```On Nov 19, 1:05 pm, "Ron N." <rhnlo...@yahoo.com> wrote:

> If the data is periodic in N and not periodic in N+Z,
> then the basis vector set of the first FFT will not be
> completely contained in the basis set of the second FFT.

Ah, yes, of course.  In other words, if the length of the FFT after
zeropadding is not an integer multiple of the period of the waveform,
then the frequency of the waveform will not fall on an FFT bin and the
magnitude will be altered according to the (effective) rectangular
window applied.

We are normalizing

N-1
|SUM F(n)*{exp[j2PIkn/N]}|
n=0

such that, when F(n) is a pure sinusoid and the period of F(n) is an
exact integer submultiple of N, the resulting magnitude equals the

N+Z-1
| SUM  F'(n)*{exp[j2PIkn/(N+Z)]}|
n=0

N-1                             N+Z-1
= |SUM  F(n)*{exp[j2PIkn/(N+Z)]}| + SUM 
n=0                              n=N

It's that (N+Z) in the denominator that changes things.

Greg
```
```On Nov 19, 3:54 am, Greg Berchin <gberc...@sentientscience.com> wrote:
> On Nov 19, 1:40 am, "Ron N." <rhnlo...@yahoo.com> wrote:
>
> > This works if the sinusoid is perfectly periodic in both
> > N and the new length after zero-padding.
>
> I had assumed from context that there was an integer number of sine
> periods in the original N.  I don't think that it's necessary that
> there be an integer number of sine periods in N+Z, where N+Z is the
> zeropadded FFT size.  You're computing the dot products of the same
> two sinusoids in both cases

The basis vector set of a DFT/FFT depends on it length.
If the data is periodic in N and not periodic in N+Z,
then the basis vector set of the first FFT will not be
completely contained in the basis set of the second FFT.
Thus the dot products won't be of the same two sinusoids
in all cases.

IMHO. YMMV.
--
rhn A.T nicholson d.0.t C-o-M
```
```On Nov 19, 6:22 am, "A.E lover" <aelove...@gmail.com> wrote:
> I read somewhere it said that zero padding doesn't increase the
> analysis resolution at all in frequency domain. What it does is to
> "interpolate" the spectrum so that we have more frequency samples and
> so yield a better look of the spectrum.

It doesn't increase the resolution in terms of being able
to resolve more closely spaced spectral peaks.  It does
interpolate accurately enough to often allow the measurement
of isolated maxima with higher resolution in the absence
of nearby spectral interference or noise.

IMHO. YMMV.
--
rhn A.T nicholson d.0.t C-o-M
```
```Thank you guys.

Hi Ron, can you please tell me more so in general case, i.e both in
both N and new length M the sinusoid is not perfectly periodic, how to
normalize the signal?
As I understand it, in this case, the reason is the angular frequency
of the original sinusoid doesn't fall exactly in a frequency bin of
DFT. Is it correct?
So is there any way to determine this scalloping losses?

On Nov 19, 1:40 am, "Ron N." <rhnlo...@yahoo.com> wrote:
> two fft's will have different scalloping losses (unless
> one does perfect interpolation), and, depending on where
> respect to the center of the dft/fft aperture might change,
> which could affect the sign of the contribution from the
> negative frequency bins.
>
> IMHO. YMMV.
> --
> rhn A.T nicholson d.0.t C-o-M
>  http://www.nicholson.com/rhn/dsp.html

analysis resolution at all in frequency domain. What it does is to
"interpolate" the spectrum so that we have more frequency samples and
so yield a better look of the spectrum. Would any one please tell me
the way to derive this interpolation methematically?

Thanks
```
```On Nov 19, 1:40 am, "Ron N." <rhnlo...@yahoo.com> wrote:

> This works if the sinusoid is perfectly periodic in both
> N and the new length after zero-padding.

I had assumed from context that there was an integer number of sine
periods in the original N.  I don't think that it's necessary that
there be an integer number of sine periods in N+Z, where N+Z is the
zeropadded FFT size.  You're computing the dot products of the same
two sinusoids in both cases -- the zeropadded section contributes
nothing to the sum.

Greg
```
```On Nov 18, 3:50 pm, Greg Berchin <gberc...@comicast.net> wrote:
> On Sun, 18 Nov 2007 15:06:45 -0800 (PST), "A.E lover"
>
> <aelove...@gmail.com> wrote:
> >Now I want to pad zeros so that the spectrum will look better, do you
> >know how to normalize fft in this way such that a sinusoid with
> >amplitude 1 will correspond with spectrum with magnitude of 1?
>
> Also divide by N/2, where N is the number of points before zeropadding.

This works if the sinusoid is perfectly periodic in both
N and the new length after zero-padding.  Otherwise the
two fft's will have different scalloping losses (unless
one does perfect interpolation), and, depending on where
respect to the center of the dft/fft aperture might change,
which could affect the sign of the contribution from the
negative frequency bins.

IMHO. YMMV.
--
rhn A.T nicholson d.0.t C-o-M
http://www.nicholson.com/rhn/dsp.html

```
```On Nov 18, 6:50 pm, Greg Berchin <gberc...@comicast.net> wrote:
> On Sun, 18 Nov 2007 15:06:45 -0800 (PST), "A.E lover"
>
> <aelove...@gmail.com> wrote:
> >Now I want to pad zeros so that the spectrum will look better, do you
> >know how to normalize fft in this way such that a sinusoid with
> >amplitude 1 will correspond with spectrum with magnitude of 1?
>
> Also divide by N/2, where N is the number of points before zeropadding.
>
> In either situation, the value at DC needs to be divided by N, not N/2.

just to add, that is if you're defining the DFT the most conventional
way:

N-1
X[k] =       SUM{ x[n] e^(-j*2*pi*n*k/N) }
n=0

with iDFT:
N-1
x[n] = (1/N)*SUM{ X[k] e^(+j*2*pi*n*k/N) }
k=0

some of us think that a more natural definition would be

DFT:
N-1
X[k] = (1/N)*SUM{ x[n] e^(-j*2*pi*n*k/N) }
n=0

iDFT:
N-1
x[n] =       SUM{ X[k] e^(+j*2*pi*n*k/N) }
k=0

and this would work:

N/2-1
x[n] =       SUM{ X[k] e^(+j*2*pi*n*k/N) }
k=-N/2

with the understanding that X[k] (as well as x[n]) are periodic:

X[k+N] = X[k]      (and x[n+N] = x[n])

i see that definition as being more natural so that (at least when
there is no zero-padding, which is IMO an active change of definition
of the signal) you need not divide by N anywhere.  the value of the
DFT bin X[k] is, by definition, the amplitude of the kth complex
frequency component (there's ambiguity about the +/- (N/2)th
components, but screw those guys, unless you wanna divide that bin
amplitude by 2 and assign each half to each frequency).

anyway, it doesn't matter where you put the 1/N factor, in the forward
or inverse DFT, or a little in both, as long as the two leading
factors, in the DFT and iDFT, themselves multiply to 1/N (so that the
round-trip gain is unity and x[n] = x[n]).

r b-j

```
```On Sun, 18 Nov 2007 15:06:45 -0800 (PST), "A.E lover"
<aelover11@gmail.com> wrote:

>Now I want to pad zeros so that the spectrum will look better, do you
>know how to normalize fft in this way such that a sinusoid with
>amplitude 1 will correspond with spectrum with magnitude of 1?

Also divide by N/2, where N is the number of points before zeropadding.

In either situation, the value at DC needs to be divided by N, not N/2.

Greg
```
```hi all,