# How would you match up these Z-transforms?

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1. May 10, 2015

### jami8337

1. The problem statement, all variables and given/known data
In Figure Q1b (on the next page), two plots display samples of the continuous-time signal uct(t) = sin(2πt) and two plots display samples of the continuous-time signal vct(t) = e −t sin(2πt). For each signal, the samples in the corresponding plots are obtained with two different sampling times. Associate each of the four plots with the corresponding Z transform in the list and justify your answer:

2. Relevant equations

Table of Z-Transforms

3. The attempt at a solution
Ok so using considering the generic denominator of the Z transform of a sinusoid:

z2+2cdz+d2

I can see that both equations 2 and 3 have a value of d=1 hence they are both undamped systems, so they are either one of (b) or (c)

Then from the generic denominator equation:

for (2) c=-2cos(ωT) =1.9842
= cos(ωT) = -0.9921

For (3) c=-2cos(ωT) =-1.9646
= cos(ωT) = 0.9823

Its at this point im stuck on what I need to do and how to see which one of b and c is the correct plot?

Thanks for any tips!

Last edited: May 10, 2015
2. May 10, 2015

### Hesch

In (b) you can count the samples per periode ( about 33.2 samples ). As one period is 360°, (b) matches a polepair = 1 / ±10.84°.

Now solve the roots in denominator in 2) and 3). See if there is a match.

3. May 24, 2015

### jami8337

Sorry for the late response, my revision swapped to a different module. That's great thank you managed to solve it for the correct answer using that