ECET345 Signals and Systems Homework #6

ECET345 Signals and Systems

Homework #6

Name of Student _________________________________________

1. Find the z-transform x(z) of x(n) = . Hint: Follow the method used in the lecture for Week 6. Also, when evaluating the numerical value of a trig function, keep in mind that the arguments of trig functions are always in radians and not in degrees.

2. Find the system transfer function of a causal LSI system whose impulse response is given by

and express the result in positive powers of z. Hint: The transfer function is just the z-transform of impulse response. However, we must first convert the power of -0.5 from (n - 1) to (n - 2) by suitable algebraic manipulation.

3. Express the following signal, x(n), in a form such that z-transform tables can be applied directly. In other words, write it in a form such that the power of 0.25 is (n-1) and the argument of sin is also expressed with a (n-1) multiplier.

Hint: Express sin(n) as sin ( (n-1+1)) = sin ( (n-1) +) and then expand using use the trig identity for Sin(A+B).

4. The transfer function of a system is given below. Find its impulse response in n-domain. Hint: First expand using partial fraction expansion and then perform its inversion using z-transform tables

5. The transfer function of a system is given by

.

To such a system we apply an input of the type . Find the response of the system in n domain using MATLAB for obtaining the partial fraction expansion and then manually inverting the output using z-transform tables.

6. A simulation diagram is shown below. We apply a unit impulse to such a system. Determine the numerical values of the first three outputs. You are free to use MATLAB where appropriate or do it entirely by hand.