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Cha ter 3: Characterization of

Communication Signals and Systems

Department of Electrical and Computer Engineering

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Spectral Characteristics of Digitally Modulated Signals

Generally, the available channel bandwidth is limited In the selection of the modulation methods, we

signals Helps to take the effect of the BW constraint into account

A digitally modulated signal is a stochastic process since

the information sequences are random ee to eterm ne power spectra ens ty o t ese

processes

transmit the information-bearing signals

Digital Communications – Chapter 3: Communication Signals & Systems 2Sem. I, 2012/13

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Spectral Characteristics of Digitally …

Considering linearly modulated band-pass signal given by

t π f 2 j cev t e s t

Where (t) is the equivalent low-pass signal Autocorrelation function of s(t) is

And its Fourier transform ields the desired ex ression for

c

f 2 j

vvss e )( Re )(

the power density spectrum ss(f) as

)( Φ )( Φ1

( )Φ

Where vv(f) is the power density spectrum of (t)


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To obtain the spectral characteristics of the bandpasssignal s(t), it suffices to determine the autocorrelation

-

pass signal (t) Consider a linear di ital modulation method for which t is

represented in the general form

nT t v t n

Where:

n

n -bit blocks into corresponding points

1/T = R/k symbols/s is the transmission rate


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The autocorrelation function of (t) is

τ )v(t (t)v E 1t)τ ,(t vv

mT)τ g(t nT)(t g I I E 21

m n

mn

Assuming the sequence of information symbols {In} is wide-

sense stationary with mean μiand autocorrelation function

Then the autocorrelation of (t) will be

mnnii I 2

(m)

n m

iivv

mT nT τ t nT t m

mT)τ g(t nT)(t g n)(mt)τ ,(t


m n

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The 2nd summation is periodic with T Thus the autocorrelation function is also periodic, i.e.,

Further the mean of (t); );( ) ,( ttTtTt vvvv

nT) g(t μ E[v(t)] i is also periodic with period T

autocorrelation function

Called a cyclostationary or a periodically stationary process

To compute the power spectral density of (t+,t), wemust eliminate the dependence on the t variable by

averaging over a single period


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2T/

2T/

2T/ vvvv

1

dt t)τ ;(t T

1 )(

nT 2T/

2T/ nm

ii

1

t mnτ g t nt g

T

m

In the above expression, the integral can be interpreted as

nT 2T/

nm T

e me au ocorre a on o e unc on g

dt τ ) g(t (t) g )( gg

So that

mT τ m1


T gg

m

vv

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Or, the spectral characteristics of (t) can be controlled bythe choice of g(t) and the correlation characteristics of the

Note also that Φii(f) is related to the autocorrelation ii(m) inthe form of an exponential Fourier series with ii(m) as the

Fourier coefficient such that

df e f T m

T

fmT jiiii

2/1

2)()(

Consider the information symbols in the sequence are realand mutually uncorrelated such that

T 2/1

)(

)( 0m

m2

2

i

2

i

ii


i

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Where i2 is the variance of the information sequence; then

π fmT 2 j22

Which is periodic with period 1/Tm

The above can be viewed as the exponential Fourier seriesof a periodic train of impulses each with an area of 1/T; i.e,

)T

mδ(f

T

μσ (f)Φ

m

i2

iii

m f δmG μG(f)σ (f)Φ

22

i22

ivv


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The first term is a continuous spectrum and its shapedepends on the spectral characteristics of signal pulse g(t)

The second expression contains discrete frequencycomponents spaced 1/T apart in frequency2

evaluated at f=m/T

If the information se uences have zero mean the discretefrequency components will vanish, i.e., μi =0

This property is most desirable for digital modulation andcan be achieved when the information sequences areequally likely & symmetrically positioned in a complex plane


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It contains zeros at multiples of 1/T in frequency and it alsodecays inversely as the square of the frequency variable

,

in (f) vanishes Thus, upon substitution for G(f)from above, we get

δ(f) μ Aπ fT

π fT T Aσ (f)Φ 2

i

2

2

22

ivv

sin


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Example 2: Consider the case where g(t) is a raised cosinepulse

T t or t T

t g

0)2

(cos12

)(


Raised cosine pulse and its energy density spectrum

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Its Fourier transform is given as

ππ π fT T sin22

)T f π fT(12

, , …….

Hence, all the spectral components, except those at zeroand f =+1/T vanish

Compared to that of the rectangular pulse, the spectrumhas a broader main lobe but the tails decay inversely as f 6


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Spectrum can also be shaped by operations performed onthe input information sequence

n

form the symbolIn= bn +bn-1

Where the {bn} are assumed to be uncorrelated random

variables, each having zero mean and unit variance e au ocorre a on o e sequence n s

I I E (m) mnnii

1(m1m


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The power spectral density of the input sequence is

π fT cos4π fT)2cos2(1(f)Φ 2

Then, the corresponding power spectral density of the (low-ass e uivalent modulated si nal becomes

fTf GT

4f 2

2

vv

cos )( )(


lect_12. spectral (1)

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