By K. K. Shukla, Arvind K. Tiwari (auth.)
Due to its inherent time-scale locality features, the discrete wavelet remodel (DWT) has got substantial awareness in signal/image processing. Wavelet transforms have first-class strength compaction features and will offer ideal reconstruction. The moving (translation) and scaling (dilation) are particular to wavelets. Orthogonality of wavelets with appreciate to dilations ends up in multigrid illustration. because the computation of DWT comprises filtering, a good filtering method is key in DWT implementation. within the multistage DWT, coefficients are calculated recursively, and also to the wavelet decomposition level, more space is needed to shop the intermediate coefficients. therefore, the final functionality relies considerably at the precision of the intermediate DWT coefficients. This paintings provides new implementation thoughts of DWT, which are effective by way of computation, garage, and with larger signal-to-noise ratio within the reconstructed signal.
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Additional info for Efficient Algorithms for Discrete Wavelet Transform: With Applications to Denoising and Fuzzy Inference Systems
Example text
This special way of obtaining a high pass from a low pass, introduced as quadrature conjugate filter (QCF) [128], has the following properties: hgh ½n; gh ½n À 2ki ¼ dk ð2:4Þ that is, the impulse response is orthogonal to its even shifts and hgg ½n; gh ½n À 2ki ¼ 0 ð2:5Þ È É or the impulse response gg ½n; gh ½n and their even shifts are mutually orthogÈ É onal. Further, gg ½n À 2k; gh ½n À 2l k; l 2 Z is an orthonormal basis for L2(Z), the space of square summable sequences. Thus, any sequence from L2(Z) can be written as follows: X X x½n ¼ ak gg ½n À 2k þ bl gh ½n À 2l ð2:6Þ k2Z l2Z where ak ¼ gg ½n À 2k; x½n and bl ¼ hgh ½n À 2l; x½ni, k and l [ Z.
In filter bank system, FCQ can result in deeper consequences. For example a QMF bank loses the alias free or PR property due to multiplier quantization. 4 demonstrates effect of coefficient quantization on pole zero and impulse response plot of a FIR filter (order 12) respectively. Selected quantizer widths are 6 and 12 bits. It is clear from the plots that quantizing the filter coefficients with 6-bit, has seriously degraded its response. With 12 bit quantizer the filter performance is very much close to that of original filter.
The filter bank has a regular structure; it is easily implemented by repeated application of identical cells Fig. 1 [106]. It is computationally efficient. The decomposition of real-time Fig. 1 Basic computational cell of (a) the DWT, (b) the inverse DWT (a) DWT Cell (b) DWT Cell 40 3 Finite Precision Error Modeling and Analysis signal into wavelet coefficients involves FIR filtering and its reconstruction involves IIR filtering. The wavelet filtering by high pass filter (h[n]) and decimation provides the wavelet coefficients at the considered octave.