Wavelets...

Hello Forum,

I have just started studying wavelets and wavelet transform. 
While the STFT breaks the signal down into equal time duration segments,
the wavelet transform uses wavelets that when they have high frequency
(compressed) give great time resolution, i.e. short time window. 
But a wavelet with a low frequency implies a stretched up wavelet and a
long time window...

It seems that wavelets can do great frequency and time resolution at high
frequencies and modest time resolution at lower frequencies...
Is that correct?

Wavelets are orthogonal to each other. could you remind me the benefit of
decomposing a signal in orthogonal basis functions aside from the fact that
each basis function gives a zero inner product with another basis function
of different frequency?
the purpose of all transforms is to find the complex coefficients to give
the basis functions for the best reconstruction of the signal...

In Fourier theory the basis functions have complex coefficients containing
amplitude and phase....do wavelets have phase information? Or just the
translation coefficient?
thanks,
Brett

Am 27.06.2012 18:16, schrieb brettcooper:

> It seems that wavelets can do great frequency and time resolution at high
> frequencies and modest time resolution at lower frequencies...
> Is that correct?

The "uncertainty" relation still holds true for wavelets, so you may ask 
about the support of the wavelet functions in time or frequency domain. 
Compared to a Fourier basis - where the basis functions have all the 
same support in the frequency domain, wavelets have a relatively small 
domain for lower frequencies (thus better resolution) and a larger 
domain (lower resolution) for higher frequencies. This implies that time 
resolution is just the inverse of that.

> Wavelets are orthogonal to each other.

No, not necessarily. Wavelets can be both orthogonal and non-orthogonal. 
Actually, the latter are the more interesting since non-orthgonal 
wavelets may have more interesting symmetries than orthogonal ones.

> could you remind me the benefit of
> decomposing a signal in orthogonal basis functions aside from the fact that
> each basis function gives a zero inner product with another basis function
> of different frequency?

That is an interesting property because it implies that you can 
implement a wavelet analysis by means of an inner product with the 
synthesis functions. For non-orthogonal filters, the relation between 
analysis and synthesis filters is not so simple.

> In Fourier theory the basis functions have complex coefficients containing
> amplitude and phase....do wavelets have phase information? Or just the
> translation coefficient?

I don't quite understand how one could define "phase" without having a 
planar wave and thus a Fourier decomposition, thus I don't quite see how 
this concept carries over directly. Still, if you have complex signals, 
you could define the "phase" in the same way, namely by representing the 
wavelet coefficients (from the wavelet analysis) as r * exp(i\phi), 
where \phi would be the phase. But it is not exactly the same type of 
"phase shift = transition in time domain" as you know it for Fourier.

Greetings,
	Thomas

Thanks Thomas.

I just read this online: "....Short Term Fourier Transform (STFT) -
provides both time and frequency information, but resolves all frequencies
equally...."

I get that.

"...Wavelet transform - provides good time resolution and poor frequency
resolution at high frequencies and good frequency resolution and poor time
resolution at low frequencies. Useful approach when signal at hand has high
frequency components for short duration and low frequency components for
long duration as in ECG....."

So a compressed, high frequency wavelets automatically have a shorter
duration, hence a good time resolution... but why do they have poor
frequency resolution?  Vice versa for the low frequency wavelets...

Poor frequency resolution means that we need wavelets of more frequencies
to suitably describe a certain event in the signal. Is it because short
duration events are usually more complicated in shape than long duration
events (smooth changes)?

for example, a chirp signal with increasing frequency: I feel that the
wavelet transform should work equally well at low and high frequencies...

Also, in communication theory, a carrier is FM modulated and demodulated to
get the instantaneous frequency (Which is related to the transmitted
message). Do they use wavelets to extract the instantaneous frequency out
of the modulated signal? If not, how do they do it?

Brett

On Thursday, June 28, 2012 9:14:01 AM UTC-5, brettcooper wrote:
> Thanks Thomas.
> 
> I just read this online: "....Short Term Fourier Transform (STFT) -
> provides both time and frequency information, but resolves all frequencies
> equally...."
> 
> I get that.
> 
> "...Wavelet transform - provides good time resolution and poor frequency
> resolution at high frequencies and good frequency resolution and poor time
> resolution at low frequencies. Useful approach when signal at hand has high
> frequency components for short duration and low frequency components for
> long duration as in ECG....."
> 
> So a compressed, high frequency wavelets automatically have a shorter
> duration, hence a good time resolution... but why do they have poor
> frequency resolution?  Vice versa for the low frequency wavelets...
> 
> Poor frequency resolution means that we need wavelets of more frequencies
> to suitably describe a certain event in the signal. Is it because short
> duration events are usually more complicated in shape than long duration
> events (smooth changes)?
> 
> for example, a chirp signal with increasing frequency: I feel that the
> wavelet transform should work equally well at low and high frequencies...
> 
> Also, in communication theory, a carrier is FM modulated and demodulated to
> get the instantaneous frequency (Which is related to the transmitted
> message). Do they use wavelets to extract the instantaneous frequency out
> of the modulated signal? If not, how do they do it?
> 
> Brett

Go to Google.com type in TIME FREQUENCY UNCERTAINITY PRINCIPAL

Am 28.06.2012 16:14, schrieb brettcooper:
> Thanks Thomas.

