If hlp(n) denotes the impulse response of a low pass filter with frequency response Hlp(ω), then what is the frequency response of the high pass filter in terms of Hlp(ω)?a)Hlp(ω-π/2)b)Hlp(ω+π/2)c)Hlp(ω-π)d)Hlp(ω+π)Correct answer is option 'C'. Can you explain this answer?

Question

If hlp(n) denotes the impulse response of a low pass filter with frequency response Hlp(&omega;), then what is the frequency response of the high pass filter in terms of Hlp(&omega;)?a)Hlp(&omega;-&pi;/2)b)Hlp(&omega;+&pi;/2)c)Hlp(&omega;-&pi;)d)Hlp(&omega;+&pi;)Correct answer is option 'C'. Can you explain this answer?

Answer

N), then the frequency response Hlp(f) can be obtained by taking the discrete Fourier transform (DFT) of the impulse response hlp(n). The DFT is a mathematical transformation that converts a sequence of discrete samples into a sequence of complex numbers representing the frequency components of the signal.

To obtain Hlp(f) from hlp(n), you can follow these steps:

1. Determine the length of the impulse response hlp(n). Let's say it has a length of N.

2. Pad the impulse response hlp(n) with zeros to a length of M, where M is a power of 2 greater than or equal to N. This is done to ensure efficient computation of the DFT.

3. Compute the DFT of the padded impulse response. This can be done using an algorithm such as the Fast Fourier Transform (FFT).

4. Normalize the DFT by dividing each complex number by the length of the impulse response, M.

5. The resulting sequence of complex numbers represents the frequency response Hlp(f) of the low pass filter. The magnitude of each complex number represents the gain of the filter at a corresponding frequency bin, and the phase represents the phase shift introduced by the filter.

Note that Hlp(f) will have a length of M, and it represents the frequency response of the low pass filter over the range of frequencies from 0 to the Nyquist frequency (which is half the sampling rate).

Keep in mind that the specific implementation details may vary depending on the programming language or software you are using.

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