The relationship between any N-length sequence x[n] and its corresponding N-point discrete Fourier transform X[k] is defined as...
Q.59 The relationship between any N-length sequence x[n] and its corresponding N-point discrete Fourier transform X[k] is defined as
X[k] = F{x[n]}.
Another sequence y[n] is formed as below
y[n] = F{F{F{F{x[n]}}}}.
For the sequence x[n] = {1, 2, 1, 3}, the value of y[0] is ___________
Solution
This problem relies on the Duality Property of the Discrete Fourier Transform (DFT).
1. The Property: Applying the DFT twice to a sequence x[n] of length N results in a time-reversed and scaled version of itself:
F{F{x[n]}} = N ⋅ x[(-n)N]
2. Deriving y[n]: The problem asks for the result of four consecutive DFT operations:
y[n] = F4{x[n]} = F2{ â„«2{x[n]} } = N ⋅ [ N ⋅ x( -(-n)N ) ] = N2 ⋅ x[n]
3. Final Calculation:
- • Length of sequence x[n] = {1, 2, 1, 3} is N = 4
- • The initial value is x[0] = 1
y[0] = 42 ⋅ x[0] = 16 ⋅ 1 = 16