Signal Design

7/29/2019 Signal Design

http://slidepdf.com/reader/full/signal-design 1/60

Wireless Information Transmission System Lab.

National Sun Yat-sen University Institute of Communications Engineering

Signal Design for Band-Limited Channels



2

IntroductionWe consider the problem of signal design when the channel isband-limited to some specified bandwidth of W Hz.

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 that

allows 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 W results in inter-symbol interference (ISI) among a number of adjacent symbols.

( ) ( )nn

v t I g t nT = −∑



3

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

where z(t ) denotes the additive noise.

( ) ( ) ( ) ( )( ) ( ) ( )t zt ct v

t zd t cvt r l

+∗=

+−= ∫∞

∞−

τ τ τ

( ) ( )t f j c

et vt s

π 2

Re=

Characterization of Band-Limited

Channels



4

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 W Hz, then C ( f ) = 0 for | f | >W .

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

| f | = W will not be passed by the channel, so we limit thebandwidth of the transmitted signal to W Hz.

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

where |C ( f )|: amplitude response.θ( f ): phase response.

The envelope delay characteristic:

( ) ( ) ( ) f je f C f C

θ =

( )( )

df

f d f

θ

π τ

2

1−=


Channels



5

A channel is said to be nondistorting or ideal if the amplituderesponse |C ( f )| is constant for all | f | ≤ W andθ( f ) is a linear

function of frequency, i.e.,τ( 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 τ( 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 W are smeared to the point that they are nolonger distinguishable as well-defined pulses at the receivingterminal. Instead, they overlap, and thus, we have inter-symbolinterference (ISI).


Channels



6

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

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


Channels



7

However, transmission of the pulse through a channel modeled as

having a linear envelope delayτ( f ) [quadratic phaseθ( f )]

results in the received pulse shown in Fig. (b), where the zero-crossings that are no longer periodically spaced.


Channels



8

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-responseof the channel by use of a filter or equalizer at the demodulator.

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

compensates for the linear distortion in the channel.


Channels



9

The equivalent low-pass transmitted signal for several different

types of digital modulation techniques has the common form

where { I n}: 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 and z(t ) is the AWGN.

( ) ( ) ( )t znT t h I t r n

nl +−= ∑∞

=0

( ) ( )∑

∞

= −= 0n

n nT t g I t v

( ) ( ) ( )

∫

∞

∞−

−= τ τ τ d t cgt h




10

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

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

point of view of signal detection is one matched to the receivedpulse. 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 receivingfilter to the input pulse h(t ).

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

( ) ( ) ( )∑∞

=

+−=0n

n t vnT t x I t y




11

If y(t ) is sampled at times t = kT +τ0 , k = 0, 1,…, we have

or, equivalently,

whereτ0: transmission delay through the channel.

The sample values can be expressed as

( ) ( ) ( )00

00 τ τ τ +++−=≡+

∑

∞

= kT vnT kT x I ykT y nnk

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

nn k

y x I I x v k x

∞

−=≠

⎛ ⎞⎜ ⎟= + + =⎜ ⎟⎜ ⎟⎝ ⎠

∑




12

We regard x0 as an arbitrary scale factor, which we arbitrarily set

equal to unity for convenience, then

where

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

: ISI

vk : additive Gaussian noise variable at the k-th

sampling instant.

k

k nn

nk nk k v x I I y ++= ∑∞

≠=

−0

∑∞

≠

=−

k n

n

nk n x I 0




13

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 signal y(t ) on thevertical 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:

The effect of ISI is to cause the eye to close.

Thereby, reducing the margin for additive noise to cause errors.




14

Effect of ISI on eye opening:

ISI distorts the position of the zero-crossings and causes areduction in the eye opening.

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

synchronization error.




15

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.”




