This chapter derives various Fourier theorems for the case of the DFT. Included are symmetry relations, the shift theorem, convolution theorem, correlation theorem, power theorem, and theorems pertaining to interpolation and downsampling. Applications related to certain theorems are outlined, including linear time-invariant filtering, sampling rate conversion, and statistical signal processing.
If is the DFT of , we say that and form a transform pair and write
If we need to indicate the length of the DFT explicitly, we will write and . As we've already seen, time-domain signals are consistently denoted using lowercase symbols such as ``,'' while frequency-domain signals (spectra), are denoted in uppercase (`` '').
Modulo Indexing, Periodic Extension
Moreover, the DFT also repeats naturally every samples, since
Definition (Periodic Extension): For any signal , we define
As an example, when indexing a spectrum , we have that which can be interpreted physically as saying that the sampling rate is the same frequency as dc for discrete time signals. Periodic extension in the time domain implies that the signal input to the DFT is mathematically treated as being samples of one period of a periodic signal, with the period being exactly seconds ( samples). The corresponding assumption in the frequency domain is that the spectrum is exactly zero between frequency samples . It is also possible to adopt the point of view that the time-domain signal consists of samples preceded and followed by zeros. In that case, the spectrum would be nonzero between spectral samples , and the spectrum between samples would be reconstructed by means of bandlimited interpolation .
In this book, an operator is defined as a signal-valued function of a signal. Thus, for the space of length complex sequences, an operator is a mapping from to :
Note that operator notation is not standard in the field of digital signal processing. It can be regarded as being influenced by the field of computer science. In the Fourier theorems below, both operator and conventional signal-processing notations are provided. In the author's opinion, operator notation is consistently clearer, allowing powerful expressions to be written naturally in one line (e.g., see Eq.(7.8)), and it is much closer to how things look in a readable computer program (such as in the matlab language).
for all sample indices . By modulo indexing, is the same as . The operator reverses the order of samples through of a sequence, leaving sample 0 alone, as shown in Fig.7.1a. Thanks to modulo indexing, it can also be viewed as ``flipping'' the sequence about the time 0, as shown in Fig.7.1b. The interpretation of Fig.7.1b is usually the one we want, and the operator is usually thought of as ``time reversal'' when applied to a signal or ``frequency reversal'' when applied to a spectrum .
(an impulse delayed one sample).
(a circular shift example).
- (another circular shift example).
The convolution of two signals and in may be denoted `` '' and defined by
Cyclic convolution can be expressed in terms of previously defined operators as
Commutativity of Convolution
In the first step we made the change of summation variable , and in the second step, we made use of the fact that any sum over all terms is equivalent to a sum from 0 to .
Convolution as a Filtering Operation
In a convolution of two signals , where both and are signals of length (real or complex), we may interpret either or as a filter that operates on the other signal which is in turn interpreted as the filter's ``input signal''.7.5 Let denote a length signal that is interpreted as a filter. Then given any input signal , the filter output signal may be defined as the cyclic convolution of and :
As discussed below (§7.2.7), one may embed acyclic convolution within a larger cyclic convolution. In this way, real-world systems may be simulated using fast DFT convolutions (see Appendix A for more on fast convolution algorithms).
Note that only linear, time-invariant (LTI) filters can be completely represented by their impulse response (the filter output in response to an impulse at time 0). The convolution representation of LTI digital filters is fully discussed in Book II  of the music signal processing book series (in which this is Book I).
Convolution Example 1: Smoothing a Rectangular Pulse
Filter input signal .
Filter output signal .
Figure 7.3 illustrates convolution of
as graphed in Fig.7.3(c). In this case, can be viewed as a ``moving three-point average'' filter. Note how the corners of the rectangular pulse are ``smoothed'' by the three-point filter. Also note that the pulse is smeared to the ``right'' (forward in time) because the filter impulse response starts at time zero. Such a filter is said to be causal (see  for details). By shifting the impulse response left one sample to get
Convolution Example 2: ADSR Envelope
Filter output signal .
