Lab 1. Exploring nonlinear signal processing with MATLAB on a PC.

Lab 1. Due 9/ 7/04.

 

Exercises to be done using MATLAB.  A PC will be available in 87MSC. We can have a discussion if it is necessary. You can do these exercises on any PC that has MATLAB. Any supplied m-files can be obtained from the class WEB site. There is an editor for writing m-files in MATLAB.

Do… FILE…NEW… m-file , then enter your code.

You have to save it in your folder.

To execute the m-file… you have to tell MATLAB where the file is. cd c:\phy735\your_name  does the job.

 

Problem 1. In order to familiarize everyone with the use of the discrete Fourier transform:

generate y(t) = 10*sin(2*pi*f*t) where f = 100Hz. Make the signal 1 second long.  Use a power of 2 for the number of samples. Note: the Nyquist theorem states you have to sample a signal at least twice as fast as the highest frequency present.  It is often better to sample at least 5Xs  faster than the highest frequency, especially, if you want a nice looking waveform to plot.

Plot the waveform, and its Fourier magnitude and phase.  Note: one variation of plotting in MATLAB uses subplot(n,m,i) where n=# of rows of plots, m=# of columns and I is the index of the current plot. E.g., if you are placing three plots on a single page and doing the second plot of the series, the MATLAB code would be:

Subplot(3,1,2)

Plot(x,y)

 

There will be an MATLAB m-file, L1P1, available as a starter for those who have difficulty with this exercise.  It can be used as a template for several of the following exercises. You can display any m-file in MATLAB by typing dbtype FileName or you can do File then Open filename. The code can be printed by selecting the lines on the screen you want with the mouse and then use the File command: followed by print selection.

 

Problem 2. Clip the waveform from Problem 1 at the following levels: 100%, 90%, 75%, 50%, 25% and 10 % of the maximum & minimum signal level, i.e., this is a symmetric clipping function. What amount of the total signal energy is present in the each of the first three harmonics for each three conditions.  Make a table and/or graph of the results.  Note that as the clipping level gets lower that the signal is essentially a square wave. When graphing this us the subplot function to place the results on a single page.

 

Problem 3. Distortion of signals results in the generation of components that are at frequencies that are not present in the input signal.  One can represent the distortion process with a polynomial function.  If x(t) is the input and y(t) the output, then we can write y = aFx + bFx2  +cFx3 +…

set a = 0, b= 1, and c=0.  Let the input consist of two sinusoids with frequencies of 1000 and 1100Hz.

What frequency components are present in the output signal?

 

Note that you can do this analytically.  It is just trigonometry.  However, if you are a bit rusty…. You can use MATLAB to compute the waveforms and then Fourier analyze them. See below.

 

Remember: Phythagorus: the Square of the hypotenuse = the sum of the squares of the sides of a right triangle.

sin2(x) + cos2(x) = 1.

sin(A ±  B) = sin(A)cos(B) ± sin(B)cos(A)

cos(A ±  B) = cos(A)cos(B) -+ sin(B) sin(A)

and if you set A = B = x  you can derive:

cos2(x) = (1 + cos(2x))/2.                                     

 

Problem. 4. With the same frequency components as in Problem 3, set a = 0, b = 0 and c = 1.

What frequency components are present in the output.  Analytically or with MATLAB FFT.

 

Problem 5. Same as Problem 4 but set a = 1 and c = .05.

What is the spectrum?  Plot it.

 

Problem 6.  A special case occurs when the ratio of the two frequency components is set to 1.5  (f2/f1 = 1.5, f1 = 1000Hz).  Set a = 1, b = c = 0.05.  Vary the phase angle between the two primary frequencies and plot the magnitude of the difference frequency, f2-f1 and the most prominent distortion product heard by animals and humans, 2f1-f2, often labeled the cubic difference tone (CDT).  We will discuss these distortion products and their generation in the auditory system in greater detail later. Use MATLAB to obtain the results.

Introduction to Matlab 6.5

 

Double click on the Matlab 6.5 icon.

At the Matlab prompt >>       Type demo

Click on matrices  (this is all Matlab really understands, i.e., matrices).

Then Run Basic matrix.   Continue through the slides.

When done click on close.

Then select numerics.

Select Fast fourier transform.   Run fast fourier transform

 Sequence the slides.

At end close.

 

Your now ready to enter a set of instructions for Matlab to execute.

Click on File on the toolbar.

   Select New and then m-file.

   You are now in an editor/debugger.

  Type in a series of instructions. A % sign indicates a comment

      e.g.  

      x=1:2:20;  % says set the x variable to 1,3,5,7,… 19

      y=x .^2; % says make y= each element of x squared.

    Plot(x,y)

Pause;  % to advance beyond  this point hit any key

  Plot(x,y,’*r’); % this says plot the curve using red asterisks.

 

%save this file with a name such as temp in your directory

%then close the editor; this returns you to Matlab

 

type cd c:\phy735\your_name

temp %this will run the series of instruction that you just saved as an m-file.

 

How to obtain the FFT of a  signal….

%fft_example.m

t=0:.025:10;

y=sin(2*pi*t*2)+.5*sin(2*pi*t)+.4;

fy=fft(y); mfy=abs(fy);

[a,n]=size(y); dc=mfy(1)/n;

fresol=1/max(t); freq=0:fresol:fresol*(n/2);

amfy(1)=dc; maxn=round(n/2);

amfy(2:maxn)=2*mfy(2:maxn)/n;

plot(freq,amfy)