Fourier Theorems for the DFT

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.

The DFT and its Inverse Restated

Let , denote an -sample complex sequence, i.e., . Then the spectrum of is defined by the Discrete Fourier Transform (DFT):

The inverse DFT (IDFT) is defined by

In this chapter, we will omit mention of an explicit sampling interval , as this is most typical in the digital signal processing literature. It is often said that the sampling frequency is . In this case, a radian frequency is in units of radians per sample.'' Elsewhere in this book, usually means radians per second.'' (Of course, there's no difference when the sampling rate is really .) Another term we use in connection with the convention is normalized frequency: All normalized radian frequencies lie in the range , and all normalized frequencies in Hz lie in the range .7.1 Note that physical units of seconds and Hertz can be reintroduced by the substitution

Notation and Terminology

If is the DFT of , we say that and form a transform pair and write

Another notation we'll use is

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

The DFT sinusoids are all periodic having periods which divide . That is, for any integer . Since a length signal can be expressed as a linear combination of the DFT sinusoids in the time domain,

it follows that the automatic'' definition of beyond the range is periodic extension, i.e., for every integer .

Moreover, the DFT also repeats naturally every samples, since

because . (The DFT sinusoids behave identically as functions of and .) Accordingly, for purposes of DFT studies, we may define all signals in as being single periods from an infinitely long periodic signal with period samples:

Definition (Periodic Extension): For any signal , we define

for every integer .

As a result of this convention, all indexing of signals and spectra7.2 can be interpreted modulo , and we may write to emphasize this. Formally,  '' is defined as with chosen to give in the range .

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 [72].

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 signal-valued function of a signal. Thus, for the space of length complex sequences, an operator is a mapping from to :

An example is the DFT operator:

The argument to an operator is always an entire signal. However, its output may be subscripted to obtain a specific sample, e.g.,

Some operators require one or more parameters affecting their definition. For example the shift operator (defined in §7.2.3 below) requires a shift amount :7.3

A time or frequency index, if present, will always be the last subscript. Thus, the signal is obtained from by shifting it samples.

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).

Flip Operator

We define the flip operator by

 (7.1)

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 .

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Shift Operator

The shift operator is defined by

and denotes the entire shifted signal. Note that since indexing is modulo , the shift is circular (or cyclic''). However, we normally use it to represent time delay by samples. We often use the shift operator in conjunction with zero padding (appending zeros to the signal , §7.2.7) in order to avoid the wrap-around'' associated with a circular shift.

Figure 7.2 illustrates successive one-sample 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

Note that this is circular convolution (or cyclic'' convolution).7.4 The importance of convolution in linear systems theory is discussed in §8.3.

Cyclic convolution can be expressed in terms of previously defined operators as

where and . This expression suggests graphical convolution, discussed below in §7.2.4.

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 :

Because the convolution is cyclic, with and chosen from the set of (periodically extended) vectors of length , is most precisely viewed as the impulse-train-response of the associated filter at time . Specifically, the impulse-train response is the response of the filter to the impulse-train signal , which, by periodic extension, is equal to

Thus, is the period of the impulse-train in samples--there is an impulse'' (a ') every samples. Neglecting the assumed periodic extension of all signals in , we may refer to more simply as the impulse signal, and as the impulse response (as opposed to impulse-train response). In contrast, for the DTFTB.1), in which the discrete-time axis is infinitely long, the impulse signal is defined as

and no periodic extension arises.

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 [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

with

to get

 (7.2)

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 [68] for details). By shifting the impulse response left one sample to get

(in which case ), we obtain a noncausal filter which is symmetric about time zero so that the input signal is smoothed in place'' with no added delay (imagine Fig.7.3(c) shifted left one sample, in which case the input pulse edges align with the midpoint of the rise and fall in the output signal).

 Filter input signal . 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 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

to get

 (7.3)

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

which is the same as . When , we say that is a matched filter for .7.7 In this case, is matched to look for a dc component,'' and also zero-padded by a factor of . The zero-padding serves to simulate acyclic convolution using circular convolution. Note from Eq.(7.3) that the maximum is obtained in the convolution output at time 0. This peak (the largest possible if all input signals are limited to in magnitude), indicates the matched filter has found'' the dc signal starting at time 0. This peak would persist in the presence of some amount of noise and/or interference from other signals. Thus, matched filtering is useful for detecting known signals in the presence of noise and/or interference [34].

