Understanding FFT in Audio Measurements

[Asinus]

And- why do we call it FAST Fourier Transform? Is there also a use for a SLOW transform?

The slow algorithm is the straight DFT which is a (NxN) * (Nx1) matrix multiplication, which means O(N^2) operations. The FFT uses a special factorization and splits the big matrix into smaller matrices with phase shifts, achieving O(N logN) operations. If you are doing offline analyisis and you have time to spare there should be no difference between FFT and DFT.

 

[scott wurcer]

Maybe this has already been answered and I just didn't catch it - what are the differences in using a Hamming vs a Blackman vs Nutall window? What advantages would each convey is various situations? Seems this has something to do with not losing transient events in a longer FFT, but FA if I can figure out how this could be.

And- why do we call it FAST Fourier Transform? Is there also a use for a SLOW transform?

Basically different windows were developed according to applications here is one table, but I'm sure a search would find a more exhaustive list.

I have even evaluated the explicit Fourier integral, as slow as it gets, in the days before non power of 2 FFT's were routinely available.

 

[KSTR]

And- why do we call it FAST Fourier Transform? Is there also a use for a SLOW transform?

There is even need for the very slow version, I just happened to write a time-domain convolver in 64bit floating point (double) because I was not happy with the results from FFT/iFFT-based convolvers with 32bit floats for an application needing very low numerical artifacts.
Code was based upon this snippet: https://www.musicdsp.org/en/latest/Filters/66-time-domain-convolution-with-o-n-log2-3.html

 

[KSTR]

Basically different windows were developed according to applications here is one table, but I'm sure a search would find a more exhaustive list.

I have even evaluated the explicit Fourier integral, as slow as it gets, in the days before non power of 2 FFT's were routinely available.View attachment 115898

Excellent! I've seen this table before but could not find it again.

 

[SIY]

Maybe this has already been answered and I just didn't catch it - what are the differences in using a Hamming vs a Blackman vs Nutall window? What advantages would each convey is various situations? Seems this has something to do with not losing transient events in a longer FFT, but FA if I can figure out how this could be.

And- why do we call it FAST Fourier Transform? Is there also a use for a SLOW transform?

Here’s a short review: https://audioxpress.com/article/pra...isition-in-audio-measurements-part-5-the-user

 

[scott wurcer]

There is even need for the very slow version, I just happened to write a time-domain convolver in 64bit floating point (double) because I was not happy with the results from FFT/iFFT-based convolvers with 32bit floats for an application needing very low numerical artifacts.
Code was based upon this snippet: https://www.musicdsp.org/en/latest/Filters/66-time-domain-convolution-with-o-n-log2-3.html

Funny you mention this, my one and only contribution to CPAN (~1994) was a patch to an FFT library for the same problem. The solution here only required computing and keeping intermediate results as doubles. Image restoration sometimes involves 100's of iterative convolutions and even 16bit data became contaminated with numerical artifacts.

 

[milosz]

I see. "FAST fourier transform" just refers to the algorithm type used for the analysis, with certain compromise choices made which gives you results which closely approximate a true Fourier transform, quickly.

I first came across the FFT when reading about FM synthesis techniques in early PC sound context. (Pre-DOS days it was....)

 

[amirm]

Does the whole Audio Community work off the same AP parameters / calibration?

No, there is a variation. And even with AP, JA at stereophile had been using the older generation with a different UI and parameters. The industry though is starting to replicate my testing to some extent so maybe standardization will get better in the future.

 

[scott wurcer]

I see. "FAST fourier transform" just refers to the algorithm type used for the analysis, with certain compromise choices made which gives you results which closely approximate a true Fourier transform, quickly.

NO!!!! Let me start by saying sorry for a strong reaction, but the results are exactly the same.

 

[Asinus]

I see. "FAST fourier transform" just refers to the algorithm type used for the analysis, with certain compromise choices made which gives you results which closely approximate a true Fourier transform, quickly.

I first came across the FFT when reading about FM synthesis techniques in early PC sound context. (Pre-DOS days it was....)

The FFT is not an approximation, it is a clever factorization. Think of the DFT as evaluating a1*x + a2*x + a1*y + a2*y, you'll need 4 multiplications and 3 additions. The FFT is evaluating (a1+a2)*(x+y), thus only two additions and a multiplication but your end result should be the same.

NO!!!! Let me start by saying sorry for a strong reaction, but the results are exactly the same.

The results between DFT and FFT should be the same within the numerical precision.
However, it is not exactly the same as evaluating the continuous Fourier Transform integral with trapezoids or Simpson's rule as there you can assign the time and frequency sample arrays with more flexibility. Still, if you match the size of the FFT and the integral arrays or scale the output accordingly you get equivalent results.

 

[scott wurcer]

Still, if you match the size of the FFT and the integral arrays or scale the output accordingly you get equivalent results.

That is what I did. Simulators have a bad habit of interpolating results to increase speed.

 

[Asinus]

But in this case these kind of errors can be measured by the sampling method like the AP. If only there would be an extra feature called 'peak error time domain" we can at least detect these kind of DSP firmware errors.

There is a feature in the AP software called the recorder, you can see it in Amir's video. If you start a SINAD measurement on the recorder you would catch a single wrapped-around sample because you would see a sudden spike on the SINAD dropping.

 

[Hayabusa]

I see. "FAST fourier transform" just refers to the algorithm type used for the analysis, with certain compromise choices made which gives you results which closely approximate a true Fourier transform, quickly.

The (D)FFT gives exacly the same output as a (D)FT, so its not an aproximation

[ Edit: Now I see this was already mentioned earlier ]

 

[Hayabusa]

There is a feature in the AP software called the recorder, you can see it in Amir's video. If you start a SINAD measurement on the recorder you would catch a single wrapped-around sample because you would see a sudden spike on the SINAD dropping.
View attachment 115911

Sounds good!

@amirm Do you use this feature in your SINAD measurements?

 

[amirm]

@amirm Do you use this feature in your SINAD measurements?

I use it for warm-up test of some amplifiers and DACs. And also to see if the difference in tweak products.

 

[KSTR]

Well, even in Adobe Audition you can monitor the FFT spectrum in realtime (when it's <= 8k points) so when you have a long recording of a DUT signal you will also see any glitch, wrapped sample, zeroed samples etc. Same for REW.

 

[JIW]

The measured square wave does not have the level displayed in the generator setting - 4 V RMS - but seems to be set such that the maxima and minima of the square are the same as for a sine with the displayed generator level, i.e. about 5.7 V. Since for a square wave, the RMS is identical to the (positive) peak value, while for a sine, the peak is greater than the RMS by a factor of the square root of 2, i.e. 1.4142, the RMS of the square would be 1.4142 times greater than the displayed value of the generator, i.e. 1.4142 x 4 V RMS = 5.66 V RMS.

 

[NTK]

The measured square wave does not have the level displayed in the generator setting - 4 V RMS ...

Yes, it is confusing, but the rationale is given in AP's Audio Measurement Handbook.

 

[Lambda]

but the rationale is given in AP's Audio Measurement Handbook.

This is stupid and confusing.
They should just change the label from RMS to peak to peak if square wave is selected.

Or just have both values there all the time and the user can see the RMS value and the PP value at the same time.

 

[Stretchneck]

Just to say thank you for this video - really helpful!