16

Assuming that the band-limited channel has ideal frequency-

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

spectral characteristic X ( f ) = |G( f )|2

, where

We are interested in determining the spectral properties of the

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

Since

the condition for no ISI is

k

k nn


≠=

−0

( )( )

( )

1 0

0 0k

k x t kT x

k

=⎧⎪= ≡ = ⎨

≠⎪⎩

( ) ( )∫−=

W

W

ft jdf e f X t x

π 2

(＊)

Design of Band-Limited Signals for

No ISI – The Nyquist Criterion



17

Nyquist pulse-shaping criterion (Nyquist condition for

zero ISI)

The necessary and sufficient condition for x(t ) to satisfy

is that its Fourier transform X ( f ) satisfy

( )( )

( )

1 0

0 0

n x nT

n

=⎧⎪= ⎨

≠⎪⎩

( ) T T m f X m

=+∑∞

−∞=





18

Proof:

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

At the sampling instant t = nT ,

( ) ( )∫∞

∞−= df e f X t x ft j π 2

( ) ( )∫∞

∞−= df e f X nT x fnT j π 2





19

Breaking up the integral into integrals covering the finite

range of 1/ T , thus, we obtain

where we define B( f ) as

( ) ( )( )( )

( )

( )

( )

2 1 2 2

2 1 2

1 22 '

1 2

1 22

1 2

1 22

1 2

' '

m T j fnT

m T m

T j f nT

T

m

T j fnT

T m

T j fnT

T

x nT X f e df

X f m T e df

X f m T e df

B f e df

π

π

π

π

∞ +

−=−∞

∞

−

=−∞∞

−=−∞

−

=

= +

⎡ ⎤= +⎢ ⎥

⎣ ⎦

=

∑ ∫

∑ ∫

∑∫

∫( ) ( )∑

∞

−∞=

+=m

T m f X f B

'

2 1 2 1: ,

2 2

1 1

' : ,2 2

'

m f f

T m m

f T T

f T T

df df

= +

− +⎡ ⎤⎢ ⎥⎣ ⎦

−⎡ ⎤

⎢ ⎥⎣ ⎦

=



(1) Periodicfunction.



20

Design of Band-Limited Signals for No

ISI–

The Nyquist CriterionObviously B( 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

( )( )

( )

Recall that the conditions for

no ISI are (from ):

1 0

0 0k

k x t kT x

k

∗

=⎧⎪= ≡ = ⎨

≠⎪⎩

( ) 2 j nfT

n

n

B f b eπ ∞

=−∞

= ∑

( )1 2

2

1 2

T nfT

n

T

b T B f e df π −

−

= ∫

( )nT Txbn −=

(2)

( ) ( )1 2

2

1 2

1T j fnT

nT

x nT B f e df bT

π −

−− = =∫

(9.2-20)



21

Therefore, the necessary and sufficient condition for

to be satisfied is that

which, when substituted into , yields

or, equivalently

This concludes the proof of the theorem.

( )

( )

0

0 0n

T nb

n

=⎧⎪= ⎨

≠⎪⎩

( ) ∑∞

−∞=

=n

nfT j

neb f Bπ 2

( ) T f B =

( ) T T m f X m

=+∑∞

−∞=

k

k nn


≠=

−

0





22

Suppose that the channel has a bandwidth of W . Then

C ( f )≣ 0 for | f | > W and X ( f ) = 0 for | f | > W .

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

Since consists of nonoverlapping

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

X ( f ) to ensure B( f )≣ T in this case and there is no way

that we can design a system with no ISI.

( ) ( )∑+∞

∞− += T n f X f B





23

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

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

In this case, there exists only one X ( f ) that results in B( f ) = T ,

namely,

which corresponds to the pulse

( )( )

( )

0

T f W X f

otherwise

⎧ <⎪= ⎨

⎪⎩

( )( )sin

sinct T t

x t t T T

π π

π

⎛ ⎞= ≡ ⎜ ⎟

⎝ ⎠


No ISI–

The Nyquist Criterion



24

Design of Band-Limited Signals for NoISI – The Nyquist Criterion

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

The difficulty with this choice of x(t ) is that it is noncausaland nonrealizable.

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

convergence to zero is slow.The tails of x(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/ t rate of decay of the pulse, and, hence, the sum of the resultingISI does not converge. 1 1 1

1 ?2 4 8

+ + + + ="1 1 1

1 ?2 3 4

+ + + + ="



25

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

replications of X ( f ) separated by 1/ T :

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

B( f )≣ T .


No ISI–




26

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

desirable spectral properties and has been widely used in

practice is the raised cosine spectrum.

Raised cosine spectrum:

β: roll-off factor. (0 ≤β≤ 1)

( )

10

2

1 1- 11 cos

2 2 2T 2

102

rc

T f T

T T X f f f

T T

f T

β

π β β β

β

β

⎧ −⎛ ⎞≤ ≤⎜ ⎟⎪

⎝ ⎠⎪⎪ ⎧ ⎫⎡ ⎤− +⎪ ⎛ ⎞ ⎛ ⎞

= + − ≤ ≤⎨ ⎨ ⎬⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦⎩ ⎭⎪

⎪+⎛ ⎞⎪ >⎜ ⎟

⎪ ⎝ ⎠⎩


No ISI–




27

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

β= 1/2 => excess bandwidth = 50 %.

