Re: How to interpolation in frequency domain will not affect time domain signal?
- From: kevin <kevinjmcgee@xxxxxxxxxxxx>
- Date: Wed, 1 Jul 2009 21:26:27 -0700 (PDT)
On Jul 1, 9:20 am, "coly" <wolon...@xxxxxxxxx> wrote:
But what puzzled me is that my interpolation method in the spectrum leads
to a distorted time data. I want to know why.
There are many different interpolation methods. You can zero pad
things, fit parabolas to the data, use a zoom technique, etc. You
could use the DFT or inverse DFT to compute as many millions of time
or frequency domain points as you want. One of the nice things about
the DFT or IDFT is that you can use fractional values of 'n' (time
index) and 'k' (frequency index).
But no matter how you go about it, you're going to have the same
fundamental problem. Let me see if I can explain it in the following
way. Astronomers routinely capture millions of data points and do
FFTs on them. They also understand that they can express what the FFT
is doing as an N equation, N known and N unknown type of problem,
where N is in the millions.
Now suppose you said to them: "I can take 100 data points in time, FFT
it to get 100 frequency domain points, then zoom interpolate my 100
frequency domain points to create a million points. Then I can inverse
transform to obtain a million points in the time domain. So you
wouldn't have to get millions of samples, just 100 of them."
They’ll probably think many things about that kind of statement, none
of them good, and all of them reflecting badly on the person making
the statement. But, being polite, they might just say: "But that
makes no sense at all. We have a million knowns (our million data
samples), and we need to solve for a million unknowns (a million
frequency domain points). It's mathematically impossible to solve for
a million unknowns when all we have is 100 knowns, unless we impose
some ridiculous assumptions on our signal processing model to reduce
the million unknowns down to 100."
So just imagine that you actually had the million point astronomy
input data. Pick any 100 of those points you want, and do whatever
interpolating you wish and try to ‘create’ or 'estimate' all those
other points from your interpolation scheme. Can you begin to
appreciate now why your interpolated results will have errors when
compared to the actual data? Unless you can impose some extraordinary
assumptions to simplify the problem, it’s an impossible task no matter
what interpolation technique you use.
As Rune pointed out, you can use other methods, such as frequency
estimation. Basically, you’d be using a different signal processing
model to obtain the needed information from the available data.
Kevin McGee
.
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