DSP - IIR filter, anlog to digital

[Hmemo]

hi,
I'm studying for a test, and found this question in one of the previous exams but there is no solution and I was not able to solve it myself.
It would be very helpful if anybody could help =]

Homework Statement

h(t) is an analog IIR filter, with the poles λ_1 = -1, λ_2 = -1+i, λ_3 = -1-i and no zeros. let h[n] be a digital IIR filter with poles which defined by e^λ_i and no zeros. is h[n] a high pass filter?

Homework Equations

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The Attempt at a Solution

I thought about using some transformation or something like that
but i got stuck, and was unable to prove/disprove.

Thanks.

 

[KataKoniK]

Hello! I'm happy to help with your question. First, let's define some terms to make sure we're on the same page.

An IIR filter, or infinite impulse response filter, is a type of filter commonly used in signal processing. The "infinite impulse response" refers to the fact that the filter's output depends on both past and present inputs, unlike a finite impulse response (FIR) filter which only depends on past inputs.

A pole is a point in the complex plane where the transfer function of a filter is undefined, meaning the output goes to infinity. In this case, the poles are defined as λ_1 = -1, λ_2 = -1+i, and λ_3 = -1-i.

The digital IIR filter, h[n], is defined by e^λ_i, which is a complex exponential function. This means that the poles of h[n] will also be complex numbers.

Now, to answer your question, whether or not h[n] is a high-pass filter depends on the values of the poles. A high-pass filter is a type of filter that allows high frequency signals to pass through while attenuating low frequency signals.

In order for h[n] to be a high-pass filter, the poles must be located in the left half of the complex plane, meaning they have a negative real part. This is because complex exponential functions decay as the real part of the exponent decreases.

From the given poles λ_1 = -1, λ_2 = -1+i, and λ_3 = -1-i, we can see that they are all located in the left half of the complex plane, with a negative real part of -1. This means that h[n] is indeed a high-pass filter.

I hope this helps with your test preparation! Let me know if you have any further questions. Good luck on your exam!