Question

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)

The question was posed to me in an online interview.

This interesting question is from LTI System as Frequency Selective Filters in section Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer 1

Right option is (c) Hlp(ω-π)

To explain: The impulse response of a high pass filter is simply obtained from the impulse response of the low pass filter by changing the signs of the odd numbered samples in hlp(n). Thus

hhp(n)=(-1)n hlp(n)=(e^jπ)^n hlp(n)

Thus the frequency response of the high pass filter is obtained as Hlp(ω-π).