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CAN A SIGNAL BE BOTH MONOCOMPONENT AND MULTICOMPONENT?

Gavin R. Putland and Boualem Boashash

Signal Processing Research Centre, Queensland University of Technology,

GPO Box 2434, Brisbane, Q 4001, Australia

Email: g.putland, b.boashash@qut.edu.au

ABSTRACT

The spectrogram of an AM or FM signal with tone modulation may show one modulated component or several stationary components, depending on the window. A two-tone

FM signal may have three “personalities” in the (t; f ) plane:

monocomponent and non-stationary, multicomponent and

non-stationary, or multicomponent (with more components)

and stationary. These observations are readily predicted by

algebraic manipulations in the time domain, and must cause

us to question whether the terms “monocomponent” and

“stationary” are well-defined.

1. INTRODUCTION

We tend to distinguish between “monocomponent” and

“multicomponent” signals, and between “stationary” and

“non-stationary” signals, as if we knew what we are talking about. But do we? Consider the full-carrier AM signal

z (t) = [1

j 2 fc t

cos 2 fm t] e

(1)

where fm is the modulating frequency and f c is the carrier

frequency. The same signal can be written

z (t) = ej 2 fc t

1

2

ej 2 [fc +fm ]t

1

2

ej 2 [fc fm ]t :

(2)

Whereas Eq. (1) appears to describe a monocomponent

signal—a modulated single carrier—Eq. (2) appears to describe a multicomponent signal, whose “components” are

commonly called the carrier, upper sideband tone and lower

sideband tone. Furthermore, the three “components” in

Eq. (2), being of constant amplitude and frequency, demand

to be described as “stationary”, whereas the modulation of

the single “component” in Eq. (1) might be interpreted as

non-stationarity.

If one of the sidebands in Eq. (2) is suppressed, the

sum of the carrier and the other sideband is equivalent

to an amplitude- and phase-modulated carrier (cf. Carlson [1], pp. 185–6). Again, we might describe the carrierplus-sideband form as multicomponent and stationary, and

the modulated-carrier form as monocomponent and nonstationary.

Further examples of apparently dual-natured signals are

readily constructed. These signals challenge us either to devise absolute criteria for deciding whether a signal is monocomponent and whether it is stationary, or to admit that no

such criteria exist. This paper considers intuitive criteria

based on time-frequency distributions (TFDs).

2. TIME-FREQUENCY EXAMPLES

In the time-frequency context, we tend to think of a monocomponent signal as one whose TFD is a single “ridge”,

i.e. a single delineated region of energy concentration. We

also stipulate that the ridge does not “fold back” in time;

that is, interpreting the crest of the “ridge” as a graph of

frequency vs. time, we require the frequency of a monocomponent signal to be single-valued. We tend to think that

a multicomponent signal is one whose TFD comprises two

or more ridges, representing the sum of two or more monocomponent signals, and that a stationary signal is one whose

TFD is independent of time.

It turns out, however, that we can construct examples in

which TFDs do not help us to decide whether the signal is

monocomponent or whether it is stationary, but merely confirm the dual natures predicted by alternative formulations

in the time domain.

2.1. Full-carrier AM, tone modulation

A full-carrier AM signal was simulated using Eq. (1) with

fc = 16=64 and fm = 3=64. For the purpose of computing TFDs, the signal was taken to be periodic with a period of 64 samples, so that 3 cycles of modulation appeared

within each time-frequency plot. Fig. 1 graphs three different TFDs of this signal; each graph has three panels showing the TFD (main panel), the amplitude spectrum (bottom

panel), and the time-domain plot (left panel).

Part (a) of Fig. 1 shows a spectrogram computed with

a short window, which gives sufficient time resolution to

show the modulation, but insufficient frequency resolution

to separate three components. The resulting TFD matches

the form of Eq. (1). Part (b) shows a spectrogram com-

(a)

Fs=1Hz N=64

Time−res=1

60

Time (seconds)

50

40

30

20

10

0.05

0.1

0.15

0.2

0.25

0.3

Frequency (Hz)

0.35

0.4

0.45

0.5

(b)

Fs=1Hz N=64

Time−res=1

puted with a long window, which resolves three components

in the frequency direction but fails to resolve the modulation in the time direction. The resulting TFD matches the

form of Eq. (2). Part (c) shows the Wigner-Ville distribution

(WVD), which includes the carrier and sidebands of Eq. (2),

the cross-term between the two sidebands (superimposed on

the carrier), and the cross-terms between the carrier and the

respective sidebands.

In Fig. 1, each spectrogram is equivalent to the WVD

convolved with a spreading function. For the short window, the spreading function is short in the time direction

and wide in the frequency direction. For the long window,

the reverse is true. In each case it is qualitatively clear how

the spreading of the WVD in part (c) gives rise to the spectrogram in part (a) or (b). Note that the spreading applies to

the cross-terms of the WVD as well as the auto-terms, and

that the modulation visible in part (a)—and in Eq. (1), from

which the signal is computed—is due to the oscillations of

the cross-terms in the WVD; for these reasons, we should

hesitate to describe the cross-terms as spurious.

