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ARTICLE IN PRESS YDSPR:642

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Digital Signal Processing ••• (••••) •••–•••

www.elsevier.com/locate/dsp

A basis for efﬁcient representation of the S-transform

R.G. Stockwell

Northwest Research Associates, Colorado Research Associates Division, 3380 Mitchell Lane, Boulder, CO 80301, USA

Abstract

The S-transform is a time-frequency representation known for its local spectral phase properties. A key feature of the S-transform

is that it uniquely combines a frequency dependent resolution of the time-frequency space and absolutely referenced local phase in-

formation. This allows one to deﬁne the meaning of phase in a local spectrum setting, and results in many desirable characteristics.

One drawback to the S-transform is the redundant representation of the time-frequency space and the consumption of computing

resources this requires (a characteristic it shares with the continuous wavelet transform, the short time Fourier transform, and Co-

hen’s class of generalized time-frequency distributions). The cost of this redundancy is ampliﬁed in multidimensional applications

such as image analysis. A more efﬁcient representation is introduced here as a orthogonal set of basis functions that localizes the

spectrum and retains the advantageous phase properties of the S-transform. These basis functions are deﬁned to have phase char-

acteristics that are directly related to the phase of the Fourier transform spectrum, and are both compact in frequency and localized

in time. Distinct from a wavelet approach, this approach allows one to directly collapse the orthogonal local spectral representation

over time to the complex-valued Fourier transform spectrum. Because it maintains the phase properties of the S-transform, one

can perform localized cross spectral analysis to measure phase shifts between each of multiple components of two time series as a

function of both time and frequency. In addition, one can deﬁne a generalized instantaneous frequency (IF) applicable to broadband

nonstationary signals. This is the ﬁrst time a channel IF has been integrated in an orthogonal local spectral representation. A direct

comparison between these basis functions and complex wavelets is performed, highlighting the advantages of this approach. The

relationship between this basis set and the fully redundant S-transform is demonstrated highlighting the ability to arbitrarily sample

the time-frequency space. The introduction of this basis set leads to efﬁcient analysis routines that may ﬁnd use in a wide range of

ﬁelds.

Keywords: S-transform; Time-frequency analysis; Wavelets

1. Introduction

In observations of the ocean and atmosphere, one ﬁnds that the environment is full of waves that are obviously not

inﬁnite in extent nor constant over time. For instance, acoustic waves, internal waves, tides and tidal harmonics, inertial

oscillations, surface and other interface waves, solitons and nonlinear wave events exhibit changing characteristics.

Many of these waves, however, have an obvious oscillatory manner, which have lead over the years to many attempts

to analyze the dynamic spectrum or local spectral nature of the observations.

One popular method is the short time Fourier transform [1] and the related Gabor transform [2]. A closely related

method is complex demodulation [3] which produces a series of band pass ﬁltered voices and is also related to the

E-mail address: stockwell@co-ra.com.

doi:10.1016/j.dsp.2006.04.006

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2 R.G. Stockwell / Digital Signal Processing ••• (••••) •••–•••

ﬁlter bank theory of wavelets. Another family of time-frequency representations is the Cohen class [4] of generalized

time-frequency distributions (GTFD). Because of the quadratic nature of this class, the presence of cross terms can

cause difﬁculty in the interpretation. Some particular choices for kernel can help to alleviate these problems. Speciﬁc

GTFDs include the Wigner distribution, the Cone–Kernel distribution [5], the Choi–Williams distribution [6] as well

as the smoothed pseudo Wigner distribution (PWD) [7] which reduce the effect of cross terms, but in doing so fail to

satisfy the marginal properties of the Wigner distribution.

One of the more popular attempts to localize the spectrum has been with the continuous wavelet transform [8]. The

basic philosophy is that any time series can be decomposed into a self-similar series of dilations and translations of a

mother wavelet. Strictly, the continuous wavelet transform gives a representation in the state space of dilation a and

translation b, but with the appropriate choice of wavelet, it can be used to measure the power spectrum locally.

