Frequency Response from a Discrete Transfer Function

[saad87]

Homework Statement

Find the DC Gain and the Frequency Response of the above discrete-time system.

The Attempt at a Solution

This isn't really a homework problem but something I've been struggling with for my exams. I am given a transfer function in terms of z and am asked to find its DC Gain. This is easy enough - I just substitue 1 in place of z and get the DC Gain. I also understand why we put in 1. Its because z = e^jw and when w = 0 (i.e DC), z = 1.

However, what I get simply stuck at is how to sketch the frequency response from a given transfer function. I understand that by drawing a pole-zero plot I can get some idea of how the frequency response will look like it but I can never really nail it. For instance, I know that if there are two zeroes on the unit circle the gain is going to fall to 0 at those points.

My problem, in this case, is how do I know what those ponts are in terms of Hz? The zeros are a complex number - how do I translate this complex number to Hz?

Secondly, often in my exams, a quick sketch is asked for the gain and phase of the filter. What's a quick way to sketch this? Am I right in assuming that I just subsitute e^jw and use some values for w and then plot the result - would this give me a plot for gain and phase?

 

[KataKoniK]

Thank you for reaching out with your question. I understand the importance of being able to accurately determine the DC gain and frequency response of a discrete-time system. I would be happy to provide some guidance on how to approach this problem.

To find the DC gain of a discrete-time system, you are correct in substituting z = 1 into the transfer function. This is because at DC (w = 0), the value of z is equal to 1. This will give you the gain at DC, which is a constant value.

To sketch the frequency response, you can use a pole-zero plot as you mentioned. The points on the unit circle correspond to frequencies in Hz. For example, if you have a zero at z = 0.5, this would correspond to a frequency of 0.5 Hz. Similarly, a pole at z = 0.5 would correspond to a frequency of -0.5 Hz. You can use this information to sketch the frequency response by plotting the gain at different frequencies.

For a quick sketch of the gain and phase, you can substitute different values for w and plot the resulting gain and phase values. This will give you an idea of how the gain and phase change with frequency. You can also use a computer program or calculator to plot the frequency response for a more accurate representation.

I hope this helps and wish you the best of luck on your exams!