TIME–FREQUENCY SPECTRA FOR FREQUENCY-MODULATED PROCESSES

Mechanical Systems and Signal Processing (1997) 11(4), 621–635

TIME–FREQUENCY SPECTRA FOR FREQUENCY-MODULATED PROCESSES S. A. D  J. K. H Institute of Sound and Vibration Research, University of Southampton, Highfield, Southampton, SO17 1BJ, U.K. (Received May 1997, accepted May 1997) Time–frequency spectra are accepted as a useful approach to characterising non-stationary phenomena. However, most applications are empirical and there is interest in constructing analytical models of such phenomena. This paper addresses the analytic modeling of a class of non-stationary stochastic processes exhibiting frequency-modulated forms. This includes acoustic phenomena with Doppler shifts and vehicle vibrational motion at variable speed. Wigner distributions are derived for specific velocity–time histories and compared with sample empirical calculations using the concept of covariance equivalence. 7 1997 Academic Press Limited

1. INTRODUCTION

This paper is concerned with the prediction of time–frequency (TF) spectra for a class of non-stationary frequency-modulated random processes. Two approaches to this are considered: a theoretical prediction of the Wigner distribution, and an empirical analysis using covariant equivalent models of the processes. The concept of non-stationarity is conceived as a result of motion which distorts of the independent variable leading to a form of frequency modulation. Such processes often arise in the physical world and examples of phenomena which may be modeled using time-variant systems in this way include Doppler shifted acoustic signals and vehicles travelling over rough terrain at variable speed [1, 2]. In this paper, some facts concerning the Wigner distribution of these processes are highlighted.

2. TIME–FREQUENCY DISTRIBUTIONS

Time–frequency methods may be split broadly into two types: (i) linear representations of a signal (e.g. short-time Fourier transform); and (ii) quadratic distributions (e.g. the Wigner distribution) describing energy distributions in the TF plane [3–6]. They have been used extensively to represent, model and characterise non-stationary processes. 2.1.  –  Linear distributions characterise the signal as a sum of simple basis functions. Members of this class include the short-time Fourier transform [3, 5], evolutionary spectra [7] and wavelets [8–10]. 1.1.1. Short-time Fourier transform (STFT) This is the simplest of the approaches to TF analysis deriving from the Fourier 0888–3270/97/040621 + 15 $25.00/0/pg970100

7 1997 Academic Press Limited

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transform. Time localisation of frequency components is obtained by windowing the signal and then transforming the segment: X(t, v) = STFTx (t, v) =

g

x(t')w(t − t') e−jvt' dt'

(1)

t'

The signal can be reconstructed from the STFT synthesis formula: x(t) =

gg

X(t', v)g(t − t') ejvt' dt' dv

(2)

v

t'

The analysis and synthesis window must satisfy f w(t)g(t) = 1. A characteristic of this transform is the resolution trade-off between time and frequency. The wavelet transform provides the possibility of controlling the resolution trade-off for a range of frequencies. 2.1.2. Evolutionary spectra A class of non-stationary random signals (oscillatory processes) may be expressed as in equation (3) below. The essence of Priestley’s idea [7] is that the basic building blocks in the representation, namely sines and cosines, are replaced by amplitude modulated sines and cosines:

g

x(t) = A(t, v)X(v) ejvt dv

(3)

The representation of a stationary process as the sum of sines and cosines is generalised for this class of processes admitting a representation which preserves orthogonality, i.e. E{X*(v1 )X(v2 )} = 0, v1 $ v2 ; this condition leads to the definition of a time-varying spectral density: Sx (t, v) = =A(t, v) =2S(v)

(4)

where S(v) is the usual spectral density arising from X(v). These two forms appear to differ significantly in that one is a double summation [equation (2)] and the other [equation (3)] a single summation. However, if equation (2) is rewritten using t − t' = u then x(t) =

gg u

X(t − u, v)g(u) ejv(t − u) du dv

v

g g

x(t) = ejvt

X(t − u, v)g(u) du dv

(5)

v

and formally,

g

v

X(t − u, v)g(u) e−jvu du = A(t, v)X(v)

