Fourier Theorems
In this section the main Fourier theorems are stated and proved. It is no small matter how simple these theorems are in the DFT case relative to the other three cases (DTFT, Fourier transform, and Fourier series, as defined in Appendix B). When infinite summations or integrals are involved, the conditions for the existence of the Fourier transform can be quite difficult to characterize mathematically. Mathematicians have expended a considerable effort on such questions. By focusing primarily on the DFT case, we are able to study the essential concepts conveyed by the Fourier theorems without getting involved with mathematical difficulties.
Linearity
Theorem: For any
and
, the DFT satisfies
Proof:
Conjugation and Reversal
Theorem: For any
,
Proof:
Theorem: For any
,
Proof: Making the change of summation variable
, we get
Theorem: For any
,
Proof:
Corollary: For any ,
Proof: Picking up the previous proof at the third formula, remembering that is real,
Thus, conjugation in the frequency domain corresponds to reversal in the time domain. Another way to say it is that negating spectral phase flips the signal around backwards in time.
Corollary: For any ,
Proof: This follows from the previous two cases.
Definition: The property
is called Hermitian symmetry
or ``conjugate symmetry.'' If
, it may be called
skewHermitian.
Another way to state the preceding corollary is
Symmetry
In the previous section, we found when is real. This fact is of high practical importance. It says that the spectrum of every real signal is Hermitian. Due to this symmetry, we may discard all negativefrequency spectral samples of a real signal and regenerate them later if needed from the positivefrequency samples. Also, spectral plots of real signals are normally displayed only for positive frequencies; e.g., spectra of sampled signals are normally plotted over the range 0 Hz to Hz. On the other hand, the spectrum of a complex signal must be shown, in general, from to (or from 0 to ), since the positive and negative frequency components of a complex signal are independent.
Recall from §7.3 that a signal is said to be even if , and odd if . Below are are Fourier theorems pertaining to even and odd signals and/or spectra.
Theorem: If
, then
re is even and
im is odd.
Proof: This follows immediately from the conjugate symmetry of for real signals
.
Theorem: If
,
is even and is odd.
Proof: This follows immediately from the conjugate symmetry of expressed
in polar form
.
The conjugate symmetry of spectra of real signals is perhaps the most important symmetry theorem. However, there are a couple more we can readily show:
Theorem: An even signal has an even transform:
Proof:
Express in terms of its real and imaginary parts by
. Note that for a complex signal to be even, both its real and
imaginary parts must be even. Then
Let even denote a function that is even in , such as , and let odd denote a function that is odd in , such as , Similarly, let even denote a function of and that is even in both and , such as , and odd mean odd in both and . Then appropriately labeling each term in the last formula above gives
Theorem: A real even signal has a real even transform:
Proof: This follows immediately from setting in the preceding
proof. From Eq.(7.5), we are left with
Instead of adapting the previous proof, we can show it directly:
Definition: A signal with a real spectrum (such as any real, even signal)
is often called a zero phase signal. However, note that when
the spectrum goes negative (which it can), the phase is really
, not 0. When a real spectrum is positive at dc (i.e.,
), it is then truly zerophase over at least some band
containing dc (up to the first zerocrossing in frequency). When the
phase switches between 0 and at the zerocrossings of the
(real) spectrum, the spectrum oscillates between being zero phase and
``constant phase''. We can say that all real spectra are
piecewise constantphase spectra, where the two constant values
are 0 and (or , which is the same phase as ). In
practice, such zerocrossings typically occur at low magnitude, such
as in the ``sidelobes'' of the DTFT of a ``zerocentered symmetric
window'' used for spectrum analysis (see Chapter 8 and Book IV
[70]).
Shift Theorem
Theorem: For any
and any integer ,
Proof:
The shift theorem is often expressed in shorthand as
Linear Phase Terms
The reason is called a linear phase term is that its phase is a linear function of frequency:
Linear Phase Signals
In practice, a signal may be said to be linear phase when its phase is of the form
Zero Phase Signals
A zerophase signal is thus a linearphase signal for which the phaseslope is zero. As mentioned above (in §7.4.3), it would be more precise to say ``0orphase signal'' instead of ``zerophase signal''. Another better term is ``zerocentered signal'', since every real (even) spectrum corresponds to an even (real) signal. Of course, a zerocentered symmetric signal is simply an even signal, by definition. Thus, a ``zerophase signal'' is more precisely termed an ``even signal''.
Application of the Shift Theorem to FFT Windows
In practical spectrum analysis, we most often use the Fast Fourier Transform^{7.15} (FFT) together with a window function . As discussed further in Chapter 8, windows are normally positive (), symmetric about their midpoint, and look pretty much like a ``bell curve.'' A window multiplies the signal being analyzed to form a windowed signal , or , which is then analyzed using an FFT. The window serves to taper the data segment gracefully to zero, thus eliminating spectral distortions due to suddenly cutting off the signal in time. Windowing is thus appropriate when is a short section of a longer signal (not a period or whole number of periods from a periodic signal).