60

2.2. Single-tone FM

50

Time (seconds)

A unit-amplitude complex FM signal has the form

z (t) = ej (t)

40

30

(3)

where (t) is real. Its instantaneous frequency (IF) may be

defined [2] as

20

fi (t) =

10

0.05

0.1

0.15

0.2

0.25

0.3

Frequency (Hz)

0.35

0.4

0.45

0.5

(4)

For simple tone modulation, we may choose

(t) = 2 fc t

(c)

0 (t)

:

2

fd

fm cos 2 fm t

(5)

so that

fi (t) = fc + fd sin 2 fm t:

(6)

Thus fc is the mean frequency and f d is the peak frequency

Fs=1Hz N=64

Time−res=1

60

deviation. Substituting Eq. (5) into Eq. (3), extracting the

factor ej 2 fc t and expanding the other factor in an exponential Fourier series, we obtain

Time (seconds)

50

40

z (t) =

30

20

10

0.05

0.1

0.15

0.2

0.25

0.3

Frequency (Hz)

0.35

0.4

0.45

0.5

Figure 1: TFDs of a full-carrier AM signal (Eq. (1)) with 64

samples, carrier frequency 0.25, modulating frequency 3/64:

(a) spectrogram, 9-point Hamming window; (b) spectrogram,

63-point rectangular window; (c) WVD.

X1

k=

1

j k Jk (fd =fm ) ej 2 [fc +kfm ]t

(7)

where Jk denotes the Bessel function of the first kind, of

order k (cf. [1], pp. 225–6). The term for k = 0, known as

the carrier, has frequency f c . The other terms, known as

sideband terms, are separated from the carrier by multiples

of fm . Although the number of sideband terms is theoretically infinite, the significant terms [1, pp. 220–37] may be

assumed to lie between the frequencies f c (fd + fm ) or,

more conservatively, f c (fd + 2fm ).

As in the amplitude-modulated example, the modulatedcarrier formulation [Eqs. (3) and (5)] appears to describe

a single modulated (non-stationary) component, whereas

the carrier-plus-sidebands form [Eq. (7)] appears to describe many unmodulated (stationary) components. To

see whether these dual natures are observable in the timefrequency domain, a simple FM signal was simulated using

Eqs. (3) and (5) with f c = 16=64 and fm = 3=64. For the

purpose of computing TFDs, the signal was taken to be periodic with a period of 64 samples, so that 3 cycles of modulation appeared within each time-frequency plot.

Fig. 2(a) shows the spectrogram computed with a short

window, which gives sufficient time resolution to show the

modulation, but insufficient frequency resolution to separate the sideband terms. The resulting TFD matches the

form of Eqs. (3) and (5). Fig. 2(b) shows a spectrogram

computed with a long window, which resolves about 7 significant components in the frequency direction but fails to

(a)

Fs=1Hz N=64

Time−res=1

60

Time (seconds)

50

40

30

20

10

0.05

0.1

0.15

0.2

0.25

0.3

Frequency (Hz)

0.35

0.4

0.45

0.5

0.05

0.1

0.15

0.2

0.25

0.3

Frequency (Hz)

0.35

0.4

0.45

0.5

(b)

Fs=1Hz N=64

Time−res=1

60

Time (seconds)

50

40

30

20

10

Figure 2: Spectrograms of a sinusoidal FM signal [Eqs. (3)

and (5)] with 64 samples, f c = 0:25 , fd = 0:1 , fm = 3=64 :

(a) 7-point rectangular window; (b) 63-point rectangular window.

resolve the modulation in the time direction. The resulting

TFD matches the form of Eq. (7).

2.3. Two-tone FM

If we take

fd1

fm1 cos 2 fm1 t

fd2

fm2 cos 2 fm2 t

(8)

fi (t) = fc + fd1 sin 2 fm1 t + fd2 sin 2 fm2 t:

(9)

(t) = 2 fc t

in Eq. (3), we find that

Thus we obtain an FM signal whose frequency deviation

contains two sinusoidal terms.

Fig. 3 shows three TFDs of a two-tone FM signal simulated using Eqs. (3) and (8). The signal was taken to be periodic with a period of 256 samples. Parameters were chosen

so that fd1 > fd2 and so that, during the 256-sample period, the modulation goes through 15 cycles for the larger

frequency deviation and 3 cycles for the smaller.

To obtain a TFD showing the FM law [Fig. 3(a)], a

WVD with a short lag window was used. 1 To obtain a TFD

showing the spectrum, a spectrogram with a long window

was used [Fig. 3(c)]. These two extremes have counterparts

in the simple FM case [Fig. 2, parts (a) and (b) respectively].