(6)

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and thus the STFT is in evolutionary spectral form. Also, substituting

g

X(v) = x(t) ejvt dt

(7)

into equation (3) yields a STFT-type equation. 2.2.     Another approach of TF analysis can be realised by considering the decomposition of signal energy. Such a decomposition is quadratic and does not necessarily reduce to the square of a linear transform [4–6]. For a stationary process, the power spectral density of the signal can be interpreted as the Fourier transform of the autocorrelation function.

g

S(v) = R(t) e−jvt dt

(8)

For a time-varying process, the autocorrelation function will be time dependent so it is logical to consider a localised autocorrelaion form, equation (9), and the family of energy distributions termed Cohen’s class [3] derives from it. The Wigner distribution (WD) is the core of this class. This generalised family of quadratic distributions is based upon a localised autocorrelation form. The power spectral density of a non-stationary process may be obtained by extending the stationary autocorrelation form to the localised time-variant equivalent:

g

C(t, v) = S(t, v) = Rt (t) e−jvt dt

(9)

Various definitions of Rt (t) lead to different distributions [4]. The Cohen class can be expressed as a smoothed version of the Wigner distribution through a 2-D convolution using a smoothing function F: Cx (t, v) =

gg

F(t' − t, v' − v)Wx (t', v') dt' dv'

(10)

2.2.1. Wigner distribution The Wigner distribution may be regarded as a prototype. Its ability to represent non-stationary signals in the time and frequency domains simultaneously has been studied extensively. It is defined as the Fourier transform of an instantaneous autocorrelation function:

g0 1 0 1

Wx (t, v) = x t +

t t −jvt x* t − e dt 2 2

(11)

where x* is the conjugate of x. For a monocomponent, asymptotic [1, 3] signal, it is desirable that the distribution should be concentrated along a well-defined instantaneous frequency. The WD satisfies this but only exactly for a linear chirp. However, it also produces negative values which cannot easily be interpreted in terms of energy. Another problem is the production of cross-terms. Suppose a two-component signal is x = x1 + x2 .

. .   . . 

624 Then the WD of x is:

Wx = Wx11 + Wx22 + Wx12 + Wx21

(12)

where WDxij relates to the sub-products. To suppress these limitations, smoothed versions of the transform are used [3] for example the pseudo Wigner distribution (PWD)

g 01 0 10 1 0 1

PWx (t, v) = h

t t t t −jvt h* − x t + x* t − e dt 2 2 2 2

(13)

which is a smoothed version in the frequency direction only and this action suppresses cross-terms preserving the time resolution. The window selection (smoothing operation) also affects the auto terms and some desirable properties of the WD are lost. This paper concentrates on the Wigner distribution and examines the properties of the distortion terms for processes exhibiting a frequency-modulated form. 3. THE MODEL OF THE PROCESS

Our objective is to model processes having a frequency-modulated form. This is achieved by considering a stationary process defined over a particular independent variable s and then introducing non-stationarity by parameterising this variable with respect to time (t), i.e. s(t). The basic model assumes a process, say h(s), which is stationary (homogeneous) when considered as a function of s which might be interpreted as a spatial variable characterised by the covariance function Rhh (s2 − s1 ) = E[h(s1 )h(s2 )]

(14)

and the corresponding wavenumber spectrum is:

g

Shh (k) = Rhh (j) e−jkj dj

where j = s2 = s1

(15)

It is convenient to consider h(s) as the output of a white noise excited filter in the s domain, i.e. process h(s) is the output of a homogeneous (stationary) shaping filter (Fig. 1). In order to create a non-stationary process, the independent variable is dilated by letting s be a function of t, i.e. s(t). One now conceives of a process h (t) in the time domain, i.e. a process equivalent to h, but with the variable s as a function of time s(t): h (t) = h[s(t)]

(16)

Then E[h(s(t1 ))h(s(t2 ))] is not, in general, a function of t2 − t1 only (i.e. h (t) is non-stationary). Our interest here is in calculating the TF description of h (t) and two approaches to this are presented. The first is a theoretical prediction arising directly from the autocorrelation function of h(s), and the second uses the concept of covariance equivalent processes to

w(s)

s-invariant filter

h(s)

Figure 1. Schematic representation of the shaping filter process h(s).