In the effort to ﬁnd an orthonormal and yet local set of basis functions, the discrete orthonormal wavelet trans-

form (DWT) has been discovered by Daubechies [9]. Orthonormal functions are those which are both orthogonal and

normalized (for basis functions u and v, u · v = 0, and u · u = v · v = 1). The DWT provides a picture of a time

series that gives very good time resolution, and some insight into the spectral components present. However, the in-

formation is sparse and the spectral regions are not unique (different dilations have overlapping spectral responses).

The DWT using Daubechies wavelets is a powerful tool useful in many applications (e.g., image compression, etc.),

however, a clearer time-frequency representation is desired. There have been many good review papers in the litera-

ture by Daubechies [9], Young [10], Rioul and Vetterli [11], Hlawatsch and Boudreaux–Bartels [7], Cohen [4], and

Qian [12].

The role of phase in wavelet analysis is not as well understood as it is for the Fourier transform, especially for

orthonormal wavelet representations. Current complex wavelets such as complex Daubechies wavelets [13], dual

tree complex wavelets [14–16], and complex Shannon wavelets [17] do not have a direct relationship to the Fourier

transform. Since “complex Daubechies wavelets are not analytic, there is not a direct equivalence between the Fourier

and wavelets as to the meaning and the role of the phase,” Lina [18].

A more recent time-frequency representation, the S-transform [19], has found application in a range of ﬁelds. It

is similar to a continuous wavelet transform in having progressive resolution but unlike the wavelet transform the

S-transform retains absolutely referenced phase information. Absolutely referenced phase information means that the

phase information given by the S-transform refers to the argument of the sinusoid at zero time (which is the same

meaning of phase given by the Fourier transform). The S-transform deﬁnes what local phase means in an intuitive

way, at a peak in local spectral amplitude (indicating a quasimonochromatic signal), as well as off peak, where the

rate of change of the phase leads to a channel instantaneous frequency analysis. The S-transform not only estimates

the local power spectrum, but also the local phase spectrum. It is also applicable to the general complex valued time

series.

One drawback to the S-transform is the size of its redundant representation of the time-frequency plane. It is

apparent that a more efﬁcient representation of the S-transform is needed, one that provides a framework on which

reduced sampling can be laid. This is the objective of this manuscript.

2. The S-transform

The continuous S-transform [19] of a function h(t) is

S(τ,f) =

∞



−∞

h(t)

|f |

√

2π

−(τ −t)

−i2πf t

dt. (1)

A voice S(τ,f

) is deﬁned as a one-dimensional function of time for a constant frequency f

, which shows how the

amplitude and phase for this exact frequency changes over time.

In the discrete case, there are computational advantages to using the equivalent frequency domain deﬁnition of the

S-transform (where H [

] is the Fourier transform of the N-point time series h[kT ])



jT,



N−1



m=0



m + n



−2π

i2πmj/N

,n= 0, (2)

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R.G. Stockwell / Digital Signal Processing ••• (••••) •••–••• 3

and for the n = 0 voice, it is equal to the constant deﬁned as

S[jT,0]=

N−1



m=0

h[mT ], (3)

where j , m, and n = 0, 1,...,N − 1. The sampling of the S-transform is such that S[jT,n/NT] has a point at each

time sample and at each Fourier frequency sample. Similar to a STFT and CWT, this is redundant. The discrete inverse

of the S-transform is found by averaging over time to get the Fourier transform spectrum, and inverting to the time

domain

h[kT ]=

N−1



n=0



N−1



j=0



jT,





i2πnk/N

. (4)

The utility of S-transform analysis of a time series has been demonstrated in a range of geophysical ﬁelds such as:

(i) analyzing internal atmospheric wave packets [20,21], (ii) atmospheric studies [22–26], (iii) characterization of seis-

mic signals [27–30], (iv) global sea surface temperature analysis [31,32], as well as: (v) electrical engineering [33,34],

(vi) mechanical engineering [35,36], and (vii) digital signal processing [37–40]. The S-transform has also been used

in the medical ﬁeld, in (viii) human brain mapping [41], (ix) cardiovascular studies [42], (x) MRI analysis [43], and

(xi) the physiological effects of drugs [44].

