Question

Show that the IDFT can be derived as a least squares signal matching problem, where $N$ points in the time domain are matched by a linear combination of $N$ sampled complex sinusoids.

   Show that the IDFT can be derived as a least squares signal matching problem, where $N$ points in the time domain are matched by a linear combination of $N$ sampled complex sinusoids.

Audio Signal Processing and Coding

Audio Signal Processing and Coding

Andreas Spanias, Ted… 1st Edition

Chapter 2, Problem 7

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The IDFT is used to convert frequency domain data back into the time domain. For a sequence of complex numbers \( X[k] \) of length \( N \), the IDFT is given by: \[ x[n] = \frac{1}{N} \sum_{k=0}^{N-1} X[k] e^{j \frac{2\pi}{N} kn} \] for \( n = 0, 1, \ldots, N-1 Show more…

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Show that the IDFT can be derived as a least squares signal matching problem, where $N$ points in the time domain are matched by a linear combination of $N$ sampled complex sinusoids.

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