The simple test of the FFT is to see the effect of an impulse:

fg impulse delta 0.1 npts 1024

w impulse.sac

*1. Test of sac2000 FFT

*Sac creates a time series with a centered impulse of height 1.0 and not

a unit area pulse.

fft

writesp am

This creates a series of the amplitude spectrum, which has values of

0.1, e.g., 1 (from the time series) * DT from the fft

ifft

This creates a time series consisting of a centered impulse with height 1.0

Thus sac2000 [8/8/2001 (Version 00.59.44)] is performing the following

operation to convert h(t) into H(f)

N-1

H(f) = SUM h(t) exp ( - j 2 pi f t ) dt

k=0

N-1

h(t) = SUM H(f)exp(+j 2 pi f t ) df

n=0

where f = n df. t = k dt , and df = 1/Ndt

*2. Constraints on testing the FFT using an analytic function*

If you wish to test an FFT, you cannot use a sine-wave since the value

of the FFT will be related to the total number of cycles in the time

window and because the Discrete Fourier Transform is an approximation to

the Fourier Transform that suffers from periodicity in both the time and

frequency domains.

A properly designed test can use the analytical Gaussian <-> Gaussian

transform pair, or perhaps some thing from Laplace transforms using

H(t) exp( - a t ) <-> 1/(s + a) where s = j 2 pi f

You must always worry about the fact there there is a limited time

length and a limited frequency band. A perfect test of an analytical

transform pair is thwarted by the uncertainty principle

*3. Other FFT's

*If you use MATLAB or MATHCAD, you will always have to read the

documentation about the definition of the FFT used.

I use the electrical engineering definition but incorporate the physical

dimensions to the integral.

Send sac-help mailing list submissions to--

sac-help<at>iris.washington.edu

To subscribe or unsubscribe via the World Wide Web, visit

http://www.iris.washington.edu/mailman/listinfo/sac-help

or, via email, send a message with subject or body 'help' to

sac-help-request<at>iris.washington.edu

You can reach the person managing the list at

sac-help-owner<at>iris.washington.edu

When replying, please edit your Subject line so it is more specific

than "Re: Contents of sac-help digest..."

Today's Topics:

1. About FFT scaling factor in SAC (wwxu)

----------------------------------------------------------------------

Message: 1

Date: Tue, 11 Mar 2008 07:53:16 GMT

From: "wwxu" <wwxu<at>mail.iggcas.ac.cn>

Subject: [SAC-HELP] About FFT scaling factor in SAC

To: sac-help<at>iris.washington.edu

Message-ID: <20080311075316.5572.eqmail<at>mail.iggcas.ac.cn>

Content-Type: text/plain; format=flowed; charset="gb2312"

Hi, ALL

We suppose that x[n] and y[k] are a DFT pairs, i.e.

y[k]=1/N*sum_n(x[n]*exp(-2*pi*j*k*n/N)) n=0,1,...,N-1

where sum_n means the sum for n=0,1,N-1(similar for sum_k),

and dt is the sampling interval, df is the sampling frequency,

N is the sampling number.

To obey Parseval's theorem, we get

sum_n(x[n]*x[n])*dt=sum_k(y[k]*y[k])*df

since sum_n(x[n]*x[n])=N*sum_k(y[k]*y[k])

and df=1/(N*dt), also consider the amplitude symmetry of the real series,

we should multiply a coefficent to y[k]:

sqrt(2)*N*dt

but in SAC software, this value is N*dt/2,

so do you have any idea?

Thanks.

------------------------------

_______________________________________________

sac-help mailing list

sac-help<at>iris.washington.edu

http://www.iris.washington.edu/mailman/listinfo/sac-help

End of sac-help Digest, Vol 33, Issue 8

***************************************

Robert B. Herrmann

Otto W. Nuttli Professor of Geophysics

Department of Earth and Atmospheric Sciences

Saint Louis University

O'Neil Hall, Room 203

3642 Lindell Boulevard

St. Louis, MO 63108

TEL: 314 977 3120

FAX: 314 977 3117

Email: rbh<at>eas.slu.edu