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Date: Wed, 10 Dec 2003 11:15:45 +0000
From: "Stewart Bryant" <stewart.bryant@virgin.net>
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Subject: Re: LF: Related Technical Query - Soundcard calibrator
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Andy

According to

http://integrals.wolfram.com/index.en.cgi

integral (sin[x^2)]/(x^2)

is

sqrt(2 pi) * FresnelC[sqrt(2/pi) * x] - sin(x^2)/x

and according to

http://functions.wolfram.com/GammaBetaErf/FresnelC/02/

FresnelC(z) = integral from 0 to z (cos (pi/2 * t ^2)) dt

But there are a number of simpler approximations including
a series approximation that looks easy to compute.

I suspect that somewhere in your lab there is a copy
of mathematica, and that is almost certainly
the best way to evaluate your integral.

73

Stewart G3YSX

Andy talbot wrote:
> Is there anyone who can answer this...
> 
> I want to make a calibrator to enable accurate (better than 0.5%) absolute 
> audio measurements using a soundcard - ie. at audio frequencies to 20kHz. 
>   Generating a square wave from CMOS logic with a precisely 5V p-p (2.5v peak) 
> waveform is trivial and can be measured accurately to doubl;e check.  The 
> levels of the odd harmonic tones are all precisely defined by :
> 
> 	RMS Amplitude = 2.5V * 4 / PI / SQRT(2) / N
> 
> Where N is the (odd) harmonic number, so the fundamental has an amplitude of 
> 2.251 Vrms which can be potted down with an accurate potential divider - 
> straightforward, all harmonics can be used at known levels...
> 
> Now, here is the complicated bit, I also want to include a calibrated noise 
> source for accurate S/N measurements and evaluation of various decoding 
> techniques.  To remove the need for accurate true RMS measurement to set up 
> this unit, I am going to use Pseudo Random Sequence generated from a shift 
> register clocked very much faster than the frequencies of interest.  It is 
> straightforward to make a 2^32-1 bit long sequence clocked at, say, 5MHz which 
> will have a repeat cycle of  14 minutes, suitable for most purposes - even Argo 
> calibration !   By filtering to a bandwidth significantly less than the clock 
> rate (20kHz max vs. 5MHz) the result will be sufficiently Gaussian to act as if 
> it were true noise.  In practice,  any real hardware filtering won't be 
> necessary as the soundcard itself will do the job; but there will be some basic 
> filtering to keep out the nasty birdies likely to be generated.
> 
> Now, here is the bit I'm less sure how to work out. How do I calculate the 
> noise density of the PN sequence in the area of interest?   Assumimg the PRN 
> Sequence has 1:1 mark/space ratio (with 2^32 bits it will be certainly near 
> enough) and assuming a perfect square edged waveform up to several 5MHz clocks 
> away (guaranteed by using HC series logic) the TOTAL power over the (almost) 
> infinite spectrum from DC to several clock multiples is defined exactly by the 
> RMS value of the waveform; and for a squarewave this is equal to the peak value 
> - here 2.5V.
> 
> This TOTAL power is spread out in a SIN(x^2)/x^2 pattern.   Since it will be 
> filtered to, essentially, just a small part of the first spectral lobe, the 
> calculation really comes down to calculating the height of this lobe which will 
> can be assumed to be sufficiently flat over the small bandwidth of interest. 
>  Once that is done, the noise power, and hence the RMS value can be defined in 
>  Volts / SQRT(Hz) which can be added to the signal / test tone of interest in 
> variouis proportions, with the S/N then known exactly.
> 
> I'm quite incapable of integrating SIN(x^2)/x^2 from first principles (or any 
> other way for that matter :-) to calculate the level of the main lobe.  Can 
> anyone point out the best way to do this calculation ?    Instinct says that 
> for a small bandwidth segment, BW,  close to zero frequency, the power will be 
> proportional to  BW / Fclock,  but I haven't a clue what scaling factors will 
> be included in this, and they are what matter for an absolute level calibrator.
> 
> Andy  G4JNT
> 
> 
> 
>  
> 
>