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John Ridges

2009-Jul-22 20:49 UTC

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

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 ?crit :
>>>
>>>       
>>>> 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
>>>>
>>>>
>>>>
>>>>         
>>>
>>>       
>>
>>     
>
>
>
>
>

Jean-Marc Valin

2009-Jul-22 21:18 UTC

head link

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

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 ?crit :
> >>>
> >>>
> >>>> 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
> >>>>
> >>>>
> >>>>
> >>>>
> >>>
> >>>
> >>
> >>
> >
> >
> >
> >
> >
>
>
>

John Ridges

2009-Jul-22 22:18 UTC

head link

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

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 ?crit :
>>>>>
>>>>>
>>>>>           
>>>>>> 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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