Q5Digital Signal Processing
Question
Q.5. State two properties of DFT.
Answer
Two key DFT properties are linearity (DFT of a weighted sum equals the weighted sum of DFTs) and circular time-shifting (a circular shift in the time domain corresponds to multiplication by a complex exponential in the frequency domain).
Linearity: if x1(n) ↔ X1(k) and x2(n) ↔ X2(k), then for any constants a, b: a·x1(n) + b·x2(n) ↔ a·X1(k) + b·X2(k), meaning the DFT of a linear combination of sequences equals the same linear combination of their individual DFTs.
Circular time shift: if x(n) ↔ X(k), then a circular shift of x(n) by m samples, x((n-m) mod N), has DFT equal to X(k)·e^(-j2πkm/N), i.e., a shift in the time domain corresponds to multiplication by a linear-phase complex exponential factor in the frequency domain, with the magnitude spectrum unchanged.