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Re: [ferret_users] Re: FFTA

Yes, the units of the result of FFTA are the same as the units of the variable.

By the way, often it's possible to do simple experiments in Ferret (or any software) to assure yourself about what it's doing.  Here, take the example in the script ef_fft_demo,

yes? USE monthly_navy_winds
yes? LET FFT_uwndtim = uwnd[x=150e:130w@ave,y=20n:40n@ave]
yes? LET/TITLE="Amplitude Spectrum" FFT_uwndfft = ffta(FFT_uwndtim)
yes? set view upper
yes? plot/thick/color=blue FFT_uwndfft

yes? let test_var = 3* FFT_uwndtim
yes? set view lower
yes? plot/thick/color=blue ffta(test_var)
It's easy to see that the result is just a multiple of the result FFT_uwndfft.  If it were units squared, for instance, that would be obvious.


On 6/5/2018 4:03 AM, Ryo Furue wrote:
Dear Yadidya and Saurabh,

On Mon, Jun 4, 2018 at 2:06 PM, saurabh rathore <rohitsrb2020@xxxxxxxxx> wrote:
Fast Fourier Transform i.e. ffta yields the same unit of the variable as it is the amplitude of the variable for various frequencies. so on y-axis it is celcius to best of my knowledge.

Without knowing Ferret's implementation of FFT, I think Saurabh's answer is most likely to be correct.  Here I would like to add a bit of background.

Normally "Fourier transform" (time -> frequency) is defined as

F(ω) = integral f(t) exp( i ω t) dt 

(give or take a factor of 1/2π or 1/(2π)^{1/2}).  Therefore, the units of the Fourier transform of temperature f(t) will be degrees × seconds. (Recall exp is dimensionless.) F(ω) is sometimes called the "spectral density".

On computers, we often talk about FFT, which is a "Fourier series expansion":

f(t) = sum{n = 0 to infinity} of A(n) exp(2 π i n t / T)

for 0 ≤ t < T.  We call these A(0), A(1), A(2), . . .  Fourier coefficients.  Because exp is dimensionless, A's and f(t) have the same units.

This distinction is sometimes important.  For example, if you compare your spectral density to those in the literature, you need to make sure you use the same units and you often need to convert your Fourier coefficients A(n) to spectral density A(n) Δt.

(Bonus exercise: Write down Parseval's relations for Fourier transform and Fourier series. Then you'll see the importance of the units.)

Best regards,

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