Question

Use the linearity of the z-transform and the transform of the exponential function to obtain the transforms of the discrete-time functions.

(a) sin(kwT)
(b) cos(kwT)

Answer 1

The z-transform can be used to find the transforms of discrete-time functions. The z-transform of sin(kwT) is (z^(k) - z^(-k))/(2iz^(wT)), while the z-transform of cos(kwT) is (z^(k) + z^(-k))/2.

Step-by-step explanation:

The z-transform is a useful tool for analysing discrete-time signals and systems. It allows us to convert a discrete-time function into the z-domain, where it can be manipulated and analysed. One of the key properties of the z-transform is linearity, which allows us to apply it to linear combinations of functions.

To find the z-transform of the function sin(kwT), we can use the identity sin(x) = (e^(ix) - e^(-ix))/(2i). We can then apply the z-transform to each term separately, using the linearity property. This gives us the z-transform of sin(kwT) as (z^(k) - z^(-k))/(2iz^(wT)).

To find the z-transform of the function cos(kwT), we can use the identity cos(x) = (e^(ix) + e^(-ix))/2. Applying the z-transform to each term separately gives us the z-transform of cos(kwT) as (z^(k) + z^(-k))/2.