Question:
The relationship between any \(N\)-length sequence \(x[n]\) and its corresponding \(N\)-point discrete Fourier transform \(X[k]\) is defined as:
\[
X[k] = \mathcal{F}\{x[n]\}.
\]
Another sequence \(y[n]\) is formed as:
\[
y[n] = \mathcal{F}\{\mathcal{F}\{\mathcal{F}\{x[n]\}\}\}.
\]
For the sequence \(x[n] = \{1, 2, 1, 3\}\), the value of \(y[0]\) is \(\_\_\_\_\).
The relationship between any \(N\)-length sequence \(x[n]\) and its corresponding \(N\)-point discrete Fourier transform \(X[k]\) is defined as:
\[
X[k] = \mathcal{F}\{x[n]\}.
\]
Another sequence \(y[n]\) is formed as:
\[
y[n] = \mathcal{F}\{\mathcal{F}\{\mathcal{F}\{x[n]\}\}\}.
\]
For the sequence \(x[n] = \{1, 2, 1, 3\}\), the value of \(y[0]\) is \(\_\_\_\_\).
Show Hint
When applying three successive Fourier transforms, the sequence is scaled by the length \(N\). Verify the scaling factor to compute the correct result.
Updated On: Jan 31, 2025
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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: \[ y[0] = 4 \times x[0] = 4 \times 1 = 4. \] Final Answer: \[ \boxed{112} \]
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: \[ y[0] = 4 \times x[0] = 4 \times 1 = 4. \] Final Answer: \[ \boxed{112} \]
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