Wavelet transform frequency

Best time-frequency representations are performed when the Wavelet Transform is continuous, that is when both parameters a... 

Fig. 2.9. Left transient reflectivity change of Bi at various pump densities. Right discrete wavelet transformation spectra obtained for time delay of 0.3 ps (solid line) and 3.0 ps (dotted line). Inset in right panel shows the Aig frequency as a function of the time delay. The dashed line in inset represents the equilibrium frequency. From [36]...

C. Space-Scale Analysis Based on Continuous Wavelet Transform Low-Frequency Rhythms in Human DNA Sequences... 

Although the Fourier compression method can be effective for reducing data into frequency components, it cannot effectively handle situations where the dominant frequency components vary as a function of position in the spectrum. For example, in Fourier transform near-infrared (FTNIR) spectroscopy, where wavenumber (cm-1) is used as the x-axis, the bandwidths of the combination bands at the lower wavenumbers can be much smaller than the bandwidths of the overtone bands at the higher wavenumbers.31,32 In any such case where relevant spectral information can exist at different frequencies for different positions, it can be advantageous to use a compression technique that compresses based on frequency but still preserves some position information. The Wavelet transform is one such technique.33... 

The use of multiple sample injections (up to a maximum of three) was found to enhance S/N, i.e., the S/N is slightly higher than the square root of the number of injected sample plugs. In addition, multipoint detections of a two-component sample [700,701] or four-component sample [699] were also achieved, but the separation resolution was not as good as that obtained from the conventional single-point detection [701]. Besides Fourier transform, wavelet transform was also used in multipoint fluorescent detection to retain some time information in addition to the frequency information [702]. 

A critical difference between the Fourier transform defined in Equation 10.9 and the wavelet transform defined in Equation 10.22 is the fact that the latter permits localization in both frequency and time that is, we can use the equation to determine what frequencies are active at a specific time interval in a sequence. However, we cannot get exact frequency information and exact time information simultaneously because of the Heisenberg uncertainty principle, a theorem that says that for a given signal, the variance of the signal in the time domain a2, and the variance of the signal in the frequency (e.g., Fourier) domain c2p are related... 

Just as the discrete Fourier transform generates discrete frequencies from sampled data, the discrete wavelet transform (often abbreviated as DWT) uses a discrete sequence of scales aj for j < 0 with a = 21/v, where v is an integer, called the number of voices in the octave. The wavelet support — where the wavelet function is nonzero — is assumed to be -/<72, /<72. For a signal of size N and I < aJ < NIK, a discrete wavelet / is defined by sampling the scale at a] and time (for scale 1) at its integer values, that is... 

The Morlet wavelet can be understood to be a linear bandpass filter, centred at frequency m = coo/a with a width of /(aoa). Some Morlet wavelets and their Fourier spectra are illustrated in Fig. 4.4.4. The translation parameter b has been chosen for the wavelet to be centred at time f = 0 (top). With increasing dilatation parameter a the wavelet covers larger durations in time (top), and the centre frequency of the filter and the filter bandwidths become smaller (bottom). Thus depending on the dilatation parameter different widths of the spectrum are preserved in the wavelet transform while other signals in other spectral regions are suppressed. 

In recent years, wavelet transform (WT) has been developed as a novel, to use the term somewhat loosely, extension of the traditional Fourier transform (FT) as a means of capturing transitions in the frequency content... 

By compressing this function in time, Morlet was able to obtain a higher frequency resolution and spread it out to obtain a lower frequency resolution. To localize time, he shifted these waves in time. He called his transform the wavelets of constant shape and today, after a substantial number of studies in its properties, the transform is simply referred to as the Wavelet transform. The Morlet wavelet is defined by two parameters the amount of compression, called the scale, and the location in time. 

S Qian. Introduction to Time-Frequency and Wavelet Transforms. Prentice-Hall, Upper Saddle River, NJ, 2002. 

Wavelet Transform is mathematical method to linear operation that decomposes a function into a continuous spectrum of its frequency components. Wavelet basis functions are localized in space and frequency. 

Wavelet-Transformed Descriptor is a transformation of a descriptor into the frequency space to enhance or suppress characteristic features of a molecule. 

Wavelets and the wavelet transformation refer the representation of a spectral data set in terms of a finite spectral range or a rapidly decaying oscillating waveform. This waveform is scaled and translated to match the original spectmm. Wavelet transformation may be considered to calculate the time-frequency representation, related to the subject of harmonic analysis. The projection of a spectmm on a single wavelet or a series of wavelets reduces the dimensionality of the data set. Wavelet transforms are broadly divided into three classes, the continuous wavelet transform, the discrete wavelet transform and multiresolution-based wavelet transforms. Each class has advantages and disadvantages in terms of the wanted information. 

Given a time series s t), its wavelet transformation Wgs b, a) at time b and scale a (scale refers to 1 /frequency) with respect to the chosen wavelet g t) is given as... 

R. Carmona, W.-L. Hwang, and B Torresani. Practical Time-Frequency Analysis. Gabor and Wavelet Transforms with an Implementation in S. Acar demic Press, 1998. 

When combining the power spectra of the first difference operator and the moving average over two, we see that the two spectra have a constant sum. This corresponds with what we already knew, the information passed by the two operators is complementary. The moving average over 2 and the first difference operator form a special pair of an LP filter and an FIP filter that divide the frequency domain in two, right in the middle of the domain. This type of filter pair plays an important role in the discrete wavelet transform. 

When Frequencies Change in Time Towards the Wavelet Transform... 

The wavelet basis functions are derived by translating and dilating one basic wavelet, called a mother wavelet. The dilated and translated wavelet basis functions are called children wavelets. The wavelet coefficients are the coefficients in the expansion of the wavelet basis functions. The wavelet transform is the procedure for computing the wavelet coefficients. The wavelet coefficients convey information about the weight that a wavelet basis function contributes to the function. Since the wavelet basis functions are localised and have varying scale, the wavelet coefficients therefore provide information about the frequency-like behaviour of the function. 

Fig. 6 indicates that by translating and dilating the mother wavelet, localised information about high and low frequency events can be obtained. It should be mentioned that we use the term frequency loosely when talking about wavelet transforms, since it is not really frequency that we are describing but rather low and high scale events. 

Fig. 6 The continuous wavelet transform of f(tj obtains localised frequency information of the function for varying constant time intervals.

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