EE313 Linear Systems & Signals - Homework 5 Hints

• Keep your work on each problem together. This makes it much easier for the grader to give you the appropriate full or partial credit due for the solution.

• Problem 5.1 The discrete-time unit step signal models a sudden change that persists, e.g. hitting the brake in one's vehicle and keeping the brake pressed down to avoid hitting an object. One can think of the unit step signal as turning on at n = 0 and staying on forever. It is denoted u[n] and has a value of 1 when n ≥ 0 and 0 otherwise. In terms of modeling, 1 means "on" and 0 means "off".

  MATLAB has a function similar to the discrete-time unit step called heaviside. The difference is that at the origin, the heaviside function has value of 1/2 instead of 1. Please don't use the heaviside function for the discrete-time unit step function.

  MATLAB had a function to implement the discrete-time unit step function u[n] called stepfun but it is obsolete. You can mimic u[n] using comparison operations in MATLAB. In MATLAB, a comparison operation will return 1 if true and 0 if false. For example,

  1 >= 0
  

  returns 1, and

  -1 >= 0
  

  returns 0. Also,

  [-1 0 1] >= 0
  

  returns [0 1 1]. We can implement u[n] where n could be a vector as

  n >= 0
  

  In MATLAB, we could express (1/2)ⁿ u[n] for -5 ≤ n ≤ 10 as follows:

  n = -5:10;
  unitstep = ( n >= 0 );
  x = ((0.5).^n) .* unitstep;
  

• Problem 5.2. A causal system depends only on current and past input values, and past output values, to compute the output signal.

• Problem 5.3 is asking to find y₂[n] which is the output of the system to the input x₂[n] = 3 u[n] - 2 u[n - 4] where u[n] is the discrete-time unit step signal. The unit step signal u[n] has value 1 for n >= 0 and 0 otherwise. See information above about the discrete-time unit step signal.

  You are asked to find the value of the output y₂[n] given the input x₂[n] based on the knowledge that the system responds with output y₁[n] = d[n] + 2 d[n - 1] - d[n - 2] to the input x₁[n] = u[n], where d[n] is the discrete-time unit impulse.

  In part (a), you're asked to determine the output y₂[n] given the input x₂[n] based on the the knowledge that system has linearity and time-invariance (LTI) properties and based on the response y₁[n] to the input x₁[n]. A system that satisfies the linearity property also satisfies the homogeneity and additivity properties.

  In part (b), you'll use deconvolution to compute the LTI FIR filter coefficients (i.e. LTI FIR filter impulse response) based on the response y₁[n] to the input x₁[n]. With knowledge of the LTI FIR filter coefficients, one can compute the output signal given any input signal. You'll then use the LTI FIR filter impulse response to compute the output y₂[n] for input x₂[n].

bevans@ece.utexas.edu