- #1

Master1022

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- Homework Statement
- Sketch the frequency magnitude characteristics of the two analogue filters whose pole-zero configurations are shown in the figure. On the basis of these curves, what can you infer about the form of their impulse responses

- Relevant Equations
- Transfer functions

Hi,

EDIT: apologies for any ambiguity, but this is for

I was attempting a problem which is shown below

.

I am not really sure how to attempt this problem, but here is my attempt. Are there general methods for tackling these types of questions?

I have assumed that the outer most poles/zeros are on the unit circle such that the system is stable.

A zero on the unit circle will create a gain of zero at that frequency, whereas a pole will create resonance at its frequency. I think this because we can write the transfer function as:

$$ G(w) = \frac{\prod_{i = 1}^{n} |z-z_i|}{\prod_{j=1}^{m} |z-p_j|} $$

Thus, for the first (left) diagram, I think there should be resonance at the frequencies of the three poles. The effect of the zero is the same for all frequencies.

For the second (right hand) diagram, I think there should be resonance at the frequency of the outermost pole (the inner one is within the unit circle) and points of zero gain at the frequencies of the two zeros.

Are these attempts correct? If so, how I do make inferences about the impulse response from these?

I know that the impulse response is the transfer function, but am not sure how to qualitatively use these plots to describe it.

Thanks in advance for any help.

EDIT: apologies for any ambiguity, but this is for

__DISCRETE systems, not analogue like the problem states.__I was attempting a problem which is shown below

.

I am not really sure how to attempt this problem, but here is my attempt. Are there general methods for tackling these types of questions?

**Attempt:**I have assumed that the outer most poles/zeros are on the unit circle such that the system is stable.

A zero on the unit circle will create a gain of zero at that frequency, whereas a pole will create resonance at its frequency. I think this because we can write the transfer function as:

$$ G(w) = \frac{\prod_{i = 1}^{n} |z-z_i|}{\prod_{j=1}^{m} |z-p_j|} $$

**Part (a)**Thus, for the first (left) diagram, I think there should be resonance at the frequencies of the three poles. The effect of the zero is the same for all frequencies.

**Part (b)**For the second (right hand) diagram, I think there should be resonance at the frequency of the outermost pole (the inner one is within the unit circle) and points of zero gain at the frequencies of the two zeros.

Are these attempts correct? If so, how I do make inferences about the impulse response from these?

I know that the impulse response is the transfer function, but am not sure how to qualitatively use these plots to describe it.

Thanks in advance for any help.

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