Daubechies wavelet transform
Looks like you need a general explanation of the discrete wavelet transform (DWT). DWT breaks a signal down into subbands distributed evenly in a logarithmic frequency scale, each subband sampled at a rate proportional to the frequencies in that band. The traditional Fourier transformation has no time domain resolution at all, or when done using many short windows on a longer data, equal resolution at all frequencies. The distribution of samples in the time and frequency domain by DWT is of form:

log f
 |XXXXXXXXXXXXXXXX  X = a sample
 |X X X X X X X X   f = frequency
 |X   X   X   X     t = time
 |X       X
 |X
 |X                
  ----------------t


Single subband decomposition and reconstruction:

    -&gt; high -&gt; decimate -------------&gt; dilute -&gt; high
   |   pass    by 2      high subband  by 2      pass \
in |                                                   + out
   |                                                  /  =in
    -&gt; low  -&gt; decimate -------------&gt; dilute -&gt; low
       pass    by 2      low subband   by 2      pass


This creates two subbands from the input signal, both sampled at half
the original frequency. The filters approximate halfband finite impulse response (FIR) filters
and are determined by the choice of wavelet. Using Daubechies wavelets
(and most others), the data can be reconstructed to the exact original
even when the halfband filters are not perfect. Note that in the above scheme, the total amount of information (samples) stays the same throughout.

Decimation by 2: ABCDEFGHIJKLMNOPQR -&gt; ACEGIKMOQ
  Dilution by 2: ACEGIKMOQ -&gt; A0C0E0G0I0K0M0O0Q0


To get the logarithmic resolution in frequency, the low subband is
re-transformed, and again, the low subband from this transformation
gets the same treatment etc.

Decomposition:

    -&gt; high -&gt; decimate --------------------------------&gt; subband0
   |   pass    by 2
in |                 -&gt; high -&gt; decimate ---------------&gt; subband1
   |                |   pass    by 2
    -&gt; low -&gt; decim |                 -&gt; high -&gt; decim -&gt; subband2
       pass   by 2  |                |   pass    by 2
                     -&gt; low -&gt; decim |
                        pass   by 2  |   .   down to what suffices
                                      -&gt; .    or if periodic data,
                                         .     until short of data


Reconstruction:

subband0 -----------------------------------&gt; dilute -&gt; high
                                              by 2      pass \
subband1 ------------------&gt; dilute -&gt; high                   + out
                             by 2      pass \                /  =in
subband2 -&gt; dilute -&gt; high                   + dilute -&gt; low
            by 2      pass \                /  by 2      pass
                            + dilute -&gt; low
Start   .                  /  by 2      pass
here!   . -&gt; dilute -&gt; low
        .    by 2      pass


In a real-time application, the filters introduce delays, so you need
to compensate them by adding additional delays to less-delayed higher
bands, to get the summation work as intended.

For periodic signals or windowed operation, this problem doesn't exist -
a single subband transformation is a matrix multiplication, with wrapping
implemented in the matrix:

Decomposition:

|L0|   |C0  C1  C2  C3                | |I0|     L = lowpass output
|H0|   |C3 -C2  C1 -C0                | |I1|     H = highpass output
|L1|   |        C0  C1  C2  C3        | |I2|     I = input
|H1| = |        C3 -C2  C1 -C0        | |I3|     C = coefficients
|L2|   |                C0  C1  C2  C3| |I4|
|H2|   |                C3 -C2  C1 -C0| |I5|
|L3|   |C2  C3                  C0  C1| |I6|
|H3|   |C1 -C0                  C3 -C2| |I7|     Daubechies 4-coef:

     1+sqrt(3)        3+sqrt(3)        3-sqrt(3)        1-sqrt(3)
C0 = ---------   C1 = ---------   C2 = ---------   C3 = ---------
     4 sqrt(2)        4 sqrt(2)        4 sqrt(2)        4 sqrt(2)


Reconstruction:

|I0|   |C0  C3                  C2  C1| |L0|
|I1|   |C1 -C2                  C3 -C0| |H0|
|I2|   |C2  C1  C0  C3                | |L1|
|I3| = |C3 -C0  C1 -C2                | |H1|
|I4|   |        C2  C1  C0  C3        | |L2|
|I5|   |        C3 -C0  C1 -C2        | |H2|
|I6|   |                C2  C1  C0  C3| |L3|
|I7|   |                C3 -C0  C1 -C2| |H3|


C0, C1, C2, C3 are the "db2" lowpass FIR filter coefficients. Highpass
coefficients you get by reversing tap order and multiplying by
sequence 1,-1, 1,-1, ... Because these are orthogonal wavelets, the
analysis and reconstruction coefficients are the same.

A coefficient set convolved by its reverse is an ideal halfband lowpass
filter multiplied by a symmetric windowing function. This creates the
kind of symmetry in the frequency domain that enables aliasing-free
reconstruction. Daubechies wavelets are the minimum-phase, minimum
number of taps solutions for a number of vanishing moments (seven in
"db7" etc.), which determines their frequency selectivity.

I was asked to show the matrices for 6 coefficients, so here they are, made a bit larger for clarity but could be the same size as before too. Decomposition:

|L0|   |C0  C1  C2  C3  C4  C5                | |I0|     
|H0|   |C5 -C4  C3 -C2  C1 -C0                | |I1|     
|L1|   |        C0  C1  C2  C3  C4  C5        | |I2|     
|H1|   |        C5 -C4  C3 -C2  C1 -C0        | |I3|     
|L2| = |                C0  C1  C2  C3  C4  C5| |I4|
|H2|   |                C5 -C4  C3 -C2  C1 -C0| |I5|
|L3|   |C4  C5                  C0  C1  C2  C3| |I6|
|H3|   |C1 -C0                  C5 -C4  C3 -C2| |I7|
|L4|   |C2  C3  C4  C5                  C0  C1| |I8|
|H4|   |C3 -C2  C1 -C0                  C5 -C4| |I9|     


Reconstruction:

|I0|   |C0  C5                  C4  C1  C2  C3| |L0|
|I1|   |C1 -C4                  C5 -C0  C3 -C2| |H0|
|I2|   |C2  C3  C0  C5                  C4  C1| |L1|
|I3|   |C3 -C2  C1 -C4                  C5 -C0| |H1|
|I4| = |C4  C1  C2  C3  C0  C5                | |L2|
|I5|   |C5 -C0  C3 -C2  C1 -C4                | |H2|
|I6|   |        C4  C1  C2  C3  C0  C5        | |L3|
|I7|   |        C5 -C0  C3 -C2  C1 -C4        | |H3|
|I8|   |                C4  C1  C2  C3  C0  C5| |L4|
|I9|   |                C5 -C0  C3 -C2  C1 -C4| |H4|


With:

C0 = 3.326705529500826159985115891390056300129233992450683597084705e-01
C1 = 8.068915093110925764944936040887134905192973949948236181650920e-01
C2 = 4.598775021184915700951519421476167208081101774314923066433867e-01
C3 = -1.350110200102545886963899066993744805622198452237811919756862e-01
C4 = -8.544127388202666169281916918177331153619763898808662976351748e-02
C5 = 3.522629188570953660274066471551002932775838791743161039893406e-02


More coefficient sets can be found here.