In this example, the input signal is a sequence of two rectangular pulses, creating a piecewise constant function, depicted in Fig.7.4(a). The filter impulse response, shown in Fig.7.4(b), is a truncated exponential.7.6
In this example, is again a causal smoothing-filter impulse response, and we could call it a ``moving weighted average'', in which the weighting is exponential into the past. The discontinuous steps in the input become exponential ``asymptotes'' in the output which are approached exponentially. The overall appearance of the output signal resembles what is called an attack, decay, release, and sustain envelope, or ADSR envelope for short. In a practical ADSR envelope, the time-constants for attack, decay, and release may be set independently. In this example, there is only one time constant, that of . The two constant levels in the input signal may be called the attack level and the sustain level, respectively. Thus, the envelope approaches the attack level at the attack rate (where the ``rate'' may be defined as the reciprocal of the time constant), it next approaches the sustain level at the ``decay rate'', and finally, it approaches zero at the ``release rate''. These envelope parameters are commonly used in analog synthesizers and their digital descendants, so-called virtual analog synthesizers. Such an ADSR envelope is typically used to multiply the output of a waveform oscillator such as a sawtooth or pulse-train oscillator. For more on virtual analog synthesis, see, for example, [78,77].
Convolution Example 3: Matched Filtering
Figure 7.5 illustrates convolution of
For example, could be a ``rectangularly windowed signal, zero-padded by a factor of 2,'' where the signal happened to be dc (all s). For the convolution, we need
As mentioned above, cyclic convolution can be written as
Note that when you multiply two polynomials together, their coefficients are convolved. To see this, let denote the th-order polynomial
where and are doubly infinite sequences, defined as zero for and , respectively.
Since decimal numbers are implicitly just polynomials in the powers of 10, e.g.,
The correlation operator for two signals and in is defined as
We may interpret the correlation operator as
Unlike all previous operators, the operator maps a length signal to a length signal, where and are integers. We use ``'' instead of ``'' as the time index to underscore this fact.
The stretch operator is used to describe and analyze upsampling, that is, increasing the sampling rate by an integer factor. A stretch by followed by lowpass filtering to the frequency band implements ideal bandlimited interpolation (introduced in Appendix D).
where , with for odd, and for even. For example,
Figure 7.7 illustrates zero padding from length out to length . Note that and could be replaced by and in the figure caption.
Note that we have unified the time-domain and frequency-domain definitions of zero-padding by interpreting the original time axis as indexing positive-time samples from 0 to (for even), and negative times in the interval .7.8 Furthermore, we require when is even, while odd requires no such restriction. In practice, we often prefer to interpret time-domain samples as extending from 0 to , i.e., with no negative-time samples. For this case, we define ``causal zero padding'' as described below.
A signal may be defined as causal when for all ``negative-time'' samples (e.g., for when is even). Thus, the signal is causal while is not. For causal signals, zero-padding is equivalent to simply appending zeros to the original signal. For example,
In practice, a signal is often an -sample frame of data taken from some longer signal, and its true starting time can be anything. In such cases, it is common to treat the start-time of the frame as zero, with no negative-time samples. In other words, represents an -sample signal-segment that is translated in time to start at time 0. In this case (no negative-time samples in the frame), it is proper to zero-pad by simply appending zeros at the end of the frame. Thus, we define e.g.,
In summary, we have defined two types of zero-padding that arise in practice, which we may term ``causal'' and ``zero-centered'' (or ``zero-phase'', or even ``periodic''). The zero-centered case is the more natural with respect to the mathematics of the DFT, so it is taken as the ``official'' definition of ZEROPAD(). In both cases, however, when properly used, we will have the basic Fourier theorem (§7.4.12 below) stating that zero-padding in the time domain corresponds to ideal bandlimited interpolation in the frequency domain, and vice versa.
Zero padding in the time domain is used extensively in practice to compute heavily interpolated spectra by taking the DFT of the zero-padded signal. Such spectral interpolation is ideal when the original signal is time limited (nonzero only over some finite duration spanned by the orignal samples).
