ECE561 Assignment 1

Digital Signal Processing Assignment 1

Submit the assignment in the prescribed format.

Do Mitra problems 2.1 through 2.5 inclusive.
Do Mitra problems 2.19, 2.23, and 2.28
Do Mitra problem 2.30. Verify your answer by plotting each sequence in Matlab over two periods.
Label a drawing of the function f(t) = A tri((t-t₀)/c). Generate an example in Matlab with A = 3, c = 5, and t₀ = 7.
Express the following functions in terms of scaled linear combinations of rect, tri, and step functions. Plot each function in Matlab using sample values of the parameters.

(a)
(b)
(c)
The decaying exponential is a basic signal in DSP because it occurs as the solution to linear-constant-coefficient difference equations.
1. Study the following Matlab function to see how it generates a discrete-time exponential signal. Then use the function to plot the exponential x[n] = (0.9)^n over the range n = 0:20.
```
function y = genexp(b, L, n0)
%GENEXP  generate an exponential signal
%    usage:  Y = genexp( B, L, N0 )
%
%       B  input scalar giving ratio between terms
%       L  length of generated signal
%      N0  starting index (integer)
%
%       Y  output signal Y(1:L)
if (L<=0)
   error('GENEXP: length not positive')
end

nn = n0 + [1:L]' - 1;  %  vector of indices
y = b.^nn;
```
2. In many derivations the exponential sequence a^n u[n] must be summed over a finite range. This sum is known in closed form:
  
  Use the results from the previous part to sum an exponential and compare the results to the formula (evaluated in Matlab)
3. Another way to generate an exponential signal is to use a recursive formula given by a difference equation. The signal y[n] = a^n u[n] is the solution to the following difference equation when the input, x[n] is an impulse:
  y[n] - a y[n-1] = x[n]
  with initial conditions y[-1] = 0. In Matlab the function filter will implement a difference equation. Use filter to generate the same sequence as in part (a)
Using the speech signal mtlb, seperate the syllables so that the phrase is "labmat" and plot the results. Reverse the signal (play it backwards) and plot the results. Remember to show the matlab operations you used for these tasks.

Maintained by John Loomis, last updated 21 Aug 2005

(a)
(b)
(c)