Graphical Convolution

As mentioned above, cyclic convolution can be written as

where and . It is instructive to interpret this expression graphically, as depicted in Fig.7.5 above. The convolution result at time is the inner product of and , or . For the next time instant, , we shift one sample to the right and repeat the inner product operation to obtain , and so on. To capture the cyclic nature of the convolution, and can be imagined plotted on a cylinder. Thus, Fig.7.5 shows the cylinder after being cut'' along the vertical line between and and unrolled'' to lay flat.

Polynomial Multiplication

Note that when you multiply two polynomials together, their coefficients are convolved. To see this, let denote the th-order polynomial

with coefficients , and let denote the th-order polynomial

with coefficients . Then we have [1]

Denoting by

we have that the th coefficient can be expressed as

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.,

it follows that multiplying two numbers convolves their digits. The only twist is that, unlike normal polynomial multiplication, we have carries. That is, when a convolution result (output digit) exceeds 10, we subtract 10 from the result and add 1 to the digit in the next higher place.

Correlation

The correlation operator for two signals and in is defined as

We may interpret the correlation operator as

which is times the coefficient of projection onto of advanced by samples (shifted circularly to the left by samples). The time shift is called the correlation lag, and is called a lagged product. Applications of correlation are discussed in §8.4.

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

Thus, to stretch a signal by the factor , insert zeros between each pair of samples. An example of a stretch by factor three is shown in Fig.7.6. The example is

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 consists of extending a signal (or spectrum) with zeros. It maps a length signal to a length signal, but need not divide .

Definition:

 (7.4)

where , with for odd, and for even. For example,

In this example, the first sample corresponds to time 0, and five zeros have been inserted between the samples corresponding to times and .

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.

Causal (Periodic) Signals

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,

Therefore, when we simply append zeros to the end of signal, we call it 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 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.,

Causal zero-padding should not be used on a spectrum of a real signal because, as we will see in §7.4.3 below, the magnitude spectrum of every real signal is symmetric about frequency zero. For the same reason, we cannot simply append zeros in the time domain when the signal frame is considered to include negative-time samples, as in zero-centered FFT processing'' (discussed in Book IV [70]). Nevertheless, in practice, appending zeros is perhaps the most common form of zero-padding. It is implemented automatically, for example, by the matlab function fft(x,N) when the FFT size N exceeds the length of the signal vector x.

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 theorem7.4.12 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 [70] 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.

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.7.9In 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

That is, is the unnormalized coefficient of projection of onto the sinusoid at frequency . When , for , we obtain the special set of spectral samples known as the DFT. For other values of , we obtain spectral points in between the DFT samples. Interpolating DFT samples should give the same result. It is straightforward to show that this ideal form of interpolation is what we call bandlimited interpolation, as discussed further in Appendix D and in Book IV [70] of this series.

Interpolation Operator

The interpolation operator interpolates a signal by an integer factor using bandlimited interpolation. For frequency-domain 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 time-limited spectral interpolation in this case).

For time-domain 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

where , and indexing of is modulo (periodic extension). Thus, the operator simply repeats its input signal times.7.10 An example of is shown in Fig.7.8. The example is

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

where is defined in §7.2.6. That is, when you stretch a signal by the factor (inserting zeros between the original samples), its spectrum is repeated times around the unit circle. The simple proof is given on page .

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 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 .

Alias Operator

Aliasing occurs when a signal is undersampled. If the signal sampling rate is too low, we get frequency-domain aliasing.

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).

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 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.

Example:

The alias operator is used to state the Fourier theorem7.4.11)

That is, when you downsample a signal by the factor , its spectrum is aliased by the factor .

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

Some of the Fourier theorems can be succinctly expressed in terms of even and odd symmetries.

Definition: A function is said to be even if .

An even function is also symmetric, but the term symmetric applies also to functions symmetric about a point other than 0.

Definition: A function is said to be odd if .

An odd function is also called antisymmetric.

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.

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 .

