Time-Frequency and Instantaneous Frequency Concepts0

CHAPTER

TIME-FREQUENCY AND INSTANTANEOUS FREQUENCY CONCEPTS0

1

INTRODUCTION AND OVERVIEW The previous chapter introduced the field of time-frequency (t, f ) signal analysis and processing (TFSAP), which concerns the analysis and processing of signals with time-varying frequency content. Such signals are best represented by a time-frequency distribution (TFD), which is intended to show how the energy of the signal is distributed over the two-dimensional (t, f ) space. Processing of the signal may then exploit the features produced by the concentration of signal energy in two dimensions (time and frequency) instead of only one (time or frequency). This chapter begins with the more formal, in-depth part of the tutorial which occupies the rest of Part I (Chapters 1–3). The present tutorial updates the one given in Ref. [1] and the first edition of this book by including more recent advances, refining terminology, and simplifying both the presentations of concepts and formulations of methodologies. This core material needs to be understood by anybody seeking to specialize in this field. Precise definitions and formulations of the key concepts needed to formulate (t, f ) methods are provided. Chapter 1 includes four sections. Section 1.1 describes in detail the justifications why timefrequency methods are preferred for a wide range of applications in which the signals have time-varying spectral characteristics or multiple components for which the variables t and f are related. Section 1.2 provides the signal models and mathematical formulations needed to describe temporal and spectral characteristics of nonstationary signals in the time-frequency domain. It defines such basic concepts as analytic signals, Hilbert transform (HT), bandwidth-duration product, and asymptotic signals. Section 1.3 defines the key quantities related to time-frequency methods, including the instantaneous frequency (IF), spectral delay (SD), and group delay (GD). Section 1.4 reinforces the material in the previous sections with a simple tutorial on defining the AM/FM characteristics of signals using the concepts of analytic signal and IF; and relating to the definition of the amplitude and phase of a nonstationary signal.

0 Author: B. Boashash, Qatar University, Doha, Qatar; University of Queensland Centre for Clinical Research, Brisbane, QLD, Australia ([email protected]). Time-Frequency Signal Analysis and Processing. http://dx.doi.org/10.1016/B978-0-12-398499-9.00001-7 Copyright © 2016 Elsevier Ltd. All rights reserved.

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CHAPTER 1 TIME FREQUENCY AND IF CONCEPTS

1.1 THE BENEFITS OF TIME-FREQUENCY DISTRIBUTIONS (TFDs) As introduced in Chapter I, the two classical representations of a signal are the time-domain representation s(t) and the frequency-domain representation S(f ). In both cases, the variables t and f are treated as mutually exclusive: to obtain a representation in terms of one variable, the other variable is “integrated out.” Consequently, each classical representation of the signal is nonlocalized w.r.t. the excluded variable; that is, the frequency representation is essentially averaged over the values of the time representation at all times, and the time representation is essentially averaged over the values of the frequency representation at all frequencies. As the conventional representations in the time or frequency domains are inadequate in situations such as those described above, an obvious solution is to seek a representation of the signal as a two-variable function or distribution whose domain is the two-dimensional (t, f ) space. Such a representation is called a time-frequency representation (TFR) and is obtained using a TFD. In the TFD, denoted by ρ(t, f ), the variables t and f are not mutually exclusive, but present together. The TFD representation is localized in both t and f subject to the interpretation and within the limits of the uncertainty principle, as mentioned in Section I.2.3.3 of Chapter I. Before proceeding further with the (t, f ) approach, for the sake of completeness, the next sections first use practical examples to present a systematic step-by-step approach linking the time domain and the frequency domain with the (t, f ) domain.

1.1.1 REPRESENTATION OF THREE REAL-LIFE SIGNALS The usefulness of representing a signal as a function of both time and frequency is illustrated by considering three signals of practical importance: 1. Sinusoidal FM signal: The sound in monophonic analog TV, and in monophonic FM radio, is transmitted on a frequency-modulated carrier. If the audio signal is a pure tone of frequency fm (the modulating frequency), then the frequency of the carrier is of the form fi (t) = fc + fd cos[2πfm t + φ],

(1.1.1)

where t is time, fi (t) is the frequency modulation law (FM law), fc is the mean (or “center”) carrier frequency, fd is the peak frequency deviation, and φ allows for the phase of the modulating signal. The amplitude of the carrier is considered constant. 2. Linear-FM signal: Consider a “sinusoidal” signal of total duration T, with constant amplitude, whose frequency increases from f0 to f0 + B at a constant rate α = B/T. If the signal begins at t = t0 , the FM law may be written fi (t) = f0 + α(t− t0 );

t0 ≤ t ≤ t0 + T.

(1.1.2)

In an electronics laboratory, such a signal is called a linear frequency sweep, and might be used in an experiment to measure the frequency response of an amplifier or filter because of the wide-band characteristics of this signal. In mineral exploration, a linear-FM signal is called a chirp or vibroseis signal, and is used as an acoustic “ping” for detecting underground formations [2].

1.1 THE BENEFITS OF TIME-FREQUENCY DISTRIBUTIONS (TFDs)

33

3. Musical performance: A musical note consists of a number of “components” of different frequencies, of which the lowest is called the fundamental and the remainder are called overtones [3, p. 56]; overtones at whole-number multiples of the fundamental frequency are also called harmonics. These components are present during a specific time interval and may vary in amplitude during that interval. In modern musical notation, each note is represented by a “head.” The position of the head in the vertical dimension (together with other information such as the clef and key signature) indicates the pitch, that is, the frequency of the most prominent component (usually the fundamental). The horizontal position of the head in relation to other symbols indicates the starting time, and the duration is specified by the shading of the head, attached bars, dots, stem and flags, and tempo markings such as Allegro. The choice of instrument—each one being characterized by its overtones and relationships with the fundamental—is indicated by using a separate stave (group of five horizontal lines) for each instrument or group of instruments, or a pair of staves for a keyboard instrument. Variations in amplitude are indicated by dynamic markings such as f or p, and crescendo. Figure 1.1.1 illustrates the system. By scanning a set of staves vertically, one can see which fundamentals are present on which instruments at any given time. By scanning the staves horizontally, one can see the times at which a given fundamental is present on a given instrument. This is clearly a (t, f ) representation. The above three signals are nonstationary; that is, each of them has a time-varying frequency or timevarying “frequency content.” Such signals are comprehensible partly because our sense of hearing readily interprets sounds in terms of variations of frequency or “frequency content” with time. However, conventional representations of a signal in the time domain or frequency domain do not facilitate such interpretation, as shown below. The following two sections revisit in more depth and more details the concepts first introduced in Sections I.1.2 and I.1.3.

FIGURE 1.1.1 Musical notation, showing the string parts at the beginning of Beethoven’s Third Symphony [4]. Roughly speaking, the horizontal dimension is time and the vertical dimension is frequency.

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CHAPTER 1 TIME FREQUENCY AND IF CONCEPTS

1.1.2 TIME-DOMAIN REPRESENTATION A signal can be described naturally as a function of time, which we may write s(t). This representation leads immediately to the instantaneous power, given by |s(t)|2 , which shows how  ∞ the energy of the signal is distributed over time; the total signal energy is expressed as: Es = −∞ |s(t)|2 dt. But the time-domain description has limitations, as may be seen by applying it to the above three examples: 1. The sinusoidal FM signal whose frequency satisfies Eq. (1.1.1) may be written as:   f s1 (t) = A cos 2πfc t + d sin[2πfm t + φ] + ψ , fm

(1.1.3)

where A is the amplitude and ψ a phase offset; the fraction fd /fm is called the modulation index and is equal to the peak phase deviation (in radians) from 2πfc t. This equation, by itself, does not clearly indicate how the frequency varies with time. Figure 1.1.2 is a graph of s1 (t) against time for A = 1, fc = 0.25 Hz, fd = 0.1 Hz, fm = (3/64) Hz, and φ = ψ = 0. The graph gives an impression of an oscillating frequency, but extracting that frequency as a function of time from the graph would be a tedious and imprecise exercise. 2. The linear-FM signal whose frequency satisfies Eq. (1.1.2) may be written 

s2 (t) = A rect

  

t − t0 − T/2 α cos 2π f0 (t − t0 ) + (t − t0 )2 + ψ , T 2

(1.1.4)

where, again, A is the amplitude and ψ a phase offset. The rect function is a rectangular pulse of unit height and unit duration, centered on the origin of time, as defined in Table I.1.1; that is, the

1

Amplitude

0.5

0

–0.5

–1 0

10

20

30 Time (s)

FIGURE 1.1.2 Time-domain representation of a sinusoidal FM signal (Eq. 1.1.3).

