Check Understanding Impulse & Frequency Response

In summary: Keep up the good work!In summary, the impulse responses for both systems are 1. To find the frequency response, we use the Fourier transform and denote it as H(w). For system 1, the frequency response is \frac{1}{N+1}\sum_{k=-N}^{N}(1-\frac{|k|}{N+1})e^{-jwk}, and for system 2, it is \frac{1}{1-a} + \frac{a}{1-a}e^{-jw}. These expressions can be simplified using geometric series.
  • #1
Poopsilon
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Homework Statement



Find the impulse and frequency responses of the following systems:

1. [itex]y(n) = \frac{1}{N+1}\sum_{k=-N}^{N}(1-\frac{|k|}{N+1})x(n-k)[/itex]

2. [itex]y(n)=ay(n-1)+(1-a)x(n)[/itex], where 0<a<1

Homework Equations


The Attempt at a Solution

Ok so for 1. I look at h(n) which is [itex]\frac{1}{N+1}[(1-\frac{N}{N+1}) + (1-\frac{N-1}{N+1}) + ... +1+ (1-\frac{1}{N+1}) + ... + (1-\frac{N}{N+1})][/itex] And then this is equal to [itex] \frac{1}{N+1}[2N+1 - \frac{2N + 2(N-1) + ... + 2}{N+1}] = 1[/itex]. Thus the impulse response is just 1.

Now for 2. I observe the recursive behavior and conclude [itex] y(n) = (1-a)x(n) + a(1-a)x(n-1) + a^2 (1-a)x(n-2)+...[/itex]. And thus [itex]h(n)= (1-a)\sum_{n=0}^{\infty} a^n = (1-a)\frac{1}{1-a} = 1[/itex].

Thus both my impulse responses are 1. And now for the frequency response I'm a bit confused, I think I find the frequency response by calculating [itex] h(w) = \sum_{m=-\infty}^{\infty}h(m)e^{-jwm}[/itex]. Is that correct? And so then h(m) would just be 1 in both cases? I'm a bit lost and we are using a Fourier analysis text and not a signal processing text so I'm having trouble matching engineering terminology with the appropriate mathematics. Thanks.
 
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  • #2


Hello! It looks like you have a good understanding of the impulse responses for both systems. For the frequency response, you are correct in using the Fourier transform. However, the notation for the frequency response is usually denoted as H(w) instead of h(w). Also, the summation should be from n=-N to N, since your system is defined over that range. So for system 1, the frequency response would be:

H(w) = \frac{1}{N+1}\sum_{k=-N}^{N}(1-\frac{|k|}{N+1})e^{-jwk}

For system 2, the frequency response would be:

H(w) = \frac{1}{1-a} + \frac{a}{1-a}e^{-jw}

You can simplify these expressions further by using geometric series and solving for H(w). Hope this helps!
 

1. What is impulse response?

Impulse response is the output of a system when it is stimulated by an impulse input. It provides information about the system's behavior and characteristics, such as its stability, linearity, and time invariance.

2. How is impulse response related to frequency response?

The frequency response of a system is the relationship between its input and output at different frequencies. It can be obtained by taking the Fourier transform of the impulse response. In other words, the impulse response is a time-domain representation of the frequency response.

3. What is the significance of understanding impulse and frequency response?

Understanding impulse and frequency response allows us to analyze and characterize a system's behavior and performance. It is essential in various fields, such as signal processing, control systems, and communication systems, to design and optimize the system for its intended purpose.

4. How can we measure impulse and frequency response experimentally?

Impulse response can be measured by sending an impulse signal to the system and recording its output. Frequency response can be obtained by applying a sinusoidal input signal at different frequencies and measuring the corresponding output amplitudes and phases.

5. Are there any applications of impulse and frequency response in real life?

Yes, there are several applications of impulse and frequency response in real life. For example, in audio engineering, the impulse response of a room can be used to simulate different acoustic environments. In medical imaging, the frequency response of a system can be used to enhance image quality. In structural engineering, the impulse response can be used to detect structural damage and monitor its health.

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