Note that the time-limited assumption directly contradicts our usual assumption of periodic extension. As mentioned in §6.7, the interpolation of a periodic signal's spectrum from its harmonics is always zero; that is, there is no spectral energy, in principle, between the harmonics of a periodic signal, and a periodic signal cannot be time-limited unless it is the zero signal. On the other hand, the interpolation of a time-limited signal's spectrum is nonzero almost everywhere between the original spectral samples. Thus, zero-padding is often used when analyzing data from a non-periodic signal in blocks, and each block, or frame, is treated as a finite-duration signal which can be zero-padded on either side with any number of zeros. In summary, the use of zero-padding corresponds to the time-limited assumption for the data frame, and more zero-padding yields denser interpolation of the frequency samples around the unit circle.
Sometimes people will say that zero-padding in the time domain yields higher spectral resolution in the frequency domain. However, signal processing practitioners should not say that, because ``resolution'' in signal processing refers to the ability to ``resolve'' closely spaced features in a spectrum analysis (see Book IV  for details). The usual way to increase spectral resolution is to take a longer DFT without zero padding--i.e., look at more data. In the field of graphics, the term resolution refers to pixel density, so the common terminology confusion is reasonable. However, remember that in signal processing, zero-padding in one domain corresponds to a higher interpolation-density in the other domain--not a higher resolution.
Using Fourier theorems, we will be able to show (§7.4.12) that zero padding in the time domain gives exact bandlimited interpolation in the frequency domain.7.9In other words, for truly time-limited signals , taking the DFT of the entire nonzero portion of extended by zeros yields exact interpolation of the complex spectrum--not an approximation (ignoring computational round-off error in the DFT itself). Because the fast Fourier transform (FFT) is so efficient, zero-padding followed by an FFT is a highly practical method for interpolating spectra of finite-duration signals, and is used extensively in practice.
Before we can interpolate a spectrum, we must be clear on what a ``spectrum'' really is. As discussed in Chapter 6, the spectrum of a signal at frequency is defined as a complex number computed using the inner product
Since is initially only defined over the roots of unity in the plane, while is defined over roots of unity, we define for by ideal bandlimited interpolation (specifically time-limited spectral interpolation in this case).
For time-domain signals , exact interpolation is similarly bandlimited interpolation, as derived in Appendix D.
Like the and operators, the operator maps a length signal to a length signal:
Definition: The repeat times operator is defined for any by
A frequency-domain example is shown in Fig.7.9. Figure 7.9a shows the original spectrum , Fig.7.9b shows the same spectrum plotted over the unit circle in the plane, and Fig.7.9c shows . The point (dc) is on the right-rear face of the enclosing box. Note that when viewed as centered about , is a somewhat ``triangularly shaped'' spectrum. We see three copies of this shape in .
The repeat operator is used to state the Fourier theorem
The operator maps a length signal down to a length signal. It is the inverse of the operator (but not vice versa), i.e.,
The stretch and downsampling operations do not commute because they are linear time-varying operators. They can be modeled using time-varying switches controlled by the sample index .
The following example of is illustrated in Fig.7.10:
Note that the term ``downsampling'' may also refer to the more elaborate process of sampling-rate conversion to a lower sampling rate, in which a signal's sampling rate is lowered by resampling using bandlimited interpolation. To distinguish these cases, we can call this bandlimited downsampling, because a lowpass-filter is needed, in general, prior to downsampling so that aliasing is avoided. This topic is address in Appendix D. Early sampling-rate converters were in fact implemented using the operation, followed by an appropriate lowpass filter, followed by , in order to implement a sampling-rate conversion by the factor .
The topic of aliasing normally arises in the context of sampling a continuous-time signal. The sampling theorem (Appendix D) says that we will have no aliasing due to sampling as long as the sampling rate is higher than twice the highest frequency present in the signal being sampled.