β= 1 => excess bandwidth = 100%.

The pulse x(t ), having the raised cosine spectrum, is

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

( ) ( ) ( )

( )

( )222

222

41

cos

sin

41

cossin

T t

T t

T t c

T t

T t

T t

T t t x

β

πβ

π

β

πβ

π

π

−=

−=


No ISI–




28

Pulses having a raised cosine spectrum:

Forβ= 0, the pulse reduces to x(t ) = sinc(πt/T ), and the

symbol rate 1/ T = 2W .

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


No ISI–






30

If the receiver filter is matched of the transmitter filter, we have X rc( f )= GT ( f ) G R( f ) = | GT ( f )|

2. Ideally,

and G R( f ) = , where t 0 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 physicalrealizability of the receiving filter.

( ) f GT

∗

( ) ( )02 ft j

rcT e f X f G π −=


No ISI–




31

Square Root Raised Cosine Filter

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 for WCDMA.

C C C

C C

C

t t t T T T

SRRC t

t t

T T

T

π β β π β

π β

μ

β

⎛ ⎞ ⎛ ⎞− + +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠=⎛ ⎞⎛ ⎞⎜ ⎟− ⎜ ⎟

⎜ ⎟⎝ ⎠⎝ ⎠≈

( )rc X f



32

Design of Band-limited Signals withControlled ISI -- Partial-Response Signals

It is necessary to reduce the symbol rate 1/T below the Nyquist

rate of 2W symbols/s to realize practical transmitting and

receiving filters.Suppose we choose to relax the condition of zero ISI and, thus,

achieve a symbol transmission rate of 2W symbols/s.

By allowing for a controlled amount of ISI, we can achieve this

symbol rate.

The condition for zero ISI is x(nT )=0 for n≠0.

Suppose that we design the band-limited signal to have controlled

ISI at one time instant. This means that we allow one additionalnonzero value in the samples { x(nT )}.

f d l d l h



33


One special case that leads to (approximately) physically

realizable transmitting and receiving filters is the duobinary signal

pulse:

Using Equation 9.2-20

When substituted into Equation 9.2-18, we obtain:

( ) ( )( )⎩

⎨⎧ ==

otherwise 0

1,0 1 nnT x

( )

( )⎩⎨⎧ =

=otherwise 0

1-,0 nT bn

( ) fT jTeT f B

π 2−+=

( )nT Txb

n−=

( ) 2 j nfT

n

n

B f b eπ

∞

=−∞

= ∑

D i f B d li i d i l i h



34


It is impossible to satisfy the above equation for 1/ T >2W .

For T =1/2W , we obtain

Therefore, x(t ) is given by:

This pulse is called a duobinary signal pulse.

( ) ( ) ( )( )

( )( )⎪⎩

⎪⎨

⎧<=

⎪⎩

⎪⎨⎧ <+=

−

−

otherwise 0

2

cos1

otherwise 0

121

2 W f W f e

W

W f eW f X

W f j

W f j

π π

π

( ) ( )1

sinc 2 sinc 22

x t Wt Wt π π ⎡ ⎤⎛ ⎞

= + −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

D i f B d li it d Si l ith



35


Time-domain and frequency-domain characteristics of a

duobinary signal.

Modified duobinary signal pulse:

( )( )( )

( )

1 -1

1 12

0 otherwise

nn

x x nT nW

=⎧⎪⎛ ⎞

= = − =⎨⎜ ⎟⎝ ⎠ ⎪

⎩

The spectrum decays

to zero smoothly.

--physically realizable

D i f B d li it d Si l ith



36


The corresponding pulse x(t ) is given as

The spectrum is given by

( )( ) ( )

sinc sinct T t T

x t

T T

π π + −= −

( )( )

⎪⎩

⎪⎨

⎧

>

≤=−=

−

W f

W f W

f

W

jee

W f X

W f jW f j

0

sin2

1 π π π

Zero at f =0.

D si f B d li it d Si ls ith



37


Other physically realizable filter characteristics are obtained by

selecting different values for the samples { x(n /2W )} and more

than two nonzero samples.

