Jerry Avins wrote:
> A global view of a long interval implies low frequency. Long delays go
> with that territory. You don't have non-causality. You simply haven't
> incorporated enough delay.

Unfortunately, I think you are right.

Damn you laws of physics! ;-) Incidentally all my schemes of an
expanding window and of a fake extension resulted in the same thing - a
fair approximation of a delayed signal.

So anyway, thanks for the help guys. I'll be back with some questions
about my upcoming project: the accelerating perpetuum mobile. ;-)

Denoir

Do-over:

Jerry Avins wrote:

> We don't need a long time to determine the presence of either component. 
> We need one to see that they are both present and to determine their 
> difference. One Hz is a low frequency, whether it is the frequency of a 
> component or the difference between two components. Resolving either 
> requires a long sample set. There's nothing about your formulation that 
> I take issue with. When discussing broad generalities, my use of "low 
> frequency" is more inclusive than yours -- maybe too inclusive.
> 
> Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Jani Huhtanen wrote:
> Jerry Avins wrote:
> 
>> Jani Huhtanen wrote:
>>> Jerry Avins wrote:
>>>
>>>> Jani Huhtanen wrote:
>>>>> Jerry Avins wrote:
>>>>>
>>>>>> lucas.denoir@gmail.com wrote:
>>>>>>
>>>>>>> ... It's really a great transform for the purpose, except for that
>>>>>>> pesky causality bit.
>>>>>> That reminds me of a Principals quip: "Running a school would be a
>>>>>> great job if it weren't for the kids."
>>>>>>
>>>>>> A global view of a long interval implies low frequency.
>>>>> I have learned that narrow bands imply long intervals. Could you
>>>>> elaborate why global view of a long interval implies low frequency?
>>>>> Perhaps I'm not entirely clear what is meant by global view here. :\
>>>> Let's suppose that a sample rate is 8 KHz, and that periods of a minute
>>>> encompass significances that would be lost by considering only shorter
>>>> intervals. That's nearly half a million samples that need to be treated
>>>> jointly.
>>> True. Is there something that implies low frequency?
>> Something whose period requires that it be represented using a minute's
>> worth of samples.
>>
> 
> Well consider sum of two sinusoids:
> 
> x[n] = sin(2*pi*f*n/Fs)) + sin(2*pi*(f+B)*n/Fs),
> 
> If B = 1 then x[n] has a period of one second. x[n] still doesn't have to
> contain any low frequencies. For example, f=Fs/4 will do.
> 
> The point is that long periods correspond to narrow bands but not
> necessarily to low frequencies.

We don't need a long time to determine the presence of either component. 
We need one to see that they are both present and to determine their 
difference. One Hz is a low frequency, whether it is the frequency of a 
component or the difference between two components. resolving either 
requires a long sample set. There's nothing about four formulation that 
I take issue with. When discussing broad generalities, my use of "low 
frequency" is more inclusive -- maybe too inclusive -- than yours.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

drick@hach.com wrote:

> lucas.denoir@gmail.com wrote:
> 
>> Apart from extending a window, I can think of another method:
>> fake-extending the signal. The basic idea being that I pad the signal
>> (symmetric extension or something like that) with the length of the
>> maximum needed delay for the original signal. I then do the transform
>> and and pick my sample from its original point in time, ignoring the
>> extended samples. I haven't tried it yet, but I suspect I'll end up
>> with the same results as with the expanding window...
> 
> Symmetric extension at the edges is a pretty common technique in
> wavelet decomposition. Waving my hands wildly, it seems to me that
> doing this nets twice the coefficient variance as for central
> coefficients that use complete input data. Proof left to the reader, of
> course.
> 
> Another approach would be to switch to asymmetric basis functions at
> the edges. That's a time-honored technique with conventional spline
> fitting. Poking through the literature on spline wavelets might yield
> something applicable.
> 
> Third idea is forget about using the coefficients that are too close to
> the boundary. If new data keeps coming in, all you have to do is wait.
> For years, folks have been using overlap techniques with FFT's to
> reduce blocking problems. What keeps you from doing the same thing with
> the DWT?
> 
> David L. Rick
> Hach Company
> 
> Note: The address in the header goes straight to the bit bucket. Actual
> humans are welcome to contact me at
> davidDOTrickAThachDOTcomREMOVE