Example: For all DFT sinusoidal frequencies ,

More generally,

for any even signal and odd signal in . In terms of inner products5.9), we may say that the even part of every real signal is orthogonal to its odd part:

Fourier Theorems

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.

Linearity

Theorem: For any and , the DFT satisfies

where and , as always in this book. Thus, the DFT is a linear operator.

Proof:

Conjugation and Reversal

Theorem: For any ,

Proof:

Theorem: For any ,

Proof: Making the change of summation variable , we get

Theorem: For any ,

Proof:

Corollary: For any ,

Proof: Picking up the previous proof at the third formula, remembering that is real,

when is real.

Thus, conjugation in the frequency domain corresponds to reversal in the time domain. Another way to say it is that negating spectral phase flips the signal around backwards in time.

Corollary: For any ,

Proof: This follows from the previous two cases.

Definition: The property is called Hermitian symmetry or conjugate symmetry.'' If , it may be called skew-Hermitian.

Another way to state the preceding corollary is

Symmetry

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.

Recall from §7.3 that a signal is said to be even if , and odd if . Below are are Fourier theorems pertaining to even and odd signals and/or spectra.

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

 (7.5)

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:

 (7.6)

Proof: This follows immediately from setting in the preceding proof. From Eq.(7.5), we are left with

Thus, the DFT of a real and even function reduces to a type of cosine transform,7.12

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 [70]).

Shift Theorem

Theorem: For any and any integer ,

Proof:

The shift theorem is often expressed in shorthand as

The shift theorem says that a delay in the time domain corresponds to a linear phase term in the frequency domain. More specifically, a delay of samples in the time waveform corresponds to the linear phase term multiplying the spectrum, where .7.13Note that spectral magnitude is unaffected by a linear phase term. That is, .

Linear Phase Terms

The reason is called a linear phase term is that its phase is a linear function of frequency:

Thus, the slope of the phase, viewed as a linear function of radian-frequency , is . In general, the time delay in samples equals minus the slope of the linear phase term. If we express the original spectrum in polar form as

where and are the magnitude and phase of , respectively (both real), we can see that a linear phase term only modifies the spectral phase :

where . A positive time delay (waveform shift to the right) adds a negatively sloped linear phase to the original spectral phase. A negative time delay (waveform shift to the left) adds a positively sloped linear phase to the original spectral phase. If we seem to be belaboring this relationship, it is because it is one of the most useful in practice.

Linear PhaseSignals

In practice, a signal may be said to be linear phase when its phase is of the form

where is any real constant (usually an integer), and is an indicator function which takes on the values 0 or over the points , . An important class of examples is when the signal is regarded as a filter impulse response.7.14 What all such signals have in common is that they are symmetric about the time in the time domain (as we will show on the next page). Thus, the term linear phase signal'' often really means a signal whose phase is linear between discontinuities.''

Zero PhaseSignals

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.

Convolution Theorem

Theorem: For any ,

Proof:

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 7.1.7.17 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.

Table 7.1: Direct versus FFT convolution times in milliseconds (convolution length = ) using Matlab 5.2 on an 800 MHz Athlon Windows PC.
 M Direct FFT Ratio 1 0.07 0.08 0.91 2 0.08 0.08 0.92 3 0.08 0.08 0.94 4 0.09 0.10 0.97 5 0.12 0.12 0.96 6 0.18 0.12 1.44 7 0.39 0.15 2.67 8 1.10 0.21 5.10 9 3.83 0.31 12.26 10 15.80 0.47 33.72 11 50.39 1.09 46.07 12 177.75 2.53 70.22 13 709.75 5.62 126.18 14 4510.25 17.50 257.73 15 19050.00 72.50 262.76 16 316375.00 440.50 718.22

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

The dual7.18 of the convolution theorem says that multiplication in the time domain is convolution in the frequency domain:

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 [70] for further discussion.

Correlation Theorem

Theorem: For all ,

where the correlation operation ' was defined in §7.2.5.

Proof:

The last step follows from the convolution theorem and the result from §7.4.2. Also, the summation range in the second line is equivalent to the range because all indexing is modulo .

