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).
Even and Odd Functions
The DFT and its Inverse Restated