Peube J-L. Fundamentals of fluid mechanics and transport phenomena

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Measurement, Representation and Analysis of Temporal Signals 373

In practice, the effective choice of a window results from the ensemble of

various considerations. For a presentation and comparison of commonly used

windows, the reader should refer to ([BAH 01], [MAD 98], [NOR 03], [STR 96]

[MAX 96]).

7.3.4.7.

Continuous wavelet transforms

7.3.4.7.1.

Introduction

The Fourier transform with a window function can also be interpreted as an

integral transform whose kernel is the product

)() of the complex

exponential

SQ2

by the window



) . In other words, we no longer perform a

Fourier transform over a limited duration; rather, we perform a transform with a

function



ettg

)(, ) that is different from the complex exponential. Such

a function is often called a wavelet.

This point of view can be generalized by considering functions of time that

contain additional parameters such as window width, which can depend on the

frequency. An example is the Gabor transform in which we decompose the signal

studied according to a basis of functions comprising the product of the complex

exponential

SQ2

and Gauss windows of width

, centered here at the origin:

 

dttxeeG

tjt

f



SQD

24/

In the Gabor transform, we thus have as an additional parameter the width

the window; we obtain, after transformation, a function with two variables which

provide redundant information for representation of the signal x (t). The frequency

representation studied in section 7.3.4.6 can be obtained by fixing the value of

window width; by varying this, we can obtain complementary information. Consider

for example a signal comprising one or two periods of a harmonic function of a

given frequency that is centered at the origin. Let us apply the Gabor transform, we

obtain a new transform which is not so different from the previous one, so long as

the analysis window has a width

of the order of that of the preceding signal, but if

this window is widened, the value of the transform will be reduced. A Gabor

transform with variable width allows the length of a signal to be identified.

7.3.4.7.2. Timescale transforms

We thus arrive at the idea that by using window width as a variable of the

analysis wavelet, we can obtain information regarding the timescales of the signal.

We thus use an analysis wavelet whose form is conserved when we dilate the

374 Fundamentals of Fluid Mechanics and Transport Phenomena

window width. It is clear that by proceeding in this way, the physical idea of

frequency is no longer so clear. If the window comprises little oscillation, then it is

its size that will be the pertinent variable for the transform.

(a) (b) (c)

t t t

Figure 7.10. Wavelets with variable width ((a) and (b))

or variable frequency ((b) and (c))

We define the transform T

(a) of the signal s(t) by the wavelet h(t/a):

 

dtathtsaaT

f



)(

2/1

in which the transform variable is the scale a.

A simple wavelet is the “Mexican hat” wavelet, which is the second derivative

of the Gauss function:



ath



Figure 7.10 shows the window variations for a timescale analysis (windows a

and b) and for a time-frequency analysis (windows b and c).

Diverse problems arise in the choice of wavelets and in particular for the

reconstruction of a signal from its transform ([ALL 04], [FLA 77], [MIS 07], [STR

96]).

7.3.5. Time-frequency (or timescale) representations

7.3.5.1.

General principles

The representation of a signal x(t) by means of an integral transform is useful so

long as this is not too different from the transform kernel: very short signals

centered at the origin will be well represented by wavelets, whereas the Fourier

Measurement, Representation and Analysis of Temporal Signals 375

representation is better suited to periodic (or nearly periodic) signals. For other

signals the preceding representations are numerically ill-conditioned.

Let us take the example of a centered signal of length 100 and which comprises

two sequences of oscillation (Figure 7.11) separated by a sequence in which the

signal is zero. Its representation by means of a Fourier series on the interval [-50,50]

(after change of time origin) is theoretically possible, but it will require a series with

so many coefficients that if will not be very useful.

x(t)

50 –50

Figure 7.11.