As we select more nonzero samples, the problem of unraveling

the controlled ISI becomes more cumbersome and impractical.

When controlled ISI is purposely introduced by selecting two or

more nonzero samples form the set { x(n /2W )}, the class of band-limited signal pulses are called partial-response signals:

( ) sinc 22 2n

n n x t x W t

W W π

∞

=−∞

⎡ ⎤⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟⎢ ⎥

⎝ ⎠ ⎝ ⎠⎣ ⎦∑

( )( )

( )⎪

⎩

⎪⎨

⎧

>

≤⎟ ⎠

⎞⎜⎝

⎛

=

−∞

−∞=∑

W f

W f eW

n x

W f X

W f jn

n

0

22

1 π



38

Data Detection for Controlled ISI

Two methods for detecting the information symbols at the

receiver when the received signal contains controlled ISI:

Symbol-by-symbol detection method.

Relatively easy to implement.

Maximum-likelihood criterion for detecting a sequence of

symbols.

Minimizes the probability of error but is a little more complex toimplement.

The following treatment is based on PAM signals, but it is easily

generalized to QAM and PSK.

We assume that the desired spectral characteristic X ( f ) for the

partial-response signal is split evenly between the transmitting

and receiving filters, i.e., |GT ( f )|=|G R( f )|=| X ( f )|1/2.



39


Symbol-by-symbol suboptimum detection

For duobinary signal pulse, x(nT )=1, for n=0,1, and is zero

otherwise.The samples at the output of the receiving filter (demodulator)

have the form

where { I m} is the transmitted sequence of amplitudes and {vm}

is a sequence of additive Gaussian noise samples.

Consider the binary case where I m=±1, Bm takes on one of

three possible values, namely, Bm =-2,0,2 with corresponding

probabilities ¼, ½, ¼.

If I m-1 is the detected symbol from the (m-1)th signaling

1m m m m m m y B v I I v−= + = + +



40



interval, its effect on Bm, the received signal in the mth

signaling interval, can be eliminated by subtraction, thusallowing I m to be detected.

Major problem with this procedure is error propagation: if

I m-1 is in error, its effect on Bm is not eliminated but, in fact,

is reinforced by the incorrect subtraction.

Error propagation can be avoided by precoding the data.

The precoding is performed on the binary data sequence

prior to modulation.

From the data sequence { Dn}, the precoded sequence {Pn}

is given by: 2,...,1 , 1 == − mP DP mmm

Modulo-2 subtraction



41



Set I m=-1 if Pm=0 and I m=1 if Pm=1, i.e., I m=2Pm-1.

The noise-free samples at the output of the receiving filterare given by

Since , it follows that the data sequence Dm

is obtained from Bm using the relation:

Consequently, if Bm=±2, then Dm=0, and if Bm=0, Dm=1.

( ) ( ) ( )1 1 12 1 2 1 2 1m m m m m m m B I I P P P P− − −= + = − + − = + −

121

1 +=+ − mmm BPP

1m m m D P P −= ⊕

( )2 mod 12

1+= mm B D



42


Binary signaling with duobinary pulses

The extension from binary PAM to multilevel PAM signalingThe M -level amplitude sequence { I m} results in a noise-free

sequence


,...2,1 ,1 =+= − m I I B mmm

reference



43



which has 2 M -1 possible equally spaced levels.

The amplitude levels are determined from the relation:

where {Pm} is the precoded sequence that is obtained from

an M -level data sequence { Dm} according to the relation

where the possible values of the sequence { Dm} are 0, 1,

2,…, M -1.In the absence of noise, the samples at the output is given

by:

( )12 −−= M P I mm

( ) M P DP mmm mod 1−=

( )[ ]12 11 −−+=+= −− M PP I I B mmmmm



44



Hence

Since Dm=Pm+Pm-1 (mod M ), it follows that

Four-level signal transmission with duobinary pulses

( )12

11 −+=+ − M BPP mmm

( ) ( ) M M B D mm mod 12

1−+=

(M=4)



45



In the case of the modified duobinary pulse, the controlled ISI

is specified by the values x(n /2W )=-1, for n=1, x(n /2W )=1, forn=-1, and zero otherwise.