40

50

60

1.1 THE BENEFITS OF TIME-FREQUENCY DISTRIBUTIONS (TFDs)

35

rect [. . .] factor in Eq. (1.1.4) is equal to unity for t0 ≤ t ≤ t0 + T, and 0 elsewhere. But again it is not immediately apparent why Eq. (1.1.4) has the required FM law. The graph of s2 (t) versus t is shown on the left side of Fig. 1.1.4(a), for A = 1, t0 = 18 s, T = 64 s, f0 = 0.1 Hz, α = (3/640) Hz s−1 , and ψ = −π/2. Although the graph gives a clear impression of a steadily increasing frequency, the exact FM law is not directly represented, and would be even further obscured if the signal were corrupted by noise (as in Fig. I.1.6 in the preceding chapter). 3. A musical performance can be represented in the time domain by variations in time of the air pressure at a particular point in space. The time-varying pressure may be converted by a microphone and amplifier into an electrical signal, say s3 (t). Indeed, music is routinely recorded and broadcast in this way. However, this signal s3 (t) is nothing like the form in which a composer would write music, or the form in which most musicians read music for the purpose of performance. Neither is it much help, by itself, to an engineer who wants to remove noise and distortion from an old “vintage” recording. The problem is that musical waveforms are far more complex than the underlying artistic ideas. As an example, Fig. 1.1.3 shows the waveform of the opening chord of a performance of Beethoven’s Third Symphony—the single staccato chord of which the string parts are shown at the left of Fig. 1.1.1. These three examples show that the time-domain representation tends to obscure information about frequency because it assumes that the two variables t and f are mutually exclusive.

1.1.3 SPECTRAL CHARACTERISTICS OF TYPICAL SIGNALS Let us consider in more  ∞detail three cases of a signal s(t) and its FT S(f ), given in Eq. (I.1.2) by  S(f ) = F {s(t)} = s(t) e−j2πft dt. For convenience, the relation between s(t) and S(f ) may be −∞

t→f

written “s(t) ←→ S(f )” or simply “s(t) ↔ S(f ).” The FT is in general complex; its magnitude is t→f

called the magnitude spectrum and its phase is called the phase spectrum. The square of the magnitude spectrum is the energy spectrum, which describes how the energy of the signal is distributed over the ∞ frequency domain; the total energy of the signal is defined in Eq. (I.1.5) as Es = −∞ |S(f )|2 df = ∞ ∗ ∗ −∞ S(f ) S (f ) df , where the superscripted asterisk ( ) denotes the complex conjugate. Although the representation S(f ) is a function of frequency only—time having been “integrated out”—the FT is a complete representation of the signal because the signal can be recovered from the FT by taking the inverse FT (IFT): s(t) = F −1 {S(f )} = t←f

∞

−∞

S(f ) ej2πft df defined in Eq. (I.1.1). But the “completeness”

of the FT representation does not make it convenient for all purposes, as may be seen by considering the same three examples: 1. Sinusoidal FM signal: If φ = ψ = 0 in Eq. (1.1.3), the expression for s1 (t) can be expanded into an infinite series (using Jacobi-Anger identity) as s1 (t) = A

∞ n=−∞

Jn (β) cos 2π(fc + nfm )t,

(1.1.5)

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CHAPTER 1 TIME FREQUENCY AND IF CONCEPTS

0.6

0.4

L+R (Amplitude)

0.2

0

–0.2

–0.4

–0.6

0

0.2

0.4

0.6

0.8

1

1.2

Time (s)

FIGURE 1.1.3 Time-domain representation of a single short E -major chord played by a full orchestra [5]. The left and right stereo channels are averaged.

where β is the modulation index (β = fd /fm ) and Jn denotes the Bessel function of the first kind, of order n [6, p. 226]. In the frequency domain, this becomes an infinite series of delta functions; one of these (the carrier) is at the mean frequency fc , and the remainder (the sideband tones) are separated from the carrier by multiples of the modulating frequency fm , as shown in Fig. 1.1.5(a). Although the number of sideband tones is theoretically infinite, the significant ones1 may be assumed to lie between the frequencies fc ± (fd + fm ) or, more conservatively, fc ± (fd + 2fm ). This information is essential if one is designing a tuning filter to isolate the TV audio carrier or separate

1 For more detail, see Carlson [6, pp. 220-237]. The above description considers only positive frequencies; similar comments apply to the negative frequencies. Theoretically, the lower sideband tones belonging to the positive-frequency carrier extend into the negative frequencies, while the corresponding sideband tones belonging to the negative-frequency carrier extend into the positive frequencies; but such “aliased” components are negligible if fc and fd are appropriately chosen.

1.1 THE BENEFITS OF TIME-FREQUENCY DISTRIBUTIONS (TFDs)

150 Magnitude

1.0

0.5 Real part

37

100 50 0

Phase (radians)

0.0

–0.5

–50 –100 –150

–1.0 –200 0

20

40

(a)

60

80

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100

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0.1

0.15

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0.2

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6.0

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4.0

100 50 0

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0

(b)

20

40

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FIGURE 1.1.4 The importance of phase: (a) time-domain representation (left) and magnitude and phase spectra (right) of a linear-FM signal (Eq. 1.1.6) with duration 64 s, starting frequency 0.1 Hz, and finishing frequency 0.4 Hz; (b) corresponding representations of another signal with the same magnitude spectrum as that in part (a). The sampling rate is 8 Hz (see also Fig. I .1.6).

one FM channel from adjacent channels. However, it is inadequate if one is designing a modulator or demodulator because its connection with the FM law is even more obscure than that of Eq. (1.1.3). 2. The linear-FM signal in Eq. (1.1.4) is the real part of the complex signal  z2 (t) = A rect

t − t0 − T/2 T



α 2 ej2π[f0 (t−t0 )+ 2 (t−t0 ) ]+ψ .

(1.1.6)

The magnitude spectrum of z2 (t) is shown by the upper trace in the right-hand graph of Fig. 1.1.4(a), for A = 1, t0 = 18 s, T = 64 s, f0 = 0.1 Hz, α = (3/640) Hz s−1 , and ψ = −π/2. It shows that the magnitude is significant in the band corresponding to the frequency-sweep range (0.1 Hz < f < 0.4 Hz) and that the energy is mostly confined to that band. But the magnitude

38

CHAPTER 1 TIME FREQUENCY AND IF CONCEPTS

600

1000

500

800

Magnitude

Magnitude

400

300

600

400

200 200 100 0 101

0 0

0.05

(a)

0.1

0.15

0.2

0.25

0.3

Frequency (Hz)

0.35

0.4

0.45

0.5

(b)

102

103

104

Frequency (Hz)

FIGURE 1.1.5 Magnitude spectra of (a) the sinusoidal FM signal in Fig. 1.1.2, and (b) the E -major chord in Fig. 1.1.3. The tall narrow spikes in (a) are computed approximations to delta functions.

spectrum fails to show that the frequency is increasing with time; it tells us what frequencies are present in the signal, but not the “times of arrival” of those frequencies. The latter information, which we call spectral delay (SD), is encoded in the phase spectrum, shown by the lower trace in the same graph. The shortcomings of the magnitude spectrum are further emphasized in Fig. 1.1.4(b), which shows a signal whose magnitude spectrum (right, upper trace) is identical to that of the linear-FM signal in part (a), but whose appearance in the time domain (left) is very different. The explanation is to be found in the phase spectrum (right, lower trace): the phase is nonlinear in part (a) but linear in part (b).2 3. Similarly, the musical performance has a magnitude spectrum, which tells us what frequencies are present, but not when they are present; the latter information is again encoded in the phase. Figure 1.1.5(b) shows the magnitude spectrum of the single chord in Fig. 1.1.3. The magnitude spectrum of a longer and more complex musical signal could have up to 120 peaks, corresponding to the notes of the chromatic scale in the audible frequency range. The relative heights of those peaks tell us something about the tonality of the music (or whether it is tonal at all), but the timing of the notes is represented in the magnitude spectrum and will not be obvious from the phase spectrum. These examples show that the frequency-domain representation “hides” information about timing, as S(f ) does not mention the variable t. We shall see in Section 1.3.2, p. 51 that the SD, which is the localization in time of the frequency components, is proportional to the derivative of the phase spectrum w.r.t. frequency [1,7]. This is similar 2 The

signal in part (b) of Fig. 1.1.4 was obtained from that in part (a) by padding the signal with zeros in the time domain, taking the fast Fourier transform (FFT), setting the phase to zero, taking the inverse FFT, and delaying the result by 50 s. The magnitude and phase spectra were then recomputed. Due to the padding and the generous over-sampling rate (8 Hz) (i.e., much greater than Shannon limit), the FFT and its inverse closely approximate the FT and its inverse.

