Selected Continuous-Time Fourier Theorems
This appendix presents Fourier theorems which are nice to know, but which do not, strictly speaking, pertain to the DFT. The differentiation theorem for Fourier Transforms comes up quite often, and its dual pertains as well to the DTFT (Appendix B). The scaling theorem provides an important basic insight into time-frequency duality. Finally, the very fundamental uncertainty principle is related to the scaling theorem.
Let denote a function differentiable for all such that and the Fourier Transforms (FT) of both and exist, where denotes the time derivative of . Then we have
Proof: This follows immediately from integration by parts:
The differentiation theorem is implicitly used in §E.6 to show that audio signals are perceptually equivalent to bandlimited signals which are infinitely differentiable for all time.
The scaling theorem (or similarity theorem) provides that if you horizontally ``stretch'' a signal by the factor in the time domain, you ``squeeze'' its Fourier transform by the same factor in the frequency domain. This is an important general Fourier duality relationship.
Theorem: For all continuous-time functions possessing a Fourier transform,
Proof: Taking the Fourier transform of the stretched signals gives
The absolute value appears above because, when , , which brings out a minus sign in front of the integral from to .
The scaling theorem is fundamentally restricted to the continuous-time, continuous-frequency (Fourier transform) case. The closest we came to the scaling theorem among the DFT theorems was the stretch theorem (§7.4.10). We found that ``stretching'' a discrete-time signal by the integer factor (filling in between samples with zeros) corresponded to the spectrum being repeated 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 using interpolation (instead of zero-fill) corresponded to the same repeated spectrum with the 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 stretch theorem for DFTs can be viewed as the discrete-time, discrete-frequency counterpart of the scaling theorem for Fourier Transforms.
The Uncertainty Principle
The uncertainty principle (for Fourier transform pairs) follows immediately from the scaling theorem. It may be loosely stated as
Time Duration Frequency Bandwidth cwhere is some constant determined by the precise definitions of ``duration'' in the time domain and ``bandwidth'' in the frequency domain.
If duration and bandwidth are defined as the ``nonzero interval,'' then we obtain , which is not very useful. This conclusion follows immediately from the definition of the Fourier transform and its inverse in §B.2.
Duration and Bandwidth as Second Moments
More interesting definitions of duration and bandwidth are obtained
for nonzero signals using the normalized second moments of the
By the DTFT power theorem, which is proved in a manner analogous to the DFT case in §7.4.8, we have . Note that writing `` '' and `` '' is an abuse of notation, but a convenient one. These duration/bandwidth definitions are routinely used in physics, e.g., in connection with the Heisenberg uncertainty principle.C.1Under these definitions, we have the following theorem [52, p. 273-274]:
Theorem: If and as , then
with equality if and only if
Proof: Without loss of generality, we may take consider to be real and normalized to have unit norm ( ). From the Schwarz inequality (see §5.9.3 for the discrete-time case),
The left-hand side can be evaluated using integration by parts:
The second term on the right-hand side of Eq.(C.3) can be evaluated using the power theorem (§7.4.8 proves the discrete-time case) and differentiation theorem (§C.1 above):
If equality holds in the uncertainty relation Eq.(C.2), then Eq.(C.3) implies
If for , then
Proof: See [52, pp. 274-5].
Time-Bandwidth Products are Unbounded Above
We have considered two lower bounds for the time-bandwidth product based on two different definitions of duration in time. In the opposite direction, there is no upper bound on time-bandwidth product. To see this, imagine filtering an arbitrary signal with an allpass filter.C.2 The allpass filter cannot affect bandwidth , but the duration can be arbitrarily extended by successive applications of the allpass filter.
Fourier Transforms for Continuous/Discrete Time/Frequency