Impulse response of FIR filter given the output is known

The correct answer is D. h[n] = Ʃ (k^2)δ[n - k]. In summary, the correct answer for the impulse response of the given FIR filter is D. h[n] = Ʃ (k^2)δ[n - k].
  • #1
Jd303
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Hi all,
I am faced with this question, I am stumped as it is presented in a different way to what I am used to:

The output from a FIR filter for a unit step input u[n] is
y[n] = δ[n] + 3 δ[n - 1] + 6 δ[n - 2] + 10 δ[n - 3] + 15 u[n]
What is the impulse response of the FIR filter?

A. h[n] = Ʃ kδ[n - k + 1]
B. h[n] = Ʃ kδ[n - k - 1]
C. h[n] = Ʃ δ[n - k + 1]
D. h[n] = Ʃ (k^2)δ[n - k]

Really don't know how to attack this problem any help would be greatly appreciated. Thanks in advance!
 
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  • #2
The correct answer is D. h[n] = Ʃ (k^2)δ[n - k].The impulse response of the FIR filter is given by the coefficients of the unit step input, which in this case are 1, 3, 6, 10, and 15. Therefore, the impulse response of the filter is h[n] = 1δ[n] + 3δ[n-1] + 6δ[n-2] + 10δ[n-3].
 

1. What is the impulse response of a Finite Impulse Response (FIR) filter?

The impulse response of an FIR filter is the output of the filter when a single impulse (a signal with amplitude 1 and duration of one sample) is applied as the input. This response is a sequence of coefficients that represent the output of the filter at each time step.

2. How is the impulse response of an FIR filter related to its frequency response?

The impulse response of an FIR filter can be used to calculate its frequency response through the Fourier transform. The frequency response represents how the filter will affect different frequencies in the input signal, and it is often used to design filters for specific frequency characteristics.

3. Can the impulse response of an FIR filter be used to determine the stability of the filter?

Yes, the stability of an FIR filter can be determined by examining its impulse response. A filter is considered stable if its impulse response decays over time and does not have any infinite or unbounded values. If the impulse response is stable, the filter will not introduce any new frequencies or amplify existing ones, making it a useful tool for signal processing.

4. How can the impulse response of an FIR filter be calculated from its transfer function?

The impulse response of an FIR filter can be calculated by taking the inverse Fourier transform of its transfer function. This process will give a sequence of coefficients that can be used to determine the output of the filter at each time step when a single impulse is applied as the input.

5. Are there any limitations to using the impulse response of an FIR filter for signal processing?

One limitation of using the impulse response of an FIR filter is that it can only be used to process signals with a finite duration. If the input signal is infinite, the impulse response will also be infinite, making it impossible to use for processing. Additionally, the accuracy of the output may be affected by the length of the impulse response, with longer responses providing a more precise output.

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