1.1 THE BENEFITS OF TIME-FREQUENCY DISTRIBUTIONS (TFDs)

39

to the relationship between group delay (GD) and phase in filter design (cf. Section 1.3.3, p. 54). Thus, the linear phase spectrum in Fig. 1.1.4(b) means constant delay (all frequencies arrive at once), whereas the increasingly negative slope of the phase spectrum in Fig. 1.1.4(a) means that the delay increases with frequency. We shall also see in Section 1.3.1, p. 49 that the IF, which is the localization in frequency of the time components, is proportional to the derivative of the instantaneous phase of the complex signal w.r.t. time. The (t, f ) approach postulates that it would be more convenient if we could infer the SD and IF from a single representation of a signal, without the need to interpret additional representations such as the phase spectrum. This is part of the motivation for devising a “sophisticated” and “practical” tool for analysis of nonstationary signals, preserving “all” the information needed for characterization and identification as this can be critical in applications requiring condition monitoring and fault detection such as medical engineering [8].

1.1.4 JOINT TIME-FREQUENCY REPRESENTATION In a TFR, a constant-t cross-section should show the frequency or frequencies present at time t, and a constant-f cross-section should show the time or times at which the frequency f is present. Key questions are: how to define a TFD to obtain a desirable TFR? Are there any required constraints? Figure 1.1.6 illustrates what is desired. The sinusoidal FM law is clear in part (a), as is the linear-FM law in part (b). Part (c) shows the appearance and decay of the components of the E -major chord. The signals in parts (b) and (d) cannot be distinguished by their magnitude spectra (cf. Fig. 1.1.4), but their (t, f ) representations show that in the former case the higher frequencies appear after the lower frequencies, whereas in the latter case all frequencies appear at once. Figure 1.1.7 shows a different TFD of a linear-FM signal (the same signal as in Fig. 1.1.4, without the delayed onset). This particular TFD, known as the Wigner-Ville distribution (WVD), is noted for its sharp resolution of linear-FM laws; note the clear indication of the frequency range and the variation in frequency with time. Such a variation may be described by the IF fi (t). A signal may have more than one component, each with its own IF; for example, Fig. 1.1.8 shows a TFD of a sum of two linearFM signals, each of which has its own IF.3 These IF features are not apparent in conventional signal representations. Nonstationary signals for which a TFD representation may be useful occur not only in broadcasting, seismic exploration, and audio, from which our three examples are taken, but also in numerous other engineering and interdisciplinary fields such as telecommunications, radar, sonar, vibration analysis, speech processing, and medical diagnosis. Time-frequency signal processing (TFSP) is the processing of such signals in the (t, f ) domain by means of TFDs.

1.1.5 DESIRABLE CHARACTERISTICS OF A TFD The use of a TFD for a particular purpose is inevitably based on particular assumptions concerning the properties of TFDs. To ensure that the conclusions of the analysis are sound, these assumptions must be identified and verified. In general, we may say that the desirable properties of TFDs are those on

3 N.B.: In subsequent graphs of TFDs in Part I, the labels on axes are similar to those in Fig. 1.1.7, but may be smaller due to space constraints.

40

CHAPTER 1 TIME FREQUENCY AND IF CONCEPTS

100 60 80

40

60

Time (s)

Time (s)

50

30

40 20 20

10

(a)

0

0.05

0.1

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0.4

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0.4

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0.15

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0.4

0.45

0.5

100 1.2 80

1

Time (s)

Time (s)

0.8 0.6

60

40 0.4 20

0.2

(c)

0 101

102

103 Frequency (Hz)

104

0

(d)

FIGURE 1.1.6 Time-frequency representations of (a) the sinusoidal-FM signal in Figs. 1.1.2 and 1.1.5; (b) the linear-FM signal in Fig. 1.1.4; (c) the E -major chord in Figs. 1.1.3 and 1.1.5; and (d) the linear-phase signal in Fig. 1.1.4(b). (The TFRs used here are called spectrograms, except in part (c), where we use the square root of the spectrogram in order to compress the dynamic range. These TFRs will be defined in the next chapter; to reproduce these plots, see toolbox in Chapter 17.)

which their most likely applications depend. The following generic applications illustrate some typical uses of TFDs, applications, and corresponding required properties [9–11]: •

• • •

Analyze the raw signal in the (t, f ) domain to identify characteristics such as number of components, relative amplitudes, IF laws, (t, f ) complexity, (t, f ) flatness, (t, f ) energy concentration areas, etc. Separate the components or (t, f ) regions from each other and from the background noise by filtering in the (t, f ) domain (see Sections 10.6 and 11.4). Synthesize the filtered TFD to reconstruct s(t) in the time domain. Estimate and analyze specific components separately, and in particular: • track the instantaneous amplitude;

1.1 THE BENEFITS OF TIME-FREQUENCY DISTRIBUTIONS (TFDs)

41

Fs = 1 Hz N = 65 Time-res = 1

60

50

Time (s)

40

30

20

10

0.05

0.1

0.15

0.2

0.25

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Frequency (Hz)

FIGURE 1.1.7 A time-frequency representation of a linear-FM signal (Eq. 1.1.4) with duration 65 samples, starting frequency 0.1 Hz, and finishing frequency 0.4 Hz (sampling rate 1 Hz). The time-domain representation appears on the left, and the magnitude spectrum at the bottom; the same pattern is followed in most TFD graphs in Part I of this book. (The TFD used in this case is the WVD option of the (t, f ) toolbox described in Chapter 17.)

•

• track the IF; and • track the instantaneous bandwidth (spread of energy about the IF). Choose a mathematical model of the signal, showing clearly the significant characteristics, such as the IF fi (t) or SD τd (f ).

Specific applications include, for example, watermarking, that is, hiding a watermark in a particular predefined secret (t, f ) location of a signal such as an audio recording [12]. These applications can be carried out using a TFD with the following properties: 1. The TFD is preferably an energy density so as to allow an energetic physical interpretation. That is, (a) the TFD is real; (b) the integral of the TFD over the entire (t, f ) plane is the total energy of the signal; (c) the integral over a rectangular region of the (t, f ) plane, corresponding to a finite bandwidth and finite time interval, is approximately the energy of the signal in that bandwidth over that interval, provided that the bandwidth and interval are sufficiently large (see Section 1.2.5).

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CHAPTER 1 TIME FREQUENCY AND IF CONCEPTS

Fs = 1 Hz N = 513 Time-res = 1 500 450 400 350

Time (s)

300 250 200 150 100 50 0.05

0.1

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0.25

0.3

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Frequency (Hz)

FIGURE 1.1.8 A time-frequency representation of two linear-FM signals with close parallel FM laws, duration 512 samples, frequency separation 0.04 Hz (sampling rate 1 Hz).

2. For a monocomponent FM signal, the peaks of the constant-time cross-sections of the TFD should give the IF law which describes the signal FM law. 3. For a multicomponent FM signal, the dominant peaks of the TFD should reflect the components’ respective FM laws; and the TFD should resolve any close components, as in Fig. 1.1.8. A simple measure of the resolution of a TFD is the concentration of energy about the IF law(s) of a signal (Section 7.4; Chapter 10). The linear-FM signal, widely used in oil exploration [2] and elsewhere because of its simplicity, is a convenient test signal for verifying properties 1 and 2. Property 3 may be tested by a sum of linear-FM signals. The above properties have further implications. For example, property 3 helps to ensure TFD robustness in the presence of noise, while properties 2 and 3 together allow discernment and discrimination of the multiple IFs of multicomponent signals. Note, however, that in some cases, such as to perform time-frequency filtering, linear time-frequency methods are useful, although they cannot be interpreted directly as energy distributions (see Chapter 11 for more details).

1.2 SIGNAL FORMULATIONS AND CHARACTERISTICS IN THE (t,f ) DOMAIN

43

1.2 SIGNAL FORMULATIONS AND CHARACTERISTICS IN THE (t,f ) DOMAIN 1.2.1 EXAMPLES OF SIGNAL MODELS In the terminology of Chapter I, a “type-5” signal (p. 6) may have the form ⎛ se (t) = ⎝

M

mk (t) ej2π

t

⎞

0 fk (τ ) dτ ⎠

+ w(t),

(1.2.1)

k=1

where mk (t) is the noisy instantaneous amplitude of the kth component, implying the presence of multiplicative noise, while fk (t) is the IF of the kth component, and w(t) is additive noise. To analyze such a general signal, we must distinguish the time-varying components from each other and from the noise. The form of Eq. (1.2.1) strongly suggests a time-frequency method of analysis. If each mk (t) is replaced by a deterministic but time-varying amplitude ak (t), the noise becomes merely additive. If w(t) is also eliminated, the entire signal becomes deterministic—and, with appropriate choices of ak (t) and fk (t), might be a passable imitation of speech or music. If the amplitudes ak (t) are then replaced by constants, the signal becomes what we have called “type 4.” If we then reduce the number of components to 1, we obtain what we have called a “type-3” signal, of which the sinusoidal-FM and linear-FM signals are examples. Even these can be advantageously represented in time-frequency form (Fig. 1.1.6). Only if the amplitudes and frequencies of all the components are constant (as in a “type-2” or “type-1” signal) does the FT give an immediately clear representation.