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ξ

h(s)

s s1

s2

Figure 2. Graph defining s1 , s2 and j.

simulate the process from empirical analysis. This has been addressed through a series of papers referred to in [1]. 3.1.       h (t) Consider two different values of s, i.e. s1 and s2 where j = s2 − s1 is the distance between the two points (Fig. 2). The autocorrelation function of the process h is defined as: Rhh (j) = Rhh (s2 − s1 ) = E[h(s1 )h*(s2 )]

(17)

Now let s be time variant, i.e. s(t). In this case s1 = s(t1 ), s2 = s(t2 ) and j = s(t2 ) − s(t1 ). Writing Rhh (j) = E[h(s(t1 ))h(s(t2 ))] = E[h (t1 )h *(t2 )] = Rh h (t1 , t2 )

(18)

and now if t1 = t − t/2 and t2 = t = t/2, the autocorrelation of h˜ is:

0 0 1 0 11

Rh h (t, t) = Rhh s t +

t t −s t− 2 2

.

(19)

The Wigner distribution of h (t) is the Fourier transform of the autocorrelation function, i.e. Wh (t, v) = Sh h (t, v) =

g

a

Rh h (t, t) e−jvt dt

(20)

−a

which using equation (19) is Sh h (t, v) =

g 0 0 1 0 11 a

s t+

−a

t t −s t− 2 2

e−jvt dt

(21)

So, knowledge of Rhh (j) and s(t) leads to Sh h (t, v). Clearly the functional form of s(t) dictates the nature of the time-variable spectrum. Some simple examples for different functions of s(t) are now presented. 3.2. -  The autocorrelation function of a first-order process may be expressed as a function of parameter a: Rhh (j) = s 2 e−a=j=

(22)

Letting s be a function of time, when j = s2 − s1 the autocorrelation Rh h (t, v) is: Rh h (t, t) = s 2 e−a=s(t + t/2) − s(t − t/2) = Consider two different functions of s(t).

(23)

626

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3.2.1. Constant velocity If s(t) = vt then the autocorrelation function is Rh h (t, t) = s 2 e−a=v(t + t/2) − v(t − t)v/2) = = s 2 e−av=t=

(24)

and the Wigner distribution is

$

Wh (t, v) = Sh h (t, v) = s 2

%

2an (an)2 + v 2

(25)

This is independent of time, implying stationarity of the process which follows directly from s = vt, which is a pure linear scaling of the independent variable. It is also a low-pass process where the bandwidth and amplitude are dependent on velocity. Compare this with the wavenumber definition, equation (15). The separation for this case is written j = vt and substituting in equation (15) results in: Shh (k) = v

g

R(vt) e−jkvt dt

(26)

b

(27)

This can be written as: Sh h (t, v) =

Shh (k) v

k=

v v

which is a stationary process. The amplitude is inversely proportional and the bandwidth proportional to the velocity, i.e. the spectrum can be obtained by scaling the wavenumber spectrum. S(k) is found from the shaping filter (Fig. 1) where the filter may be expressed in differential form: dh(s) + ah(s) = w(s) ds

(28)

and wavenumber response function form: H(k) +

1 a + jk

(29)

These notions are well known [11] but are a useful starting point for our generalisation which follows. 3.2.2. Constant acceleration Now let s(t) be a quadratic function so that s˙ is now linear and s¨ (acceleration) is constant. For this case the distance s as a function of time is: q s(t) = vt + t 2 2

