ECE 381: Laboratory 10

Winter 2006

Discrete Time difference equation and convolution.

April 04

    

Assignment

 

I.  Numerical solutions of Discrete-time Difference Equations using MATLAB

 

1.      Solve the following difference equation numerically using MATLAB.  Show the stem plot for the result for n=-5:1:10

 

            y[n] -1.1y[n-1] +0.3y[n-2]=0;  y[-1]=9, y[-2]=49/3

 

      Refer to Computer Example C3.3 on page 274 of the course textbook for help.

 

2.      Because the difference equation in (1) has no input, the output y[n] is also the zero-input response. 

       The zero-input response to this equation was derived in class as

 

            y[n]=(2.*(.5).^n+3.*(0.6).^n).*u[n]

 

      Show this result on a stem plot also to show that this equation matches the result found for part (1).  Note, however, that the closed form  expression of y[n] that is given here only applies for n>=0, so you must append values to y to also account for values of n in -5<=n<=-1.

 

II. The MATLAB filter function

 

3.      MATLAB includes a "filter" function that allows one to provide the a and b coefficients from a difference equation and produce the filtered output of x[n] that results from using that transfer function.  Recall that the a coefficients would come from Q(E) in

1.             Q(E)y[n]=P(E)x[n] and b's from P(E). 

 

The form of filter is "h = filter(b,a,x(n))" where b and a are vectors of coefficients b = [b0 b1 b2 ...] and a = [1 a1 a2 ...].  Use "help filter" in MATLAB if more information is needed.

 

      Show the output of the MATLAB filter function for the following difference equation.

 

            y[n] - 0.6y[n-1] -0.16y[n-2] = 5x[n]

     x = inline('n==0') which means x is the DT impulse function

 

2.      Using the approach you used in problem (1), show the output of the difference equation in problem (3) and show

      that the results are the same.

 

III. Convolution using MATLAB

 

3.      Refer to Computer Example C3.7 in the course textbook in page 299.  This describes the DT convolution function that is available in MATLAB.  Show a stem plot for the convolution of the following two signals.  Make sure to appropriately label the time axis n.

 

                        x[n] = (0.8).^n.*(u[n] - u[n-20])

          h[n] = u[n-3]-u[n-20]

 

4.      Refer to the lecture notes for Section 3.8 were a convolution example very similar to this was given.  An error was made in the computation, so the answer should really have been -5.*((0.8).^(n-2)-1).*u[n-3].  Plot that result that for 0<=n<=40 and compare it to the plot for problem (5).  Explain where and why the plots are similar and where and why the plots are different.

 

IV. Using MATLAB to Find the Zero-state Response

 

5.      Now as a final exercise we wish to use MATLAB to find the zero-state response of a system.  This involves (1) finding the impulse response, and (2) finding yzs[n] as the convolution of h[n] and x[n].  Find and show the stem plot for zero-state response of the following difference equation.

 

                        y[n] - 5y[n-1] +6y[n-2] = x[n-1] + 5x[n-2]

          x[n] = (3.*n+5).*(n>=0)

 

(a)   Show the stem plot for the impulse response.

(b)   Show the stem plot for the zero-state response yzs[n] = x[n] * h[n]

 

6.      Produce the result for (7) using the MATLAB filter command.