1.2.2 ANALYTIC SIGNALS As discussed in Section I.1.4, a signal s(t) is real if and only if (iff) S(−f ) = S∗ (f ),

(1.2.2)

where S(f ) is the FT of s(t). In other words, a real signal is one that exhibits Hermitian symmetry between the positive-frequency and negative-frequency components, allowing the latter to be deduced from the former. Hence the negative-frequency components of a real signal may be eliminated from the signal representation without losing information by forming the analytic signal. In the case of a real low-pass signal, removal of negative frequencies has two beneficial effects. First, for narrowband signals, it halves the total bandwidth, allowing the signal to be sampled at half the usual Nyquist rate without aliasing [13,14]. Second, it avoids the appearance of some interference terms generated by the interaction of positive and negative components in quadratic TFDs (to be treated in detail in Section 3.1.2). (Moreover, there is an argument that negative frequencies have no clear physical meaning that directly reflects the real world). Note that, although an analytic signal contains no negative frequencies, it may have a spectral component at zero frequency (DC). Theorem 1.2.1. Let us consider two real signals s(t) and y(t); the signal z(t) = s(t) + jy(t)

(1.2.3)

44

CHAPTER 1 TIME FREQUENCY AND IF CONCEPTS

is analytic with a real DC component, iff Y(f ) = (−j sgn f )S(f ),

(1.2.4)

where S(f ) and Y(f ) are the FTs of s(t) and y(t), respectively, and where ⎧ ⎨ −1  0 sgn ξ = ⎩ +1

if if if

ξ < 0; ξ = 0; ξ > 0.

(1.2.5)

Proof and (Exercise). Take the FT of Eq. (1.2.3) and use Eq. (I.1.3).

1.2.3 HILBERT TRANSFORM (HT) AND ANALYTIC ASSOCIATE If the FTs of s(t) and y(t) are related according to Eq. (1.2.4), we say that y(t) is the Hilbert transform (HT) of s(t), and we write y(t) = H {s(t)} .

(1.2.6)

Hence, we may restate Theorem 1.2.1 as follows: A signal is analytic with a real DC component iff its imaginary part is the HT of its real part. By invoking the “if” form of Theorem 1.2.1 and restating the sufficient condition in terms of Eq. (1.2.3), we may now see the practical significance of the theorem and the HT: Given a real signal s(t), we can construct the complex signal z(t) = s(t) + jH {s(t)}

(1.2.7)

and know that z(t) is analytic. This z(t) is called the analytic signal “corresponding to” or “associated with” the real signal s(t). In this book, for convenience, we shall usually call z(t) the analytic associate of s(t). Other advantages of using z(t) instead of s(t) are described in Chapters 2 and 6 [12,15]. By taking the IFT of Eq. (1.2.4) and applying Eq. (1.2.6), we arrive at the following concise definition of the HT: Definition 1.2.2. The Hilbert transform of a signal s(t), denoted by H {s(t)}, may be expressed using the FT F {·} of s(t) as:   −1 H {s(t)} = Ft←f (−j sgn f ) F {s(t)} .

(1.2.8)

t→f

In other words, the HT of s(t) can be evaluated as follows: 1. take the FT S(f ) of s(t); 2. multiply S(f ) by −j for f > 0, by +j for f < 0, and by 0 for f = 0; and 3. take the IFT. According to Step 2 of the above procedure, a Hilbert transformer introduces a phase lag of 90◦ (as −j = e−jπ/2 ), producing a signal in quadrature to the input signal. The effect is well illustrated by the following result, which is easily verified using Definition 1.2.2 and a table of transforms: Example 1.2.3. If f0 is a positive constant, then H {cos(2πf0 t)} = sin(2πf0 t)

(1.2.9)

1.2 SIGNAL FORMULATIONS AND CHARACTERISTICS IN THE (t,f ) DOMAIN

H {sin(2πf0 t)} = − cos(2πf0 t).

45

(1.2.10)

It would be convenient for constructing the analytic signal if the pattern of Example 1.2.3 were applicable to modulated signals so that, for example, we could have H {a(t) cos φ(t)} = a(t) sin φ(t),

(1.2.11)

which would imply that the analytic associate of a real signal s(t) = a(t) cos φ(t) is z(t) = a(t) cos φ(t) + jH {a(t) cos φ(t)} = a(t) cos φ(t) + ja(t) sin φ(t) = a(t) ejφ(t) .

(1.2.12)

The condition for Eqs. (1.2.11) and (1.2.12) to hold is that the variation of a(t) is sufficiently slow to ensure “spectral disjointness” [15], that is, to avoid overlap between the spectrum of a(t) and the spectrum of cos φ(t). More details appear in Section 1.4.3 and Ref. [16]. Equation (1.2.4) indicates that the transfer function of a Hilbert transformer is −j sgn f . The corresponding impulse response is F −1 {−j sgn f } = t←f

1 . πt

(1.2.13)

Definition 1.2.4. Using the above result and applying the convolution property to Eq. (1.2.8), we obtain a definition of the Hilbert transform in the time domain: 1 πt  ∞  s(τ ) 1 = p.v. dτ , π −∞ t − τ

H {s(t)} = s(t) ∗

(1.2.14) (1.2.15)

where p.v. {·} denotes the Cauchy principal value of the improper integral (i.e., to account for the t = τ situation) [17]; the p.v. is given in this case by1  t−ε   ∞ s(τ ) s(τ ) dτ + dτ . ε→0 −∞ t − τ t+ε t − τ lim

(1.2.16)

1.2.4 DURATION, BANDWIDTH, AND BT PRODUCT If a signal s(t) has the FT S(f ), the duration of the signal is the range of time outside which s(t) = 0, while the bandwidth of the signal is the range of frequencies outside which S(f ) = 0. These definitions, as we shall see, lead to the conclusion that a finite duration implies infinite bandwidth and vice versa.

1 In practice, Eq. (1.2.14) is rarely used in actual computations and implementations because the FT properties make it easier to work in the frequency domain.

46

CHAPTER 1 TIME FREQUENCY AND IF CONCEPTS

In practice, however, signals are observed for finite periods of time, and measured or processed by devices with finite usable bandwidths.2 In practice, therefore, the literal definitions need to be relaxed in some way.

1.2.4.1 Finite duration vs finite bandwidth

A time-limited signal, of duration T centered at t = 0, can be expressed as sT (t) = s(t) rect [t/T] ,

(1.2.17)

where the subscript T indicates the duration (see Table I.1.1). The FT of sT (t) is ST (f ) = S(f ) ∗ T sinc fT, f

(1.2.18)

where ∗ denotes convolution in frequency. Thus, the bandwidth of ST (f ) is infinite. f

If, in order to avoid the effects of discontinuities, we replace rect [t/T] with a smoother window w(t) of the same duration T, we can write sT (t) = s(t) rect [t/T] w(t),

(1.2.19)

whose FT still involves a convolution with sinc fT and therefore still gives an infinite bandwidth. In analogy to the time-limited case, a band-limited signal, of bandwidth B centered at the origin, can be expressed in the frequency domain as SB (f ) = S(f ) rect [f /B] .

(1.2.20)

In the time domain, the signal is given by sB (t) = s(t) ∗ B sinc Bt, t

(1.2.21)

which has an infinite duration. Thus, under the “literal” definitions, finite T implies infinite B and vice versa.

1.2.4.2 Practical measures of bandwidth and duration If there is no finite bandwidth containing all the energy of the signal, there may still be a finite bandwidth containing most of the energy. Hence, for example, the minimum bandwidth containing 99% of the signal energy might be accepted as a useful measure of the signal bandwidth. If the nominated fraction of the signal energy were confined between the frequencies fmin and fmax , the bandwidth would be B = fmax − fmin . Similarly, the minimum duration containing a nominated fraction of the signal energy might be accepted as a useful measure of the signal duration. A less arbitrary but less conservative measure of bandwidth, published by Gabor3 in 1946 [18], is the so-called effective bandwidth Be , defined by

signal acquisition and measuring systems are low-pass as their |f | must be bounded. Gabor (1900-1979), a Hungarian-born electrical engineer who settled in Britain, is best known for the invention of holography (1947-1948), for which he was awarded the Nobel Prize for Physics in 1971.

2 All

3 Dennis

1.2 SIGNAL FORMULATIONS AND CHARACTERISTICS IN THE (t,f ) DOMAIN

B2e =

 ∞ 1 f 2 |S(f )|2 df , Es −∞

47

(1.2.22)

where S(f ) is the FT of the signal, and Es is the total energy of the signal: Es =

 ∞

−∞

|S(f )|2 df .

