From Table 2-3,

\[
y[\cos a k T]=\frac{z(z-\cos a T)}{z^{2}-2 z \cos a T+1}
\]

(a) Find the conditions on the parameter \(a\) such that \({ }_{z}[\cos a k T]\) is first order (pole-zero cancellation occurs).

(b) Give the first-order transfer function in part (a).

(c) Find \(a\) such that \(y[\cos a k T]=y[u(k T)]\), where \(u(k T)\) is the unit step function.

Table 2-3

Transcribed Image Text:

TABLE 2-3 z-Transforms Sequence (k - n) Transform Z k ak kak z-1 Z (z-1) z(z+1) (z - 1) Z za az (z - a) sin ak cos ak a sin bk akcos bk z sina z2 2z cosa +1 z(z - cos a) z22z cos a +1 az sin b 22az cos ba z - az cos b z - 2az cos b + a