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{F{x[n]}}}}.
For the sequence x[n] = {1, 2, 1, 3}, the value of Y[0] is___.
X[k] = F(x[n]}.
Another sequence y[n] is formed as below
y[n] = F F{F{F{F{x[n]}}}}.
For the sequence x[n] = {1, 2, 1, 3}, the value of Y[0] is___.
Show Hint
Correct Answer: 112
Solution and Explanation
Step 1: Define the DFT of \(x[n]\).
The Discrete Fourier Transform (DFT) of a sequence \(x[n]\) is given by: \[ X[k] = \mathcal{F}\{x[n]\}. \] Step 2: Effect of three successive Fourier transforms.
When a sequence \(x[n]\) of length \(N\) undergoes three successive Fourier transforms, the resulting sequence \(y[n]\) is scaled by \(N\), the length of the original sequence: \[ y[n] = N \cdot x[n]. \] Step 3: Compute the scaled sequence.
For \(N = 4\) and \(x[n] = \{1, 2, 1, 3\}\), the scaled sequence is: \[ y[n] = 4 \cdot \{1, 2, 1, 3\} = \{4, 8, 4, 12\}. \] Step 4: Calculate \(y[0]\).
The value of \(y[0]\) is: \[ y[0] = 16 \times x[0] = 16 \times 7 = 112. \] Final Answer: \[ \boxed{112} \]
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