ECE300 Communication Theory

Problem Set VII: Digital Baseband Processing

1. Consider a digital communication scheme using 8-PSK, a bit rate of 6M bps, digital

processing with 16 samples per symbol, and employing p

RC pulses with 20% rollo§.

(a) Compute the symbol rate.

(b) Compute the sampling rate for the digital processor.

(c) Compute the bandwidth of the pulses.

2. Here you will use MATLAB to explore pulse design. Take BPSK (symbols 1) with

symbol rate Rs = 1M bps, L = 16 samples per symbol. We want an FIR approximation

to a p

RC Ölter with rollo§ 30%, that spans 4 symbols.

In MATLAB, the following will generate the coe¢ cient vector of the FIR Ölter (i.e.,

the underlying pulse shape):

p = r cos design(beta; span; L;0

sqrt0

);

where here span = 4. Let g [n] denote the pulse shape at the output of the matched

Ölter, which is obtained by convolving p [n] with its áip.

(a) Generate the p vector as above. Let N denote its length. Obtain a stem plot of

p (horizontal axis from 0 to N 1). Find the index where the peak occurs, say

kp_peak. You will note the pulse extends out so that interference comes from

indices kp_peak L and 2L.

(b) Obtain a stem plot of the g vector, as you did for the p vector (careful: it is

longer than p!). Let kg_peak denote the index where the peak occurs. The

ISI at the output of the matched Ölter is determined by indices that are o§set

by L; 2L; 3L; 4L from this peak index. Observe the peak value of g is 1;

determine the maximum (magnitude) of any single interferer; and compute the

signal-to-interference ratio (SIR) in decibels for this case.

(c) Use freqz to compute jP (f)j and jG (f)j, the magnitude spectra of p [n] and g [n],

respectively, at 1000 frequency points from 0 to Rs. Plot jP (f)j on a linear scale

(horizontal axis in Hertz). Obtain a separate plot in which you show jG (f)j,

jG (Rs f)j, and jG (f)j + jG (Rs f)j superimposed (di§erent colors). Then

repeat this plot (of the jGj spectra) with the magnitude on a decibel scale; for

the case of the decibel scale, set the vertical axis limits so the maximum versus

minimum is no more than 30dB.