Power Theorem

Theorem: For all ,

Proof:

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

Normalized DFTPower Theorem

Note that the power theorem would be more elegant if the DFT were defined as the coefficient of projection onto the normalized DFT sinusoids

That is, for the normalized DFT6.10), the power theorem becomes simply

(Normalized DFT case)

We see that the power theorem expresses the invariance of the inner product between two signals in the time and frequency domains. If we think of the inner product geometrically, as in Chapter 5, then this result is expected, because and are merely coordinates of the same geometric object (a signal) relative to two different sets of basis signals (the shifted impulses and the normalized DFT sinusoids).

Rayleigh Energy Theorem (Parseval's Theorem)

Theorem: For any ,

I.e.,

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.

Stretch Theorem (Repeat Theorem)

Theorem: For all ,

Proof: Recall the stretch operator:

Let , where , . Also define the new denser frequency grid associated with length by , and define as usual. Then

But

Thus, , and by the modulo indexing of , copies of are generated as goes from 0 to .

Downsampling Theorem (Aliasing Theorem)

Theorem: For all ,

Proof: Let denote the frequency index in the aliased spectrum, and let . Then is length , where is the downsampling factor. We have

Since , the sum over becomes

using the closed form expression for a geometric series derived in §6.1. We see that the sum over effectively samples every samples. This can be expressed in the previous formula by defining which ranges only over the nonzero samples:

Since the above derivation also works in reverse, the theorem is proved.

An illustration of aliasing in the frequency domain is shown in Fig.7.12.

Illustration of the Downsampling/Aliasing Theorem in Matlab

>> 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


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

where was defined in Eq.(7.4), followed by the definition of .

Proof: Let with . Then

Thus, this theorem follows directly from the definition of the ideal interpolation operator . See §8.1.3 for an example of zero-padding in spectrum analysis.

The dual of the zero-padding theorem states formally that zero padding in the frequency domain corresponds to periodic interpolation in the time domain:

Definition: For all and any integer ,

 (7.7)

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).

Relation to Stretch Theorem

It is instructive to interpret the periodic interpolation theorem in terms of the stretch theorem, . To do this, it is convenient to define a zero-centered rectangular window'' operator:

Definition: For any and any odd integer we define the length even rectangular windowing operation by

Thus, this zero-phase rectangular window,'' when applied to a spectrum , sets the spectrum to zero everywhere outside a zero-centered interval of samples. Note that is the ideal lowpass filtering operation in the frequency domain. The cut-off frequency'' is radians per sample. For even , we allow to be passed'' by the window, but in our usage (below), this sample should always be zero anyway. With this notation defined we can efficiently restate periodic interpolation in terms of the operator:

Theorem: When consists of one or more periods from a periodic signal ,

In other words, ideal periodic interpolation of one period of by the integer factor may be carried out by first stretching by the factor (inserting zeros between adjacent samples of ), taking the DFT, applying the ideal lowpass filter as an -point rectangular window in the frequency domain, and performing the inverse DFT.

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 last step is provided by the zero-padding theorem7.4.12).

Bandlimited Interpolation of Time-Limited Signals

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

1. replacing the rectangular window with a smoother spectral window , and
2. 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 smoother spectral window can be thought of as the frequency response of the FIR7.22 filter used as the bandlimited interpolation kernel in the time domain. The number of zeros needed in the zero-padding of in the time domain is simply length of minus 1, and the number of zeros to be appended to is the length of minus 1. With this much zero-padding, the cyclic convolution of and implemented using the DFT becomes equivalent to acyclic convolution, as desired for the time-limited signals and . Thus, if denotes the nonzero length of , then the nonzero length of is , and we require the DFT length to be , where is the filter length. In operator notation, we can express bandlimited sampling-rate up-conversion by the factor for time-limited signals by

 (7.8)

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 :

where now the lowpass filter frequency response must be close to zero for all . While such a sampling-rate conversion algorithm can be made more efficient by using an FFT in place of the DFT (see Appendix A), it is not necessarily the most efficient algorithm possible. This is because (1) out of output samples from the IDFT need not be computed at all, and (2) has many zeros in it which do not need explicit handling. For an introduction to time-domain sampling-rate conversion (bandlimited interpolation) algorithms which take advantage of points (1) and (2) in this paragraph, see, e.g., Appendix D and [72].

DFT Theorems Problems

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