Assignment: Signals and Systems

Find the z-transform x(z) of x(n) = Cos(0.45n+0.25)u(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.

Find the system transfer function of a causal LSI system whose impulse response is given by h[n]=(-0.5)?^(n-1) sin[0.5(n-2) ]u[n-2] 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.

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.

x[n]= (0.25)^n sin(π/2 n)u[n-1]

Hint: Express sin(π/2n) as sin (π/2 (n-1+1)) = sin (π/2 (n-1) +π/2) and then expand using use the trig identity for Sin(A+B).

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

The transfer function of a system is given by

H(z)= Z/((z^2-0.8z+ 0.15)).

To such a system we apply an input of the type x[n]=e^(-0.4n) "forn"≥0. Find the response of the system in ndomain using MATLAB for obtaining the partial fraction expansion and then manually inverting the output using z-transform tables.

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.