# Number of Daubechies coefficients

I am wondering about the correlation between input size and number of coefficients given by a discrete wavelet transform.

I am using Daubechies wavelets to describe a 1D function and I'm using PyWavelets to implement it (which is analogous to the MATLAB toolbox).

I started by implementing it using Haar wavelets, which gave correct results and I understand exactly how it works. Let's say my input function has 16 datapoints, if I use Haar, what I get from a multilevel decomposition (wavedec) is something like this (the number of shifts in brackets):

V1, W1, W2, W3, W4


This is all well and good. The V1 gives me the scaling function and the W1-W5 wavelets of different scale and dilation. My problem is when I use the next Daubechies (referred to as 'db2' in the toolbox, which is called the D4), and I get

V1, W1, W2


I lose all my intuition. I have no idea where 6, 6 and 9 come from, and they change depending on the level I specify (not even sure what it means to specify a level) and of course the input size. How can I predict the number of coefficients, and what are some good resources for gaining better understanding of why?

Thanks!

EDIT: Clarification on V and W:

$V_n$ usually denotes the span of a certain scaling function, $\phi$, i.e. $\{\phi_{n,k}\}$, where $k$ is the shift and $n$ the scaling. $W_n$ is the same except for the wavelet function, $\psi$. I might have abused the notation a bit by referring to the vectors of coefficients by V and W though.

EDIT2: Code

Here is the MATLAB code to produce the above:

>> [C, L] = wavedec(1:16, 4, 'db1'); L
L =
1     1     2     4     8    16
>> [C, L] = wavedec(1:16, 2, 'db2'); L
L =
6     6     9    16


I actually used PyWavelets, where it looked like this:

>>> import pywt
>>> map(len, pywt.wavedec(range(16), 'db1')) # defaults to level = 4
[1, 1, 2, 4, 8]
>>> map(len, pywt.wavedec(range(16), 'db2')) # defaults to level = 2
[6, 6, 9]

• What are V and W? Jul 3, 2012 at 13:38
• @Phonon I added a clarification in the question. Jul 3, 2012 at 14:26
• So by V1, W1, W2 you mean that you get a scaling function of length 6, and two wavelet functions of lengths 6 and 9? Or are these the numbers of coefficients of the different level of your transformed signal? MATLAB code to get these would be very useful as well. Jul 3, 2012 at 14:36
• @Phonon The latter. Check the code I added. Thanks! Jul 3, 2012 at 15:24
• Yeah, this is helpful. I'll dig around in MATLAB docs. Jul 3, 2012 at 15:35

According to MATLAB documentation on wavedec,

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 the coefficients $cA_1$ and $cD_1$ are of length

$$\text{floor}\left( \frac{n-1}{2} \right) + N$$

Here, $n=16$ is the length of your signal, and $N=2$ is the Daubechies number.

Putting all that together, your detail coefficients at the second level should be of length

$$\text{floor}\left( \frac{16-1}{2} \right) + 2 = 7+2 = 9.$$

At the second level, you coefficients should be of length

$$\text{floor}\left( \frac{9-1}{2} \right) + 2 = 4+2 = 6.$$

If you're wondering why this must be the case, imagine the filtering-decimation procedure. Scaling and wavelet function for $\text{db}m$ wavelets are of length $2m$. When you convolve length $n$ signal with length $2m$ signal, you get back a signal of length $l_0=n+2m-1$. If you take every second sample of this resulting signal, you get back something of length $l_1 = \frac{n+2m-1}{2} = \frac{l_0}{2}$. Of course, if $l_0$ is odd ($n$ is even), then we cannot split the signal exactly into two parts. With clever mathematics that I will not go into here, you can show that this last non-paired coefficient is redundant anyway (carries no information that we don't yet know), so you can just omit it. Therefore, we will always have the resulting decimated signal of length

$$l_1 = \text{floor}\left( \frac{l_0}{2} \right) = \text{floor}\left( \frac{n+2N-1}{2} \right) = \text{floor}\left( \frac{n-1}{2} \right)+N.$$

• Exactly the kind of derivation I was looking for. Thanks a lot! Jul 3, 2012 at 16:59
• Can you please explain that what are F and G signals?
– Weam
Feb 23, 2016 at 18:55