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### FFT scaling for periodic and aperiodic signals

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```Hi all
I have a question on FFT scaling difference  between periodic and  a periodic signals.

if I have a periodic signal x1 (sinewave), the FFT of this signal is normalized by the number of points N to get the correct amplitude spectrum scale < abs(fft(x1))/N >.

However if there is aperiodic signal e.g. x2 (square pulse), the normalization by the number of points representing pulse amplitude (k) gives amplitude spectrum with a maximum of 1.
what I know is that the spectrum of a square pulse is a sinc waveform of amplitude (A*w). where A is the pulse amplitude and w is the pulse width.

does this mean the normalization is different for periodic and aperiodic signals?
and if so, how does spectrum analyzers know how to normalize different signals?

Any hints?
many thanks

N = 4096;                % number of FFT points
ts = 1e-3;               % Sampling time
t = [0:N-1]*ts;             % Time Vector

x1 = sin(2*pi*1*t);  % Sinewave
X1 = abs(fft(x1))/N; % Amplitude spectrum of the sinewave

k = 2000;    % number of points representing pulse width
x2 = [zeros(1,1000) ones(1,k) zeros(1,1096)]; % square pulse of width 2 seconds
X2 = abs(fft(x2))/k; % Amplitude Spectrum of the square pulse

subplot(2,1,1),plot(X1)
subplot(2,1,2),plot(X2)
```
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```On Oct 30, 11:14=A0am, "abees Tero" <abee...@hotmail.com> wrote:
> Hi all
>...
>
> does this mean the normalization is different for periodic and aperiodic =
signals?
Yes
> and if so, how does spectrum analyzers know how to normalize different si=
gnals?
Spectrum analyzers don't know. In fact, the signal being analyzed may
have components that should be scaled in different ways.
>
> Any hints?
> many thanks
> ...

The three cases are signals that have a power spectrum (tone like),
signals that have a power spectral density (PSD) (noise like) and
signals that have an energy spectral density (ESD)(transients).

As I have posted here before:
-begin quote-
The PSD, ESD and power spectrum can be calculated via fft based
methods in Matlab.
Manufacturers of dynamic signal analyzers have provided these
functions, properly scaled, for years. Some have been nice enough to
accurately document their functions and make and keep the
documentation available.

Take a look at "Choose your Units!" from B&K:

http://www.bksv.com/doc/bo0438.pdf

and for more detail, "Signals and Units" on page 29 of:

http://www.bksv.com/doc/bv0031.pdf

For an discussion of the signal processing,
consider pages 5-21 of:

Particularly "3 Introduction" on page 5 and "9 Scaling the results" on
page 15.
I consider "13 Testing the Algorithm" on page 21 and on as a more
practically oriented discussion of pwelch than the Matlab docs.
-end quote-

Dale B. Dalrymple
```
 0

```dbd <dbd@ieee.org> wrote in message <1bded1a9-6241-4f2c-85be-d9020e5a2b78@t13g2000yqm.googlegroups.com>...
> On Oct 30, 11:14 am, "abees Tero" <abee...@hotmail.com> wrote:
> > Hi all
> >...
> >
> > does this mean the normalization is different for periodic and aperiodic signals?
> Yes
> > and if so, how does spectrum analyzers know how to normalize different signals?
> Spectrum analyzers don't know. In fact, the signal being analyzed may
> have components that should be scaled in different ways.
> >
> > Any hints?
> > many thanks
> > ...
>
> The three cases are signals that have a power spectrum (tone like),
> signals that have a power spectral density (PSD) (noise like) and
> signals that have an energy spectral density (ESD)(transients).
>
> As I have posted here before:
> -begin quote-
> The PSD, ESD and power spectrum can be calculated via fft based
> methods in Matlab.
> Manufacturers of dynamic signal analyzers have provided these
> functions, properly scaled, for years. Some have been nice enough to
> accurately document their functions and make and keep the
> documentation available.
>
> Take a look at "Choose your Units!" from B&K:
>
> http://www.bksv.com/doc/bo0438.pdf
>
> and for more detail, "Signals and Units" on page 29 of:
>
> http://www.bksv.com/doc/bv0031.pdf
>
> For an discussion of the signal processing,
> consider pages 5-21 of:
>
>
> Particularly "3 Introduction" on page 5 and "9 Scaling the results" on
> page 15.
> I consider "13 Testing the Algorithm" on page 21 and on as a more
> practically oriented discussion of pwelch than the Matlab docs.
> -end quote-
>
> Dale B. Dalrymple

Dear Dale,,,
I would like to thank you for your kind, fast, really useful response.
you cleared all my doubts.
thanks again.
regards,,,,
```
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