Theorem: Real symmetric FFT windows are linear phase.
Proof: Let denote the window samples for
.
Since the window is symmetric, we have
for all .
When is odd, there is a sample at the midpoint at time
. The midpoint can be translated to the time origin to
create an even signal. As established on page ,
the DFT of a real and even signal is real and even. By the shift
theorem, the DFT of the original symmetric window is a real, even
spectrum multiplied by a linear phase term, yielding a spectrum
having a phase that is linear in frequency with possible
discontinuities of radians. Thus, all oddlength real
symmetric signals are ``linear phase'', including FFT windows.
When is even, the window midpoint at time lands halfway between samples, so we cannot simply translate the window to zerocentered form. However, we can still factor the window spectrum into the product of a linear phase term and a real spectrum (verify this as an exercise), which satisfies the definition of a linear phase signal.
Convolution Theorem
Theorem: For any
,
Proof:
This is perhaps the most important single Fourier theorem of all. It is the basis of a large number of FFT applications. Since an FFT provides a fast Fourier transform, it also provides fast convolution, thanks to the convolution theorem. It turns out that using an FFT to perform convolution is really more efficient in practice only for reasonably long convolutions, such as . For much longer convolutions, the savings become enormous compared with ``direct'' convolution. This happens because direct convolution requires on the order of operations (multiplications and additions), while FFTbased convolution requires on the order of operations, where denotes the logarithmbase2 of (see §A.1.2 for an explanation).
The simple matlab example in Fig.7.13 illustrates how much faster convolution can be performed using an FFT.^{7.16} We see that for a length convolution, the fft function is approximately 300 times faster in Octave, and 30 times faster in Matlab. (The conv routine is much faster in Matlab, even though it is a builtin function in both cases.)
N = 1024; % FFT much faster at this length t = 0:N1; % [0,1,2,...,N1] h = exp(t); % filter impulse reponse H = fft(h); % filter frequency response x = ones(1,N); % input = dc (any signal will do) Nrep = 100; % number of trials to average t0 = clock; % latch the current time for i=1:Nrep, y = conv(x,h); end % Direct convolution t1 = etime(clock,t0)*1000; % elapsed time in msec t0 = clock; for i=1:Nrep, y = ifft(fft(x) .* H); end % FFT convolution t2 = etime(clock,t0)*1000; disp(sprintf([... 'Average directconvolution time = %0.2f msec\n',... 'Average FFTconvolution time = %0.2f msec\n',... 'Ratio = %0.2f (Direct/FFT)'],... t1/Nrep,t2/Nrep,t1/t2)); % =================== EXAMPLE RESULTS =================== Octave: Average directconvolution time = 69.49 msec Average FFTconvolution time = 0.23 msec Ratio = 296.40 (Direct/FFT) Matlab: Average directconvolution time = 15.73 msec Average FFTconvolution time = 0.50 msec Ratio = 31.46 (Direct/FFT) 
A similar program produced the results for different FFT lengths shown in Table 7.1.^{7.17} In this software environment, the fft function is faster starting with length , and it is never significantly slower at short lengths, where ``calling overhead'' dominates.

A table similar to Table 7.1 in Strum and Kirk [79, p. 521], based on the number of real multiplies, finds that the fft is faster starting at length , and that direct convolution is significantly faster for very short convolutions (e.g., 16 operations for a direct length4 convolution, versus 176 for the fft function).
See Appendix A for further discussion of FFT algorithms and their applications.
Dual of the Convolution Theorem
The dual^{7.18} of the convolution theorem says that multiplication in the time domain is convolution in the frequency domain:
Theorem:
Proof: The steps are the same as in the convolution theorem.
This theorem also bears on the use of FFT windows. It implies that windowing in the time domain corresponds to smoothing in the frequency domain. That is, the spectrum of is simply filtered by , or, . This smoothing reduces sidelobes associated with the rectangular window, which is the window one is using implicitly when a data frame is considered time limited and therefore eligible for ``windowing'' (and zeropadding). See Chapter 8 and Book IV [70] for further discussion.
Correlation Theorem
Theorem: For all
,
Proof:
The last step follows from the convolution theorem and the result from §7.4.2. Also, the summation range in the second line is equivalent to the range because all indexing is modulo .