Intermittent signal comprising two different sequences

The use of a Fourier transform in the same window will hardly be more

satisfactory. The use of a Fourier transform with window functions would be even

worse on account of signal attenuation towards the window edges. Furthermore, the

(complex) transforms will not indicate in a simple manner the fact that the signal

comprises two sequences of different frequencies at distinct instants, rather than a

uniformly distributed frequency content. We can also think of performing two

Fourier transforms on the separate intervals, but such a procedure is directly related

to the signal structure and so cannot be easily generalized.

The most direct representation thus consists of recording the signal by means of

a window function which is centered on the instant

, and then performing a short-

time Fourier transform. We are thus led to perform sliding transforms, which allow

the identification of the spectral content of signals contained in the window ) (t -

This sliding Fourier transform is expressed:

dtet

)

)()( ) , (



f



)



The transform ),(

F is a function of two variables

and

which we will

qualify as a time-frequency representation. The spectral distributions become

functions of time. However, such a continuous representation contains much

redundant information concerning the signal x(t) since knowledge of a single

transform on the time interval of the signal suffices for a reconstruction of the initial

signal.

376 Fundamentals of Fluid Mechanics and Transport Phenomena

7.3.5.2.

Practical applications

A natural solution for reducing the size of the time-frequency representation thus

consists of performing a partition of the total study interval into n temporal

segments each of a length 't in the order of the window width, and only conserving

the Fourier transforms performed at the center of the segments. We thus obtain a

succession of spectra, each of which can be attributed to a given instant.

Figure 7.12.

Spectrogram

The representation of these spectra in the form of a succession of spectra as a

function of time is known as a spectrogram (Figure 7.12). However, a spectrogram

can quickly become quite complex and difficult to read for long signals. We often

prefer a representation, known as a sonogram, where the values are shown in the

time-frequency plane where the use of color or grayscale allows the indication of the

signal level. In this way the information contained in the signal can be visualized in

terms of the evolution of the frequencies and amplitudes.

Figure 7.13.

Sonogram of the word “shah” pronounced by a person

Measurement, Representation and Analysis of Temporal Signals 377

Sonograms (often known as spectrograms) are very often used for the analysis of

complex signals; Figure 7.13 shows the sonogram of an acoustic signal

corresponding to the word “shah”. We can note that the ensemble of letters “sh”

corresponds to higher frequencies than the ensemble “ah”.

The considerations of section 7.3.4.6, relating to the time-frequency uncertainty,

here applies and the Heisenberg-Gabor inequality indicated earlier remains valid: a

precise temporal localization and an accurate analysis are incompatible. This

phenomenon is illustrated by Figure 7.14. which presents sonograms of a signal

composed of a silence of 0.05 second duration, followed by a harmonic oscillation

of frequency 4 kHz during 0.05 seconds, and then, once again a silence. The

analyses performed by means of the software COOL EDIT with windows of width

respectively equal to 0.031 and 0.002 seconds show that the first value allows a

suitable identification of the frequency, whereas the second allows a better temporal

localization of the portions of the signal. The Blackmann-Harris window used gave

better results than the other windows that were available; however, the reader

should remember that the choice of window depends on empirical considerations

([ALL 04], [JUR 08], [MAD 98], [STR 96], [WAI 90]).

Figure 7.14.

Influence of the width

of the Fourier window

on the sonogram (4,000 Hz)

The explanations which have just been provided for the time-frequency analysis

can be transposed to timescale analysis. This domain is more common in image

treatment; the interested reader should see specialized texts on this subject. The

principle of time-frequency representation can be applied in two dimensions, the

frequency or scale spectra being thus functions of two variables.

378 Fundamentals of Fluid Mechanics and Transport Phenomena

7.3.5.3.

Regarding multiple timescales

The fact that spatial or temporal variations of a phenomenon take place at two

completely different scales has important but variable consequences depending on

the mathematical form of the variations. If these are relatively uniform and non-

oscillatory, we have a singular perturbation problem (see section 6.4.3). For

oscillations of a linear system, the preceding time-frequency analysis is only

meaningful if the notion of frequency can be defined, in order to allow separate

identification of the rapid component (short oscillation period) and the slow

component (characteristic time of the amplitude and phase variation, etc. (section

7.3.4.4)).