(1.2.23)

Thus B2e is the normalized second moment of |S(f )|2 w.r.t. frequency, about the origin (f = 0). For brevity, we call B2e the “2nd moment of the signal w.r.t. f .” Similarly, the so-called effective duration Te is defined as Te2 =

 ∞ 1 t2 |s(t)|2 dt, Es −∞

(1.2.24)

that is, the normalized 2nd moment of |s(t)|2 w.r.t. time, about the origin (t = 0). For brevity, we refer to Te2 as “the 2nd moment of the signal w.r.t. time.” As an aid to remembering these definitions, note that if f were a random variable and |S(f )|2 were its probability density function (p.d.f.), then we would have Es = 1 so that B2e would be the variance of f if the mean of f were 0. Thus the effective bandwidth Be is analogous to the standard deviation of f . Similarly, the effective duration Te is analogous to the standard deviation of t. When we consider how little of a typical probability distribution falls within an interval of one standard deviation, we realize that the “effective” bandwidth or duration is only a mathematical construction, not an estimate of the actual bandwidth or duration required to measure or process the signal. The bandwidth or duration needed for sufficiently accurate processing of a signal is known as the significant or essential bandwidth or duration. For a given signal s(t) whose FT is S(f ), let us define a “time-truncation” signal sˆ(t) that satisfies ⎧ ⎨ 0 s(t) sˆ(t) = ⎩ 0

if t < t1 if t1 ≤ t ≤ t2 if t > t2 ,

(1.2.25)

where t2 > t1 , so that the duration of sˆ(t) is t2 − t1 . Let a “frequency-truncation” be defined similarly in the frequency domain. Then the “essential” duration of s(t) is the duration of the shortest timetruncation sˆ(t) that is an acceptable substitute for s(t), while the “essential” bandwidth of s(t) is the ˆ ) that is an acceptable substitute for bandwidth of the most narrow-band frequency-truncation S(f S(f ). Notice that the previous definitions of effective duration/bandwidth do not rely on the idea of “truncation.” What is “acceptable” depends on the application. According to Slepian’s definition [19], an acceptable time- or frequency-truncation is one that the detecting or measuring apparatus cannot distinguish from the original signal. Given B and T, the bandwidth-duration product BT is self-explanatory.4 If we use conservative (essential) values of B and T, the BT product has a simple practical significance: because a signal of total bandwidth B can be reconstructed from samples at the sampling rate B, the product BT is the

4 If

the bandwidth is written |f | < W, then B = 2W. Hence, “BT” is also known as “2WT.”

48

CHAPTER 1 TIME FREQUENCY AND IF CONCEPTS

total number of samples required to represent the signal.5 If, due to noise or other sources of error, each sample has an accuracy of n bits, then the signal can encode nBT bits of information. Thus BT is proportional to the information richness of the signal: one cannot extract large amounts of information from a signal with a small BT product [16]. 2 2 Example 1.2.5. For the Gaussian signal s(t) = e−α t , the effective duration is Te =

The FT of s(t) is S(f ) =

√ π α

e−π

2 f 2 /α 2

1 . 2α

(1.2.26)

so that the effective bandwidth is Be =

α . 2π

(1.2.27)

1 From Eqs. (1.2.26) and (1.2.27), Be Te = 4π . The Gaussian signal is the only signal for which this equality holds, and that for all other signals 1 Be Te > 4π as per Eq. (I.2.4) [18] (see also p. 20).

1.2.5 ASYMPTOTIC SIGNALS For signals with significant information content, it is desired to know not only the overall bandwidth B but also the distribution of energy through the bandwidth, for example, the frequencies present, their relative amplitudes, and the times during which they are significant. Similarly we may want to know not only the overall duration T but also the distribution of energy throughout the duration, for example, the times during which the signal is present, the relative amplitudes at those times, and the significant frequencies present during those times. Such signals may be modeled by the class of asymptotic signals. Definition 1.2.6. A signal s(t) is asymptotic iff it has the following properties: (a) (b) (c) (d)

The essential duration T, as defined previously, is finite. The essential bandwidth B, as defined previously, is finite. The product BT is large (e.g., >10). ∞ The amplitude is bounded, and the energy Es = −∞ |s(t)|2 dt is finite. As an example, the two-component linear-FM signal in Fig. 1.1.8 (p. 42) is asymptotic because:

(a) from the time-domain graph (left) or the TFD, we see that the duration is just under 400 s; (b) from the magnitude spectrum (bottom) or the TFD, we see that the signal is almost entirely between 0.15 and 0.35 Hz, giving an essential bandwidth of about 0.2 Hz if negative frequencies are not considered significant (or 0.7 Hz if they are considered significant); (c) if we accept the lower estimate of the essential bandwidth, the BT product is almost 80; and (d) from the time-domain graph or the TFD, we get a clear impression that the amplitude is bounded. Asymptotic signals allow useful approximations for deriving simple signal models (e.g., to express analytic signals).

5 In the language of statistics, BT is the number of degrees of freedom in the signal. In the language of linear algebra, BT is the dimension of the signal, considered as a vector.

1.3 INSTANTANEOUS FREQUENCY (IF) AND SPECTRAL DELAY (SD)

49

1.2.6 MONOCOMPONENT VS MULTICOMPONENT SIGNALS A monocomponent signal is described in the (t, f ) domain by one single “ridge,” corresponding to an elongated region of energy concentration. Furthermore, interpreting the crest of the “ridge” as a graph of IF versus time, we require the IF of a monocomponent signal to be a single-valued function of time [16, p. 527]. Figure 1.1.7 shows an example. Such a monocomponent signal s(t) has an analytic associate of the form z(t) = a(t) ejφ(t) ,

(1.2.28)

where • • •

a(t), known as the instantaneous amplitude, is real and positive; φ(t), known as the instantaneous phase, is differentiable; and a(t) and ejφ(t) are spectrally disjoint.

Without the requirement of spectral disjointness, any complex signal would qualify as monocomponent because any complex signal can be written in the form of Eq. (1.2.28). If, in addition, s(t) itself is real and asymptotic with amplitude modulation a(t), it can also be expressed approximately as [16] s(t) = a(t) cos φ(t).

(1.2.29)

A multicomponent signal may be described as the sum of two or more monocomponent signals (having analytic associates of the form Eq. (1.2.28)) such that: z(t) =

M

ai (t) ejφi (t) .

(1.2.30)

i=1

The decomposition into components is not necessarily unique [16, p. 535]. This model allows the extraction and separation of components from a given multi-component signal using (t, f ) filtering methods [20]. Further in-depth discussion of monocomponent and multicomponent signals can be found in Refs. [1,10]. Figure 1.1.8 (p. 42) shows an example of a multicomponent signal with two components.

1.3 INSTANTANEOUS FREQUENCY (IF) AND SPECTRAL DELAY (SD) 1.3.1 INSTANTANEOUS FREQUENCY Definition 1.3.1. The instantaneous frequency of a monocomponent signal is fi (t) =

1  1 dφ φ (t) ≡ (t), 2π 2π dt

(1.3.1)

where φ(t) is the instantaneous phase of the signal, and the prime ( ) indicates differentiation. This formulation is justified below by considering first a constant-frequency signal, then a variablefrequency signal. Consider the amplitude-modulated signal x(t) = a(t) cos(2πfc t + ψ),

(1.3.2)

50

CHAPTER 1 TIME FREQUENCY AND IF CONCEPTS

where fc and ψ are constant. As t increases by the increment 1/fc , the argument of the cosine function increases by 2π and the signal passes through one cycle. So the period of the signal is 1/fc , and the frequency, being the reciprocal of the period, is fc . The same signal can be written as x(t) = a(t) cos φ(t),

(1.3.3)

φ(t) = 2πfc t + ψ,

(1.3.4)

1  1 dφ φ (t) ≡ (t). 2π 2π dt

(1.3.5)

where

from which we obtain fc =

Although the left-hand side of this equation (the frequency) has been assumed constant, the right-hand side would be variable if φ(t) were a nonlinear function. So, let us check whether this result can be extended to a variable frequency. Consider a monocomponent signal s(t) whose analytic associate is z(t) = a(t) ejφ(t) ,

(1.3.6)

s(t) is then given by Eq. (1.2.29), where a(t) and φ(t) are real and a(t) is positive; then a(t) is called the instantaneous amplitude and φ(t) is called the instantaneous phase. Let φ(t) be evaluated at t = t1 and t = t2 , where t2 > t1 . By the mean value theorem of elementary calculus, if φ(t) is differentiable, there exists a time instant t between t1 and t2 such that φ(t2 ) − φ(t1 ) = (t2 − t1 ) φ  (t).

(1.3.7)

Let pi be the period of one particular oscillation of s(t), and let fi = 1/pi . If we let t2 = t1 + pi , then φ(t2 ) = φ(t1 ) + 2π so that Eq. (1.3.7) becomes 2π = pi φ  (t);

(1.3.8)

φ  (t) 1 dφ ≡ (t). 2π 2π dt

(1.3.9)

that is, fi =

Now t is an instant during a cycle of oscillation and fi is the frequency (inverse of the period) of that oscillation, suggesting that the right-hand side be defined as the IF at time t, as in Definition 1.3.1 above. Comparing Eqs. (1.2.29) and (1.3.6), we see that s(t) = Re {z(t)} ,

(1.3.10)

where Re {·} denotes the real part. Now let us define 

y(t) = Im {z(t)} = a(t) sin φ(t),

where Im {·} denotes the imaginary part.