(30)

Then the derivative of s (velocity) is: s˙ (t) = v + qt

(31)

– 

627

0 1 0 1

(32)

and the separation j is: j=s t+

t t − s t − = (v + qt)t 2 2

Substituting this into equation (22), the autocorrelation function is: Rh h (t, t) = s 2 e−a=vt + qtt=

(33)

Using equation (21), the Wigner distribution is: Sh h (t, v) =

g

a

Rh h (t, t) e−jvt dt =

−a

g

a

s 2 e−a=vt + qtt= e−jvt dt

(34)

−a

Assuming (v + qt) q 0 Sh h (t, v) = s 2

$

2a(v + qt) a(v + qt)2 + v 2

%

(35)

Clearly the Wigner distribution is now time dependent, but it is important to note that for this linear velocity variation the constant velocity in equation (25) is replaced by the instantaneous velocity. The predicted spectrum tracks the ‘conventional’ stationary spectrum exactly. A feature is that as time evolves the bandwidth increases and the amplitude decreases. A similar result can be obtained for (v + tq) Q 0. However, this result does not hold if the velocity variation is non-linear. Further, the Wigner distribution of the equivalent process cannot be easily computed analytically and is addressed later. 3.3.      Consider a general stationary process and linear velocity variation. The autocorrelation function R(j) which corresponds to this stationary process yields a wavenumber spectrum expressed in equation (15). Let s be time dependent, then for a linear variation of s˙ (t) = v + qt the separation is j = s(t + t/2) − s(t − t/2) = vt + qtt and the corresponding Wigner distribution is:

g

Sh h (t, v) = R((n + qt)t) e−jvt dt

(36)

which can be written as: Sh h (t, v) =

1 v + qt

g

v

R(j) e−jv + qtj dj

(37)

This is the scaled wavenumber transform for k = v(v + qt), i.e.

0

Sh h (t, v) = SShh k =

1

v 1 v + qt v + qt

(38)

Thus, the Wigner distribution for linear velocity variation is obtained by scaling (in frequency and amplitude) the conventional (stationary) wavenumber spectrum. The scaling factor depends on the instantaneous velocity. It is important to note that

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628

(v + qt) q 0, i.e. t q −v/q for this result, i.e. the velocity is positive. If (v + qt) Q 0 then that implies running backwards through the field. These conceptional ideas can be accommodated but, for convenience, consideration is restricted to s˙ (t) q 0. The method allows the evaluation of the TF response function under different motion scenarios (only linear) simply given the corresponding stationary spectrum. Example. Consider the shaping filter h(s) in Fig. 1 described by a second-order function. Then the autocorrelation function of the process h(s) is Rhh (j) = s 2 e−a=j= cos (b =j =)

Velocity (m/s)

Frequency (rad)

16

(39)

(a)

14

70

12

60

10

50

8

40

6

30

4

20

2

10

0 10 3

5

0 10

20

40

60

80

2

4

6

8

100

120

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200

10

12

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100

120

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160

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10

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2 5

1 0

5

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Time (s)

Time (s)

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Velocity (m/s)

Frequency (rad)

(b) 14

70

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5

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0 20 15 10 5

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60

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2

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Time (s)

Figure 3. Prediction of the Wigner distribution for a second-order shaping filter process. The TF spectrum is shown for a constant velocity in (a) and the non-stationary spectrum due to acceleration (linear velocity variation) in (b) together with the instantaneous spectra. The non-stationary spectrum has been produced by scaling the wavenumber spectrum of the stationary process.