In this chapter, we are considering only discrete-time signals, in order to keep the math as simple as possible. Aliasing in this context occurs when a discrete-time signal is downsampled to reduce its sampling rate. You can think of continuous-time sampling as the limiting case for which the starting sampling rate is infinity.
An example of aliasing is shown in Fig.7.11. In the figure, the high-frequency sinusoid is indistinguishable from the lower-frequency sinusoid due to aliasing. We say the higher frequency aliases to the lower frequency.
Undersampling in the frequency domain gives rise to time-domain aliasing. If time or frequency is not specified, the term ``aliasing'' normally means frequency-domain aliasing (due to undersampling in the time domain).
Like the operator, the operator maps a length signal down to a length signal. A way to think of it is to partition the original samples into blocks of length , with the first block extending from sample 0 to sample , the second block from to , etc. Then just add up the blocks. This process is called aliasing. If the original signal is a time signal, it is called time-domain aliasing; if it is a spectrum, we call it frequency-domain aliasing, or just aliasing. Note that aliasing is not invertible in general. Once the blocks are added together, it is usually not possible to recover the original blocks.
Figure 7.12 shows the result of applied to from Figure 7.9c. Imagine the spectrum of Fig.7.12a as being plotted on a piece of paper rolled to form a cylinder, with the edges of the paper meeting at (upper right corner of Fig.7.12a). Then the operation can be simulated by rerolling the cylinder of paper to cut its circumference in half. That is, reroll it so that at every point, two sheets of paper are in contact at all points on the new, narrower cylinder. Now, simply add the values on the two overlapping sheets together, and you have the of the original spectrum on the unit circle. To alias by , we would shrink the cylinder further until the paper edges again line up, giving three layers of paper in the cylinder, and so on.
Figure 7.12b shows what is plotted on the first circular wrap of the cylinder of paper, and Fig.7.12c shows what is on the second wrap. These are overlaid in Fig.7.12d and added together in Fig.7.12e. Finally, Figure 7.12f shows both the addition and the overlay of the two components. We say that the second component (Fig.7.12c) ``aliases'' to new frequency components, while the first component (Fig.7.12b) is considered to be at its original frequencies. If the unit circle of Fig.7.12a covers frequencies 0 to , all other unit circles (Fig.7.12b-c) cover frequencies 0 to .
In general, aliasing by the factor corresponds to a sampling-rate reduction by the factor . To prevent aliasing when reducing the sampling rate, an anti-aliasing lowpass filter is generally used. The lowpass filter attenuates all signal components at frequencies outside the interval so that all frequency components which would alias are first removed.
Conceptually, in the frequency domain, the unit circle is reduced by to a unit circle half the original size, where the two halves are summed. The inverse of aliasing is then ``repeating'' which should be understood as increasing the unit circle circumference using ``periodic extension'' to generate ``more spectrum'' for the larger unit circle. In the time domain, on the other hand, downsampling is the inverse of the stretch operator. We may interchange ``time'' and ``frequency'' and repeat these remarks. All of these relationships are precise only for integer stretch/downsampling/aliasing/repeat factors; in continuous time and frequency, the restriction to integer factors is removed, and we obtain the (simpler) scaling theorem (proved in §C.2).
Some of the Fourier theorems can be succinctly expressed in terms of even and odd symmetries.
Definition: A function is said to be odd if .
Note that every finite odd function must satisfy .7.11 Moreover, for any with even, we also have since ; that is, and index the same point when is even.
Theorem: Every function can be decomposed into a sum of its even part and odd part , where
Proof: In the above definitions, is even and is odd by construction. Summing, we have
Theorem: The product of even functions is even, the product of odd functions is even, and the product of an even times an odd function is odd.
Proof: Readily shown.
Since even times even is even, odd times odd is even, and even times odd is odd, we can think of even as and odd as :
Example: , , is an even signal since .
Example: is an odd signal since .
Example: is an odd signal (even times odd).
Example: is an even signal (odd times odd).
Theorem: The sum of all the samples of an odd signal in is zero.
Proof: This is readily shown by writing the sum as , where the last term only occurs when is even. Each term so written is zero for an odd signal .
