## 1D- Discrete Wavelet Transform

### 1D- Discrete Wavelet Transform

When I decompose a signal into 7 levels, can I say that the a7
sub-band signal best reflects the original signal, since according to
the wavelet theory, the approximation signal at level n is the
aggregation of the approximation at level n-1 plus the detail at
level n-1 (MRA algorithm) ?

### 1D- Discrete Wavelet Transform

You mean:
approx at n = approx at n+1 + detail at n+1
(ie not n-1, but n+1)

a7 "best reflects the original signal" makes no sense to me.
a7 + sum of details 1 to 7 = original signal

### 1D- Discrete Wavelet Transform

Greetings.

Thanks NZ. Agreed on what you said. But one more question.
People say wavelet provides time-frequency information about a
signal. In 1D-DWT, I notice that the frequency information is at the
sub-bands, but what about the time information?

Regards,
Mike.

### 1D- Discrete Wavelet Transform

Each detail is a function of time. Its length is the same as the
original signal.
Each detail represents the frequency content of the band between the
detail before and the detail after.
But I think it's easier to think in terms of "timescale", not
frequency, where timescale = 1/frequency.
The timescales double with each detail.

In my experience with many real datasets, the timescale of the 1st
detail is 3*dt, where dt is the interval between data, though the
textbooks say it is 2*dt. I've come to this by doing zero-crossing
analyses on details from all sorts of data. Thus, in practice, I've
found the timescales go 3, 6, 12, 24, 48, 96, ...*dt, never mind what
the mathematicians say.

One other point about time information.
If you take the variance of each detail and plot it against timescale,
you will get a spectrum of sorts. It's not the same as a Fourier
spectrum because the intervals between data are not equal in
frequency, so there's a problem of how to scale the thing. However,
if you sum the variances, you'll get the variance of the original
signal, just as with taking the area under a Fourier spectrum. Now,
when you take the variance of the details, you're removing the time
dependence, just as a Fourier spectrum does. And that's the
difference. Wavelet details have their energy distributed with time.

### 1D- Discrete Wavelet Transform

Thanks NZ, for the details.

Speaking of variance, I know that if the signal mean is zero or
negligible, the variance = average power of the discrete signal,
which is
N-1
P = 1/N sum x^2
n=0
N-1
and the energy is E = sum x^2
n=0

What is the unit for this average power? I read a few articles, they
said we normally dont give unit to this power and energy. Is it so ?

By the way, before I do the 1D-DWT, I analyse my signal in time
domain, where I found out that the signal total energy = sum (x^2)/fs
(unit = V^2.sec). This signal energy in time domain, what has it got
to do with the above-mentioned energy in wavelet tranformation ?

Hoping you can clear my doubt.

Thanks very much...

Regards,
Mike.