These notions of fast and slow scales are not however so easy to define. Let us

consider the example of a signal s(t), which is the sum of two harmonic signals of

close frequencies NN G

and NN G

 

tNtNtNNtNNts GSS GSGS 2cos.2cos

2cos2cos)(

000

This expression shows that s(t) can be considered as a harmonic signal

2cos S with fast variations, whose amplitude is slowly modulated by the

function tNGS2cos . The two timescales are here clearly 1/N and 1/GN. The

amplitude modulation can be any broadband slow signal a(t) as in the transmission

of radio waves.

Frequency modulation is also a system with two timescales, with a low-

frequency carrier signal N

and a slow variation of frequency GN:



.2cos)(

ttNNts GS

The spectrum of this signal modulated with frequency tNtN

cos)((

not very different from the amplitude modulated spectrum: in addition to the two

close frequencies



and



, it contains only a small number of

weak peaks.

The amplitude spectra of amplitude and frequency modulated signals are nearly

similar, but the temporal signals and the phase spectra have very different structures.

This comparison illustrates the difficulty of reconstructing, with its detailed

characteristics, the time signal which corresponds to an amplitude or energy

spectrum: theoretically only the temporal and complex spectral representations are

completely equivalent. More generally, a signal similar to a harmonic signal of

Measurement, Representation and Analysis of Temporal Signals 379

frequency N

can be studied with Hilbert transform, amplitude and frequency

modulations being defined from its analytical signal (section 7.3.4.4).

In practice, signals used in telecommunications are modulated in amplitude or in

frequency by a low frequency signal, which contains information to be transmitted

and decoded. In mechanical or energetic systems, these modulations are the result,

either of the presence of two close frequencies, or of a slow oscillatory variation of

the physical characteristics of an oscillator (stiffness or length of a vibrating system,

for example, or interaction between structures of different frequencies in fluid

mechanics or acoustics).

We will re-encounter the problem of multiple time (and space) scales in

turbulent fluids, at the heart of which are fluctuations of central importance. An

everyday example can be found for flow in the atmosphere: the wind presents

instantaneous fluctuations with very short periods (a couple of seconds for gusts of

wind), and longer periods (from many hours to a day for perturbations (in the

meteorological sense)) which are relatively individual structures at the atmospheric

scale. These two categories of scale are generally distinct and we can thus separate

meteorological and turbulence phenomena by performing statistics for the mean

values at the scale of the turbulent fluctuations. Meteorological predictions and data

provide the mean velocity, the effect of fluctuations due to gusts being smoothed at

timescales of a fraction of an hour.

On the other hand, turbulence frequencies have a continuous spectrum and the

temporal separation of phenomena is hazardous in proportion to the degree of non-

linearity of the turbulence mechanisms. Furthermore, the corresponding flows can

nonetheless be subject to temporal variations at a slower scale due to influences

other than that of the turbulence.

7.3.5.4.

Study of intermittency

We call intermittency that property of a signal that comprises structures of the

same nature, which occur at intervals that may be more or less regular or random.

The signal is comprised of “packets” with a specific structure different from the

characteristics of the rest of the signal (Figure 7.15).

(

)

Figure 7.15.

Example of intermittent signal

380 Fundamentals of Fluid Mechanics and Transport Phenomena

Intermittency is encountered in many flows; for example, Figure 7.16 shows that

a fixed observer at the horizontal level N will perform measurements characterized

by intermittency for liquid packets in a water jet (a), the external turbulent zones of

a boundary layer (b), or ascending currents in clouds which are carried by the wind

(c).

U U

(a) (b) (c)

Figure 7.16.

