Exercise · ZW-DISCRETE-FOURIER-TRANSFORM-009 · ★★☆ Intermediate

For N=4, what is \(W_4^3\)?

Choose one option.

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A. j

Explanation

Since \(W_4=-j\), we get \(W_4^3=(-j)^3=j\).

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?