(1.3.11)

1.3 INSTANTANEOUS FREQUENCY (IF) AND SPECTRAL DELAY (SD)

51

Using Eq. (1.3.1), we can easily confirm that the signals described in the time domain by Eqs. (1.1.3) and (1.1.4) have the IFs given in Eqs. (1.1.1) and (1.1.2), respectively (sinusoidal FM and linear FM). Definition 1.3.1 is strictly meaningful only for a monocomponent signal; a multicomponent signal ought to have a separate IF for each component. In general, the IF of the sum of two signals is not the sum of their two IFs. Further details regarding the IF can be found in Ref. [21]. The IF is a detailed description of the frequency characteristics of a signal. This contrasts with the notion of “average frequency” defined next. Definition 1.3.2. The average frequency of a signal is ∞ f |S(f )|2 df , f0 = 0∞ 2 0 |S(f )| df

(1.3.12)

where S(f ) is the FT of the signal. In other words, f0 is the first moment of |S(f )|2 w.r.t. frequency; that is, we define the “average” frequency as if |S(f )|2 were the p.d.f. of the frequency. Notice that if |S(f )|2 is replaced by a TFD, f0 becomes a function of time, suggesting that perhaps the first moment of a TFD w.r.t. frequency is a measure of IF. The conditions under which this is true will be explained in due course (and summarized in the table on p. 122). For now we simply note that any reasonable (t, f ) representation of a signal should contain information about the IF laws of the components. Ideally, a TFD should show the signal energy concentrated along the IF. This criterion is often used for adapting the parameters of the analysis window of the Spectrogram [22]. The IF estimate obtained as derivative of phase can be related to the first-order moment of a quadratic TFD (QTFD) [23]. Specifically, the IF estimate obtained using the first-order moment of the TFD is equivalent to applying a low-pass filtering function to the direct estimator based on the central finite differencing (CFD) of the phase of the analytic signal [23]. In particular, it would be most convenient if the crests of the ridges in the (t, f ) domain represented the IF laws.

1.3.2 SPECTRAL DELAY The IF of a signal indicates the dominant frequency of the signal at a given time. Let us now seek a dual or “inverse” of the IF, indicating the dominant time when a given frequency occurs. If z(t) is an analytic signal with FT Z(f ) = A δ(f − fi ),

(1.3.13)

where A is a real constant, the dominant frequency is fi . Taking the IFT of Z(f ) gives z(t) = A ej2πfi t .

(1.3.14)

The instantaneous phase of z(t), denoted by φ(t), is φ(t) = arg z(t) = 2πfi t

(1.3.15)

so that fi =

1  1 dφ φ (t) ≡ (t). 2π 2π dt

(1.3.16)

52

CHAPTER 1 TIME FREQUENCY AND IF CONCEPTS

Although this result has been obtained for constant frequency, it is also valid for a variable-frequency signal, as explained earlier. More specifically, for a variable frequency fi (t), the phase can be written as t φ(t) = 2π 0 fi (τ ) dτ + ψ, where ψ is the initial phase (often considered to be zero). The real signal then becomes: s(t) = A cos φ(t).

(1.3.17)

Now let us repeat the argument with the time and frequency variables interchanged. The signal z(t) = a δ(t − τd ),

(1.3.18)

is an impulse at time τd , and a is in general complex. If we ask what is the “delay” of this signal, the only sensible answer is τd . The FT of this signal is Z(f ) = a e−j2πf τd

(1.3.19)

θ(f ) = arg Z(f ) = −2πf τd + arg a

(1.3.20)

and its phase, denoted by θ (f ), is

so that τd = −

1  1 dθ θ (f ) ≡ − (f ). 2π 2π df

(1.3.21)

Again, although this result has been obtained for a constant τd , the right-hand side of Eq. (1.3.21) is well defined even if θ  (f ) varies with f . Definition 1.3.3. If z(t) is an analytic signal with FT Z(f ), and θ (f ) = arg Z(f ) then the spectral delay of z(t), denoted by τd (f ), is τd (f ) = −

1  1 dθ θ (f ) ≡ − (f ). 2π 2π df

(1.3.22)

Notice that the definitions of τd and fi are similar, except that time and frequency are interchanged and Eq. (1.3.22) has an extra minus sign; hence, we say that SD is the dual of IF.1 Figure 1.3.1 illustrates these concepts. Seeing that the IF fi (t) is a function assigning a frequency to a given time, whereas the SD τd (f ) is a function assigning a time to a given frequency, we may well ask whether the two functions are inverses of each other. In fact, they are not always inverses because the IF function may not be invertible. So, let us restrict the question to invertible signals, that is, monocomponent signals whose IFs are monotonic functions of time. One example of an invertible signal is the generalized Gaussian signal, that is, a linear-FM signal with a Gaussian envelope. Let such a signal peak at time t0 , with peak amplitude A, center frequency fc ,

1 The SD τ is often called the “time delay.” This term, although well established, is tautological in that “delay” implies d “time.” The term instantaneous frequency is a quantity with the dimension of frequency modified by an adjective indicating localization in time. The dual of this term should therefore be a quantity with the dimension of time modified by an adjective indicating localization in frequency, for example, “frequency-localized delay” or, more succinctly, spectral delay. This term is used as a standard terminology in this book.

1.3 INSTANTANEOUS FREQUENCY (IF) AND SPECTRAL DELAY (SD)

53

120

IF

Time (s)

100

80

60

Spectral delay 40

20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Frequency (Hz)

FIGURE 1.3.1 Illustration of instantaneous frequency (IF) and spectral delay (SD): LFM signal of length 128 samples, starting frequency = 0.1 Hz, finishing frequency = 0.4 Hz with sampling rate = 1; at time 80, the value of the IF is about 0.3; in other words, for the frequency value 0.3, the SD is 80.

sweep rate α, and decay constant β, and suppose that the IF is positive while the envelope is significant. Then the analytic associate of the signal is   α + jβ 2 z(t) = A exp j2π fc [t − t0 ] + [t − t0 ] 2 

(1.3.23)

  and, from Eq. (1.3.16) with a(t) = A exp −πβ(t − t0 )2 , its IF is fi (t) = fc + α[t − t0 ].

(1.3.24)

To find the inverse function of fi (t), we simply solve for t, obtaining t = t0 +

fi (t) − fc , α

(1.3.25)

which suggests that the SD τd (f ) of z(t) can be estimated by τˆd (f ) = t0 +

f − fc . α

(1.3.26)

In general, τd and τˆd are not equal, but converge to each other as the BT product increases. As an example, Fig. 1.3.2 shows the IF (solid line) and SD (dotted line) of the generalized Gaussian signal z(t) for two values of Be Te , where the subscript “e” means “effective.” Note that the two curves are closer for a larger value of Be Te .

54

CHAPTER 1 TIME FREQUENCY AND IF CONCEPTS

IF (solid) and group delay (dotted) 35 Fs = 1 Hz N = 33 Time-res = 1

30 30 25 Time or group delay (s)

25

Time (s)

20

15

20

15

10 10 5 5 0.05

0.1

0.15

0.2 0.25 0.3 Frequency (Hz)

0.35

0.4

0.45

0.5 0

0

0.05

0.1

0.15

0.2 0.25 0.3 IF or frequency (Hz)

(a)

0.35

0.4

0.45

0.5

0.4

0.45

0.5

IF (solid) and group delay (dotted) 70 Fs = 1 Hz N = 65 Time-res = 1

60 60

50 Time or group delay (s)

50

Time (s)

40

30

40

30

20

20 10

10 0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Frequency (Hz)

0

(b)

0

0.05

0.1

0.15

0.2 0.25 0.3 IF or frequency (Hz)

0.35

FIGURE 1.3.2 Instantaneous frequency and spectral delay for a linear-FM signal with a Gaussian envelope: (a) total duration T = 33 s, Be Te = 0.1806; (b) T = 65 s, Be Te = 0.3338. For each signal, the left-hand graph shows the time trace, spectrum, and TFD (WVD, defined later in Eq. (2.1.17)), while the right-hand graph shows the IF (solid line) and SD (dotted line). The vertical dotted segments are caused by truncation of the frequency range (to avoid finite-precision effects).