I believe basis functions adapted to the edges are part of the so called
second generation wavelets. See Wim Swelden's papers about lifting scheme:
http://cm.bell-labs.com/who/wim/papers/papers.html
especially his paper "The lifting scheme: A construction of second
generation wavelets"

-- 
Jani Huhtanen
Tampere University of Technology, Pori

Scott Seidman wrote:

> Jani Huhtanen <jani.huhtanen@kolumbus.fi> wrote in news:H46Dg.588$n02.156
> @reader1.news.jippii.net:
> 
>> If B = 1 then x[n] has a period of one second. x[n] still doesn't have to
>> contain any low frequencies. For example, f=Fs/4 will do.
> 
> You mean Fs=4f will do.

Usually one defines the frequency of the sinusoids and not the sampling
frequency. Latter is often given as a constant.

> 
>> The point is that long periods correspond to narrow bands but not
>> necessarily to low frequencies.
>> 
> 
> By Narrow bands, you mean better resolution in the frequency domain.

Pretty much yes. Although the implication is wrong way around (in what I
said). In my earlies post the implication was right. Better spectral
resolution implies that ones has to use longer periods of signal.

> 
> If you don't have good resolution in the frequency domain, though, you
> might have trouble resolving low frequencies.  For me, this usually turns
> out to be one case where you're not trading one off over the other--to get
> good enough resolution in the frequency domain to resolve low frequencies,
> I like to collect at least 1.5-2 cycles of the lowest frequency I'm trying
> to resolve.  Practically, it seems easier to get sufficient resolution in
> the higher frequencies just by zero-padding your data, but this can really
> distort your low frequency spectrum.   So if you need high frequency
> resolution but care about low frequencies, the easiest way to go is to
> make
> sure you sample for long enough.  Alternatively, if you're not interested
> in the high frequencies, you can often filter then downsample to spread
> out your low frequency spectrum.
> 

All this is true. Only thing I was arguing was that long periods don't
necessarily imply low frequency. Low frequecies may imply long periods.

For example, one can define a dyadic wavelet decomposition which branches
always from the highpass filter. Such filter bank has a fine spectral
resolution and coarse time resolution in high frequencies. "Low"
frequencies have coarse spectral resolution but high time resolution. Here
low frequencies mean everything below Fs/4. One can synthesize a signal
which contains information in high frequencies which has to be resolved
from long periods of the signal.

This is what caused me to object that long periods imply automatically low
frequencies.

-- 
Jani Huhtanen
Tampere University of Technology, Pori

Jani Huhtanen <jani.huhtanen@kolumbus.fi> wrote in news:H46Dg.588$n02.156
@reader1.news.jippii.net:

> If B = 1 then x[n] has a period of one second. x[n] still doesn't have to
> contain any low frequencies. For example, f=Fs/4 will do.

You mean Fs=4f will do.  

> The point is that long periods correspond to narrow bands but not
> necessarily to low frequencies.
> 

By Narrow bands, you mean better resolution in the frequency domain.

If you don't have good resolution in the frequency domain, though, you 
might have trouble resolving low frequencies.  For me, this usually turns 
out to be one case where you're not trading one off over the other--to get 
good enough resolution in the frequency domain to resolve low frequencies, 
I like to collect at least 1.5-2 cycles of the lowest frequency I'm trying 
to resolve.  Practically, it seems easier to get sufficient resolution in 
the higher frequencies just by zero-padding your data, but this can really 
distort your low frequency spectrum.   So if you need high frequency 
resolution but care about low frequencies, the easiest way to go is to make 
sure you sample for long enough.  Alternatively, if you're not interested 
in the high frequencies, you can often filter then downsample to spread out 
your low frequency spectrum.


-- 
Scott
Reverse name to reply

lucas.denoir@gmail.com wrote:

> Apart from extending a window, I can think of another method:
> fake-extending the signal. The basic idea being that I pad the signal
> (symmetric extension or something like that) with the length of the
> maximum needed delay for the original signal. I then do the transform
> and and pick my sample from its original point in time, ignoring the
> extended samples. I haven't tried it yet, but I suspect I'll end up
> with the same results as with the expanding window...

