[Speex-dev] A technical question about the speex preprocessor.

[John Ridges]

I got the approximation from a Google book:

http://books.google.com/books?id=2CAqsF-RebgC&pg=PA385

Page 392, formula (10.33)

Using this formula, you're right, hypergeom_gain() would *not* converge 
to 1 for large x, but would instead be gamma(1.25)/sqrt(sqrt(x)) which 
would approach zero. Now if the formula for the hypergeometric gain were 
instead gamma(1.5) * M(-.5;1;-x) / sqrt(x) that *would* approach 1, but 
that's just me noodling around with the formula to get something that 
approaches 1. Since I don't know how the hypergoemetric gain was derived 
(or even really what it means) I don't know if that's useful or not. Can 
you tell me what the source was for the original table values?

John Ridges


Jean-Marc Valin wrote:
> Something looks odd without your values (or the doc) because hypergeom_gain()
> should really approach 1 as x goes to infinity. But in the end, an
> approximation is probably OK because denoising is anything but an exact science
> :-)
>
>    Jean-Marc
>
> Quoting John Ridges <jridges at masque.com>:
>
>   
>> By my reckoning the confluent hypergoemetric functions should have the
>> following values:
>>
>> M(-.25;1;-.5) = 1.11433
>> M(-.25;1;-1) = 1.21088
>> M(-.25;1;-1.5) = 1.29385
>> M(-.25;1;-2) = 1.36627
>> M(-.25;1;-2.5) = 1.43038
>> M(-.25;1;-3) = 1.48784
>> M(-.25;1;-3.5) = 1.53988
>> M(-.25;1;-4) = 1.58747
>> M(-.25;1;-4.5) = 1.63134
>> M(-.25;1;-5) = 1.67206
>> M(-.25;1;-5.5) = 1.71009
>> M(-.25;1;-6) = 1.74579
>> M(-.25;1;-6.5) = 1.77947
>> M(-.25;1;-7) = 1.81136
>> M(-.25;1;-7.5) = 1.84167
>> M(-.25;1;-8) = 1.87056
>> M(-.25;1;-8.5) = 1.89818
>> M(-.25;1;-9) = 1.92466
>> M(-.25;1;-9.5) = 1.95009
>> M(-.25;1;-10) = 1.97456
>>
>> Which would give table values of:
>>
>>    static const float table[21] = {
>>       0.82157f, 0.91549f, 0.99482f, 1.06298f, 1.12248f, 1.17515f, 1.22235f,
>>       1.26511f, 1.30421f, 1.34025f, 1.37371f, 1.40495f, 1.43428f, 1.46195f,
>>       1.48815f, 1.51305f, 1.53679f, 1.55948f, 1.58123f, 1.60212f, 1.62223f};
>>
>> There is also a formula which asymptotically approaches M(a;b;-x) for
>> high values of x that is:
>>
>> (gamma(b)/gamma(b-a))*(x^-a)
>>
>> Which for M(-.25;1;-x) is sqrt(sqrt(x))*1.10326
>>
>> at x=10 this gives a value of 1.96191 (vs. what I think is the true
>> value of 1.97456).
>>
>> The reason I've gotten into all this is I'm trying to vectorize with SSE
>> intrinsics some of the slower loops in the preprocessor, and the
>> hypergeom_gain function is the only thing stopping me from removing all
>> the branches in the loops. I don't know how critical the accuracy of the
>> function is to the performance of the preprocessor, but if that
>> aforementioned approximation was good enough for all the values of x, it
>> would really speed the loops up.
>>
>> Let me know what you think. I hope I'm helping out here (and not just
>> confusing things).
>>
>> John Ridges
>>
>>
>> Jean-Marc Valin wrote:
>>     
>>> OK, so the problem is that the table you see does not match the definition?
>>> y = gamma(1.25)^2 * M(-.25;1;-x) / sqrt(x)
>>> Note that the table data has an interval of .5 for the x axis.
>>>
>>> How far are your results from the data in the table?
>>>
>>> Cheers,
>>>
>>>    Jean-Marc
>>>
>>> Quoting John Ridges <jridges at masque.com>:
>>>
>>>
>>>       
>>>> Thanks for the confirmation  Jean-Marc. I kind of suspected from the
>>>> comments that it was the confluent hypergoemetric function, which I was
>>>> trying to evaluate using Kummer's equation, namely:
>>>>
>>>> M(a;b;x) is the sum from n=0 to infinity of (a)n*x^n / (b)n*n!
>>>> where (a)n = a(a+1)(a+2) ... (a+n-1)
>>>>
>>>> But when I use Kummer's equation, I don't get the values in the
>>>> "hypergeom_gain" table. Did you use a different solution to the
>>>> confluent hypergoemetric function when you created the table?
>>>>
>>>> John Ridges
>>>>
>>>>
>>>> Jean-Marc Valin wrote:
>>>>
>>>>         
>>>>> Hi John,
>>>>>
>>>>> M(;;) is the confluent hypergeometric function.
>>>>>
>>>>> 	Jean-Marc
>>>>>
>>>>> John Ridges a écrit :
>>>>>
>>>>>
>>>>>           
>>>>>> Hi,
>>>>>>
>>>>>> I've been trying to re-create the table in the function "hypergeom_gain"
>>>>>> in preprocess.c, and I just simply can't get the same values. I get the
>>>>>> same value for the first element, so I know I'm computing gamma(1.25)^2
>>>>>> correctly, but I can't get the same numbers for M(-.25;1;-x), which I
>>>>>> assume is Kummer's function. Is it possible that the comment is out of
>>>>>> date and the values of Kummer's function used to make the table were
>>>>>> different? Any help would be greatly appreciated.
>>>>>>
>>>>>> John Ridges
>>>>>>
>>>>>>
>>>>>> _______________________________________________
>>>>>> Speex-dev mailing list
>>>>>> Speex-dev at xiph.org
>>>>>> http://lists.xiph.org/mailman/listinfo/speex-dev
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>>             
>>>>>           
>>>>         
>>>
>>>
>>>
>>>       
>>
>>     
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