Sinc weirdness

Cool.

Tim

-- Seven Transistor Labs, LLC Electrical Engineering Consultation and Design Website:

On a sunny day (Tue, 29 Oct 2019 21:44:28 +0100) it happened Jeroen Belleman wrote in :

Interesting, never knew that, because did not pay attention in math class as I was sitting..

And only fairly recently discovered too 2001. It converges to pi provided that the sum of the reciprocals of 1/(2n+1) stays under 1.

A more accessible intuitive proof that isn't behind a paywall is at:

Thanks for pointing this one out. I would never have guessed that it broke down after working OK for so many terms.

Wildly oscillatory integrals can be very tricky if you don't choose your path wisely.

Martin Brown wrote in news:qpbit1$7pv$ snipped-for-privacy@gioia.aioe.org:

Yes.. one could end up back in 1955 at a high school dance.

Are you related to Emmett Brown? :-)

Interesting, thanks! (that goes for Jeroen too.)

George H

What is the significance, if any, of pi? And who encounters these oddball i ntegrals of products in any practical application of signal processing? Wha t good is this useless factoid? Answer: none. Best I can find is it might h ave something to do with research mathematicians falling into some psycholo gical trap about the value of induction based on limited evidence, for what that's worth.

The math essentially describes the effect of successive, repeated moving-average filters. This is quite common in signal processing and therefor perfectly relevant.

Jeroen Belleman

Multiple successive moving-average filters, sounds odd, can you describe a few examples of that?

Thinking about math, especially oddball math, keeps an engineer's brain in tune. Obsessing on gloom and doom might not.

Do you have no use for pi?

As soon as the kids are finished with this big laser controller project, they can do the FPGA for my alternator simulator. I'm picking off currents and voltages with the ADUM isolated delta-sigma converter, and they will have to filter the bit stream into 16-bit integers, which will be feedbacks into a control loop. One common delta-sigma recovery filter is called sinc3, which is made out of rectangular summers, basically what Jeroen is discussing. It has the bouncing-ball frequency response, not my favorite thing to put inside a control loop.

Wiki talks about sinc filters.

I prefer integrator-based IIR filters, but they are trickier inside an FPGA.

Weird. Is this the normalised or non-normalised sinc function?

Clifford Heath.

sinc(x/N) has transform N rect(x/N).

If you think about it in the frequency domain, convolving a wide rectangle with a succession of narrower rectangles leaves the DC value the same until the sum of the widths of the narrow rectangles is more than half the width of the wide one. The DC value of the convolution is the integral over all X of the product of the functions.

Not rocket science--anybody Bracewell ever taught Fourier to would get that in three seconds.

Cheers

Phil Hobbs

N rect(NX), that is.

Cheers

Phil Hobbs

...

..and three minutes :-)

I wouldn't say it was common but it is sometimes done.

Unweighted boxcar moving average is about the simplest to implement hardware filter that there is. And if you nest a few of them the same then the result is a fair approximation to Gaussian convolution (and would still be for all engineering purposes even in the failing case being discussed here).

It is still an interesting and curious result though.

Truncated and Gaussian weighted sinc was used in the past for regridding frequency domain data onto a rectangular grid prior to using the FFT. It has artefacts so modern methods use better functions which have the nice property that like a Gaussian they are their own FT but only when truncated to a fixed length in one of the domains. This gives very nice antialiasing properties at the expense of needing a guard band around the edges. Prolate spheroidal Bessel functions are the canonical one.

One possibility that we are considering for our control loop is to dump a 20 MHz delta-sigma bit stream into a 16-bit shift register and use that as the feedback signal into our P+I controller.

Even more extreme would be to take a 1 and translate that to 0x7FFF, and take a 0 and treat that as 0x8000 into the loop.

Might work.

iptechnology.com:

it is possible to do all you processing directly on the fast 1 bit stream, remember doing a bit on filters working like that at uni. The rationale bei ng that multiplier become simple multiplexers, but since it also has to run at the much higher rate and filters will have to be that much longer I'm not sure there's much to gain other than the headache of making it work

I somehow ended up with two copies of "Probability and Information Theory, with Applications to Radar" by Phillip Woodward, 1953.

It seems improbable.