– 

629

. s

ϑ v t 0 Figure 4. Transitional velocity variation.

with wavenumber spectrum: Shh (k) =

1 (v02 − k 2 )2 + 4z 2v02 k 2

(40)

where a = zv0 and b = v0 z1 − z 2. Using equation (38) the WD can be evaluated. This is shown in Fig. 3. The magnitude of the conventional stationary spectrum for constant velocity [Fig. 3(a)] is shown together with the non-stationary spectrum after scaling of the conventional spectrum (due to constant acceleration) [Fig. 3(b)]. It can be seen that the original stationary spectrum results in a non-stationary frequency-modulated process in which the amplitude decreases and the bandwidth broadens as time increases. It is important to emphasise that S(k) can have many resonances, i.e. can be multicomponent, but for a linear velocity variation for the stochastic case there are no interaction terms. In any empirical analysis using a single realisation of the process, interaction terms would appear but they could be suppressed by smoothing (preferably by ensemble averaging). So for this linear velocity variation the Wigner distribution is derived very simply directly from the conventional spectrum. This is analogous to the Wigner distribution being perfect for linear chirps in the deterministic case [4]. 4. NON-LINEAR VELOCITY VARIATION

For arbitrary motion (i.e. s˙ not necessarily linear) it is necessary to include non-linear and transitional cases. Consider what effect a non-linear velocity variation has on the Wigner distribution as computed from the wavenumber spectrum. Clearly the simple mapping of S(k) no longer applies. Some specific functional forms of s˙ are now addressed. Recall that the integral required is equation (21). To be specific, the question might be asked: what is the difference in the spectral shape due to a transition from constant to linear velocity variation? Consider a simple case as a combination of linear and constant velocity variation for t q 0 and t Q 0, respectively (Fig. 4): s˙ (t) = n v + qt

tQ0 tq0

(41)

Away from t = 0 it might be expected that the spectrum is equivalent to what has been predicted above, but what happens near t = 0? To answer this, Rh h (t, t) needs to be evaluated and thus the behaviour of Rhh (s2 − s1 ) must be examined for various combinations of s1 and s2 assuming that s(0) = 0:

0 0 1 1 11

Rh h (t, t) = Rhh (s2 − s1 ) = Rhh s t +

t t −s t− 2 2

(42)

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630

s

vt + ϑ t2/2

s2 = s(t + τ /2)

s1 = s(t – τ /2)

vt

τ –2t

–t

0

t

2t

Figure 5. The distance parameter as a function of t or t q 0.

and consequently the aim is to evaluate s(t + t/2) − 2(t − t/2) for the above-mentioned combination. This can be done algebraically and the functional forms of s1 and s2 are shown in Fig. 5 for t q 0 and in Fig. 6 for t Q 0. In general, for s1 or s2 positive, the functions are quadratic, and for s1 and s2 negative, are linear. From this and the known analytical form we return to our original aim to evaluate Rh h (t, t): For t Q −2t:

0 1 0 1 0 1 $ 0 1 0 1%

j=s t+

=v−

t t t t q t −s t− =v t+ − v t− + t− 2 2 2 2 2 2

0 1

q t t− 2 2

2

2

for −2t Q t Q 2t:

0 1 0 1 $ 0 1 0 1%

j=v t+

t q t + t+ 2 2 2

2

− v t−

t q t + t− 2 2 2

2

= vt − qtt

(43)

s

2t

s1 = s(t – τ /2)

t

0 –vt

–t

–2t

τ

s2 = s(t + τ /2)

Figure 6. The distance parameters as a function of t for t Q 0.

– 

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Velocity (m/s)

Frequency (rad)

16 14

70

12

60

10

50

8

40

6

30

4

20

2

10

0 15 10

5

10

0 15

20

40

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–4

–3

–2

–1

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140

160

180

200

0

1

2

3

4

5

10

5

5 0 –5

0 5 –5

0

Time (s)

Time (s)

Figure 7. Theoretical Wigner distribution for a second-order shaping filter process due to non-linear velocity transition (q = 1.4 m/s2, v0 = 1 m/s, a = 0.3, b = 1.2).

and for t q 2t:

0 1 $ 0 1 0 1%

z=v t+

t t q t − v t− + t+ 2 2 2 2

2

= vt +

0 1

q t t+ 2 2

2

For values of t close to zero, function j is quadratic while for values away from zero it is linear (the transient effect is fading). The Wigner distribution of the process is Sh h (t, v) =

g

a

Rhh (t, t) e−jvt dt

(44)

