The fastest fast Fourier transform in not just the west, but the world,
now for the most popular toy ISA.
On a high level, it follows the design of the AVX2 version closely,
with the exception that the input is slightly less permuted as we don't have
to do lane switching with the input on double 4pt and 8pt.
On a low level, the lack of subadd/addsub instructions REALLY penalizes
any attempt at writing an FFT. That single register matters a lot,
and reloading it simply takes unacceptably long.
In x86 land, vendors would've noticed developers need this.
In ARM land, you get a badly designed complex multiplication instruction
we cannot use, that's not present on 95% of devices. Because only
compilers matter, right?
Future optimization options are very few, perhaps better register
management to use more ld1/st1s.
All timings below are in cycles:
A53:
Length | C | New (lavu) | Old (lavc) | FFTW
------ |-------------|-------------|-------------|-----
4 | 842 | 420 | 1210 | 1460
8 | 1538 | 1020 | 1850 | 2520
16 | 3717 | 1900 | 3700 | 3990
32 | 9156 | 4070 | 8289 | 8860
64 | 21160 | 9931 | 18600 | 19625
128 | 49180 | 23278 | 41922 | 41922
256 | 112073 | 53876 | 93202 | 101092
512 | 252864 | 122884 | 205897 | 207868
1024 | 560512 | 278322 | 458071 | 453053
2048 | 1295402 | 775835 | 1038205 | 1020265
4096 | 3281263 | 2021221 | 2409718 | 2577554
8192 | 8577845 | 4780526 | 5673041 | 6802722
Apple M1
New - Total for len 512 reps 2097152 = 1.459141 s
Old - Total for len 512 reps 2097152 = 2.251344 s
FFTW - Total for len 512 reps 2097152 = 1.868429 s
New - Total for len 1024 reps 4194304 = 6.490080 s
Old - Total for len 1024 reps 4194304 = 9.604949 s
FFTW - Total for len 1024 reps 4194304 = 7.889281 s
New - Total for len 16384 reps 262144 = 10.374001 s
Old - Total for len 16384 reps 262144 = 15.266713 s
FFTW - Total for len 16384 reps 262144 = 12.341745 s
New - Total for len 65536 reps 8192 = 1.769812 s
Old - Total for len 65536 reps 8192 = 4.209413 s
FFTW - Total for len 65536 reps 8192 = 3.012365 s
New - Total for len 131072 reps 4096 = 1.942836 s
Old - Segfaults
FFTW - Total for len 131072 reps 4096 = 3.713713 s
Thanks to wbs for some simplifications, assembler fixes and a review
and to jannau for giving it a look.
Convert the input from a scatter to a gather instead,
which is faster and better for SIMD.
Also, add a pre-shuffled exptab version to avoid
gathering there at all. This doubles the exptab size,
but the speedup makes it worth it. In SIMD, the
exptab will likely be purged to a higher cache
anyway because of the FFT in the middle, and
the amount of loads stays identical.
For a 960-point inverse MDCT, the speedup is 10%.
This makes it possible to write sane and fast SIMD
versions of inverse MDCTs.
This commit does some refactoring to make defining assembly codelets
smaller, and fixes compiler redefinition warnings. It also allows
for other assembly versions to reuse the same boilerplate code as
x86.
Finally, it also adds the out_of_place flag to all assembly codelets.
This changes nothing, as out-of-place operation was assumed to be
available anyway, but this makes it more explicit.
This commit rewrites the internal transform code into a constructor
that stitches transforms (codelets).
This allows for transforms to reuse arbitrary parts of other
transforms, and allows transforms to be stacked onto one
another (such as a full iMDCT using a half-iMDCT which in turn
uses an FFT). It also permits for each step to be individually
replaced by assembly or a custom implementation (such as an ASIC).
This commit adds a pure x86 assembly SIMD version of the FFT in libavutil/tx.
The design of this pure assembly FFT is pretty unconventional.
On the lowest level, instead of splitting the complex numbers into
real and imaginary parts, we keep complex numbers together but split
them in terms of parity. This saves a number of shuffles in each transform,
but more importantly, it splits each transform into two independent
paths, which we process using separate registers in parallel.
This allows us to keep all units saturated and lets us use all available
registers to avoid dependencies.
Moreover, it allows us to double the granularity of our per-load permutation,
skipping many expensive lookups and allowing us to use just 4 loads per register,
rather than 8, or in case FMA3 (and by extension, AVX2), use the vgatherdpd
instruction, which is at least as fast as 4 separate loads on old hardware,
and quite a bit faster on modern CPUs).
Higher up, we go for a bottom-up construction of large transforms, foregoing
the traditional per-transform call-return recursion chains. Instead, we always
start at the bottom-most basis transform (in this case, a 32-point transform),
and continue constructing larger and larger transforms until we return to the
top-most transform.
This way, we only touch the stack 3 times per a complete target transform:
once for the 1/2 length transform and two times for the 1/4 length transform.
The combination algorithm we use is a standard Split-Radix algorithm,
as used in our C code. Although a version with less operations exists
(Steven G. Johnson and Matteo Frigo's "A modified split-radix FFT with fewer
arithmetic operations", IEEE Trans. Signal Process. 55 (1), 111–119 (2007),
which is the one FFTW uses), it only has 2% less operations and requires at least 4x
the binary code (due to it needing 4 different paths to do a single transform).