1.3.3 MEAN IF AND GROUP DELAY Let z(t) be a band-pass analytic signal with center frequency fc and FT: Z(f ) = M(f − fc ) ejθ(f ) ,

(1.3.27)

where the magnitude M(f − fc ) and phase θ (f ) are real. If the signal has linear phase in support of Z(f ), that is, if θ (f ) is a linear function of f wherever Z(f ) is nonzero, we can let θ(f ) = −2πτp fc − 2πτg [f − fc ],

(1.3.28)

1.3 INSTANTANEOUS FREQUENCY (IF) AND SPECTRAL DELAY (SD)

55

where τp and τg are real-time constants. Equation (1.3.27) then becomes Z(f ) = M(f − fc ) e−j(2πτp fc +2πτg [f −fc ]) = e−j2πfc τp M(f − fc ) e−j2πτg [f −fc ] .

(1.3.29) (1.3.30)

Taking the IFT of Z(f ), we find z(t) = m(t − τg ) ej2πfc [t−τp ] ,

(1.3.31)

where m(t) is the IFT of M(f ). Since M(f ) is real, m(t) is Hermitian (i.e., m(−t) = m∗ (t)) so that |m(t)| is even. Hence τg is the time about which the envelope function is symmetrical; for this reason, τg is called the GD [24, pp. 123-124]. The phase of the oscillatory factor is −2πfc τp , wherefore τp is called the phase delay. These observations and Eq. (1.3.28) lead to the definition below: Definition 1.3.4. If an analytic signal has the FT Z(f ) = |Z(f )| ejθ(f ) , then the group delay of the signal reduces to the spectral delay, that is, 1  θ (f ) 2π

(1.3.32)

1 θ(f ). 2πf

(1.3.33)

τg (f ) = −

and the phase delay of the signal is τp (f ) = −

Equation (1.3.32) is found by differentiating Eq. (1.3.28) w.r.t. f , and Eq. (1.3.33) is found by putting fc = f in Eq. (1.3.28). Whereas Eq. (1.3.28) assumes linear phase, the above definition is meaningful whether the phase is linear or not. Thus, the GD (Eq. 1.3.32) equals the SD (Eq. 1.3.22), although the physical interpretations are different; the SD applies to an impulse, whereas the GD applies to the envelope of a narrowband signal. Now consider the dual of the above argument. Let z(t) be a time-limited signal centered on t = tc , and let z(t) = a(t − tc ) ejφ(t) ,

(1.3.34)

where a(t) and φ(t) are real. If the signal has constant IF in support of z(t), that is, if φ(t) is a linear function of t wherever z(t) is nonzero, we can let φ(t) = 2πf0 tc + 2πfm [t − tc ],

(1.3.35)

where f0 and fm are real frequency constants. Equation (1.3.34) then becomes z(t) = a(t − tc ) ej(2πf0 tc +2πfm [t−tc ])

(1.3.36)

= ej2πf0 tc a(t − tc ) ej2πfm [t−tc ] .

(1.3.37)

Taking the FT of z(t), we find Z(f ) = A(f − fm ) e−j2π[f −f0 ]tc ,

(1.3.38)

where A(f ) is the FT of a(t). Now because a(t) is real, A(f ) is Hermitian so that |A(f )| is even. Hence, fm is the frequency about which the amplitude spectrum is symmetrical; for this reason, fm is called the mean IF. Differentiating Eq. (1.3.35) w.r.t. t leads to the following definition:

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CHAPTER 1 TIME FREQUENCY AND IF CONCEPTS

Definition 1.3.5. For the signal z(t) = |z(t)| ejφ(t) ,

(1.3.39)

the mean IF is fm (t) =

1  1 dφ φ (t) ≡ (t). 2π 2π dt

(1.3.40)

Thus the mean IF is the same as the IF defined earlier (Eq. 1.3.1), but the physical interpretations are different. The IF has been derived for a tone (and earlier for a modulated sinusoid), whereas the mean IF applies to the spectrum of a short-duration signal.

1.3.4 RELAXATION TIME, DYNAMIC BANDWIDTH For a linear-FM signal, the instantaneous phase φ(t) is quadratic. So φ(t) can be expanded in a Taylor series about t = t0 : 1 φ(t) = φ(t0 ) + φ  (t0 )[t − t0 ] + φ  (t0 )[t − t0 ]2 2 1 = φ(t0 ) + 2πfi (t0 )[t − t0 ] + 2πfi  (t0 )[t − t0 ]2 . 2

(1.3.41) (1.3.42)

The relaxation time Tr , as defined by Rihaczek [25, p. 374], is the duration over which the instantaneous phase deviates no more than π/4 from linearity. That is,    1  2πfi  (t0 )T 2 /4 = π/4. r  2

(1.3.43)

Solving this equation leads to the following definition: Definition 1.3.6. The relaxation time of a signal is    df (t) −1/2 Tr (t) =  i  , dt

(1.3.44)

where fi (t) is the IF. The dual of relaxation time, known as dynamic bandwidth, is the bandwidth over which the phase spectrum, assumed to be a quadratic function of frequency, deviates no more than π/4 from linearity. The result is as follows: Definition 1.3.7. The dynamic bandwidth of a signal is    dτ (f ) −1/2 Bd (f ) =  d  , df

(1.3.45)

where τd (f ) is the SD.

1.3.4.1 Interpretation As the relaxation time is a measure of the time needed to observe significant variations in IF, so the dynamic bandwidth is a measure of the bandwidth needed to observe significant variations in SD.

1.3 INSTANTANEOUS FREQUENCY (IF) AND SPECTRAL DELAY (SD)

57

Definition 1.3.8. The instantaneous bandwidth (IB) of a signal s(t) is a measure of frequency spread at a given time, and it can be expressed as [26,27]: ∞ (f − fi (t))2 ρ(t, f ) df . Bi (t) = σf2 (t) = −∞  ∞ −∞ ρ(t, f ) df

(1.3.46)

Heuristic Proof . By analogy with the PDF, the classical bandwidth of a stationary signal may be defined as the deviation of frequency about the mean frequency of the signal and it can be expressed as: ∞

2 2 −∞ (f − fc ) |S(f )| df ∞ . 2 −∞ |S(f )| df

2 (t) = B = σBW

(1.3.47)

For nonstationary signals, the square magnitude spectrum representing an energy density is replaced by the TFD and the concept of bandwidth can be extended by replacing the mean frequency fc by the IF so that the IB of a monocomponent signal represents the deviations about the IF for a given time, resulting in Eq. (1.3.46). For a multicomponent signal, at a certain time, the IB represents the difference between the highest value of subcomponent IF and the lowest value of subcomponent IF, assuming the subcomponents IB are negligible. Otherwise, a corrective factor needs to be introduced.

1.3.4.2 Illustration Consider a newborn EEG signal which is multicomponent and nonstationary in nature with subcomponents IFs shown in Fig. 1.3.3 [9]. One can observe that the multicomponent signal IB at the 4th second

20

Time (s)

16

12 Bi10 8 Bi4

4

0

0

1

2

3

4

5

Frequency (Hz)

FIGURE 1.3.3 IF laws and IB of an epoch of a multicomponent newborn EEG signal. (The signal IB has two distinct values at t = 4 and t = 10; not to be confused with the subcomponents IB.)

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CHAPTER 1 TIME FREQUENCY AND IF CONCEPTS

is different from the multicomponent signal IB at the 10th second. As the IFs change over time, the IB will also vary. This IB measure is another important characteristic of a nonstationary multicomponent signal, useful in some applications [26].

1.4 DEFINING AMPLITUDE, PHASE, AND IF USING THE ANALYTIC SIGNAL This section revisits material discussed earlier and further elaborates on the equivalence between a signal x(t) and the two characteristics of amplitude and phase.

1.4.1 AMPLITUDE AND PHASE FORMULATION FOR THE AM/FM MODEL Following the approach taken in Section 1.3, let us consider a monochromatic signal a cos (2πft + ψ) with amplitude a, frequency f , and initial phase ψ. The argument of the cosine function is the linear function φ(t) = 2πft + ψ so that f = φ  (t)/(2π). Hence, for a more general amplitude- and frequencymodulated signal x(t) = a(t) cos φ(t),

(1.4.1)

where a(t) ≥ 0 and φ(t) is not necessarily linear, it is natural to define the instantaneous amplitude as a(t), the instantaneous phase as φ(t), and the IF as fi (t) = φ  (t)/(2π). Unfortunately, these widely used “natural” definitions are sometimes ambiguous. Consider, for example, the problem of defining the amplitude and phase of the signal y(t) = cos φ1 (t) cos φ2 (t). One solution could be a(t) = cos φ1 (t) and φ(t) = φ2 (t). Another could be a(t) = cos φ2 (t) and φ(t) = φ1 (t). By specifying a(t) and φ(t), we uniquely specify x(t) in Eq. (1.4.1). But the inverse problem—finding a(t) and φ(t), given x(t)—has infinitely many solutions [28]: if φ2 (t) is any function such that cos φ2 (t) has the same sign as x(t), we can find a2 (t) ≥ 0 such that x(t) = a2 (t) cos φ2 (t).

