# Appendix D: The Similarity Theorem

## GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Appendix D: The Similarity Theorem

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# Appendix D: The Similarity Theorem

The similarity theorem is fundamentally restricted to the continuous-time case. It says that if you ''stretch'' a signal by the factor in the time domain, you ''squeeze'' its Fourier transform by the same factor in the frequencydomain. This is such a fundamental Fourier relationship, that we include it here rather than leave it out as a non-DFT result.

The closest we came to the similarity theorem among the DFT theorems was the the Interpolation Theorem. We found that ''stretching'' adiscrete-time signal by the integer factor (filling in between samples with zeros) corresponded to the spectrum beingrepeated times around the unit circle. As a result, the ''baseband'' copy of the spectrum ''shrinks'' in width (relative to ) by the factor . Similarly, stretching a signal usinginterpolation (instead of zero-fill) corresponded to the repeated spectrum with all spurious spectral copies zeroed out. The spectrum of the interpolated signal can therefore be seen as having been stretched by the inverse of the time-domain stretch factor. In summary, the Interpolation DFT Theorem can be viewed as the discrete-time counterpart of the similarity Fourier Transform (FT) theorem.

Theorem: For all continuous-time functions possessing a Fourier transform, where and is any nonzero real number (the abscissa scaling factor).

Proof: The absolute value appears above because, when , , which brings out a minus sign in front of the integral from to .

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