to close.Thereby, reducing the margin for additive noise to cause errors.


Si l D i f B d Li i d


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Effect of ISI on eye opening:

ISI distorts the position of the zero-crossings and causes a

reduction in the eye opening.

Thus, it causes the system to be more sensitive to a

synchronization error.


Si l D i f B d Li it d


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For PSK and QAM, it is customary to display the eye pattern as

a two-dimensional scatter diagram illustrating the sampled values

{yk} that represent the decision variables at the sampling instants.

Two-dimensional digital eye patterns.


D i f B d Li i d i l f


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Assuming that the band-limited channel has ideal frequency-

response, i.e., C(f) = 1 for |f| W, then the pulsex(t) has a

spectral characteristicX(f) = |G(f)|2, where

We are interested in determining the spectral properties of the

pulsex(t), that results in no inter-symbol interference.

Since

the condition for no ISI is

k

kn

n

nknkk vxIIy ++=

=

0

( )( )

( )

1 0

0 0k

kx t kT x

k

== =

( ) ( )=W

W

ftjdfefXtx

2

()

Design of Band-Limited Signals forNo ISI The Nyquist Criterion

D i f B d Li i d Si l f


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Nyquist pulse-shaping criterion (Nyquist condition for

zero ISI)

The necessary and sufficient condition forx(t) to satisfy

is that its Fourier transformX(f) satisfy

( )( )

( )

1 0

0 0

nx nT

n

==

( ) TTmfXm

=+

=


D i f B d Li it d Si l f


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Proof:

In general,x(t) is the inverse Fourier transform ofX(f). Hence,

At the sampling instant t= nT,

( ) ( )

= dfefXtx ftj 2

( ) ( )

= dfefXnTx fnTj 2


D i f B d Li it d Si l f


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Breaking up the integral into integrals covering the finite

range of 1/T, thus, we obtain

where we defineB(f) as

( ) ( )( )( )

( )

( )

( )

2 1 2

22 1 2

1 22 '

1 2

1 22

1 2

1 22

1 2

' '

m T

j fnT m T

m

Tj f nT

Tm

Tj fnT

Tm

Tj fnT

T

x nT X f e df

X f m T e df

X f m T e df

B f e df

+

=

=

=

=

= +

= +

=

( ) ( )

=

+=m

TmfXfB

'

2 1 2 1: ,

2 2

1 1' : ,2 2

'

mf f

Tm m

fT T

f T T

df df

= +

+

=


(1)

D si f B d Li it d Si ls f


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ObviouslyB(f) is a periodic function with period 1/T, and,

therefore, it can be expanded in terms of its Fourier series

coefficients {bn} as

where

Comparing (1) and (2), we obtain

( ) 2j nfT nn

B f b e

=

=

( )1 2

2

1 2

TnfT

nT

b T B f e df

=

( )nTTxbn =

(2)

( )( )

( )

Recall that the conditions forno ISI are (from ):

1 0

0 0k

kx t kT x

k

== =



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Design of Band Limited Signals for


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Suppose that the channel has a bandwidth ofW. Then

C(f) 0 for |f| > WandX(f) = 0 for |f| > W.

When T< 1/2W(or 1/T> 2W)

Since consists of nonoverlapping

replicas ofX(f), separated by 1/T, there is no choice for

X(f) to ensureB(f) Tin this case and there is no way

that we can design a system with no ISI.

( ) ( )+

+= TnfXfB




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When T= 1/2W, or 1/T= 2W(the Nyquist rate), the

replications ofX(f), separated by 1/T, are shown below:

In this case, there exists only oneX(f) that results inB(f) = T,

namely,

which corresponds to the pulse

( )( )

0

T f WX f

otherwise


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The smallest value ofTfor which transmission with zero ISI ispossible is T= 1/2W, and for this value,x(t) has to be a sincfunction.

The difficulty with this choice ofx(t) is that it is noncausal andnonrealizable.

A second difficulty with this pulse shape is that its rate of

convergence to zero is slow.The tails ofx(t) decay as 1/t; consequently, a small mistimingerror in sampling the output of the matched filter at thedemodulator results in an infinite series of ISI components.

