A.-H. NAJMI AND J. SADOWSKY APPLIED RESEARCH The Continuous Wavelet Transform and Variable Resolution Time–Frequency Analysis Amir-Homayoon Najmi and John Sadowsky Wavelet transforms have recently emerged as a mathematical tool for multiresolution decomposition of signals. They have potential applications in many areas of signal processing that require variable time–frequency localization. The continuous wavelet transform is presented here, and its frequency resolution is derived analytically and shown to depend exclusively on one parameter that should be carefully selected in constructing a variable resolution time–frequency distribution for a given signal. Several examples of application to synthetic and real data are shown. (Keywords: Continuous wavelets, Time–frequency analysis, Signal processing.) TIME–FREQUENCY DECOMPOSITION OF SIGNALS AND IMPLEMENTATION OF MORLET WAVELET TRANSFORM ∞ ⌠ In most signal processing applications, we are inter- Sf,(ttp )= stwt ( ) ( ± )exp(±2 iftdt ) . ⌡ (2) ested in constructing a transformation that represents ±∞ signal features simultaneously in time t and frequency f. Standard Fourier analysis decomposes signals into Gabor actually used a Gaussian window, but in general frequency components but does not provide a time any window function can be used. The transformation history of when the frequencies actually occur. When can be thought of as an expansion in terms of basis the frequency content of a signal is time-varying, the functions, which are generated by modulation, and by Fourier transform S(f) of a signal s(t), translation of the window w(t), where f and t are the modulation and translation parameters, respectively. The main problem with the Gabor transform is that ∞ ⌠ the fixed-duration window function is accompanied by Sf( )= st ( )exp(±2p iftdt ) , ⌡ (1) a fixed frequency resolution. Thus, this transform allows ±∞ only a fixed time–frequency resolution. Furthermore, let us define the time width and the frequency width is incapable of capturing any local time variations and of the window function by st and sf, respectively: so would not be suitable for the analysis of nonstation- ary signals. A partial solution to this problem was pro- ∞22 ∞22 1 ∫tw() tdt ∫fW() fdf vided by Gabor, who in 1946 described the short-time ss2=±∞ , 2=±∞ , t∞ f∞ (3) Fourier transform in which a fixed-duration window ∫wtdt2() ∫Wfdf2() over the time function extracts all the frequency con- ±∞ ±∞ tent in that time interval. Denoting the window func- tion by w(t) and its midpoint position by t, the Gabor where W(f ) is the Fourier transform of the window transform is given by function. It is well known (the uncertainty principle) 134 JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 18, NUMBER 1 (1997) CONTINUOUS WAVELET TRANSFORM –1 that stsf $ (2p) , with the minimum achieved for a ∞ 1⌠ tb± Gaussian window function. Increasing t simply amounts Wa,b()=st ()c dt , (4) to translating the window (also known as the mother a⌡±∞ a function) in time while its spread is kept fixed. Sim- ilarly, as the modulation parameter f increases, the transform translates in frequency, retaining a constant where a > 0. Mother wavelet functions of interest are width. Thus, the resolution cells in the time–frequency bandpass filters that are oscillatory in the time domain. Thus, for large values of a, the basis function becomes plane have dimensions st and sf, which are fixed for all t and f. Figure 1 shows the constant-resolution cells a stretched version of the mother wavelet, i.e., a low- for the short-time Fourier transform in which a sliding frequency function, whereas for small values of a, the time window is centered at integral multiples of t and basis function is a contracted version of the mother the transforms are evaluated at bin frequencies centered wavelet, which is a short-duration, high-frequency function. Parameter b defines a translation of the wave- at integral multiples of f0. The wavelet transform, on the other hand, is based let and provides for time localization. 2 on a set of basis functions formed by dilation (as op- The transformation is invertible if and only if the posed to modulation) and translation of a prototype following admissibility condition holds: mother function c(t). The dilation of the mother func- tion produces short-duration, high-frequency and long- ∞ ⌠ 2 duration, low-frequency functions. These basis func- C()f c ≡ df < ∞ , tions are clearly better suited for representing short c (5) ⌡ f bursts of high-frequency or long-duration slowly vary- 0 ing signals. Mathematically, if C(f) is the Fourier trans- form of the mother function c(t), then the dilated (and which implies that the DC component C(0) must van- normalized) function ()1/atac() / will have aafC() ish. Thus, c(t) is a bandpass signal that should decay as its Fourier transform, where a is the scale parameter. sufficiently