That version also has other issues which prevent it from being implemented
with SIMD code as efficiently, which makes it lose the marginal gains it offered,
and cannot be performed bottom-up, requiring many recursive call-return chains,
whose overhead adds up.
We go through a lot of effort to minimize load/stores by keeping as much in
registers in between construcring transforms. This saves us around 32 cycles,
on paper, but in reality a lot more due to load/store aliasing (a load from a
memory location cannot be issued while there's a store pending, and there are
only so many (2 for Zen 3) load/store units in a CPU).
Also, we interleave coefficients during the last stage to save on a store+load
per register.
Each of the smallest, basis transforms (4, 8 and 16-point in our case)
has been extremely optimized. Our 8-point transform is barely 20 instructions
in total, beating our old implementation 8-point transform by 1 instruction.
Our 2x8-point transform is 23 instructions, beating our old implementation by
6 instruction and needing 50% less cycles. Our 16-point transform's combination
code takes slightly more instructions than our old implementation, but makes up
for it by requiring a lot less arithmetic operations.
Overall, the transform was optimized for the timings of Zen 3, which at the
time of writing has the most IPC from all documented CPUs. Shuffles were
preferred over arithmetic operations due to their 1/0.5 latency/throughput.
On average, this code is 30% faster than our old libavcodec implementation.
It's able to trade blows with the previously-untouchable FFTW on small transforms,
and due to its tiny size and better prediction, outdoes FFTW on larger transforms
by 11% on the largest currently supported size.
This sadly required making changes to the code itself,
due to the same context needing to be reused for both versions.
The lookup table had to be duplicated for both versions.
This commit refactors the power-of-two FFT, making it faster and
halving the size of all tables, making the code much smaller on
all systems.
This removes the big/small pass split, because on modern systems
the "big" pass is always faster, and even on older machines there
is no measurable speed difference.
out[lut[i]] = in[i] lookups were 4.04 times(!) slower than
out[i] = in[lut[i]] lookups for an out-of-place FFT of length 4096.
The permutes remain unchanged for anything but out-of-place monolithic
FFT, as those benefit quite a lot from the current order (it means
there's only 1 lookup necessary to add to an offset, rather than
a full gather).
The code was based around non-power-of-two FFTs, so this wasn't
benchmarked early on.
This commit adds support for in-place FFT transforms. Since our
internal transforms were all in-place anyway, this only changes
the permutation on the input.
Unfortunately, research papers were of no help here. All focused
on dry hardware implementations, where permutes are free, or on
software implementations where binary bloat is of no concern so
storing dozen times the transforms for each permutation and version
is not considered bad practice.
Still, for a pure C implementation, it's only around 28% slower
than the multi-megabyte FFTW3 in unaligned mode.
Unlike a closed permutation like with PFA, split-radix FFT bit-reversals
contain multiple NOPs, multiple simple swaps, and a few chained swaps,
so regular single-loop single-state permute loops were not possible.
Instead, we filter out parts of the input indices which are redundant.
This allows for a single branch, and with some clever AVX512 asm,
could possibly be SIMD'd without refactoring.
The inplace_idx array is guaranteed to never be larger than the
revtab array, and in practice only requires around log2(len) entries.
The power-of-two MDCTs can be done in-place as well. And it's
possible to eliminate a copy in the compound MDCTs too, however
it'll be slower than doing them out of place, and we'd need to dirty
the input array.
This patch adds support for arbitrary-point FFTs and all even MDCT
transforms.
Odd MDCTs are not supported yet as they're based on the DCT-II and DCT-III
and they're very niche.
With this we can now write tests.
INT32_MAX (2147483647) isn't exactly representable by a floating point
value, with the closest being 2147483648.0. So when rescaling a value
of 1.0, this could overflow when casting the 64-bit value returned from
lrintf() into 32 bits.
Unfortunately the properties of integer overflows don't match up well
with how a Fourier Transform operates. So clip the value before
casting to a 32-bit int.
Should be noted we don't have overflows with the table values we're
currently using. However, converting a Kaiser-Bessel window function
with a length of 256 and a parameter of 5.0 to fixed point did create
overflows. So this is more of insurance to save debugging time
in case something changes in the future.
The macro is only used during init, so it being a little slower is
not a problem.
Required minimal changes to the code so made sense to implement.
FFT and MDCT tested, the output of both was properly rounded.
Fun fact: the non-power-of-two fixed-point FFT and MDCT are the fastest ever
non-power-of-two fixed-point FFT and MDCT written.
This can replace the power of two integer MDCTs in aac and ac3 if the
MIPS optimizations are ported across.
Unfortunately the ac3 encoder uses a 16-bit fixed point forward transform,
unlike the encoder which uses a 32bit inverse transform, so some modifications
might be required there.
The 3-point FFT is somewhat less accurate than it otherwise could be,
having minor rounding errors with bigger transforms. However, this
could be improved later, and the way its currently written is the way one
would write assembly for it.
Similar rounding errors can also be found throughout the power of two FFTs
as well, though those are more difficult to correct.
Despite this, the integer transforms are more than accurate enough.
Simply moves and templates the actual transforms to support an
additional data type.
Unlike the float version, which is equal or better than libfftw3f,
double precision output is bit identical with libfftw3.
Lynne
Andreas Rheinhardt
Anton Khirnov