In this problem, we will develop some of the properties of a class of discrete-time syst...

In this problem, we will develop some of the properties of a class of discrete-time systems called frequency-sampling filters. This class of filters has system functions of the form

where z_k = for k = 0, 1, . . . , N − 1.

(a) System functions such asH(z) can be implemented as a cascade of an FIR system whose system function is (1−z −N) with a parallel combination of 1st-order IIR systems. Draw the signal flow graph of such an implementation.

(b) Show that H(z) is an (N − 1)st-degree polynomial in z ⁻¹. To do this, it is necessary to show that H(z) has no poles other than z = 0 and that it has no powers of z ⁻¹ higher than (N −1).What do these conditions imply about the length of the impulse response of the system?

(c) Show that the impulse response is given by the expression

Hint: Determine the impulse responses of the FIR and the IIR parts of the system, and convolve them to find the overall impulse response.

(d) Use l’Hôpital’s rule to show that

m= 0, 1, . . . , N − 1;

i.e., show that the constants [m] are samples of the frequency response of the system, H(e^jω), at equally spaced frequencies ω_m = (2π/N)m for m = 0, 1, . . . , N − 1. It is this property that accounts for the name of this class of FIR systems.

(e) In general, both the poles zk of the IIR part and the samples of the frequency response [k] will be complex. However, if h[n] is real, we can find an implementation involving only real quantities. Specifically, show that if h[n] is real and N is an even integer, then H(z) can be expressed as

where H(e^jω) = |H(e^jω)|e^jθ(ω). Draw the signal flow graph representation of such a system when N = 16 and H(e^jωk ) = 0 for k = 3, 4, . . . , 14.