fast to provide good time resolution. The Thus, a contraction in time results in an expansion in Parseval relation for the wavelet transform is frequency and vice versa. This procedure of dilating and translating is analogous to constant Q filters in which ∞∞ ∞ the ratio of the root-mean-square bandwidth to center ⌠ ⌠ 2db da ⌠ 2 frequency of all dilated functions is a constant; i.e., each Wab(,)=cstdt () . (6) ⌡ ⌡ 2 c⌡ dilated wavelet will have a spread in the frequency ab=0 =∞±±a ∞ domain equal to (sf/a) and a center frequency of (f0/ a), which will have a constant ratio of (sf/f0) for all the The orthonormal wavelet transform preserves energy dilated functions. between the different scales, which are parametrized by The continuous wavelet transform of a signal s(t) is a, in the sense that then defined by ∞ ∞ 2 ⌠ 2 ⌠ = 1 tb± sf cc()tdt dt . (7) ⌡ ∞ a ± ⌡±∞ a 4t st The Morlet wavelet3 is a good example of a mother function for the construction of the continuous wavelet 3t transform.4 It is defined by 2t 2 222t Time = cpps(tiftf ) exp(±220 ) ± exp() ±0 exp ± . (8) 2s2 t Its Fourier transform is 0 0 f 2 f 0 0 C(fff )=22ps222{ exp[] ± p s ( ± )2 Frequency 0 (9) Figure 1. Resolution cells for the fixed-duration, time-window, ± exp()() ±22ps222ff exp ± ps 222 } , short-time Fourier transform. 0 JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 18, NUMBER 1 (1997) 135 A.-H. NAJMI AND J. SADOWSKY which satisfies the admissibility condition C(0) = 0. DISCRETE WAVELET TRANSFORM VS. With this choice of mother function, the continuous THE CONTINUOUS WAVELET TRANSFORM wavelet transform upon time discretization t = nDT, The signal transform computed in the article is the con- where DT is the sampling time in seconds, becomes tinuous wavelet transform (CWT), even though a discrete- time formulation is used. This formulation differs, however, from the discrete wavelet transform (DWT). In the article, time has been discretized to correspond with the sampling DDT ∞ (±)nT b2 Wab(,)= ∑sn( )exp ± of the physical signals, and the summation form follows 22 sp2 an=∞± 2as directly from the Riemann sum approximation to the inte- gral in the CWT definition. The CWT displays the distri- nTD± b (10) bution of signal amplitude and phase in two variables, time ×exp ±2ifp . 0 a and scale. The DWT, on the other hand, plays a role similar to Fourier series coefficients. One can create a set of basis functions from a mother wavelet by rescaling the wavelet We choose a number of octaves and voices within each over octaves (powers of 2) and translating the wavelet over discrete, scale-dependent time steps. The DWT computes octave. Denoting the product of the number of octaves the coefficients of a signal with respect to this basis. Thus, k/V and the number of voices by M, we use ak =2 , where in effect, it acts as a rectangular array of filters centered V is the number of voices and 1 # k # M. We further about specific times and scales (frequencies). Although discretize the translation parameter b and finally obtain these coefficients can provide an approximation to the CWT, in a way similar to the approximation of a Fourier transform using a discrete Fourier series, they should more properly be multiplied by the associated basis functions to DT ∞ (±)n l2 Wkl( , )= ∑sn ( )exp ± obtain an approximation to the CWT. The DWT is an k/M 12+(/)kM 22 important structure to study in its own right, both in the sp22 n=∞± 2 f s s theory of wavelets and in applications such as signal and f (11) image compression, subband coding, and pattern recogni- ×exp ±22ip0 ±/kM (nl ± ) , f tion. Many features of the DWT make it an attractive tool s for signal processing, such as the existence of Mallat’s linear- time algorithm to compute the DWT, which is more effi- ∞ ∞ cient than the Nlog(N) fast Fourier transform algorithm, where 1 # k # M, – # l # , and fs is the sampling and a generalization of representations in bases to represen- frequency in hertz. Although this equation represents tations in overdetermined sets of functions, called frames, a discretization of the dilation and the translation pa- in which uniqueness of representation is sacrificed in favor rameters, it is not the discrete wavelet transform.5 The of more localized influence of coefficients. latter is defined by a fixed set of coefficients that rep- resent the bandpass signal at all scales (see the boxed insert on this page). width and a time width denoted by sf and st, respec- The dilation operation for the wavelet transform tively, the dilated wavelet at scale a has a frequency also divides the time–frequency plane into resolution width and time width equal to sf/a and ast, respectively. cells, analogous to the division of the time–frequency Thus, the wavelet transform provides a variable resolu- plane by the operation of modulation for the short-time tion in the time–frequency plane, as shown in Fig.