All FFT's need double precision - not

Article: PhysX87: Software Deficiency
By: Michael S (already5chosen.delete@this.yahoo.com), July 21, 2010 2:35 pm
Room: Moderated Discussions
Ungo (a@b.c.d.e) on 7/21/10 wrote:
---------------------------
>Michael S (already5chosen@yahoo.com) on 7/20/10 wrote:
>---------------------------
>>Of course, nearly all FFTs used in signal processing (spectrum estimation, FIR
>>filtering, crosscorrelation etc) are fine with single precision. In fact, in majority
>>of cases 24-bit mantissa is an overkill. HiFi audio is the only notable exception I could think about.
>
>24-bit mantissa is usually overkill there, too. It turns out that when you do
>double blind testing, essentially nobody can tell the difference between 24-bit/192KHz
>and the same audio sample resampled to 16/44.1. The few who can are young people
>with truly exceptional high frequency hearing extending a little above 20 KHz.
>
>You could provide enough performance for such people with 48 KHz sampling. 96
>KHz sampling is dumb, and 192K is absurd. 24-bit sample depth is... meh. There's
>not much need for it, and the ADCs/DACs don't deliver true 24-bit dynamic range anyways.
>

We were talking about FFT.
According to my understanding FFT of size N calculated with m-bit precision would on average deliver only m1=m-k good bit, where k=0.5*log2(log2(N))+1.
So, for example, for m=24,N=1024 => m1=21.3

Not knowing much about Hi FI audio processing I'd think that FFT is not widely used here anyway, but, from pure theoretical point of view, if you have a desire to use FFT then IEEE single precision is probably insufficient.

Traditionally, most floating-point DSPs that were marketed for high end audio had 40-bit FPU registers with 31b or 32b of mantissa. As long as in-memory format is still 32b (storing all 40 bit in memory tends to be twice slower) you probably don't gain much of precision in FFT. My back-of-envelop estimate suggests gain of 1 bit for Radix-2 FFT or 1.5 bit for Radix-4. However for other popular algorithms, in particular for FIR filtering, extended precision accumulators help to easily retain full 24b precisions of result.
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                  All FFT's need double precision - notMichael S2010/07/20 01:16 AM
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