Lab 10: Fourier Matlab

Table of Contents

1. Introduction
2. Procedure
3. Feedback

1. Introduction

This exercise show cases the digital signal processing toolbox of MATLAB. A Fourier series is an expansion of any periodic function in terms of an infinite weighted sum of sine waves. Here the Fourier series is been approximated from a finite weighted sum of sine waves instead of infinite sum.

2. Procedure

2.1. Introductory example

In this part we would form a square wave using a sum of odd harmonics.

Using Fourier series to approach square wave

Figure 1. Adding sine waves to produce square wave

As we can see, adding more weighted odd harmonics waves the result would approach to a square wave.

The heard square wave is more bright then the sine wave and the sine wave is more smooth to hear.

2.2. Approximation of a square wave using the Fourier series

CODE 2 implements the Fourier series by computing the coefficients of the series via symbolic integrator of MATLAB, then using a for-loop to approach the infinite sum.

Observing the plot we can see the N_TERMS = 7 approximation has the greatest error.

Figure 2. Approximation of square wave using various number of terms

Table 1. Overshoot for different number of terms of the FS

Number of terms	Overshoot
7	0.0921
20	0.0899
30	0.0897
50	0.0896

Table 2. Computational time for different number of terms of the FS

Number of terms	Computational time
7	0.055059
20	0.097536
30	0.160673
50	0.260833
100	0.529958

Basically, the balance between the precision and computational cost for using Fourier series is determined by the sample rate used for the problem. With a higher sample rate there needs more computational power for better result, with lower sample rate the precision can be sacrificed.

2.3. Approximation of a triangular wave using the Fourier series

To produce the triangular wave:

  |t=-10:0.0001:10;
  |x=(t/(2*pi) - floor(t/(2*pi)));
  |plot(t,x)

Figure 3. Approximation of triangular wave using various number of terms

3. Feedback

This lab shows how to use the symbolic integrator of MATLAB combined with array computation to implement Fourier series, it also demonstrates how to produce common signals from a sum of harmonic waves.

For Part B, the code used to compute the coefficient is unnecessarily inefficient, since it would not only cause computes the integration for an and bn each N_TERMS times, it is also using fractions and symbolic terms instead of floating point numbers which makes the computation slow, and the result is symbolic which makes computing the overshoot with code much slower if not taking a different approach.

I used the following program to computer the overshoot, note that an is essentially zero:

   |L = pi; N_TERMS = 50; n = 1:N_TERMS;
   |t = -10:0.0001:10;;
   |a0 = 1.0;
   |% an = (1/L)*int;
 5 |% bn = (1/L)*int(1*sin(n*x), x, 0, pi);
   |f = 0;
   |for n = 1:N_TERMS
   |f = f+0+(1/L)*((2*sin((pi*n)/2)^2)/n)*sin(n*t));
   |end
10 |f_approx = (a0/2)+ f;
   |err = max(f_approx) - 1