Umal-qura university

Lab # 9

OBJECTIVES OF THE LAB

This lab aims at the understanding of:

• 
  Power of Continuous & Discrete time Signals
  
• 
  Synthesis of Square Wave
  
• 
  Synthesis of Triangular Wave
  

1. 
   SIGNAL POWER
   

Average power of continuous time signal can be calculated using the formula:

To carry out the integral, Euler Approximation can be used. It simply tells that a definite integral can be approximated using a sum i.e.
In this method, the region over which integral is carried out is divided into N parts or intervals, each of duration t, such that function stays constant over those short intervals. Approximating function in this way is shown below. Note that as the number of intervals N is increased, the approximation gets better.

Approximating integrals using sums is a deep subject of numerical analysis by itself; therefore its further detail is out of scope. It is enough to know that Euler’s formula is easy to implement and produces good results for almost all the signals that will be studied here as long as N is selected large enough.

Example – Power of Continuous Time Cosine

clc clear all close all

t = ‐1:0.005:0.995; % time duration of given signal; xt = cos(2*pi*t/2); % generate signal

plot(t, xt); % plot signal
xlabel('time, t'); ylabel('Amplitude, A'); title('Continuous Time Cosine');

abs_xt_2 = abs(xt).^2; % take absolute square of signal

T = 2; % length of interval
delta_t = 0.005; % interval duration
pxt = sum(abs_xt_2)*delta_t/T % power of given signal
pxt =
0.5000

2. FOURIER SERIES

Fourier series theory states that a periodic wave can be represented as a summation of sinusoidal waves with different frequencies, amplitudes and phase values.

1. Synthesis of Square wave

The square wave for one cycle can be represented mathematically as: x(t) = 1 0 <= t < T/2
‐1 T/2 <= t < T The Complex Amplitude is given by:
^Xk ^{= (4/j*pi*k) for k=±1, ±3, ±5…..}

0 for k=0,±2, ±4, ±6…..

For f = 1/T = 25Hz, only the frequencies ±25, ±50, ±75 etc. are in the spectrum.

1.