The noise-free sampled output from the receiving filter is

given as:

Where the M -level sequence { I m} is obtained by mapping a

precoded sequence according to

and

2−−= mmm I I B

( ) M P DP mmm mod 2−⊕=

( )12 −−= M P I mm



46



From these relations, it is easy to show that the detection

rule for recovering the data sequence { Dm} from { Bm} inthe absence of noise is

The precoding of the data at the transmitter makes it possibleto detect the received data on a symbol-by-symbol basis

without having to look back at previously detected symbols.

Thus, error propagation is avoided.

The symbol-by-symbol detection rule is not the optimum

detection scheme for partial-response signals. Nevertheless, it

is relatively simple to implement.

( ) M B D mm mod 2

1=



47


Maximum-likelihood Sequence Detection

Partial-response waveforms are signal waveforms with

memory. This memory is conveniently represented by a trellis.The trellis for the duobinary partial-response signal for binary

data transmission is illustrated in the following figure.

The first number on the left is the new data bit and the number on the

right is the received signal level.



48



The duobinary signal has a memory of length L=1. In general,

for M -ary modulation, the number of trellis states is M L

.The optimum maximum-likelihood sequence detector selects

the most probable path through the trellis upon observing the

received data sequence { ym} at the sampling instants t =mT ,

m=1,2,….

The trellis search is performed by the Viterbi algorithm.

For the class of partial-response signals, the received sequence

{ ym,1≤m≤ N } is generally described statistically by the jointPDF p(y N |I N ), where y N =[ y1 y2 … y N ]’ and I N =[ I 1 I 2 … I N ]’

and N > L.



49



When the additive noise is zero-mean Gaussian, p(y N |I N ) is a

multivariate Gaussian PDF, i.e.,

where B N =[ B1 B2 … B N ]＇ is the mean of the vector y N and Cis the N X N covariance matrix of y N .

The ML sequence detector selects the sequence through the

trellis that maximizes the PDF p(y N |I N ).

Taking the natural logarithms of p(y N |I N ):

( )( )

( ) ( )1

2

1 1| exp

22 det N N N N N N N

pπ

−′⎡ ⎤= − − −⎢ ⎥⎣ ⎦

y I y B C y BC

( ) ( ) ( ) ( ) N N N N N N N p ByCByCIy −′

−−−= −1

2

1det2ln

2

1|ln π



50



Given the received sequence { ym}, the data sequence { I m} that

maximizes ln p(y N |I N ) is identical to the sequence { I N } thatminimizes (y N -B N )＇C-1(y N -B N ), i.e.,

The metric computations in the trellis search are complicated

by the correlation of the noise samples at the output of the

matched filter for the partial-response signal.

In the case of the duobinary signal waveform, the correlation

of the noise sequence {vm} is over two successive signal

samples.

( ) ( )⎥⎦⎤

⎢⎣⎡ −′−= −

N N N N N

N

ByCByII

1minargˆ



51



Hence, vm and vm+k are correlated for k =1 and uncorrelated for

k >1.If we isgnore the noise correlation by assuming that

E (vmvm+k )=0 for k >0, the computation can be simplified to

where

and xk = x(kT ) are the sampled values of the partial-response

signal waveform.

( ) ( )

2

1 0

ˆ arg min arg min N N

N L

N N N N N m k m k

m k

y x I −= =

⎡ ⎤⎛ ⎞′⎡ ⎤= − − = −⎢ ⎥⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠⎢ ⎥⎣ ⎦∑ ∑I I

I y B y B

∑=

−= L

k

k mk m I x B0



52

Signal Design for Channels with Distortion

In this section, we perform the signal design under the condition

that the channel distorts the transmitted signal.

We assume that the channel frequency-response C ( f ) is know for

| f |≤W and that C ( f )=0 for | f |>W .

The filter responses GT ( f ) and G R( f ) may be selected to

minimize the error probability at the detector.

The additive channel noise is assumed to be Gaussian with powerspectral densityΦnn( f ).



53


For the signal component at the output of the demodulator, we

must satisfy the condition

where X d ( f ) is the desired frequency response of the casecade of

the modulator, channel, and demodulator, and t 0 is a time delay

that is necessary to ensure the physical realizability of themodulation and demodulation filter.

The desired frequency response X d ( f ) may be selected to yield

either zero ISI or controlled ISI at the sampling instants.

We shall consider the case of zero ISI by selecting X d ( f ) =

X rc( f ), where X rc( f ) is the raised cosine spectrum with an

arbitrary roll-off factor.

( ) ( ) ( ) ( )02

,

j ft

T R d G f C f G f X f e f W

π −= ≤



54


The noise at the output of the demodulation filter may be

expressed as

where n(t ) is the input to the filter.

