All-real FFT for FPGA

On 2/14/2017 3:22 PM, Tim Wescott wrote:
> On Tue, 14 Feb 2017 20:42:12 +0100, Christian Gollwitzer wrote:
>
>> Am 14.02.17 um 20:21 schrieb rickman:
>>> I don't know that -(a * b) wouldn't be exactly the same result as (a *
>>> -b).
>>>
>>>
>> It is bitexact the same. In IEEE float, the sign is signalled by a
>> single bit, it doesn't use two's complement or similar. Therefore, -x
>> simply inverts the sign bit, and it doesnt matter if you do this before
>> or after the multiplication. The only possible corner cases re special
>> values like NaN and Inf; I believe that even then, the result is
>> bitexact the same, but I'm too lazy to check the standard.
>
> While the use of floats in FPGAs is getting more and more common, I
> suspect that there are still loads of FFTs getting done in fixed-point.
>
> I would need to do some very tedious searching to know for sure, but I
> suspect that if there's truncation, potential rollover, or 2's compliment
> values of 'b1000...0 involved, then my condition may be violated.
>
> And I don't know how much an optimizer is going to analyze what's going
> on inside of a DSP block -- if you instantiate one in your design and
> feed it all zeros, is the optimizer smart enough to take it out?

There is such a thing as "hard" IP which means a block of logic that is 
already mapped, placed and routed.  But they are rare, but not subject 
to any optimizations.  Otherwise yes, if you feed a constant into any 
block of logic synthesis will not only replace that logic with constants 
on the output, it will replace all the registers that may be used to 
improve the throughput of the constants.  There are synthesis commands 
to avoid that, but otherwise the tool does as it sees optimal.

The significant optimizations in an FFT come from recognizing all the 
redundant computations which the optimizer will *not* be able to see as 
they are dispersed across columns and rows or time.  Logic optimizations 
are much more obvious and basic.  Otherwise you could just feed the tool 
a DFT and let it work out the details.  Heck, that might get you some 
significant savings.  In a DFT done all at once, the tool might actually 
spot that you are multiplying a given input sample by the same 
coefficient multiple times.  But that's still not an FFT which is much 
more subtle.

-- 

Rick C

rickman  <gnuarm@gmail.com> wrote:

>On 2/14/2017 4:56 PM, Steve Pope wrote:

>> rickman  <gnuarm@gmail.com> wrote:

>>> On 2/14/2017 11:39 AM, Steve Pope wrote:

>>>> Tim Wescott  <tim@seemywebsite.com> wrote:

>>>>> On Tue, 14 Feb 2017 06:52:32 +0000, Steve Pope wrote:

>>>>>> If for example you compute (a * b) and also compute (a * -b),
>>>>>> the synthesizer is smart enough to know there are not two full
>>>>>> multipliers needed.

Note: I did not say that in an HDL, -(a * b) is equal in a bit-exact
sense to (a * -b).  

>>>>> I'm pretty sure that -(a * b) is not necessarily (a * -b).

We agree

>>> If the tool is going to use a
>>> single multiplier for calculating (a * b) and (a * -b),

>> Okay so far

>>> that implies it calculates (a * b)  and then negates that to get
>>> (a * -b) as -(a * b), no?

>> Not necessarily exactly this.

>If not this, then what?  Why are you being so coy?

Because one would have to look at the HDL language definition,
the declared bit widths and declared data types and possibly other stuff.

But you still wouldn't need two multpliers to compute the two
values in my example; just one multiplier plus some other logic
(much less than the gate count of a second multiplier) to take care of 
the extremal cases to make the output exactly match.

Steve

On 2/14/2017 5:48 PM, Steve Pope wrote:
> rickman  <gnuarm@gmail.com> wrote:
>
>> On 2/14/2017 4:56 PM, Steve Pope wrote:
>
>>> rickman  <gnuarm@gmail.com> wrote:
>
>>>> On 2/14/2017 11:39 AM, Steve Pope wrote:
>
>>>>> Tim Wescott  <tim@seemywebsite.com> wrote:
>
>>>>>> On Tue, 14 Feb 2017 06:52:32 +0000, Steve Pope wrote:
>
>>>>>>> If for example you compute (a * b) and also compute (a * -b),
>>>>>>> the synthesizer is smart enough to know there are not two full
>>>>>>> multipliers needed.
>
> Note: I did not say that in an HDL, -(a * b) is equal in a bit-exact
> sense to (a * -b).
>
>>>>>> I'm pretty sure that -(a * b) is not necessarily (a * -b).
>
> We agree
>
>>>> If the tool is going to use a
>>>> single multiplier for calculating (a * b) and (a * -b),
>
>>> Okay so far
>
>>>> that implies it calculates (a * b)  and then negates that to get
>>>> (a * -b) as -(a * b), no?
>
>>> Not necessarily exactly this.
>
>> If not this, then what?  Why are you being so coy?
>
> Because one would have to look at the HDL language definition,
> the declared bit widths and declared data types and possibly other stuff.
>
> But you still wouldn't need two multpliers to compute the two
> values in my example; just one multiplier plus some other logic
> (much less than the gate count of a second multiplier) to take care of
> the extremal cases to make the output exactly match.

It was a *long* way around the woods to get here.  I seriously doubt any 
logic synthesis tool is going to substitute two multipliers and an adder 
with a single multiplier and a bunch of other logic.  Have you seen a 
tool that was that smart?  If so, that is a pretty advanced tool.

BTW, what *are* the extremal[sic] cases?

-- 

Rick C

rickman  <gnuarm@gmail.com> wrote:

>On 2/14/2017 5:48 PM, Steve Pope wrote:

>> But you still wouldn't need two multpliers to compute the two
>> values in my example; just one multiplier plus some other logic
>> (much less than the gate count of a second multiplier) to take care of
>> the extremal cases to make the output exactly match.

>It was a *long* way around the woods to get here.  I seriously doubt any 
>logic synthesis tool is going to substitute two multipliers and an adder 
>with a single multiplier and a bunch of other logic.  

I very strongly disagree ... minimizing purely combinatorial logic
is the _sine_qua_non_ of a synthesis tool.  

Lots of other things synthesizers try to do are more intricate and less
predictable, but not this.

Steve

On 2/14/2017 7:35 PM, Steve Pope wrote:
> rickman  <gnuarm@gmail.com> wrote:
>
>> On 2/14/2017 5:48 PM, Steve Pope wrote:
>
>>> But you still wouldn't need two multpliers to compute the two
>>> values in my example; just one multiplier plus some other logic
>>> (much less than the gate count of a second multiplier) to take care of
>>> the extremal cases to make the output exactly match.
>
>> It was a *long* way around the woods to get here.  I seriously doubt any
>> logic synthesis tool is going to substitute two multipliers and an adder
>> with a single multiplier and a bunch of other logic.
>
> I very strongly disagree ... minimizing purely combinatorial logic
> is the _sine_qua_non_ of a synthesis tool.
>
> Lots of other things synthesizers try to do are more intricate and less
> predictable, but not this.

I hear you, but I don't think the tools will be able to "see" the 
simplifications as they are not strictly logic simplifications.  Most of 
it is trig.

Take a look at just how the FFT works.  The combinations of 
multiplications work out because of properties of the sine function, not 
because of Boolean logic.

-- 

Rick C