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) ?

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

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.

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.

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.

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.

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.

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