Adaptive Filter | ADPCM | ADSP | ADSP-2181 | Aliasing | AMR | Anti-Aliasing | ARMA | Autocorrelation | AutoCovariance | Beamforming | Bessel | Blackfin | Butterworth | C6713 | CCS | Chebyshev | CIC Filter | Circular Convolution | Code Composer Studio | Comb Filter | Compression | Convolution | Cross Correlation | DCT | Decimation | Deconvolution | Demodulation | DM642 | DSP Boards | DSP/BIOS | DTMF | Echo Cancellation | Equalization | Equalizer | ETSI | EZLITE (Ez-kit Lite) | FFT | FFTW | FIR Filter | Fixed Point | FSK | G.711 | G.723 | G.729 | Gaussian Noise | Goertzel | GPIO | Hilbert Transform | IFFT | IIR Filter | Interpolation | Invariance | JTAG | Kalman | Laplace Transform | Levinson | LPC | McBSP | MIPS | Modulation | MPEG | Multirate | Notch Filter | Nyquist | OFDM | Oversampling | Pink Noise | Pitch | PLL | Polyphase | QAM | QDMA | Quantization | Quantizer | Radar | Random Noise | Reed Solomon | Remez | Resampling | RTDX | Sampling | Sharc | TI C6711 | Undersampling | Viterbi | Wavelets | White Noise | Wiener Filter | Windowing | XDS510PP | Z Transform

OK:  I'm posting this because I'm lazy.  I haven't even _tried_ to look 
this up yet, or to attempt to choke an answer out of Maxima.  If there 
_is_ an answer, I'm sure it'll be a Bessel function, and those things 
always hit me like headlights do a deer.

So, anyone feel like some math hand-holding?  Here's the integral I'm 
trying to find an answer for:

integral from -pi to pi, e^(-a + b * cos(theta)) d theta,

a > b > 0.

This seems to be the class of problem that I can answer by myself, but 
only after I've beat my head against the wall for ages or held myself 
out for public ridicule -- I'm in a hurry, so I'm going to embarrass 
myself _first_, in hopes of finding the solution quicker.

-- 
Tim Wescott
Control system and signal processing consulting
www.wescottdesign.com

______________________________
New DSP Code Snippets Section now Live.   Learn more about the reward program for contributors here.

Tim Wescott wrote:
> OK:  I'm posting this because I'm lazy.  I haven't even _tried_ to look 
> this up yet, or to attempt to choke an answer out of Maxima.  If there 
> _is_ an answer, I'm sure it'll be a Bessel function, and those things 
> always hit me like headlights do a deer.
> 
> So, anyone feel like some math hand-holding?  Here's the integral I'm 
> trying to find an answer for:
> 
> integral from -pi to pi, e^(-a + b * cos(theta)) d theta,
> 
> a > b > 0.
> 
> This seems to be the class of problem that I can answer by myself, but 
> only after I've beat my head against the wall for ages or held myself 
> out for public ridicule -- I'm in a hurry, so I'm going to embarrass 
> myself _first_, in hopes of finding the solution quicker.
> 
Worse, maybe it's _not_ covered by Bessel functions, and I'm _really_ on 
my own.  Argh.

(Proceeding to plug away at the math).

-- 
Tim Wescott
Control system and signal processing consulting
www.wescottdesign.com

______________________________
New DSP Code Snippets Section now Live.   Learn more about the reward program for contributors here.

Tim Wescott <t...@seemywebsite.now> wrote:
 
> So, anyone feel like some math hand-holding?  Here's the integral I'm 
> trying to find an answer for:
 
> integral from -pi to pi, e^(-a + b * cos(theta)) d theta,
 
> a > b > 0.

Well, first I would factor out exp(-a), as that is constant.

Next I put it into http://integrals.wolfram.com/

which says that "Mathematica could not find a formula for your
integral."

But that only does indefinite integrals, and this seems like a
case where the definite integral might be possible to evaluate
even though the indefinte integral doesn't have an analytical
solution.  (Not that you can find a numeric solution for an
indefinite integral.)

-- glen 

______________________________
New DSP Code Snippets Section now Live.   Learn more about the reward program for contributors here.

Tim Wescott wrote:
> OK:  I'm posting this because I'm lazy.  I haven't even _tried_ to look 
> this up yet, or to attempt to choke an answer out of Maxima.  If there 
> _is_ an answer, I'm sure it'll be a Bessel function, and those things 
> always hit me like headlights do a deer.
> 
> So, anyone feel like some math hand-holding?  Here's the integral I'm 
> trying to find an answer for:
> 
> integral from -pi to pi, e^(-a + b * cos(theta)) d theta,
> 
> a > b > 0.
> 
> This seems to be the class of problem that I can answer by myself, but 
> only after I've beat my head against the wall for ages or held myself 
> out for public ridicule -- I'm in a hurry, so I'm going to embarrass 
> myself _first_, in hopes of finding the solution quicker.