−a

Using the above-mentioned functions, the power spectrum has been evaluated numerically for a second-order system with autocorrelation function Rhh (j) = s 2 e−a=j=cos (b=j=)

(45)

and parameters a = 0.3 and b = 1.2. In the Wigner distribution (Fig. 7) it is seen that distortion (interaction) terms are present close to the transition point (t = 0) and also that the bandwidth broadens. This method, which accurately estimates the Wigner distribution, enables detailed analysis of transients and non-linear cases. In comparison to scaling the wavenumber spectrum, this method is, of course, more general but usually requires the numerical evaluation of an integral. A second example has been produced for a sinusoidal velocity variation s˙ (t) = v(1 + cos (t)) again for the second-order system, equation (45). It is seen that the spectrum follows the evolution of velocity. Additionally the bandwidth broadens for high velocity and the amplitude decreases in a similar manner to the previous linear case. Distortion terms are present and can be seen clearly when the velocity is low (Fig. 8).

. .   . . 

632

5. COVARIANCE EQUIVALENT MODELS

The objective of this section is to generate time histories for empirical analysis of frequency-modulated processes by a method which does not dilate the axis of a spatially stationary process. The covariance equivalent principle [1, 2] is used, and is outlined briefly below. Suppose a stationary (homogeneous) process can be expressed by the differential equation: dh(s) + ah(s) = w(s) ds

(46)

where w(s) is a white noise process E[w(s1 )w(s2 )] = d(s1 − s2 ). Now create a non-stationary process h (t) by letting s be time variant, s(t), where h (t) = h[s(t)]. Thus the differential is written: 1 dh[s(t)] + ah[s(t)] = w[s(t)] s˙ dt

(47)

The dilation of the s-axis results non-linearly in a frequency-modulated process. Using results from the covariance equivalence theory (see [1, 2]) it can be argued that if w˜ (t) = w[s(t)]

w'(t)

and

(48)

zs˙ (t)

[where w'(t) is a stationary white noise process with E[w'(t1 )w'(t2 ) = d(t1 − t2 )], then their processes are equivalent in that their autocovariances are indistinguishable. This result implies that w(s(t)) can be replaced by w'(t)/zs˙ (t); then h (t) = h(s(t)) is replaced by a covariant equivalent h'(t). Therefore the differential equation (47) may be written in the covariant equivalent form: 1 dh' w'(t) + ah' = s˙ dt zs˙ (t)

(49)

14 60 12 50

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Frequency (rad)

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4 2 0

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10

Time (s)

Figure 8. Theoretical Wigner distribution for a second-order shaping filter process due to non-linear velocity variation (sinusoidal) (v0 = 2 m/s, a = 0.3, b = 1.2).

– 

633

ν

h(s)

h(s)

s Figure 9. Model of a moving point.

This expresses a non-stationary process as a time-variant filtered stationary white noise input w'(t). The process h'(t) is a non-stationary frequency-modulated process that is covariant equivalent to h (s(t)), i.e. h (t) = h(s(t)) and h'(t) are indistinguishable in terms of their autocovariance functions. Since h'(t) is a linear time-variable filtered version of w'(t) it is possible to predict evolutionary spectra [1]. This paper considers Wigner distributions. Example. For illustrative purposes a resonant filter is more effective than the simple filter above. Consider a second-order process modeled by the shaping filter (Fig. 9): dh 2(s) h(s) + 2zv0 + v02 h(s) = w(s) ds 2 ds

(50)

In state space form, equation (50) becomes d ds

& '

%&

h(s) 0 1 dh = −v02 −2zv0 ds

$

'

h(s) 0 w(s) dh + 1 ds



%$(51)

If we write h = dh/ds then the differential equation for h is: dh = Ah(s) + bw(s) ds where A =

