6.2 - The power Spectrum of a Digital PAM Signal

A digtal PAM signal at the input to a communication channl

scale factor (where 2d is the “Euclidean distance”)

pulse waveform

reciprocal of the symbol rate

to determine the spectral characteristics of the random process V(t), we must evaluate the power spectrum.

mean value

autocorrelation function

average autocorrelation function

The power spectrum of V(t) is the Fouriertransform of the average autocorrelation function.

The power spectrum of amplitude sequence an

Problem 6.1

Answerip_06_01MATLAB.lnk

Determine the power spectrum of V(t) when {an}is an uncorrelated sequence and g(t) is the followingrectangular pulse.

The Fourier trasform of g(t) is

and

This power spectrum is illustrated in the followingFigure:

Problem 6.2

Answerip_06_02MATLAB.lnk

otherwise

m

m

mRa

,

,,

,

)(

0

112

1

01

Suppose the autocorrelation function of the sequence {an} is

and g(t) is the rectangular pulse.shown at the previousproblem. Evaluate Sv(f) in this case.

SOLUTIONThe power spectrum of the PAM signal was given by

from

and using also the data we can replace...

Consequently,

This power spectrum is illustrated in the followingFigure:

6.3 - Characterization of bandlimited channels and channels distortion

We can characterize many communication channels as bandlimited linear filters, described by their frequency response

Amplitude responsephase response

:

we can also express it as an envelope delay (group delay)

nondistorting (ideal) channelAmplitude response -constantphase response - linear function of frequncy(envelope delay -constant)

distorting channelAmplitude distortion -Amplitude response isn’t constantdelay distortion -envelope delay isn’t constant)

Effect of channel distortionchannelinput

channeloutput

equalizer

Average amplitude and delay characteristicsof a medium-range telephone channel,

Impulse response of the average channelwith amplitude and delay shown before

Problem 6.3

Answerip_06_03MATLAB.lnk

T.B.D

Impulse response

Frequency responseof linear phase FIR filter

Problem 6.4

Window method

Impulse response

if the desired channel responseis C(f) for |f|<W and C(f)=0 for|f|>W then

for example: if channel is ideal, then C(f)=1, |f|<Wand hence

Answerip_06_04MATLAB.lnk

Let us design an FIR filter with:W=2000 Hz and Fs =1/Ts=10KHZ.(this digital filter may be implemented by sampling h(t) at t=nTs, where Ts is the sampling interval and n = 0,+/-1,+/-2,...)

Samples of h(n)

Frequency response (rectangular window)

Frequency response (Hanning window)

Problem 6.5

The impulse response of a two-path(multipath) radio channel can expressed as:

random processes

the delay between 2 paths

Answerip_06_05MATLAB.lnk

simulate such a channel on the computer