Symmetric extension at the edges is a pretty common technique in
wavelet decomposition. Waving my hands wildly, it seems to me that
doing this nets twice the coefficient variance as for central
coefficients that use complete input data. Proof left to the reader, of
course.

Another approach would be to switch to asymmetric basis functions at
the edges. That's a time-honored technique with conventional spline
fitting. Poking through the literature on spline wavelets might yield
something applicable.

Third idea is forget about using the coefficients that are too close to
the boundary. If new data keeps coming in, all you have to do is wait.
For years, folks have been using overlap techniques with FFT's to
reduce blocking problems. What keeps you from doing the same thing with
the DWT?

David L. Rick
Hach Company

Note: The address in the header goes straight to the bit bucket. Actual
humans are welcome to contact me at
davidDOTrickAThachDOTcomREMOVE

Jerry Avins wrote:

> Jani Huhtanen wrote:
>> Jerry Avins wrote:
>> 
>>> Jani Huhtanen wrote:
>>>> Jerry Avins wrote:
>>>>
>>>>> lucas.denoir@gmail.com wrote:
>>>>>
>>>>>> ... It's really a great transform for the purpose, except for that
>>>>>> pesky causality bit.
>>>>> That reminds me of a Principals quip: "Running a school would be a
>>>>> great job if it weren't for the kids."
>>>>>
>>>>> A global view of a long interval implies low frequency.
>>>> I have learned that narrow bands imply long intervals. Could you
>>>> elaborate why global view of a long interval implies low frequency?
>>>> Perhaps I'm not entirely clear what is meant by global view here. :\
>>> Let's suppose that a sample rate is 8 KHz, and that periods of a minute
>>> encompass significances that would be lost by considering only shorter
>>> intervals. That's nearly half a million samples that need to be treated
>>> jointly.
>> 
>> True. Is there something that implies low frequency?
> 
> Something whose period requires that it be represented using a minute's
> worth of samples.
> 

Well consider sum of two sinusoids:

x[n] = sin(2*pi*f*n/Fs)) + sin(2*pi*(f+B)*n/Fs),

If B = 1 then x[n] has a period of one second. x[n] still doesn't have to
contain any low frequencies. For example, f=Fs/4 will do.

The point is that long periods correspond to narrow bands but not
necessarily to low frequencies.

-- 
Jani Huhtanen
Tampere University of Technology, Pori

Jani Huhtanen wrote:
> Jerry Avins wrote:
> 
>> Jani Huhtanen wrote:
>>> Jerry Avins wrote:
>>>
>>>> lucas.denoir@gmail.com wrote:
>>>>
>>>>> ... It's really a great transform for the purpose, except for that
>>>>> pesky causality bit.
>>>> That reminds me of a Principals quip: "Running a school would be a great
>>>> job if it weren't for the kids."
>>>>
>>>> A global view of a long interval implies low frequency.
>>> I have learned that narrow bands imply long intervals. Could you
>>> elaborate why global view of a long interval implies low frequency?
>>> Perhaps I'm not entirely clear what is meant by global view here. :\
>> Let's suppose that a sample rate is 8 KHz, and that periods of a minute
>> encompass significances that would be lost by considering only shorter
>> intervals. That's nearly half a million samples that need to be treated
>> jointly.
> 
> True. Is there something that implies low frequency?

Something whose period requires that it be represented using a minute's 
worth of samples.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Jerry Avins wrote:

> Jani Huhtanen wrote:
>> Jerry Avins wrote:
>> 
>>> lucas.denoir@gmail.com wrote:
>>>
>>>> ... It's really a great transform for the purpose, except for that
>>>> pesky causality bit.
>>> That reminds me of a Principals quip: "Running a school would be a great
>>> job if it weren't for the kids."
>>>
>>> A global view of a long interval implies low frequency.
>> 
>> I have learned that narrow bands imply long intervals. Could you
>> elaborate why global view of a long interval implies low frequency?
>> Perhaps I'm not entirely clear what is meant by global view here. :\
> 
> Let's suppose that a sample rate is 8 KHz, and that periods of a minute
> encompass significances that would be lost by considering only shorter
> intervals. That's nearly half a million samples that need to be treated
> jointly.

True. Is there something that implies low frequency?

-- 
Jani Huhtanen
Tampere University of Technology, Pori