0
424views
Determine the z-transform of the signals (a) x(n)=(cosω0n)u(n) (b) x(n)=(sinωk,n)u(n)
1 Answer
0
7views

Solution:

(a) By using Euler's identity,

the signal x(n) can be expressed as,

x(n)=(cosω0n)μ(n)=12ejmannu(n)+12ejωpnu(n)

that,

X(z)=12Z{ejmovnu(n)}+12Z{ejmovnu(n)}

If we set α=e±jω0(|α|=|e±ja0|=1) in (3.2.2).

we obtain,

ejman)nu(n)11e/ω0z1 ROC: |z|>1

and

Thus

X(z)=1211ejω0z1+1211ejω1z1ROC:|z|>1

(b) From Euler's identity.

Thus,

x(z)=12j(11esinz1111etanz1) ROC: |z|>1

Time shifting. If,

x(n)zX(z)

then,

x(nk)zkX(z)

The ROC of zkX(z) is the same as that of X(z) except for z=0 if k>0 and z= if k<0. The proof of this property follows immediately from the definition of the z-transform.

The properties of linearity and time shifting are the key features that make the z-transform extremely useful for the analysis of discrete-time LTI systems.

Please log in to add an answer.