I'm too rusty to see why iterated integration by parts won't work.

Jerry
-- 
Discovery consists of seeing what everybody has seen, and thinking what
nobody has thought.    .. Albert Szent-Gyorgi
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

______________________________
New DSP Code Snippets Section now Live.   Learn more about the reward program for contributors here.

On Mar 31, 2:24=A0pm, Tim Wescott <t...@seemywebsite.now> wrote:


>
> integral from -pi to pi, e^(-a + b * cos(theta)) d theta,


Yes, it is a Bessel function (times a constant).
Look at Abramowitz and Stegun, 9.6.16 which says
that

I_0(z) =3D (1/pi) integral from 0 to pi e^(z cos theta) d theta

where I_0(z) is a modified Bessel function of the first kind
and order 0.  Shows up a lot in studies of noncoherent
FSK, DPSK and the like and in the Ricean pdf which
is the pdf of the amplitude of a sinusoid plus narrowband
Gaussian noise.

Hope this helps.

--Dilip Sarwate

______________________________
New DSP Code Snippets Section now Live.   Learn more about the reward program for contributors here.

Glen wrote:
>Tim Wescott <t...@seemywebsite.now> wrote:
> 
>> So, anyone feel like some math hand-holding?  Here's the integral I'm 
>> trying to find an answer for:
> 
>> integral from -pi to pi, e^(-a + b * cos(theta)) d theta,
> 
>> a > b > 0.
>
>Well, first I would factor out exp(-a), as that is constant.
>
>Next I put it into http://integrals.wolfram.com/
>
>which says that "Mathematica could not find a formula for your
>integral."
>
>But that only does indefinite integrals, and this seems like a
>case where the definite integral might be possible to evaluate
>even though the indefinte integral doesn't have an analytical
>solution.  (Not that you can find a numeric solution for an
>indefinite integral.)


Well, ignoring the exp(-a) as Glen did, I put it in Mathematica and
obtained:

Integrate[Exp[b*Cos[th]], {th, -pi, pi}]

(where I've typed out "pi" for your newsreader's sake)

And also ran:

FullSimplify[%, Im[b] == 0 && Re[b] > 0]

The result was:

2 pi (BesselI[0, b] - StruveL[0, b])

(...I'll post again if I see any font problems in google...)

HTH,
Michael

______________________________
New DSP Code Snippets Section now Live.   Learn more about the reward program for contributors here.

"glen herrmannsfeldt" <g...@ugcs.caltech.edu> schrieb im Newsbeitrag 
news:hp096e$blq$4...@naig.caltech.edu...
> Tim Wescott <t...@seemywebsite.now> wrote:
>
>> So, anyone feel like some math hand-holding?  Here's the integral I'm
>> trying to find an answer for:
>
>> integral from -pi to pi, e^(-a + b * cos(theta)) d theta,
>
>> a > b > 0.
>
> Well, first I would factor out exp(-a), as that is constant.
>
> Next I put it into http://integrals.wolfram.com/
>
> which says that "Mathematica could not find a formula for your
> integral."
>
> But that only does indefinite integrals, and this seems like a
> case where the definite integral might be possible to evaluate
> even though the indefinte integral doesn't have an analytical
> solution.  (Not that you can find a numeric solution for an
> indefinite integral.)
>


>> integral from -pi to pi, e^(-a + b * cos(theta)) d theta,

For definite integral use Gaussian Integration:

* http://home.arcor.de/janch/janch/_news/20100401-numerical-integration/

EXAMPLE: Page 1
PROGRAM: Page 2


-- 
Regards JCH




u
> -- glen

  

______________________________
New DSP Code Snippets Section now Live.   Learn more about the reward program for contributors here.

Tim Wescott <t...@seemywebsite.now> writes:

> OK:  I'm posting this because I'm lazy.  I haven't even _tried_ to
> look this up yet, or to attempt to choke an answer out of Maxima.  If
> there _is_ an answer, I'm sure it'll be a Bessel function, and those
> things always hit me like headlights do a deer.
>
> So, anyone feel like some math hand-holding?  Here's the integral I'm
> trying to find an answer for:
>
> integral from -pi to pi, e^(-a + b * cos(theta)) d theta,
>
> a > b > 0.
>
> This seems to be the class of problem that I can answer by myself, but
> only after I've beat my head against the wall for ages or held myself
> out for public ridicule -- I'm in a hurry, so I'm going to embarrass
> myself _first_, in hopes of finding the solution quicker.

Mathematica 7.0:

In[1]:= Integrate[E^(-a+b*Cos[theta]),{theta,-Pi,Pi},Assumptions->a>b>0]

        2 Pi BesselI[0, b]
Out[1]= ------------------
                 a
                E

Scott
-- 
Scott Hemphill	h...@alumni.caltech.edu
"This isn't flying.  This is falling, with style."  -- Buzz Lightyear

______________________________
New DSP Code Snippets Section now Live.   Learn more about the reward program for contributors here.