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.
Theorem: For any and , the DFT satisfies
Theorem: For any ,
Theorem: For any ,
Proof: Making the change of summation variable , we get
Theorem: For any ,
Corollary: For any ,
Proof: Picking up the previous proof at the third formula, remembering that is real,
Corollary: For any ,
Proof: This follows from the previous two cases.
Another way to state the preceding corollary is
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 negative-frequency spectral samples of a real signal and regenerate them later if needed from the positive-frequency 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.
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 zero-phase over at least some band containing dc (up to the first zero-crossing in frequency). When the phase switches between 0 and at the zero-crossings of the (real) spectrum, the spectrum oscillates between being zero phase and ``constant phase''. We can say that all real spectra are piecewise constant-phase spectra, where the two constant values are 0 and (or , which is the same phase as ). In practice, such zero-crossings typically occur at low magnitude, such as in the ``side-lobes'' of the DTFT of a ``zero-centered symmetric window'' used for spectrum analysis (see Chapter 8 and Book IV ).
Theorem: For any and any integer ,
The shift theorem is often expressed in shorthand as
A zero-phase signal is thus a linear-phase signal for which the phase-slope is zero. As mentioned above (in §7.4.3), it would be more precise to say ``0-or--phase signal'' instead of ``zero-phase signal''. Another better term is ``zero-centered signal'', since every real (even) spectrum corresponds to an even (real) signal. Of course, a zero-centered symmetric signal is simply an even signal, by definition. Thus, a ``zero-phase 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 Transform7.15 (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 odd-length real symmetric signals are ``linear phase'', including FFT windows.
When is even, the window midpoint at time lands half-way between samples, so we cannot simply translate the window to zero-centered 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.
Theorem: For any ,
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 FFT-based convolution requires on the order of operations, where denotes the logarithm-base-2 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 built-in function in both cases.)
N = 1024; % FFT much faster at this length t = 0:N-1; % [0,1,2,...,N-1] 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 direct-convolution time = %0.2f msec\n',... 'Average FFT-convolution time = %0.2f msec\n',... 'Ratio = %0.2f (Direct/FFT)'],... t1/Nrep,t2/Nrep,t1/t2)); % =================== EXAMPLE RESULTS =================== Octave: Average direct-convolution time = 69.49 msec Average FFT-convolution time = 0.23 msec Ratio = 296.40 (Direct/FFT) Matlab: Average direct-convolution time = 15.73 msec Average FFT-convolution time = 0.50 msec Ratio = 31.46 (Direct/FFT)
A similar program produced the results for different FFT lengths shown in Table 220.127.116.11 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 length-4 convolution, versus 176 for the fft function).
See Appendix A for further discussion of FFT algorithms and their applications.
Dual of the Convolution 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 zero-padding). See Chapter 8 and Book IV  for further discussion.
Theorem: For all ,
Theorem: For all ,
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
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.
Theorem: For all ,
Proof: Recall the stretch operator:
Downsampling Theorem (Aliasing Theorem)
Theorem: For all ,
Since , the sum over becomes
Since the above derivation also works in reverse, the theorem is proved.
>> 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 time-limited signals):
Theorem: For any
Proof: Let with . Then
Periodic Interpolation (Spectral Zero Padding)
Definition: For all and any integer ,
where zero padding is defined in §7.2.7 and illustrated in Figure 7.7. In other words, zero-padding a DFT by the factor in the frequency domain (by inserting zeros at bin number corresponding to the folding frequency7.21) 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 time-aliased 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 non-periodic signals, which is almost always the case in practice, bandlimited interpolation should be used instead (Appendix D).
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.,
The previous result can be extended toward bandlimited interpolation of which includes all nonzero samples from an arbitrary time-limited 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 zero-padding 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 high-quality sampling-rate 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 up-sampled blocks using overlap-add. 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 time-domain signal for downsampling, say by :
Example Applications of the DFT
Derivation of the Discrete Fourier Transform (DFT)