2.1. Why use the S-transform approach?

What distinguishes the S-transform from the many time-frequency representations available is that the S-transform

uniquely combines progressive resolution with absolutely referenced phase information. Daubechies has stated that

progressive resolution gives a fundamentally more sound time-frequency representation [9]. Absolutely referenced

phase means that the phase information given by the S-transform is always referenced to time t = 0, which is also true

for the phase given by the Fourier transform. This is true for each S-transform sample of the time-frequency space.

This is in contrast to a wavelet approach, where the phase of the wavelet transform is relative to the center (in time) of

the analyzing wavelet. Thus as the wavelet translates, the reference point of the phase translates. This is called “locally

referenced phase” to distinguish it from the phase properties of the S-transform.

From one point of view, local spectral analysis is a generalization of the global Fourier spectrum. In fact, since no

record of observations is inﬁnite, all discrete spectral analysis ever performed on measured data have been local (i.e.,

restricted to the time of observation). Thus there must be a direct relationship between a local spectral representation

and the global Fourier spectrum. This philosophy can be stated as the fundamental principle of S-transform analysis:

The time average of the local spectral representation should result identically in the complex-valued global Fourier

spectrum.

This leads to phase values of the local spectrum that are obvious and meaningful. While there can be several important

deﬁnitions of “proper phase,” a natural one would be as follows. Consider a signal h(t ) = A exp(i2πf

t + φ).The

Fourier transform spectrum of this signal at the frequency f

would return the amplitude of A and the phase constant φ.

In order to carry this understanding of phase into the realm of wavelet theory, time-frequency representations and

local spectra, it would require a transform that would return a voice (function of time) for the frequency f

that had a

constant amplitude A and a constant phase φ. Such is the case with the S-transform, and thus it can be described as a

generalization of the Fourier transform to the case of nonstationary signals.

In summary the S-transform has the following unique properties. It uniquely combines frequency dependent reso-

lution with absolutely reference phase, and therefore the time average of the S-transform equals the Fourier spectrum.

It simultaneously estimates the local amplitude spectrum and the local phase spectrum, whereas a wavelet approach is

only capable of probing the local amplitude/power spectrum. It independently probes the positive frequency spectrum

and the negative frequency spectrum, whereas many wavelet approaches are incapable of being applied to a complex

time series. It is sampled at the discrete Fourier transform frequencies unlike the CWT where the sampling is arbi-

trary. Because of the absolutely referenced phase of the S-transform, it is possible to deﬁne a channel instantaneous

frequency function for each voice [19]. It is possible to perform a localized cross spectral analysis as shown in [20,21]

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4 R.G. Stockwell / Digital Signal Processing ••• (••••) •••–•••

where transient wavelike motions were detected in the time series measured by spaced receivers (physically separated

by a known distance). By analyzing phase shifts in the S-transforms of the two time series, the phase speed of the

wavelike motion was deduced. Such a straightforward analysis is not possible with a wavelet type approach.

The goal of this paper is to deﬁne a discrete orthonormal S-transform (DOST) representation that encompasses

these properties.

3. The basis functions of the discrete orthonormal S-transform (DOST)

There are several reasons to desire an orthonormal time-frequency version of the S-transform. An orthonormal

transformation takes an N-point time series to an N -point time-frequency representation, thus achieving the max-

imum efﬁciency of representation. Also, each point of the result is linearly independent from any other point. The

transformation matrix (taking the time series to the DOST representation) is orthogonal, meaning that the inverse

matrix is equal to the complex conjugate transpose. By being an orthonormal transformation, the vector norm is pre-

served. Thus a Parseval theorem applies, stating that the norm of the time series equals the norm of the DOST. An

orthonormal transform is referred to as an energy preserving transform.

The efﬁcient representation of the S-transform can be deﬁned as the inner products between a time series h[kT ]

and the basis functions deﬁned as a function of [kT ], with the parameters ν (a frequency variable indicative of the

center of a frequency band and analogous to the “voice” of the wavelet transform), β (indicating the width of the

frequency band), and τ (a time variable indicating the time localization).



h[kT ]



= S



τT,



k=N−1



k=0

h[kT ]S

[ν,β,τ]

[kT ]. (5)

These basis functions S

[ν,β,τ]

[kT ] for the general case are deﬁned as

[ν,β,τ]

[kT ]=

−iπτ

√

−i2π(k/N−τ/β)(ν−β/2−1/2)

−e

−i2π(k/N−τ/β)(ν+β/2−1/2)

}

2sin[π(k/N − τ/β)]

. (6)

At this point, the sampling of the time-frequency space has not yet been determined. Rules must be applied to the

sampling of the time-frequency space to ensure orthogonality. These rules are as follows:

• Rule 1. τ = 0, 1,...,β −1.

