Fourier Theorems for the DTFT
This section states and proves selected Fourier theorems for the DTFT. A more complete list for the DFT case is given in .3.4Since this material was originally part of an appendix, it is relatively dry reading. Feel free to skip to the next chapter and refer back as desired when a theorem is invoked.
We say that is the spectrum of .
Linearity of the DTFT
where are any scalars (real or complex numbers), and are any two discrete-time signals (real- or complex-valued functions of the integers), and are their corresponding continuous-frequency spectra defined over the unit circle in the complex plane.
Proof: We have
One way to describe the linearity property is to observe that the Fourier transform ``commutes with mixing.''
For any complex signal , , we have
Arguably, should include complex conjugation. Let
denote such a definition. Then in this case we have
In the typical special case of real signals ( ), we have so that
That is, time-reversing a real signal conjugates its spectrum.
Most (if not all) of the signals we deal with in practice are real signals. Here we note some spectral symmetries associated with real signals.
The previous section established that the spectrum of every real signal satisfies
In other terms, if a signal is real, then its spectrum is Hermitian (``conjugate symmetric''). Hermitian spectra have the following equivalent characterizations:
- The real part is even, while the imaginary part is odd:
- The magnitude is even, while the phase is odd:
Real Even (or Odd) Signals
If a signal is even in addition to being real, then its DTFT is also real and even. This follows immediately from the Hermitian symmetry of real signals, and the fact that the DTFT of any even signal is real:
This is true since cosine is even, sine is odd, even times even is even, even times odd is odd, and the sum over all samples of an odd signal is zero. I.e.,
If is real and even, the following are true:
Similarly, if a signal is odd and real, then its DTFT is odd and purely imaginary. This follows from Hermitian symmetry for real signals, and the fact that the DTFT of any odd signal is imaginary.
where we used the fact that
Shift Theorem for the DTFT
where is any integer ( ). Thus, is a right-shift or delay by samples.
The shift theorem states3.5
or, in operator notation,
Note that is a linear phase term, so called because it is a linear function of frequency with slope equal to :
The shift theorem gives us that multiplying a spectrum by a linear phase term corresponds to a delay in the time domain by samples. If , it is called a time advance by samples.
Convolution Theorem for the DTFT
This is sometimes called acyclic convolution to distinguish it from the cyclic convolution used for length sequences in the context of the DFT . Convolution is cyclic in the time domain for the DFT and FS cases (i.e., whenever the time domain has a finite length), and acyclic for the DTFT and FT cases.3.6
That is, convolution in the time domain corresponds to pointwise multiplication in the frequency domain.
Proof: The result follows immediately from interchanging the order of summations associated with the convolution and DTFT:
Correlation Theorem for the DTFT
The correlation theorem for DTFTs is then
From the correlation theorem, we have
Note that this definition of autocorrelation is appropriate for signals having finite support (nonzero over a finite number of samples). For infinite-energy (but finite-power) signals, such as stationary noise processes, we define the sample autocorrelation to include a normalization suitable for this case (see Chapter 6 and Appendix C).
From the autocorrelation theorem we have that a digital-filter impulse-response is that of a lossless allpass filter  if and only if . In other words, the autocorrelation of the impulse-response of every allpass filter is impulsive.
Power Theorem for the DTFT
The inner product of two spectra may be defined as
Note that the frequency-domain inner product includes a normalization factor while the time-domain definition does not.
That is, the inner product of two signals in the time domain equals the inner product of their respective spectra (a complex scalar in general).
When we consider the inner product of a signal with itself, we have the special case known as the energy theorem (or Rayleigh's energy theorem):
where denotes the norm induced by the inner product. It is always real.
In other terms, we stretch a sampled signal by the factor by inserting zeros in between each pair of samples of the signal.
In the literature on multirate filter banks (see Chapter 11), the stretch operator is typically called instead the upsampling operator. That is, stretching a signal by the factor of is called upsampling the signal by the factor . (See §11.1.1 for the graphical symbol ( ) and associated discussion.) The term ``stretch'' is preferred in this book because ``upsampling'' is easily confused with ``increasing the sampling rate''; resampling a signal to a higher sampling rate is conceptually implemented by a stretch operation followed by an ideal lowpass filter which moves the inserted zeros to their properly interpolated values.
Note that we could also call the stretch operator the scaling operator, to unify the terminology in the discrete-time case with that of the continuous-time case (§2.4.1 below).
where denotes the radian frequency variable after applying the repeat operator.
The repeat operator maps the entire unit circle (taken as to ) to a segment of itself , centered about , and repeated times. This is illustrated in Fig.2.2 for .
Since the frequency axis is continuous and -periodic for DTFTs, the repeat operator is precisely equivalent to a scaling operator for the Fourier transform case (§B.4). We call it ``repeat'' rather than ``scale'' because we are restricting the scale factor to positive integers, and because the name ``repeat'' describes more vividly what happens to a periodic spectrum that is compressively frequency-scaled over the unit circle by an integer factor.
Using these definitions, we can compactly state the stretch theorem:
As traverses the interval , traverses the unit circle times, thus implementing the repeat operation on the unit circle. Note also that when , we have , so that dc always maps to dc. At half the sampling rate , on the other hand, after the mapping, we may have either ( odd), or ( even), where .
The stretch theorem makes it clear how to do ideal sampling-rate conversion for integer upsampling ratios : We first stretch the signal by the factor (introducing zeros between each pair of samples), followed by an ideal lowpass filter cutting off at . That is, the filter has a gain of 1 for , and a gain of 0 for . Such a system (if it were realizable) implements ideal bandlimited interpolation of the original signal by the factor .
Downsampling and Aliasing
The downsampling operator selects every sample of a signal:
where the operator is defined as
for . The summation terms for are called aliasing components.
In z transform notation, the operator can be expressed as 
where is a common notation for the primitive th root of unity. On the unit circle of the plane, this becomes
The frequency scaling corresponds to having a sampling interval of after downsampling, which corresponds to the interval prior to downsampling.
The aliasing theorem makes it clear that, in order to downsample by factor without aliasing, we must first lowpass-filter the spectrum to . This filtering (when ideal) zeroes out the spectral regions which alias upon downsampling.
Note that any rational sampling-rate conversion factor may be implemented as an upsampling by the factor followed by downsampling by the factor [50,287]. Conceptually, a stretch-by- is followed by a lowpass filter cutting off at , followed by downsample-by- , i.e.,
In practice, there are more efficient algorithms for sampling-rate conversion [270,135,78] based on a more direct approach to bandlimited interpolation.
where we have chosen to keep frequency samples in terms of the original frequency axis prior to downsampling, i.e., for both and . This choice allows us to easily take the limit as by simply replacing by :
Replacing by and converting to -transform notation instead of Fourier transform notation , with , yields the final result.
denote the derivative of with respect to . Then we have
where denotes the DTFT of .
Proof: Using integration by parts, we obtain
An alternate method of proof is given in §B.3.
Corollary: Perhaps a cleaner statement is as follows:
This completes our coverage of selected DTFT theorems. The next section adds some especially useful FT theorems having no precise counterpart in the DTFT (discrete-time) case.
Continuous-Time Fourier Theorems
Fourier Transform (FT) and Inverse