(1.4.2)

To obtain an inverse problem with a unique solution, we must impose further conditions. There are numerous ways to do this. The most classical approach uses the HT and the analytic signal (AS), as presented in Section 1.3. Let us now further explore these concepts and some of their implications concerning the definitions and properties of the IF.

1.4.2 ANALYTIC SIGNAL, HT, AND AM/FM MODEL Let us seek to define a one-to-one correspondence between the real signal x(t), and the pair of real functions [a(t), φ(t)] comprising the instantaneous amplitude and phase, with fi (t) = φ  (t)/(2π); in other words, find a one-to-one correspondence between x(t) and a complex signal z(t), whose modulus and argument are [a(t), φ(t)]. Let X(f ) be the FT of the real signal x(t). As in Eq. (1.2.2), X(f ) satisfies the Hermitian symmetry X(−f ) = X ∗ (f ). The analytic signal (AS) z(t) is then obtained from x(t) by applying Z(f ) = 0 for f < 0 to find the FT of z(t) as Z(f ) = (1 + sgn f ) X(f ).

(1.4.3)

1.4 DEFINING AMPLITUDE, PHASE, AND IF USING THE ANALYTIC SIGNAL

59

Hence the FT of z∗ (t) is Z ∗ (−f ) = (1 − sgn f ) X ∗ (−f ) = (1 − sgn f ) X(f ). Adding this result to Eq. (1.4.3) and taking IFTs returns x(t) = (1/2)[z(t) + z∗ (t)] as the real part of the AS. Thus for any x(t), we have specified z(t) and vice versa: a one-to-one correspondence, as previously defined in Eq. (1.2.3). As per Eq. (1.2.6), the HT of the real signal x(t), H{x(t)}, defines the imaginary part y(t) of the AS. Let its FT be Y(f ); then z(t) = x(t) + jy(t), so that Z(f ) = X(f ) + jY(f ), which satisfies Eq. (1.4.3) iff Y(f ) = −jX(f ) sgn f (cf. Eq. 1.2.4 and Example 1.2.3). Then, taking the IFT yields y(t). Equation (1.4.3) indicates that z(t) and y(t) are deduced from x(t) through linear filters with frequency responses 1 + sgn f and −j sgn f , respectively. As Z(f ) does not have Hermitian symmetry, z(t) cannot be real. Let the modulus (non-negative) of z(t) be a(t) and the argument (modulo 2π) be φ(t) so that z(t) = a(t) exp[jφ(t)].

(1.4.4)

Then the real signal x(t) is associated with a unique pair [a(t), φ(t)], which may be called its canonical pair [29]. This is a consequence of Eq. (1.4.3). There are questions on the physical meaning of the AS, some of which are discussed in Ref. [28]. It is shown in Refs. [15,30] that if we make some a priori physical assumptions, the only possible definition of the instantaneous amplitude and phase is the one given above; and it allows easy calculations and interpretations.

1.4.3 CHARACTERIZATION OF AM/FM CANONICAL PAIRS [a(t ), φ(t )] An arbitrary pair [a(t), φ(t)] is not necessarily canonical in the sense defined above, given the requirement that the complex function z(t) defined by Eq. (1.4.4) is an AS (i.e., its FT is zero for negative frequencies). Let us discuss the consequences. As the characterization of an AS is deduced from the spectral condition (zero FT for negative frequencies), it is tempting to translate the same idea for a(t) and φ(t). Noting that most applications of AM/FM concern narrow-band signals, it is appropriate in such cases to separate the contribution of the carrier frequency by using, instead of Eq. (1.4.1), a signal of the form x(t) = a(t) cos[2πf0 t + ψ(t)].

(1.4.5)

This is the real part of w(t) exp(j2πf0 t), where w(t) = a(t) exp[jψ(t)] is called the complex envelope of x(t). We therefore seek conditions on [a(t), ψ(t)] which ensure that w(t) exp(j2πf0 t) is an AS, as a generalization of Eq. (1.2.11). An elementary answer, introducing the idea of an asymptotic AS (see below), is that any physical signal w(t) is approximately band-limited and may therefore be made approximately an AS by shifting it far enough in the positive frequency direction—that is, by multiplying it by exp(j2πf0 t) for a sufficiently high f0 [16,31]. This reasoning applies rigorously if the signal is strictly band-limited. For amplitude modulation, the instantaneous amplitude-phase pair is [a(t), 2πf0t + ψ], where ψ is constant. It is easy to verify that this pair is canonical iff a(t) is a positive band-limited signal whose FT is zero for |f | > f0 [28]. Let us now try to use spectral methods for the characterization of more general pairs of functions [a(t), φ(t)].

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CHAPTER 1 TIME FREQUENCY AND IF CONCEPTS

Saying that a(t) exp[jφ(t)] is an AS implies (or is equivalent to saying) that the HT of a(t) cos φ(t) equals a(t) sin φ(t), which is ensured by Bedrosian theorem [32] on the HT of a product of two real functions x1 (t) and x2 (t). A simple derivation of this theorem and some extensions can be found in Ref. [28]. The main result is as follows: let X1 (f ) and X2 (f ) be the FTs of x1 (t) and x2 (t), respectively. If x1 (t) and x2 (t) are low- and high-frequency signals such that X1 (f ) = 0 for f > B and X2 (f ) = 0 for f < B, then H{x1 (t) x2 (t)} = x1 (t) H{x2 (t)}.

(1.4.6)

It follows that if a(t) is a low-frequency signal and cos φ(t) a high-frequency signal as defined above, then H{a(t) cos φ(t)} = a(t)H{cos φ(t)} = a(t) sin φ(t).

(1.4.7)

The estimation and computational aspects of the HT, analytic signal, and IF are discussed in Refs. [15,33] and Section 6.5, with applications presented in Ref. [16].

1.5 SUMMARY AND DISCUSSION Many practical signals are characterized by a variation of frequency content with time. A TFR of a signal shows this variation directly. The magnitude spectrum (magnitude of the FT) does not show this variation at all. In principle, the signal can be completely represented as a function of time or as a complex FT; but in practice, neither of these is as effective as a TFR in showing how the frequency content varies with time. The bandwidth-duration (BT) product of a signal is proportional to its information content. A large BT product is among the characteristics of an asymptotic signal. The use of the analytic associate of a given real signal, rather than the signal itself, is essential for obtaining an unambiguous IF, and useful for reducing the required sampling rate. The analytic associate contains only non-negative frequencies. It is related to the HT, which removes the zero-frequency component, changes the signs of the negative-frequency components, and introduces a phase lag of π/2. The analytic associate of a monocomponent asymptotic signal is fully characterized by its instantaneous amplitude and instantaneous phase, from which we can determine its IF fi (t). IF estimation methods can be found in Refs. [15,34] and Chapter 10. An understanding of IF and its dual, SD, is necessary for the interpretation of TFDs. The concept of the analytic signal z(t) is related to the definition of the canonical pairs [a(t), φ(t)] for a given real signal s(t). Other applications of the HT appear in [35–37]. The next chapter introduces various formulations of TFDs and demonstrates the importance of using the analytic associate.

1.6 ADDITIONAL EXERCISES This section provides additional exercises that cover important topics from this chapter and prepares the reader for the next chapters in the book. It is recommended to use the (t, f ) toolbox described in Chapter 17 as an aid for doing these exercises. In a formal class setting, the instructor may wish to guide the student by adding intermediate questions with focus on one or two assessment criteria.

REFERENCES

61

Exercise 1.6.1. Synthesize an LFM signal with starting frequency 10 Hz, end frequency 90 Hz, and duration 10 s. 1. Plot the signal in the time domain. 2. Plot the signal in the frequency domain. Exercise 1.6.2. Compute the analytic signal associated with the one synthesized in the previous exercise. 1. Show the frequency domain representation of the analytic signal. 2. Write a code to estimate the IF and GD from the analytic signal and its frequency domain representation. Note that before implementing a derivative operation in Matlab, you need to unwrap the phase of the signal. Exercise 1.6.3. Illustrate the (t, f ) representation of the above analytic signal using the WVD. 1. Compute the WVD of this analytic signal and estimate the IF and GD from the location of peaks in the (t, f ) domain; then relate IF to GD and SD. 2. Compare your results with the results obtained using the phase derivative algorithm using the TFSAP toolbox described in Chapter 17. Exercise 1.6.4. Repeat Exercises 1.6.2 and 1.6.3 for the same signal corrupted by noise with SNR-0, 5, and 10 dB. Find which algorithm is more robust to noise, in terms of mean square error. Exercise 1.6.5. Generate and plot the following signals: 1.  s(t) =

cos (0.1πt) + cos (2π(0.35t − 0.005t2 )), 0,

0 ≤ t ≤ 255 otherwise.

2. The test signal given in Eq. (1.1.3), p. 34 with the same values given in the text below the equation. Exercise 1.6.6. Using time-frequency representations, find out whether the signals given in Exercise 1.6.5 are multi-component or mono-component. Justify your answer.

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