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
|
