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### Log-time sequences

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```Consider a finite-length discrete-time signal where, instead of
traditional linearly-spaced sample points, we have sample points on a
time grid of
t = ln(k),
k an integer from 1 to N.

Such a sequence will have widely-spaced samples for small k, with
increasing sample density as k increases.

I would like to take the Fourier transform of such a sequence. I could
obviously do this using a modified DFT by correlating the sequence
with logarithmically-spaced samples of sin and cos waves. However, I
wonder if there is not a "fast" method of doing this that would save
computation. I'm pretty sure there is no existing soulution for this
problem. Any takers?

Regards

```
 0

```"Liz" <robert.w.adams@verizon.net> wrote in message
> Consider a finite-length discrete-time signal where, instead of
> traditional linearly-spaced sample points, we have sample points on a
> time grid of
>  t = ln(k),
> k an integer from 1 to N.
>
> Such a sequence will have widely-spaced samples for small k, with
> increasing sample density as k increases.
>
> I would like to take the Fourier transform of such a sequence. I could
> obviously do this using a modified DFT by correlating the sequence
> with logarithmically-spaced samples of sin and cos waves. However, I
> wonder if there is not a "fast" method of doing this that would save
> computation. I'm pretty sure there is no existing soulution for this
> problem. Any takers?

First, assure that the sampling rate is adequate for the large sample
intervals.  This is a big deal, so don't blow it off without thinking about
it carefully.  The log-sampled sinusoids for correlation would yield some
interesting results which might demand some sort of weighting.  Well, in
fact, log sampling the sinusoids would probably be unreasonable - a guess.

Then, why not interpolate to linear samples at the large sample intervals?

You should ask yourself what it is you expect the Fourier Transform to look
like in this situation?

Fred

```
 0

```robert.w.adams@verizon.net (Liz) wrote in message news:<9dec5a83.0312252059.71a97e5f@posting.google.com>...
> Consider a finite-length discrete-time signal where, instead of
> traditional linearly-spaced sample points, we have sample points on a
> time grid of
>  t = ln(k),
> k an integer from 1 to N.
>
> Such a sequence will have widely-spaced samples for small k, with
> increasing sample density as k increases.
>
> I would like to take the Fourier transform of such a sequence. I could
> obviously do this using a modified DFT by correlating the sequence
> with logarithmically-spaced samples of sin and cos waves. However, I
> wonder if there is not a "fast" method of doing this that would save
> computation. I'm pretty sure there is no existing soulution for this
> problem. Any takers?

Check out the writings of Barbarossa and Petrone at the University
of Roma. They did some work on polynomial-phase signals in the early
1990ies. These are signals where the exponent in the Fourier transform
varies non-linearly.

Rune
```
 0