Single-level discrete 1-D wavelet transform
dwt command performs a single-level one-dimensional wavelet decomposition with respect to either a particular wavelet ('
for more information) or particular wavelet decomposition filters (
Hi_D) that you specify.
[cA,cD] = dwt(X,'
) computes the approximation coefficients vector
cA and detail coefficients vector
cD, obtained by a wavelet decomposition of the vector
X. The string '
wname' contains the wavelet name.
[cA,cD] = dwt(X,Lo_D,Hi_D) computes the wavelet decomposition as above, given these filters as input:
Hi_D must be the same length.
lx = the length of
lf = the length of the filters
length(cA) = length(cD) = la where
la = ceil(lx/2), if the DWT extension mode is set to periodization. For the other extension modes,
For more information about the different Discrete Wavelet Transform extension modes, see
[cA,cD] = dwt(...,'
,MODE) computes the wavelet decomposition with the extension mode
MODE that you specify.
MODE is a string containing the desired extension mode.
% The current extension mode is zero-padding (see
dwtmode). % Construct elementary original one-dimensional signal. randn('seed',531316785) s = 2 + kron(ones(1,8),[1 -1]) + ... ((1:16).^2)/32 + 0.2*randn(1,16); % Perform single-level discrete wavelet transform of s by haar. [ca1,cd1] = dwt(s,'haar'); subplot(311); plot(s); title('Original signal'); subplot(323); plot(ca1); title('Approx. coef. for haar'); subplot(324); plot(cd1); title('Detail coef. for haar'); % For a given wavelet, compute the two associated decomposition % filters and compute approximation and detail coefficients % using directly the filters. [Lo_D,Hi_D] = wfilters('haar','d'); [ca1,cd1] = dwt(s,Lo_D,Hi_D); % Perform single-level discrete wavelet transform of s by db2 % and observe edge effects for last coefficients. % These extra coefficients are only used to ensure exact % global reconstruction. [ca2,cd2] = dwt(s,'db2'); subplot(325); plot(ca2); title('Approx. coef. for db2'); subplot(326); plot(cd2); title('Detail coef. for db2'); % Editing some graphical properties, % the following figure is generated.
Starting from a signal s, two sets of coefficients are computed: approximation coefficients CA1, and detail coefficients CD1. These vectors are obtained by convolving s with the low-pass filter Lo_D for approximation and with the high-pass filter Hi_D for detail, followed by dyadic decimation.
More precisely, the first step is
The length of each filter is equal to 2N. If n =
length(s), the signals F and G are of length n + 2N - 1, and then the coefficients CA1 and CD1 are of length
To deal with signal-end effects involved by a convolution-based algorithm, a global variable managed by
dwtmode is used. This variable defines the kind of signal extension mode used. The possible options include zero-padding (used in the previous example) and symmetric extension, which is the default mode.
dwtmode, idwt, wavedec, waveinfo
Daubechies, I. (1992), Ten lectures on wavelets, CBMS-NSF conference series in applied mathematics. SIAM Ed.
Mallat, S. (1989), "A theory for multiresolution signal decomposition: the wavelet representation," IEEE Pattern Anal. and Machine Intell., vol. 11,
no. 7, pp. 674-693.
Meyer, Y. (1990), Ondelettes et opérateurs, Tome 1, Hermann Ed. (English translation: Wavelets and operators, Cambridge Univ. Press. 1993.)