what is the difference between db1, db2,db3,... in wavelet analysis?

[alison]

Hi everyone,
Please help me to find the meaning of 'difference between db1,db2,db3,db.. in wavelet analysis of 1D discrete signal'.
With thanks,
aye

[Wayne King]

"alison " <ayem...@e.ntu.edu.sg> wrote in message <i2hq98$t5a$1...@fred.mathworks.com>...

Hi Aye, db refers to a particular family of wavelets. They are technically speaking called the Daubechies extremal phase wavelets. The number refers to the number of vanishing moments. Basically, the higher the number of vanishing moments, the smoother the wavelet (and longer the wavelet filter). The length of the wavelet (and scaling) filter is two times that number, so

[LoD,HiD] = wfilters('db2');
length(LoD) %answer is 4=2*2
[LoD,HiD] = wfilters('db3');
length(LoD) %answer is 6=2*3

and so on.

The theoretical significance of this is that if your signal of interest is exhibiting behavior on an interval consistent with a polynomial of degree at most N and you are using a wavelet with N vanishing moments, the wavelet coefficients will be zero on that interval. The wavelet with N vanishing moments is orthogonal to polynomials of degree at most N. So a 'db1' will return wavelet coefficients of zero on an interval if the signal is a polynomial of degree at most 1 on that interval.

For example:

t = linspace(0,1,512);
dwtmode('per');
y = t+1; % polynomial of degree 1
[C1,L1] = wavedec(y,2,'db1');
d1 = detcoef(C1,L1,1);
plot(d1) %detail coefficients are all approximately zero
y1 = t.^2+1; % polynomial of degree 2
[C2,L2] = wavedec(y1,2,'db1');
d1 = detcoef(C2,L2,1);
plot(d1) % detail coefficients are not all zero

Hope that helps,
Wayne