Signal Operators
It will be convenient in the Fourier theorems of §7.4 to make use of the following signal operator definitions.
Operator Notation
In this book, an operator is defined as a signalvalued 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 signalprocessing 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).
Flip Operator
We define the flip operator by
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 .
Shift Operator
The shift operator is defined by
Figure 7.2 illustrates successive onesample delays of a periodic signal having first period given by .
Examples

(an impulse delayed one sample).

(a circular shift example).
 (another circular shift example).
Convolution
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
Convolution (cyclic or acyclic) is commutative, i.e.,
Proof:
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, realworld systems may be simulated using fast DFT convolutions (see Appendix A for more on fast convolution algorithms).
Note that only linear, timeinvariant (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 [68] of the music signal processing book series (in which this is Book I).
Convolution Example 1: Smoothing a Rectangular Pulse
Filter
input signal .
Filter impulse response .
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 threepoint average'' filter. Note how the corners of the rectangular pulse are ``smoothed'' by the threepoint 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 [68] for details). By shifting the impulse response left one sample to get
Convolution Example 2: ADSR Envelope
Filter impulse response .
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 smoothingfilter 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 timeconstants 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, socalled virtual analog synthesizers. Such an ADSR envelope is typically used to multiply the output of a waveform oscillator such as a sawtooth or pulsetrain oscillator. For more on virtual analog synthesis, see, for example, [78,77].
Convolution Example 3: Matched Filtering
Figure 7.5 illustrates convolution of
to get
For example, could be a ``rectangularly windowed signal, zeropadded by a factor of 2,'' where the signal happened to be dc (all s). For the convolution, we need
Graphical Convolution
As mentioned above, cyclic convolution can be written as
Polynomial Multiplication
Note that when you multiply two polynomials together, their coefficients are convolved. To see this, let denote the thorder polynomial
Denoting by
where and are doubly infinite sequences, defined as zero for and , respectively.
Multiplication of Decimal Numbers
Since decimal numbers are implicitly just polynomials in the powers of 10, e.g.,
Correlation
The correlation operator for two signals and in is defined as
We may interpret the correlation operator as
Stretch Operator
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.
A stretch by factor is defined by
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).
Zero Padding
Zero padding consists of extending a signal (or spectrum) with zeros. It maps a length signal to a length signal, but need not divide .
Definition:
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 timedomain and frequencydomain definitions of zeropadding by interpreting the original time axis as indexing positivetime 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 timedomain samples as extending from 0 to , i.e., with no negativetime samples. For this case, we define ``causal zero padding'' as described below.
Causal (Periodic) Signals
A signal may be defined as causal when for all ``negativetime'' samples (e.g., for when is even). Thus, the signal is causal while is not. For causal signals, zeropadding is equivalent to simply appending zeros to the original signal. For example,
Causal Zero Padding
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 starttime of the frame as zero, with no negativetime samples. In other words, represents an sample signalsegment that is translated in time to start at time 0. In this case (no negativetime samples in the frame), it is proper to zeropad by simply appending zeros at the end of the frame. Thus, we define e.g.,
In summary, we have defined two types of zeropadding that arise in practice, which we may term ``causal'' and ``zerocentered'' (or ``zerophase'', or even ``periodic''). The zerocentered 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 zeropadding in the time domain corresponds to ideal bandlimited interpolation in the frequency domain, and vice versa.
Zero Padding Applications
Zero padding in the time domain is used extensively in practice to compute heavily interpolated spectra by taking the DFT of the zeropadded 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 timelimited 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 timelimited unless it is the zero signal. On the other hand, the interpolation of a timelimited signal's spectrum is nonzero almost everywhere between the original spectral samples. Thus, zeropadding is often used when analyzing data from a nonperiodic signal in blocks, and each block, or frame, is treated as a finiteduration signal which can be zeropadded on either side with any number of zeros. In summary, the use of zeropadding corresponds to the timelimited assumption for the data frame, and more zeropadding yields denser interpolation of the frequency samples around the unit circle.
Sometimes people will say that zeropadding 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 [70] for details). The usual way to increase spectral resolution is to take a longer DFT without zero paddingi.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, zeropadding in one domain corresponds to a higher interpolationdensity in the other domainnot a higher resolution.
Ideal Spectral Interpolation
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.^{7.9}In other words, for truly timelimited signals , taking the DFT of the entire nonzero portion of extended by zeros yields exact interpolation of the complex spectrumnot an approximation (ignoring computational roundoff error in the DFT itself). Because the fast Fourier transform (FFT) is so efficient, zeropadding followed by an FFT is a highly practical method for interpolating spectra of finiteduration 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
Interpolation Operator
The interpolation operator interpolates a signal by an integer factor using bandlimited interpolation. For frequencydomain signals , , we may write spectral interpolation as follows:
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 timelimited spectral interpolation in this case).
For timedomain signals , exact interpolation is similarly bandlimited interpolation, as derived in Appendix D.
Repeat Operator
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 frequencydomain 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 rightrear 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
Downsampling Operator
Downsampling by (also called decimation by ) is defined for as taking every th sample, starting with sample zero:
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 timevarying operators. They can be modeled using timevarying 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 samplingrate 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 lowpassfilter is needed, in general, prior to downsampling so that aliasing is avoided. This topic is address in Appendix D. Early samplingrate converters were in fact implemented using the operation, followed by an appropriate lowpass filter, followed by , in order to implement a samplingrate conversion by the factor .
Alias Operator
Aliasing occurs when a signal is undersampled. If the signal sampling rate is too low, we get frequencydomain aliasing.
The topic of aliasing normally arises in the context of sampling a continuoustime 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 discretetime signals, in order to keep the math as simple as possible. Aliasing in this context occurs when a discretetime signal is downsampled to reduce its sampling rate. You can think of continuoustime 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 highfrequency sinusoid is indistinguishable from the lowerfrequency sinusoid due to aliasing. We say the higher frequency aliases to the lower frequency.
Undersampling in the frequency domain gives rise to timedomain aliasing. If time or frequency is not specified, the term ``aliasing'' normally means frequencydomain aliasing (due to undersampling in the time domain).
The aliasing operator for sample signals is defined by
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 timedomain aliasing; if it is a spectrum, we call it frequencydomain 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.
Example:
The alias operator is used to state the Fourier theorem (§7.4.11)
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.7.12bc) cover frequencies 0 to .
In general, aliasing by the factor corresponds to a samplingrate reduction by the factor . To prevent aliasing when reducing the sampling rate, an antialiasing 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).
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Even and Odd Functions
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The DFT and its Inverse Restated