Fourier Theorems for the DTFTThis 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.
As introduced in §2.1 above, the Discrete-Time Fourier Transform (DTFT) may be defined as
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
signal , , we have
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.
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) SignalsIf 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:
Shift Theorem for the DTFTWe define the shift operator for sampled signals by
where is any integer ( ). Thus, is a right-shift or delay by samples. The shift theorem states3.5
or, in operator notation,
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 DTFTThe convolution of discrete-time signals and is defined as
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 The convolution theorem is then
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 DTFTWe define the correlation of discrete-time signals and by
signal is simply the cross-correlation of with itself:
From the correlation theorem, we have
Power Theorem for the DTFTThe inner product of two signals may be defined in the time domain by 
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. Using inner-product notation, the power theorem (or Parseval's theorem ) for DTFTs can be stated as follows:
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 the time domain by
In other terms, we stretch a sampled signal by the factor by inserting zeros in between each pair of samples of the signal. 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).
repeat operator in the frequency domain as a scaling of frequency axis by some integer factor :
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.
Downsampling and AliasingThe downsampling operator selects every sample of a signal:
The aliasing theorem states that downsampling in time corresponds to aliasing in the frequency domain:
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.
Theorem: Let denote a signal with DTFT , and let
denote the derivative of with respect to . Then we have
Proof: Using integration by parts, we obtain
Continuous-Time Fourier Theorems
Fourier Transform (FT) and Inverse