Exercise · ZW-DISCRETE-FOURIER-TRANSFORM-010 · ★★★ Advanced
Let \(x[n]=\{1,2,1,0\}\) with N=4. What is \(X[2]\)?
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Explanation
With \(W_4^{2n}=(-1)^n\), we get \(X[2]=1-2+1-0=0\), the alternating sum of the sequence values.
Review the concept: Discrete Fourier transform
The discrete Fourier transform (DFT) maps a finite sequence of N complex values \(x[n]\) to N values \(X[k]\) representing its components at the frequencies \(2\pi k/N\).
\[X[k]=\sum_{n=0}^{N-1}x[n]\,e^{-j2\pi kn/N}\]
More practice problems: discrete fourier transform
- ZW-DISCRETE-FOURIER-TRANSFORM-001 How many complex values does the DFT of a sequence of length N produce?
- ZW-DISCRETE-FOURIER-TRANSFORM-002 What is the period of the sequence X[k] of an N-point DFT?
- ZW-DISCRETE-FOURIER-TRANSFORM-003 Which expression defines the factor W_N used in the DFT?
- ZW-DISCRETE-FOURIER-TRANSFORM-004 What is the transformation called that recovers the original sequence from its values X[k]?
- ZW-DISCRETE-FOURIER-TRANSFORM-005 Which DFT property states that the transform of a sum of sequences is the sum of their transforms?
- ZW-DISCRETE-FOURIER-TRANSFORM-006 Compute X[0] for the sequence x[n]=\2,1,3\ of length N=3.
- ZW-DISCRETE-FOURIER-TRANSFORM-007 For N=4, what is W_4^2?
- ZW-DISCRETE-FOURIER-TRANSFORM-008 What is the relation between X[0] and the mean of the values of x[n] in an N-point DFT?