• Rule 2. ν and β must be selected such that each Fourier frequency sample is used once and only once.

Implicit in this deﬁnition is the phase correction of the S-transform that distinguishes it from the wavelet or ﬁlter

bank approach. Here the parameters ν, β, τ are integers deﬁned such that the functions do form a basis. For each voice,

there are one or more local time samples (τ ), this number being equal to β (see Rule 1) thus the wider the frequency

resolution (large β), the more samples in time (large τ ). This can be seen as a consequence of the uncertainty principle.

Examples and methods for determining these parameters are described below.

Distinct from a wavelet function, these basis functions have no vanishing moments (in fact the functions are nor-

malized to unit area). These basis functions are not translations of a single function, and they are not self-similar.

3.1. Orthonormal basis functions with octave sampling

In order to compare the DOST with orthonormal wavelet transforms and with the S-transform, an octave sampling

of the time-frequency domain is illustrated. This has the property of progressive resolution that both the S-transform

and wavelet transforms share. Octave sampling implies that the voice bandwidth doubles for each increasing voice (as

sampling allows).

By imposing speciﬁc rules on the basis functions (here octave sampling) it implies a strict deﬁnition for ν and β.

By introducing a new variable p which corresponds to the octave number p = 0, 1, 2,...,log

(N) −1 one can deﬁne

all the parameters (ν, β, τ ) of equation 6 in terms of p as follows:

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R.G. Stockwell / Digital Signal Processing ••• (••••) •••–••• 5

Fig. 1. Selected DOST basis functions of increasing time with octave sampling for N =100. Note that the DOST functions are not translates of a

single function. Each plot is a basis function with a different time index τ for the same voice frequency. Note that the time index parameter τ is

scaled by β/N. Thus τ = 10 corresponds to a shift of t =τN/β = 10 ×100/16 = 62.5 which is where the center of that basis function (f ) occurs.

The solid line is the real component, the dotted line is the imaginary part, and the light grey trace is the amplitude.

For p>1, we have:

p =2,...,log

(N) −1, (7)

ν = 2

(p−1)

(p−2)

, (8)

β = 2

(p−1)

, (9)

τ =0, 1,...,2

(p−1)

−1. (10)

For the case p = 0, then ν = 0, β = 1, and τ = 0. Also when p = 1, then ν =1, β =1, and τ = 0.

Thus the DOST basis functions for octave sampling of a time series is given as follows (by application of (7)–(10)

into (6)):

[p,τ]

[kT ]=

−iπτ

√

(p−1)/2

−i2π(k/N−τ/2

p−1

)(2

p−1

−1/2)

−e

−i2π(k/N−τ/2

p−1

)(2

−1/2)

}

2sin[π(k/N − τ/2

p−1

)]

. (11)

The application of the DOST with octave sampling for the case where N = 100 is illustrated in Figs. 1 and 2. In

Fig. 1, octave sampled DOST basis functions are shown for a speciﬁc DOST voice (p = 5 and therefore ν = 24 and

bandwidth β = 16), for six different time samples (τ = 5,...,10). These functions are not simple translations, as is

most evident in a comparison of Figs. 1d and 1f.

Figure 2 also shows six octave sampled DOST basis functions, but now the order p varies (as does voice frequency

ν and the bandwidth β). The time sample τ is chosen simply to center the function. Again the lack of the self-similar

property is evident. No basis function shown in Fig. 2 is a dilation of any other basis function.

While octave partitioning has the familiar frequency dependent resolution seen in the S-transform and wavelets,

the formalism deﬁned in Eq. (6) allows for any arbitrary sampling of the time-frequency space. Often the resolution

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