Exercise · ZW-DISCRETE-FOURIER-TRANSFORM-008 · ★★☆ Intermediate
What is the relation between \(X[0]\) and the mean of the values of \(x[n]\) in an N-point DFT?
Choose one option.
Show answer
- B. \(X[0]\) equals \(N\) times the mean of \(x[n]\)
Explanation
\(X[0]\) is the sum of all values of \(x[n]\), that is, \(N\) times their mean; this is why it is the DC component.
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-009 For N=4, what is W_4^3?
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