# DSP: z-transforms question

Discussion in 'Homework Help' started by dewasiuk, Apr 1, 2011.

1. ### dewasiuk Thread Starter New Member

Feb 14, 2011
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0
So I am currently in the z-transforms unit of my DSP course in school and I'm not sure how to sketch the frequency response of a particular question.

Question: A transfer function H(z), has one zero at the origin of the complex z-plane, another zero at (√2)/2 on the real axis, and a pair of complex conjugate poles at (√2)/2 + j(√2)/2.

a) Draw a pole-zero plot of the transfer function.
b) Without matlab, sketch the magnitude vs. frequency response of the system.
c) Write the transfer function, H(z).
d) Write the discrete-time function, h[n].
e) Write the difference equation for the impulse of the system.
f) Given a sampling frequency of 16kHz, calculate the frequency of oscillation for the system.

The only part I am unsure of is part b). I was able to solve the other questions and I included some solved information in the attached image:

http://img8.imageshack.us/i/ztransformhelp.jpg/

How can I solve for the approximate magnitudes of poles or zeros so I can roughly sketch the response?

2. ### Georacer Moderator

Nov 25, 2009
5,151
1,266
I don't think I can be of much help, except from giving you the bode plot as produced by Matlab for sampling frequencies of 1Hz (normalized) and 16000Hz.

It seems I need to revise my control systems theoretical basis.

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3. ### dewasiuk Thread Starter New Member

Feb 14, 2011
24
0
One thing I am familiar with is that usually for magnitude plots, we can take 20log(H(z)) substiting z with certain points. However, I'm not sure what we susbtitute z for. I know that in the s-plane, you can find out the magnitude at certain points because for example, τs + 1 = jω/ωc +1. However I do not think that we have learned enough material to directly see how this releates to complex z.

4. ### guitarguy12387 Active Member

Apr 10, 2008
359
12
Check this out:
http://cnx.org/content/m10548/latest/

Short answer is that you can traverse the unit circle from w = 0 to w = pi and for poles, you amplify and zeros you drop gain. That's just how to do a very rough sketch of the magnitude. You can plug in some values and calculate the magnitude at given frequencies once you have the transfer function.

Feb 14, 2011
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6. ### guitarguy12387 Active Member

Apr 10, 2008
359
12
Oh i don't think it works for me either haha. Sorry about that, I more meant the article. It does a good job of describing it!

7. ### dewasiuk Thread Starter New Member

Feb 14, 2011
24
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I got the article to load on a third computer haha. After reading the article, it basically confirmed what I know on the rough estimation how steep/shallow/etc. a pole or zero would look like. I do not understand how it tells us how to solve for the approximate magnitude of each pole in dB though.

8. ### guitarguy12387 Active Member

Apr 10, 2008
359
12
Can you elaborate your question a bit more... i'm not sure I understand.

My interpretation: You know the location of the poles... magnitude of pole = sqrt(RE{p}^2+IM{p}^2)

9. ### dewasiuk Thread Starter New Member

Feb 14, 2011
24
0
When I entered the information in matlab, the frequency responce I got showed a sharp peak of about 74 dB at the normalized frquency of pi/4. How do you calculate that 74 dB?