State University of New York at Buffalo

CSE 473/573 Fall 2011 Homework Set #1

Date: Monday September 12, 2011; Due: Monday September 26, 2011 at the start of Class

Name: ____________________________ Student Number: ______________________ Problem (1) (Convolution) 20%

Let the input signal be and let the system impulse response be , which are defined as follows:

2 1 3 and

2 1 2 1 . in which the unit impulse function is defined as:

0,1,

00

Compute and hand plot each of the following convolutions: (a) ∗ (b) 2 ∗ (c) ∗ 2

Problem (2) (Fourier Series) 20%

Consider the following three continuous-time signals with a fundamental period of 12:

cos 4 , sin 4 ,

(a) Determine the Fourier Series coefficients of . (b) Determine the Fourier Series coefficients of . (c) Use results of parts (a) and (b), along with the multiplication property of the Fourier

Series, to determine the Fourier Series coefficients of . (d) Determine the Fourier Series coefficients of through direct expansion of in

trigonometric form, and compare your result with that of part (c).

Problem (3) (MATLAB) 60% In this problem, we will consider a technique known as block convolution, which is often used in real-time implementation of digital filters for speech processing and image processing in which short processing delay may be desired. This technique is particularly useful when processing long input sequence with a relatively short duration filter. The input sequence is broken into short blocks, each of which can be independently processed with relatively little delay. The linearity of convolution guarantees that the superposition of the outputs from all individual blocks will be equal to the convolution of the entire sequence with the impulse response of the filter. For this problem, you will perform each of the small convolutions using the MATLAB function conv. To illustrate the procedure, assume that you have a filter with a finite-length impulse response which is nonzero only on the interval 0 1. Also assume that the input sequence is 0 for 0 and that the length of is significantly greater than . You can break the signal into segments of length ,

,

where , and ,

00 1,

otherwise,

For this problem, we define the unit step function as:

0,1,

00

(a) For 0.9 10 and cos sin 2 5⁄ , compute

∗ for 0 99 directly using conv. Make a plot of over this range using stem.

(b) For 50, break the sequence into two sequences, each of length 50. Compute ∗ , and ∗ , where contains the first 50

samples of and contains the second 50 samples of . The form of the output is given by:

∗ Determine the appropriate value of to use, and note that and will both be of length 1. When and are added together, there will generally be a region where both are nonzero. It is for this reason that this method of block convolution is called the overlap-add method. Compute in this manner, and plot over the range 0 99. Is your result the same as what you found in Part (a)?