![]() Figure 3.2(b) shows one period of a sine wave, x( t) contained within a window. It is easy to see that the convolution of the block pulse with a signal ( equation (3.14)) results in a local weighted averaging of the signal over the nonzero portion of the pulse. Figure 3.2(a) shows a simple scaling function, a block pulse, at scale index 0 and location index 0: ϕ 0,0( t) = ϕ( t)-the father function-together with two of its corresponding dilations at that location. This continuous approximation approaches x( t) at small scales, i.e. Where x m( t) is a smooth, scaling-function-dependent, version of the signal x( t) at scale index m. We will see later in this chapter how, given an initial discrete input signal, which we treat as an initial approximation to the underlying continuous signal, we can compute the wavelet transform and inverse transform discretely, quickly and without loss of signal information. We can then sum the DWT coefficients (equation (3.10a)) to get the original signal back exactly. On the other hand, for the DWT, as defined in equation (3.9), the transform integral remains continuous but is determined only on a discretized grid of a scales and b locations. How close an approximation to the original signal is recovered depends mainly on the resolution of the discretization used and, with care, usually a very good approximation can be recovered. The inverse continuous wavelet transform is also computed as a discrete approximation. a summation) computed on a discrete grid of a scales and b locations. The discretization of the continuous wavelet transform, required for its practical implementation, involves a discrete approximation of the transform integral (i.e. Using the dyadic grid wavelet of equation (3.7a), the discrete wavelet transform (DWT) can be written as:īefore continuing it is important to make clear the distinct difference between the DWT and the discretized approximations of the CWT covered in chapter 2. An orthonormal basis has component vectors which, in addition to being able to completely define the signal, are perpendicular to each other.) The discrete dyadic grid wavelet lends itself to a fast computer algorithm, as we shall see later. (A basis is a set of vectors, a combination of which can completely define the signal, x( t). Ortho-normal wavelets have frame bounds A = B = 1 and the corresponding wavelet family is an orthonormal basis. This can be seen from equation (3.8) as, when m = m′ and n = n′, the integral gives the energy of the wavelet function equal to unity. In addition to being orthogonal, orthonormal wavelets are normalized to have unit energy. This means that the information stored in a wavelet coefficient T m, n is not repeated elsewhere and allows for the complete regeneration of the original signal without redundancy. those which are translated and/or dilated versions of each other) are zero. That is to say, the product of each wavelet with all others in the same dyadic system (i.e. The nearly tight Mexican hat wavelet frame with these parameters ( a 0 = 2 1/2 and b 0 = 0.5) is shown in figure 3.1 for two consecutive scales m and m + 1 and at three consecutive locations, n = 0, 1 and 2. Thus discretizing a Mexican hat wavelet transform using these scale and location parameters results in a highly redundant representation of the signal but with very little difference between x( t) and x′( t). The closer this ratio is to unity, the tighter the frame. (This fractional discretization, v, of the power-of-two scale is known as a voice.) For example, setting a 0 = 2 1/2 and b 0 = 0.5 for the Mexican hat leads to A = 13.639 and B = 13.673 and the ratio B/A equals 1.002. It has been shown, for the case of the Mexican hat wavelet, that if we use a 0 = 2 1/ v, where v ≥ 2 and b 0 ≤ 0.5, the frame is nearly tight or ‘snug’ and for practical purposes it may be considered tight. The error becomes acceptably small for practical purposes when the ratio B/A is near unity. Where x′( t) is the reconstruction which differs from the original signal x( t) by an error which depends on the values of the frame bounds. M+1 respectively, and three consecutive b locations separated by a/2. Three consecutive locations of the Mexican hat wavelet for scale indices m (top) and m + 1 (lower) and location indices n, n + 1, n + 2. The nearly tight Mexican hat wavelet frame with aĠ = 0.5.
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