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Simulation in Digital Communication

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Simulation in

Digital Communication

chapter # 6

Digital Transmission Through Bandlimited Channels

6.2 - The power Spectrum of a Digital PAM Signal

A digtal PAM signal at the input to a communication channl

reciprocal of the symbol rate

pulse waveform

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

to determine the spectral characteristics of

the random process V(t), we must evaluate

the power spectrum.

1)mean value

2)autocorrelation function

3)average autocorrelation function

The power spectrum of V(t) is the Fourier

transform of the average autocorrelation function.

The power spectrum of amplitude sequence an

Problem 6.1

Answer

ip_06_01

Determine the power spectrum of V(t) when {an}

is an uncorrelated sequence and g(t) is the following

rectangular pulse.

The Fourier trasform of g(t) is

and

This power spectrum is illustrated in the following

Figure:

Problem 6.2

Answer

ip_06_02

Suppose the autocorrelation function of the sequence

{an} is

and g(t) is the rectangular pulse.shown at the previous

problem. Evaluate Sv(f) in this case.

SOLUTION

The 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 following

Figure:

6.3 - Characterization of bandlimited channels and channels distortion

We can characterize many communication

channels as bandlimited linear filters, described

by their frequency response

phase response

Amplitude response

we can also express it as an

envelope delay (group delay)

:

nondistorting (ideal) channel

Amplitude response -constant

phase response - linear function of frequncy

(envelope delay -constant)

distorting channel

Amplitude distortion -

Amplitude response isn’tconstant

delaydistortion-

envelope delayisn’tconstant)

Effect of channel distortion

channel

input

channel

output

equalizer

Average amplitude and delay characteristics

of a medium-range telephone channel,

Impulse response of the average channel

with amplitude and delay shown before

Problem 6.3

Answer

ip_06_03

T.B.D

Impulse response

Frequency response

of linear phase FIR filter

Problem 6.4

Window method

if the desired channel response

is C(f) for |f|<W and C(f)=0 for

|f|>W then

Impulse response

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

and hence

Answer

ip_06_04

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

Answer

ip_06_05

simulate such a channel on the computer