Wavelets+and+Filter+Banks.ppt

Inverse filtering Y(z)=H(z)X(z) Problem: if H(z) is not invertible, how to get Y? Ill-posed problem: most modern signal or image processing problems are ill-posed. Mathematically, can not be solved. Physically, we must and can solve them. Another path: filter bank. 注：物理可解，就一定有数学解 Filter bank （滤波器组）=Lowpass + Highpass (互补), Simplest idea: H0 and H1, where H0 is a lowpass filter, and H1 is a highpass filter, the lost information in the process of lowpass filtering can be fund in the output of the highpass filter. Some problems: How to reconstruct the signal? How to find such filter bank? How to reduce the computation? How to reduce the cost (storage and/or hardware)? Any more properties beside reconstruction? Delay （延时、延迟） Sx(n)=x(n-1) Advance（提前） S-1x(n)=x(n+1) SS-1=S-1S=I, where I represent the unit operator. Time-invariant filters: H is a linear filter, if H(Sx)=S(Hx): a shift of the input produces a shift of the output. Ideal filters, （理想滤波器） Ideal lowpass: Ideal Highpass: For ideal lowpass fitler: But in practice, we must use finite filter for computation, the first idea is to use finite part of this filter, it leads to Gibbs phenomenon。 -11k11 -21k21 -41k41 -81k81 -201k201 Overshot: 0.089490…. --Gibbs, J. Willard, Fourier Series. Nature 59, 200 (1898) and 606 (1899). Traditional filter design methods: Firstly, we note that we only need to construct lowpass filter, and shift in phase by π we can get corresponding highpass filters. Window method（窗口法）: h(n)=hI(n)w(n) Hamming window: Hanning window: Kaiser window: Equiripple method（等纹波方法）: The filter with the smallest maximum error in passband and stopband is an equiripple filter. Means the ripples in passband and stopband is equal height. Remez exchange algorithm Weighted least squares（加权最小二乘） (eigenfilters（特征滤波器）) This method is to minimize the function: 纹波减弱，或者控制在特定形式下，但永在。 Heisenberg’s Uncertainty Principle: Define two window width:time and frequency: Then: if ||f||=1, we have If f is the