Since n(t ) is zero-mean Gaussian, v(t ) is zero-mean Gaussian,

with a power spectral density

For simplicity, we consider binary PAM transmission. Then, the

sampled output of the matched filter is

where x0 is normalized to unity, I m=±d , and vm represents the

noise term.

( ) ( ) ( ) Rv t n t g d τ τ τ ∞

−∞= −∫

( ) ( ) ( )2

vv nn R f f G f Φ = Φ

0m m m m m y x I v I v= + = +



55


vm is zero-mean Gaussian with variance

The error probability is given by

The probability of error is minimized by maximizing the ratiod 2 / σ2

v .

There are two possible solutions for the case in which the

additive Gaussian noise is white so thatΦnn( f )= N 0 /2.1st solution: pre-compensate for the total channel distortion at the

transmitter, so that the filter at the receiver is matched to the

received signal.

( ) ( )22

v nn R f G f df σ ∞

−∞= Φ∫

22

/ 2

2 2 /

1

2 v

y

d v

d P e dy Q

σ σ π

∞−

⎛ ⎞= = ⎜ ⎟

⎜ ⎟⎝ ⎠

∫

SNR

l D f Ch l h D



56


The transmitter and receiver filters have the magnitude

characteristics

The phase characteristic of the channel frequency response

C ( f ) may also be compensated at the transmitter filter.

For these filter characteristics, the average transmitted power

is

( ) ( )( )

( ) ( )

,

,

rcT

R rc

X f G f f W

C f

G f X f f W

= ≤

= ≤

( )( ) ( )

( )

( )

2 2 222

2

W W W m rc

av T T W W W

E I X f d d P g t dt G f df df

T T T C f − − −

= = =∫ ∫ ∫

(9.2-71)

Si l D i f Ch l ith Di t ti



57


Hence,

The noise at the output of the receiver filter isσ2v= N 0 /2 and,

hence, the SNR at the detector is

( )

( )

1

2

2

W rc

avW

X f d P T df

C f

−

−

⎡ ⎤⎢ ⎥=⎢ ⎥

⎣ ⎦

∫

( )( )

1

2

22

0

2 W rcav

W v

X f P T d df N C f σ

−

−⎡ ⎤⎢ ⎥=⎢ ⎥⎣ ⎦∫

(9.2-73)

(9.2-74)




58


2nd solution: As an alternative, suppose we split the channel

compensation equally between the transmitter and receiver filters,

i.e.,

The phase characteristic of C ( f ) may also be split equally

between the transmitter and receiver filter.

The average transmitter power is

( ) ( )( )

( )

( )

( )

1/ 2

1/ 2

,

,

rcT

rc

R

X f G f f W

C f

X f

G f f W C f

= ≤

= ≤

( )

( )

2W

rc

avW

X f d P df

T C f −= ∫

(9.2-75)




59


The noise variance at the output of the receiver filter is

The SNR at the detector is

From Equations 9.2-73 and 9.2-78, we observe that when we

express the SNR d 2 / σ2

v

in terms of the average transmitter power

Pav, there is a loss incurred due to channel distortion.

( )

( )

2 0

2

W rc

vW

X f N df

C f

σ −

= ∫

( )

( )

22

2

0

2 W rcav

W v

X f P T d df

N C f σ

−

−

⎡ ⎤= ⎢ ⎥

⎢ ⎥⎣ ⎦∫ (9.2-78)

Si n l D si n f Ch nn ls ith Dist ti n



60


In the case of the filters given by Equation 9.2-71, the loss is

In the case of the filters given by Equation 9.2-75, the loss is

When C ( f )=1 for | f |≤W , the channel is ideal and

so that no loss is incurred.

When there is amplitude distortion, |C ( f )|<1 for some range of frequencies in the band | f |≤W and there is a loss in SNR.

It can be shown that the filters given by 9.2-75 result in thesmaller SNR loss.

( )

( )

210log

W rc

W

X f df

C f −∫

( )

( )

2

210log

W rc

W

X f df

C f −

⎡ ⎤⎢ ⎥

⎢ ⎥⎣ ⎦

∫

( ) 1W

rcW

X f df −

=∫

Date post:	04-Apr-2018
Category:	Documents
Upload:	abdkabeer-akande
View:	218 times
Download:	0 times

Signal Design

Documents