CTU: EE 343 – Signals and Systems: Lab 2: Convolution in MATLAB 1

Colorado Technical University EE 343 – Signals and Systems

Lab 2: Convolution May 2010

Loren Schwappach

ABSTRACT: This lab report was completed as a course requirement to obtain full course credit in EE343, Signals and Systems at Colorado Technical University. Given several input responses this lab report uses MATLAB to examine the principle of convolution.

If you have any questions or concerns in regards to this laboratory assignment, this laboratory report, the process used in designing the indicated circuitry, or the final conclusions and recommendations derived, please send an email to LSchwappach@yahoo.com. All computer drawn figures and pictures used in this report are of original and authentic content.

I. INTRODUCTION

MATLAB is a powerful program and is useful in the

visualization of mathematics, physics, and applied

engineering. In this lab exercise MATLAB will be used to

determine the output response of discrete convolution

problems.

Given the following impulse response:

.

Use MATLAB to find the output response, y[n], by convolving

several input responses with the impulse response. The main

system impulse response is shown in figure 1 below.

MATLAB code for impulse response:

>> n = [0:10];

>> h = 3 * (((2/3)*ones(1,11)).^n);

>> stem(n,h)

Figure 1: Impulse Response h[n]

II. PROCEDURE / RESULTS

After defining our impulse response in MATLAB we can now calculate various output responses by convolving their input function with the defined impulse response.

Our first demonstration of convolution involves the

main impulse response convolved with the shifted impulse response below:

.

The MATLAB code need to create this shifted impulse response is below as is it’s stem graph. >> x = [0 1 0 0 0 1 0 0 0 0 -1]; >> stem(n,x)

Figure 2: Shifted Input Impulse Response x[n] Now the impulse response h[n] must be convoluted with the shifted impulse response x[n] to produce the output response y[n]. This is accomplished with the conv() function in MATLAB. The results of this convolution and MATLAB code follow.

CTU: EE 343 – Signals and Systems: Lab 2: Convolution in MATLAB 2

MATLAB Code: >> n = [0:20]; >> y=conv(x,h); >> stem(n,y)

Figure 3: Output response y[n] Next a unit step response x[n] = u[n] is represented in MATLAB and convolved with the input response h[n] to produce a second output response y[n]. MATLAB Code: >> x = [ones(1,11)]; >> y=conv(x,h); >> n = [0:10]; >> stem(n,x)

Figure 4: Step Response x[n] = u[n]

>> n = [0:20];

>> stem(n,y)

Figure 5: Output Response y[n]

Finally, the input of a pulse response x[n] defined below is convolved with the original impulse response h[n].

MATLAB Code: >> x = [0 0 1 1 1 1 0 0 0 0 0]; >> n = [0:10]; >> stem(n,x)

Figure 6: Rectangular Pulse Response x[n] >> y=conv(x,h); >> n=[0:20]; >> stem(n,y)

CTU: EE 343 – Signals and Systems: Lab 2: Convolution in MATLAB 3

Figure 7: Output Response y[n]

III. EVALUATION

. By hand calculating the output response results from the convolution of the impulse response h[n] with the input of the shifted impulse response:

. You can see (Hand calculations are attached to this report). the output response y[n] is simply the summation of the impulse response h[n] occurring at each of the impulses defined in x[n]. This is the beauty of impulse functions, and verifies that are MATLAB data is correct.

IV. CONCLUSIONS

. MATLAB is a great utility for representing complex

concepts visually and can easily be manipulated to show

signals in various formats. This lab project was successful in

demonstrating MATLABs powerful features in a quick and

easy method, and demonstrating how MATLAB can be used

for convolving discrete-time signals.

REFERENCES

[1] Haykin, S., “Signals and Systems 2nd Edition” McGraw-Hill, New York, NY, 2007.