Intermittency phenomena in: (a) a liquid jet; (b) in the frontier zone of a

turbulent boundary layer; (c) in atmospheric flow with cumulus

Figure 7.17 shows the sonogram of a sound recording of a water flow that issues

from a floater tap in which a rapid liquid jet bursts in an air pocket which is trapped

at the top of a vertical pipe before filling a reservoir. The impact of the liquid

packets on the walls is particularly loud and is translated by dark regions on the

sonogram. When the air pocket is purged, all that remains is the sound produced by

the turbulence in the water which is much less noisy, and the sonogram is thus a

relatively uniform shade of gray.

Figure 7.17.

Sonogram of the noise of a flow of water with

air pockets downstream a tap

Measurement, Representation and Analysis of Temporal Signals 381

In general, the emission of sound in a flow is always associated with unsteady

structures. This is a frequent nuisance, easily identified by an attentive ear, that we

seek to eliminate. The analysis of flow noise with the signal processing techniques

discussed above, and in particular time-frequency analysis, often allows the

identification and improvement of noisy regions of a flow. These procedures merit

broader use, but they require relatively complete knowledge in the domains of fluid

mechanics and acoustics.

7.3.6. Discretized signals

7.3.6.1. Evolution of techniques

While physical quantities are represented by real or complex numbers with

continuous values, the results of experiments are truncated decimal approximations,

which are subject to the uncertainties of the experiment. Since the earliest scientific

developments, it is the experimenter himself who discretizes the values of the

measurements he performs. In the same way numerical applications of the

equations, which result from exact or empirical theories, can only be used with

discretized, truncated numerical values. The introduction of computer technology

has completely modified both experimental techniques and the practice of system

modeling.

The evolution of experimental and measurement methods has been marked by

two types of devices, depending on the nature of the electronic treatment:

– analog devices in which the input signal (voltage or electric current) is a

continuous function of time which is then transformed by electronic circuits into an

electrical output signal, which is itself a continuous function of time; in these

devices, mathematical discontinuities don’t really exist. These measurement devices

generally have a relative accuracy associated with the scale chosen for the

measurement;

– digital devices, which use information technology, and operate on signals

constituted of a finite sequence of truncated numerical values (which are encoded in

a number of “bits” or “bytes”). Here, once again, digitization errors (quantification)

are in direct relation with the maximum value chosen for the representation of

numbers (12 or 16 bits for example).

The interest of digital techniques lies in the ease of adaptation and the near

infinite possibilities: any modification in the treatment of a signal is performed

through the modification of a computer program, on the contrary, the modification

of an electric analog circuit can only be achieved through modifying the physical

properties of the components, some of which may be variable by construction

anyway.

382 Fundamentals of Fluid Mechanics and Transport Phenomena

The techniques used in practice are often mixed for different reasons:

– as the initial signals are often continuous, it is necessary to construct a table of

numerical values which we wish to retain by sampling (this is the job of the analog-

to-digital converter);

– the signals which result from a treatment are often destined to be used to act on

analog materials which require a certain power (power amplifiers, actuators,

loudspeakers, etc.); the discontinuities of the numerical values must therefore be

interpolated by means of a digital-to-analog converter which delivers a continuous

electrical signal.

7.3.6.2.

Sampling of continuous signals

The representation of continuous signals by tables of numerical values is only

useful if the initial signal can be reconstructed exactly without any loss of

information. We replace the continuous signal by a sequence of values that are

generally recorded at regular intervals. It is clear that the signal should not have

varied too much between two successive values, such that its reconstruction in a

continuous form can be performed with suitable precision.

The temporal representation of a signal s(t) amounts to its decomposition into a

basis of Dirac functions according to the following property:

 

WWGW

dtftf 

f

f

)(

Sampling a signal s(t) comes down to multiplying it by the sampling function

)(

III

(also known as Dirac comb or impulse train) defined by the sum of Dirac

impulses positioned at the points kT (k integer):



integer,)(

kt-kTkT

f

f

III

We thus have:



() (),integer

skT  tkTstdtk

d

d

d





The sampling of temporal signals is a complex operation ([BAH 01], [BEL 00],

[CAS 06], [HIG 93], [MAD 98], [MAX 96]).