Question

If cx(n) is the complex cepstrum sequence obtained from the inverse Fourier transform of ln X(ω), then what is the expression for cθ(n)?

(a) \(\frac{1}{2π} \int_0^π \theta(ω) e^{jωn} dω\)

(b) \(\frac{1}{2π} \int_{-π}^π \theta(ω) e^{-jωn} dω\)

(c) \(\frac{1}{2π} \int_0^π \theta(ω) e^{jωn} dω\)

(d) \(\frac{1}{2π} \int_{-π}^π \theta(ω) e^{jωn} dω\)

The question was posed to me in an interview.

My enquiry is from Frequency Analysis of Discrete Time Signal topic in chapter Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer 1

Right choice is (d) \(\frac{1}{2π} \int_{-π}^π \theta(ω) e^{jωn} dω\)

The explanation is: We know that,

cx(n)=\(\frac{1}{2π} \int_{-π}^π ln⁡(X(ω))e^{jωn} dω\)

If we express X(ω) in terms of its magnitude and phase, say

X(ω)=|X(ω)|e^jθ(ω)

Then ln X(ω)=ln |X(ω)|+jθ(ω)

=> cx(n)=\(\frac{1}{2π} \int_{-π}^π[ln|X(ω)|+jθ(ω)]e^{jωn} dω\) => cx(n)=cm(n)+jcθ(n)(say)

=> cθ(n)=\(\frac{1}{2π} \int_{-π}^πθ(ω) e^{jωn} dω\)