Power Theorem
Theorem: For all
,
Proof:
As mentioned in §5.8, physical power is energy per unit time.^{7.19} For example, when a force produces a motion, the power delivered is given by the force times the velocity of the motion. Therefore, if and are in physical units of force and velocity (or any analogous quantities such as voltage and current, etc.), then their product is proportional to the power per sample at time , and becomes proportional to the total energy supplied (or absorbed) by the driving force. By the power theorem, can be interpreted as the energy per bin in the DFT, or spectral power, i.e., the energy associated with a spectral band of width .^{7.20}
Normalized DFT Power Theorem
Note that the power theorem would be more elegant if the DFT were defined as the coefficient of projection onto the normalized DFT sinusoids
Rayleigh Energy Theorem (Parseval's Theorem)
Theorem:
For any
,
Proof: This is a special case of the power theorem.
Note that again the relationship would be cleaner ( ) if we were using the normalized DFT.
Stretch Theorem (Repeat Theorem)
Theorem: For all
,
Proof:
Recall the stretch operator:
Downsampling Theorem (Aliasing Theorem)
Theorem: For all
,
Proof: Let
denote the frequency index in the
aliased spectrum, and
let
. Then is length ,
where is the downsampling factor. We have
Since , the sum over becomes
Since the above derivation also works in reverse, the theorem is proved.
An illustration of aliasing in the frequency domain is shown in Fig.7.12.
Illustration of the Downsampling/Aliasing Theorem in Matlab
>> N=4; >> x = 1:N; >> X = fft(x); >> x2 = x(1:2:N); >> fft(x2) % FFT(Downsample(x,2)) ans = 4 2 >> (X(1:N/2) + X(N/2 + 1:N))/2 % (1/2) Alias(X,2) ans = 4 2
Zero Padding Theorem (Spectral Interpolation)
A fundamental tool in practical spectrum analysis is zero padding. This theorem shows that zero padding in the time domain corresponds to ideal interpolation in the frequency domain (for timelimited signals):
Theorem: For any
Proof: Let with . Then
Thus, this theorem follows directly from the definition of the ideal interpolation operator . See §8.1.3 for an example of zeropadding in spectrum analysis.
Periodic Interpolation (Spectral Zero Padding)
The dual of the zeropadding theorem states formally that zero padding in the frequency domain corresponds to periodic interpolation in the time domain:
Definition: For all
and any integer ,
where zero padding is defined in §7.2.7 and illustrated in Figure 7.7. In other words, zeropadding a DFT by the factor in the frequency domain (by inserting zeros at bin number corresponding to the folding frequency^{7.21}) gives rise to ``periodic interpolation'' by the factor in the time domain. It is straightforward to show that the interpolation kernel used in periodic interpolation is an aliased sinc function, that is, a sinc function that has been timealiased on a block of length . Such an aliased sinc function is of course periodic with period samples. See Appendix D for a discussion of ideal bandlimited interpolation, in which the interpolating sinc function is not aliased.
Periodic interpolation is ideal for signals that are periodic in samples, where is the DFT length. For nonperiodic signals, which is almost always the case in practice, bandlimited interpolation should be used instead (Appendix D).
Relation to Stretch Theorem
It is instructive to interpret the periodic interpolation theorem in terms of the stretch theorem, . To do this, it is convenient to define a ``zerocentered rectangular window'' operator:
Definition: For any
and any odd integer we define the
length even rectangular windowing operation by
Theorem: When
consists of one or more periods from a periodic
signal
,
Proof: First, recall that
. That is,
stretching a signal by the factor gives a new signal
which has a spectrum consisting of copies of
repeated around the unit circle. The ``baseband copy'' of in
can be defined as the sample sequence centered about frequency
zero. Therefore, we can use an ``ideal filter'' to ``pass'' the
baseband spectral copy and zero out all others, thereby converting
to
. I.e.,
Bandlimited Interpolation of TimeLimited Signals
The previous result can be extended toward bandlimited interpolation of which includes all nonzero samples from an arbitrary timelimited signal (i.e., going beyond the interpolation of only periodic bandlimited signals given one or more periods ) by
 replacing the rectangular window with a smoother spectral window , and
 using extra zeropadding in the time domain to convert the cyclic convolution between and into an acyclic convolution between them (recall §7.2.4).
The approximation symbol `' approaches equality as the spectral window approaches (the frequency response of the ideal lowpass filter passing only the original spectrum ), while at the same time allowing no time aliasing (convolution remains acyclic in the time domain).
Equation (7.8) can provide the basis for a highquality samplingrate conversion algorithm. Arbitrarily long signals can be accommodated by breaking them into segments of length , applying the above algorithm to each block, and summing the upsampled blocks using overlapadd. That is, the lowpass filter ``rings'' into the next block and possibly beyond (or even into both adjacent time blocks when is not causal), and this ringing must be summed into all affected adjacent blocks. Finally, the filter can ``window away'' more than the top copies of in , thereby preparing the timedomain signal for downsampling, say by :
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DFT Theorems Problems
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Even and Odd Functions