Such a series is not absolutely summable because of the 1/trate of decay of the pulse, and, hence, the sum of the resultingISI does not converge.




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When T> 1/2W,B(f) consists of overlapping replications of

X(f) separated by 1/T:

In this case, there exist numerous choices forX(f) such that

X(f) such thatB(f) T.


Design of Band-Limited Signals for


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A particular pulse spectrum, for the T> 1/2Wcase, that has

desirable spectral properties and has been widely used in

practice is the raised cosine spectrum.

Raised cosine spectrum:

: roll-off factor. (0 1)

( )

10

21 1- 1

1 cos2 2 2T 2

10

2

rc

T f

TT T

X f f f T T

fT

+ = +

+ >




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The bandwidth occupied by the signal beyond the Nyquistfrequency 1/2Tis called the excess bandwidth and is usuallyexpressed as a percentage of the Nyquist frequency.

= 1/2 => excess bandwidth = 50 %.= 1 => excess bandwidth = 100%.

The pulsex(t), having the raised cosine spectrum, is

x(t) is normalized so thatx(0) = 1.

( )( ) ( )

( ) ( )222

222

41cossin

41

cossin

TtTtTtc

Tt

Tt

Tt

Tttx

=

=




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Pulses having a raised cosine spectrum:

For= 0, the pulse reduces tox(t) = sinc(t/T), and the

symbol rate 1/T= 2W.

When= 1, the symbol rate is 1/T= W.

Design of Band Limited Signals forNo ISI The Nyquist Criterion



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In general, the tails ofx(t) decay as 1/t3 for> 0.

Consequently, a mistiming error in sampling leads to a series

of ISI components that converges to a finite value.

Because of the smooth characteristics of the raised cosine

spectrum, it is possible to design practical filters for the

transmitter and the receiver that approximate the overall desired

frequency response.

In the special case where the channel is ideal, i.e., C(f) = 1,

|f| W, we have

where GT(f) and GR(f) are the frequency responses of the two

filters.

( ) ( ) ( )fGfGfX RTrc =




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If the receiver filter is matched of the transmitter filter, we haveXrc(f)= GT(f) GR(f) = | GT(f)|

2. Ideally,

and GR(f) = , where t0 is some nominal delay that isrequired to ensure physical realizability of the filter.

Thus, the overall raised cosine spectral characteristic is splitevenly between the transmitting filter and the receiving filter.

An additional delay is necessary to ensure the physical realization

of the receiving filter.

( )fGT

( ) ( )02 ftjrcT efXfG

=



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Implementation ofSquare Root Raise Cosine Filter


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Simplified System Architecture

Transmitter

SRRC-TX DACLPF

DAC SRRC-RX

Receiver


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Design of SRRC

Operation of the SRRC-TX module includes:

Over-sampling

Over-sampling rate is a system design issue (performancevs. cost).

For a over-sampling rate = 4, we pad three zeros between

each sample.Finite-impulse-response (FIR) filter

The FIR filter acts as a low pass filter.

The FIR filter is a square root raised cosine filter with roll-off factor= 0.22.


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Detail of FIR Filter Module

h(0) h(1) h(2)

Input

0

h(n/2)


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Raised Cosine Filter

Raised cosine spectrum:

The pulsex(t), having the raised cosine spectrum, is

( )

10

2

1 1- 11 cos

2 2 2T 21

02

rc

T fT

T TX f f f

T T

fT

+

= +

+ >

( )

( ) ( )

( )

( )2 2 2 2 2 2

sin cos cossin

1 4 1 4

t T t T t T x t c t T

t T t T t T

= =


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The cosine roll-off transfer function can be achieved by using

identical square root raised cosine filter at the

transmitter and receiver.The pulse SRRC(t), having the square root raised cosine

spectrum, is

( )( ) ( )

2

sin 1 4 cos 1

1 4

where is the inverse of chip rate ( 0.2604167 s)

and = 0.22.

C C C

C C

C

t t t

T T TSRRC t

t tT T

T

+ + =

( )rcX f


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Because SRRC(t) is non-causal, it must be truncated, and pulse

shaping filter are typically implemented for6TCabout the t= 0

point for each symbol.

For an over-sampling rate of 4, n is equal to 48.

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