# Math Help - Z transform of a periodic function

1. ## Z transform of a periodic function

Can anybody help me about the following problem?
I need to obtain the z-transform of a periodic function. Function f(t) is "t/a" for t values in (0,a], "1" in (a,3a], "4 - t/a" in(3a,5a], "-1" in(5a,7a], "t/a - 8" in(7a,8a] where a=pi/4. Note that this is one period and the period is 2*pi. The function repeats itself until infinity. I need to find the z transform of this time function as a rational fraction format (i.e. A(z)/B(z)). Sampling time for this continuous function is T=10ms. Is there anyone who can help me?

2. Originally Posted by ercan
Can anybody help me about the following problem?
I need to obtain the z-transform of a periodic function. Function f(t) is "t/a" for t values in (0,a], "1" in (a,3a], "4 - t/a" in(3a,5a], "-1" in(5a,7a], "t/a - 8" in(7a,8a] where a=pi/4. Note that this is one period and the period is 2*pi. The function repeats itself until infinity. I need to find the z transform of this time function as a rational fraction format (i.e. A(z)/B(z)). Sampling time for this continuous function is T=10ms. Is there anyone who can help me?

It would help if the sampling interval divided the period.

CB

3. Thanks for your interest, CaptainBlack. But I couldn't understand. Do you mean that I have to divide the period to T (2*pi/0.01). What is its function?

4. Originally Posted by ercan
Thanks for your interest, CaptainBlack. But I couldn't understand. Do you mean that I have to divide the period to T (2*pi/0.01). What is its function?
What I mean is that to compute the z-transform it would help if the period of the waveform were a multiple of the sampling interval. For example if the frequency were 1 Hz (period 1 second) then a sampling frequency of 100 Hz (sampling interval of 10ms) would give exactly 100 samples in a single cycle of the waveform. Then we take the first 100 terms of the z-transform series and sum them, and every following set of 100 terms is a $z^{-100}$ times the sum of the previous set, and se we have a geometric series which we can sum (for $z<1$) to give the final answer. (this is the equivalent to the sum of time shifted basic waveforms)

An alternative approach is to take the Laplace transform for the signal, and then use the relationship between the ZT and LT to move from the s-domain to the z-domain.

CB