> "...Wavelet transform - provides good time resolution and poor frequency
> resolution at high frequencies and good frequency resolution and poor time
> resolution at low frequencies. Useful approach when signal at hand has high
> frequency components for short duration and low frequency components for
> long duration as in ECG....."
>
> So a compressed, high frequency wavelets automatically have a shorter
> duration, hence a good time resolution... but why do they have poor
> frequency resolution?  Vice versa for the low frequency wavelets...

Because you have a fundamental inequality between time resolution and 
frequency resolution that forbids you to have both at the same time. 
Actually, it is the same inequality in physics where the "frequency 
space" is also the "momentum space" of quantum particles, and the "time 
space" is the "position space" of the particle, where the Fourier 
transform transforms between the two representations. It is there (in 
physics) called the "uncertainty relation" because it implies that you 
cannot measure both the position *and* the momentum of a quantum 
particle sharply.

The same relation holds in signal processing as well, simply because the 
mathematical transformation is the same:

To define a frequency sharply, you need an infinitely extended "planar 
wave", i.e. f(x) = exp(i k x), which has a fixed 'k', but is obviously 
extended all over the space. Vice versa, if you want a sharp position in 
the time domain, the delta "function" (actually, distribution) does 
that. Its image in the Fourier domain, after Fourier transformation, is 
of course the plane wave again, \hat{f}(k) = exp(-i k x) where x is now 
the parameter where the "peak" is and k the argument of the function 
(the other way around than above). It is hence infinitely extended in 
'k' (frequency, momentum) space, but located in the time domain.

Wavelets are compromises between the two extreme, neither localized in 
time nor in the frequency domain.

> Poor frequency resolution means that we need wavelets of more frequencies
> to suitably describe a certain event in the signal.

How many basis functions you need is entirely up to the signal you want 
to describe. One cannot say, in general, you need "more" or "less". It 
is the nature of the signal that tells you. If the signal is a slightly 
distorted plane wave, a Fourier representation would be more suitable.

> Is it because short
> duration events are usually more complicated in shape than long duration
> events (smooth changes)?


Signals of short duration have a more complex spectrum, i.e. have a less 
sharp frequency distribution. Shorter signals = "more overtones", and 
"purer signals" require longer samplings, i.e. are not localized in time 
domain.

> for example, a chirp signal with increasing frequency: I feel that the
> wavelet transform should work equally well at low and high frequencies...

That cannot be answered in general. For such a signal, i.e. a slow 
variation in the time domain, I would believe that a Fourier description 
is more adequate. I don't think one can compute the spectrum of such a 
signal explicitly, but I haven't tried either. An approximation would be 
feasible.

> Also, in communication theory, a carrier is FM modulated and demodulated to
> get the instantaneous frequency (Which is related to the transmitted
> message). Do they use wavelets to extract the instantaneous frequency out
> of the modulated signal? If not, how do they do it?

Sorry, I don't think I got this question. Wavelets are not a tool for FM 
modulation, at least I wouldn't know how to apply them to this problem 
without thinking more carefully about it. Wavelets are, just to give one 
example, a good model to describe natural images. JPEG 2000 uses for 
example a wavelet decomposition to describe the image data. It is here a 
good model since the frequency resolution of wavelets gets lower for 
high frequencies and higher at low frequencies, which is very much how 
the eye-brain visual system behaves. So it is a reasonably good model of 
how we "see" the world.

So long,
	Thomas

>Thanks Thomas.
>
>I just read this online: "....Short Term Fourier Transform (STFT) -
>provides both time and frequency information, but resolves all
frequencies
>equally...."
>
>I get that.
>
>"...Wavelet transform - provides good time resolution and poor frequency
>resolution at high frequencies and good frequency resolution and poor
time
>resolution at low frequencies. Useful approach when signal at hand has
high
>frequency components for short duration and low frequency components for
>long duration as in ECG....."
>
>So a compressed, high frequency wavelets automatically have a shorter
>duration, hence a good time resolution... but why do they have poor
>frequency resolution?  Vice versa for the low frequency wavelets...
>
>Poor frequency resolution means that we need wavelets of more frequencies
>to suitably describe a certain event in the signal. Is it because short
>duration events are usually more complicated in shape than long duration
>events (smooth changes)?
>
>for example, a chirp signal with increasing frequency: I feel that the
>wavelet transform should work equally well at low and high frequencies...
>
>Also, in communication theory, a carrier is FM modulated and demodulated
to
>get the instantaneous frequency (Which is related to the transmitted
>message). Do they use wavelets to extract the instantaneous frequency out
>of the modulated signal? If not, how do they do it?
>
>Brett
>

http://en.wikipedia.org/wiki/Wavelet_packet_decomposition

Thank you again....

i guess I got confused with the fact that a FM modulated signal, with the
concept of instantaneous frequency, can have a specific single frequency at
every instant in time (that is its instantaneous freq.) 
But the Fourier transform of a FM modulated signal clearly shows a wide
bandwidth....

The concept of frequency has multiple definitions. for instance, a pure
sinusoid with a Gaussian envelope forms a finite time durations signal with
a nonzero frequency bandwidth...
But the interzero distance in this signal is constant, i.e. the
instantaneous frequency is one and constant while the Fourier frequency is
not just one but a whole bandwidth...

Brett