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Introduction

1.1 From Fourier transform to wavelet transform

1.1.1 Fourier transform [1]

Information in the time domain and frequency domain is the basic characteristic of the description of a signal x(t). Information in the time domain is easy to observe, whereas information in the frequency domain is not observable unless the signal transforms. Fourier first proposed the method to get frequency domain information when he was researching an equation of heat conduction. Moreover, he suggested transforming the equation of heat conduction from the time domain to the frequency domain, which is the famous Fourier transform concept. The definition of Fourier transform of a continuous signal x(t) is

Fourier transform established the relation between the time domain and frequency domain of a signal.

With the development of computer technology, all the computing problems in science and engineering now relate to computers inextricably. A typical feature of computer computing is discretization. Fourier transform, defined in Equation (1.1), is essentially integral computation, which reflects continuous characteristics. Meanwhile, discrete sampling in applications obtains signals. Equation (1.1) needs to be discretized with high efficiency and high accuracy in order to sample information through discretization and compute Fourier transform effectively using computers. Thus, one must derive the definition of discrete Fourier transform (DFT).

Let x(t) be a limited signal over the interval [-p,p], so Fourier transform of x(t) can be simplified as

Moreover, let the signal be equidistant sampled. The sampling number is N, the input signal in the time domain is xk, and the output signal required in the frequency domain is Xk. To get more accurate output Xk of Equation (1.1) using sampling points of input xk, DFT is a polynomial of best approximation S(t) fitting from x(t), according to xk, and having S(t) instead of x(t) in Equation (1.1) to get Xk. The following paragraph briefly discusses the solution of S(t) and Xk.

Using a given group of orthogonal basis () to verify the inner product relations of the vectors: , in which IN is an identity matrix for N dimensions, and .

Let , and use orthogonal basis {Fk}, to solve the least-square problem:

Solution of Equation (1.3):

Use definition of S(t) and coefficient ck from Equation (1.4) to approach Xk.

Its DFT is a translation by coefficient ck of the polynomial S(t) of x(t):

Except for 2p, DFT is defined by Equation (1.5), where the input xn and output Xl are the time domain information and frequency domain information of the signal, respectively.

1.1.2 Short-time Fourier transform [1, 2]

Although Fourier transform and DFT have been applied to signal processing, especially time-frequency analysis, extensively, the Fourier integral cannot be localized in both the time and frequency neighborhoods at one time. For example, according to Equation (1.1), Fourier transform cannot analyze the time evolution of such spectral components and the signal cycle. That is to say, Fourier transform cannot localize in the frequency neighborhoods. Actually, the signal is transformed from the time domain to frequency domain by Fourier transform, where was added at all points in time and was the limitation of frequency. Thus, Fourier transform is not a survey of frequency domain information of a signal in a period. In contrast, for signal processing, especially nonstationary signal processing (voice, seismic signal, etc.), local frequency of signal and its period should be realized. Because standard Fourier transform has the ability of local analysis in the frequency domain rather than time domain, Dennis Gabor proposed short-time Fourier transform (STFT) in 1946. The basic idea behind STFT is dividing the signal into many time intervals to analyze each of them and determine their frequencies. Figure 1.1 is a sketch of signal analysis using Fourier transform.

Figure 1.1 Diagram of short-time Fourier transform: (a) window in time domain, (b) time-frequency plane division

We assume that we are only interested in the frequency of x(t) neighboring t = t, which is the value of Equation (1.1) in a certain period It. We have

With |It| representing the length of the period It, we definite the square wave function gt(t) as

So Equation (1.6) can be written as

with i representing the whole real axis. According to Equations (1.1), (1.7), and (1.8), to analyze the local frequency domain information at time t, Equation (1.6) is the windowed function gt(t) of function x(t). Obviously, the smaller the length |It| of the window is, the more easily the function could reflect the local frequency domain information of signals.

The definition of STFT is given in Reference [2].

For a given signal , STFT can be written as

with

and

The windowed function g(t) should be symmetrical. The meaning of STFT is as follows. Add a window function g(t) to x(t). The time variable of x(t) and g(t) changes from t to t. The windowed signal transforms to get the Fourier transform of the signal at time t. A moving time t, which means a moving central position of the windowed function, can derive the Fourier transform at different times. The set of these Fourier transforms is STFTx(t, ?), as shown in Figure 1.2. STFTx(t, ?) is a two-dimensional function of variables (t, ?).

Figure 1.2 Diagram of STFT

The windowed function g(t) is finite supported in the time domain, and ei?t is a line spectrum in the frequency domain. Thus, the basis function of STFT is finite supported in both the time and frequency domains. In this way, the inner product of Equation (1.9) can realize the function of time-frequency locating for x(t). Fourier transform on both sides of Equation (1.10):

in which ? is the equivalent frequency variable the same as ?.

Thus,

The equation indicates that windowed x(t) in the time domain (which is ) is equal to windowed X(?) in the frequency domain (which is ).

STFT solved the problem that standard Fourier transform has the ability of local analysis only in the frequency domain rather than the time domain. However, STFT is itself flawed because a fixed windowed function whose form and shape don't change determine the resolution of STFT. To change the resolution is to choose a different windowed function. STFT is effective when you use it to analyze segment-wise stationary signals or approximate-stationary signals. But nonstationary signals, which change dramatically, demand high time resolution. When the signal is relatively flat, such as a low-frequency signal, it demands a windowed function with high-frequency resolution. STFT cannot balance the demand of both frequency resolution and time resolution.

1.1.3 Time-frequency analysis and wavelet transform

In view of the analysis in Section 1.1.2, Fourier transform can reveal a signal's frequency domain feature and the energy feature of a stationary signal. That is to say, Fourier transform is an overall transformation of signal, which is completely in the time domain or frequency domain. Therefore, it cannot reveal the law of the time-varying signal spectrum. For a nonstationary signal, because of large variations in a spectrum, analysis method is demanded to reflect local time-varying spectrum features of signal instead of simply those in the time domain or frequency domain. To make up the shortcoming of Fourier transform, a novel method that can realize time-frequency localization for signals is essential. Consequently, the time-frequency analysis was proposed, which can represent signals in the time and frequency domain at the same time.

During the development of Fourier transform theory, people realized its shortcomings (mentioned in this chapter). Therefore, in 1946, Gabor proposed representing one signal with both a time axis and frequency axis. The Gabor expansion of signal x(t) is [3-5]:

The windowed function...