The two-tone FM signal also admits a third representation having no counterpart in the simple FM case. The

FM law in Eq. (9) is equivalent to modulating the carrier

according to the higher-frequency term of the FM law, converting the signal to carrier-plus-sidebands form, and then

modulating all the components thereof according to the

lower-frequency term in the FM law; this may be verified

by reworking the argument of Eqs. (3) to (7) and applying

the convolution properties of the exponential Fourier series.

Numerical confirmation is given by the magnitude spectrum (appearing below each TFD in Fig. 3), which shows

the carrier and sidebands due to the higher modulating frequency, albeit with each component replaced by a carrierplus-sidebands pattern corresponding to the lower modulating frequency. But the clearest confirmation is given by

Fig. 3(b), in which we see the carrier and sidebands for

the higher modulating frequency, all frequency-modulated

at the lower modulating frequency; the TFD in this case is

a spectrogram whose window spans several cycles of the

higher modulating frequency but less than one cycle of the

lower modulating frequency.

In Fig. 3, part (a) seems to be monocomponent and nonstationary, part (b) multicomponent and non-stationary, and

part (c) multicomponent (with more components) and stationary.

1 A spectrogram with a short window shows the FM law but also

shows a spurious amplitude modulation, the apparent amplitude being inversely related to the frequency sweep rate.

3. DISCUSSION

(a)

Fs=1Hz N=256

Time−res=1

250

Time (seconds)

200

150

100

50

0.05

0.1

0.15

0.2

0.25

0.3

Frequency (Hz)

0.35

0.4

0.45

0.5

0.05

0.1

0.15

0.2

0.25

0.3

Frequency (Hz)

0.35

0.4

0.45

0.5

(b)

Fs=1Hz N=256

Time−res=1

250

Time (seconds)

200

150

100

50

(c)

Fs=1Hz N=256

Time−res=1

250

If a TFD has strong frequency support, i.e. if the TFD is

zero wherever the spectrum is zero, then it cannot produce

plots like Fig. 2(a), Fig. 3(a) and Fig. 3(b), in which the IFs

are seen to vary continuously while the magnitude spectra

admit only discrete frequencies. As the notion of IF is of

great practical importance, these examples must cause us to

question the desirability of strong frequency support.

It may be argued that the “TFDs” shown in Figs. 1(b),

2(b) and 3(c) are not true TFDs, but only spectra disguised as TFDs, inasmuch as they do not admit any timedependence of frequency content. But no such claim can be

made concerning part (b) of Fig. 3. Part (b), like part (a),

shows a time-varying frequency content, but disagrees with

part (a) as to the number of components and the FM

laws thereof. Together, parts (a) and (b) show that timefrequency analysis does not uniquely determine the number

of components in an arbitrary signal.

Ambiguity in the number of components is not necessarily a vice. For example, when analyzing or designing

an FM receiver, we prefer a carrier-plus-sidebands signal

model (multicomponent and stationary) for the tuning filters, but a modulated-carrier model (monocomponent and

non-stationary) for the demodulator.

Intuitively, a monocomponent signal is a complex sinusoid possibly modulated in amplitude and/or frequency, i.e.

a signal of the form a(t)e j (t) , where a(t) is real and positive and (t) is real. But any complex signal can be written

in that form by defining a(t) as the modulus and (t) as the

argument. We may try to tighten the definition by requiring

the signal to be analytic, (t) to be differentiable and increasing, a(t) to be low-pass, and e j (t) to be high pass and

spectrally disjoint with a(t). But all the signals considered

in this paper comply with these constraints and still have

multicomponent aspects.

200

Time (seconds)

4. CONCLUSION

150

100

50

0.05

0.1

0.15

0.2

0.25

0.3

Frequency (Hz)

0.35

0.4

0.45

0.5

That the same signal can have both monocomponent and

multicomponent aspects is demonstrated not only by algebraic identities, but also by alternative numerical timefrequency representations. In some cases the multicomponent form of the signal is stationary while the monocomponent form is not, implying that the same signal can have

both stationary and non-stationary aspects.

5. REFERENCES

Figure 3: TFDs of the two-tone FM signal [Eqs. (3) and (8)]

with 256 samples, f c = 0:25, fd1 = 0:1, fm1 = 15=256 ,

fd2 = 0:01,

fm2 = 3=256 :

(a) WVD, 9-point Hamming

lag window; (b) spectrogram, 75-point Hamming window;

(c) spectrogram, 255-point rectangular window. Plotted time

resolution is 1 sample for (a), 5 samples for (b) and (c).

[1] A. B. Carlson, Communication Systems. Tokyo: McGrawHill, second ed., 1975.

[2] B. Boashash, “Estimating and interpreting the instantaneous

frequency of a signal—Part 1: Fundamentals,” Proceedings of

the IEEE, vol. 80, pp. 520–538, April 1992.

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