$

%

(52)



%$0 1 0 and b = . −v02 −2zv0 1

Now using the substitution of s which is written as a function of time s(t), then h(s(t)):h(s(t)) = h (s(t)) and (52) can be written in the form: dh = s˙ Ah + bs˙ w(s(t)) dt

(53)

Now using covariance equivalence we introduce the process h' which is equivlent of h (t) where: dh' = s˙ Ah ' + bzs˙ w'(t) dt

(54)

This differential equation has been solved numerically (Runge–Kutta method) and the Wigner distribution of the covariant equivalent process h' for this motion scenario is shown in Fig. 10(a). Ensemble averages have been used. The velocity v is a combination of constant and linear variation. It is shown that the spectrum which corresponds to the constant velocity part is constant but when the velocity is varying the

. .   . . 

634

spectrum has a linear frequency-modulated form where the bandwidth is broadening and the amplitude is decreasing (the spectrum is spreading around the modulated frequency). A second example is presented for the second-order system [equation (50)] with parameters (v0 = 1, z = 0.03) using a different motion scenario. In this case the velocity is considered as a combination of constant sinusoidal varying velocity: t Q 20

s˙ (t) = v

(55)

= v(1 + cos (ct)) t q 20

40

3.5

(a) 35

3

30

Frequency (rad)

2.5

25 2 20

1.5 15 1

10

Velocity (m/s)

0.5

5

0 3 10

2

0

1

2 1.5

5

50

100

150

200

250

300

350

400

450

1 0.5

0

10

20

30 40 0

5

10

15

Time (s)

20

25

30

35

40

Time (s)

3.5

40

(b) 35

3

30

Frequency (rad)

2.5

25 2 20 1.5 15 1 10

Velocity (m/s)

0.5

5

0 0.6 0.4 4

0.2

0

0.6

3

0.4

2

0.2

1

0

10 20 30 40 0

Time (s)

100

5

200

10

300

15

400

20

500

25

600

30

700

35

40

Time (s)

Figure 10. Numerical simulation of the covariant equivalent model due to a combination of (a) constant and linear velocity variation and (b) sinusoidal velocity variation. The Wigner distribution of the covariant equivalent process is shown together with the instantaneous spectra.

– 

635

where c is a constant. The differential equation (54) has been solved numerically and the Wigner distribution of the result is shown in Fig. 10(b). The spectrum which corresponds to the constant velocity part is constant but when the velocity varies the spectrum follows this variation and has a sinusoidal frequency-modulated form. It is interesting to compare these results with the prediction of the Wigner distribution in Figs 7 and 8. The spectrum in Fig. 7 is similar to the one in Fig. 10(a). In Figs 8 and 10(b), the sinusoidal velocity variations which have been used are slightly different [in Fig. 8, 0 Q v Q 4 and in Fig. 10(b), 1 Q v Q 3] and thus the TF distributions are different but both have a sinusoidal frequency-modulated form. As might have been expected, distortion terms are present for both cases because the velocity variation is non-linear. Strong distortion components are present when the velocity is low.

6. CONCLUSIONS

Frequency-modulated processes due to velocity variations can be modeled using analytical models and analysed by TF spectra. The non-stationarity due to motion is obtained by distortion of the independent variable axis. From knowledge of the autocorrelation function of a process and the velocity function we can produce the Wigner distribution of the process. The functional form of velocity dictates the form of the distribution through a velocity variable. For a purely linear velocity variation, the Wigner distribution is obtained directly from the wavenumber spectrum by scaling of the bandwidth and amplitude and no distortion occurs for multicomponent spectra (there are no interaction terms). Non-linear velocity variations can also be accommodated but the integral usually needs numerical evaluation. The distortions of the Wigner distribution due to non-linear velocity are apparent. By introducing the concept of covariance equivalence these frequency-modulated processes may be simulated and empirical analysis employing numerical methods and ensemble averaging is shown to support the theoretical prediction.

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