Discrete wavelet transform 2
Other forms of discrete wavelet transform include the non- or undecimated wavelet transform (where downsampling is omitted), the Newland transform (where an orthonormal basis of wavelets is formed from appropriately constructed top-hat filters in frequency space). Wavelet packet transforms are also related to the discrete wavelet transform. Complex wavelet transform is another form.
The Haar DWT illustrates the desirable properties of wavelets in general. First, it can be performed in operations; second, it captures not only a notion of the frequency content of the input, by examining it at different scales, but also temporal content, i.e. the times at which these frequencies occur. Combined, these two properties make the Fast wavelet transform (FWT) an alternative to the conventional Fast Fourier Transform (FFT). Time Issues
Due to the rate-change operators in the filter bank, the discrete WT is not time-invariant but actually very sensitive to the alignment of the signal in time. To address the time-varying problem of wavelet transforms, Mallat and Zhong proposed a new algorithm for wavelet representation of a signal, which is invariant to time shifts. According to this algorithm, which is called a TI-DWT, only the scale parameter is sampled along the dyadic sequence 2^j (j∈Z) and the wavelet transform is calculated for each point in time. Applications
The discrete wavelet transform has a huge number of applications in science, engineering, mathematics and computer science. Most notably, it is used for signal coding, to represent a discrete signal in a more redundant form, often as a preconditioning for data compression. Practical applications can also be found in signal processing of accelerations for gait analysis, in digital communications and many others.  It is shown that discrete wavelet transform (discrete in scale and shift, and continuous in time) is successfully implemented as analog filter bank in biomedical signal processing for design of low-power pacemakers and also in ultra-wideband (UWB) wireless communications.
Comparison with Fourier transform
To illustrate the differences and similarities between the discrete wavelet transform with the discrete Fourier transform, consider the DWT and DFT of the following sequence: (1,0,0,0), a unit impulse. The DFT has orthogonal basis (DFT matrix):
1 1 1 1
1 0 –1 0
0 1 0 –1
1 –1 1 –1
while the DWT with Haar wavelets for length 4 data has orthogonal basis in the rows of: 1 1 1 1
1 1 –1 –1
1 –1 0 0
0 0 1 –1
(To simplify notation, whole numbers are used, so the bases are orthogonal but not orthonormal.) Discrete wavelet transform 3
Preliminary observations include:
• Wavelets have location – the (1,1,–1,–1) wavelet corresponds to “left side” versus “right side”, while the last two wavelets have support on the left side or the right side, and one is a translation of the other. • Sinusoidal waves do not have location – they spread across the whole space – but do have phase – the second and third waves are translations of each other, corresponding to being 90° out of phase, like cosine and sine, of which these are discrete versions.
Decomposing the sequence with respect to these bases yields: The DWT demonstrates the localization: the (1,1,1,1) term gives the average signal value, the (1,1,–1,–1) places the signal in the left side of the domain, and the (1,–1,0,0) places it at the left side of the left side, and truncating at any stage yields a downsampled version of the signal:
The sinc function, showing the time domain
artifacts (undershoot and ringing) of truncating a
The DFT, by contrast, expresses the sequence by the interference of waves of various frequencies – thus truncating the series yields a low-pass filtered version of the series:
Notably, the middle approximation (2-term) differs. From the frequency domain perspective, this is a better approximation,...
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