# Archive for category math

### Microscope objectives – NA, cost, and parabolas

Posted by Michael Beach in math, metrology, optics, tech notes on December 14th, 2009

We were looking at various microscope objectives – those lenses on the turrets that aim toward the slides. Or, if you’re like me, the expensive silver thing that just went ‘crunch’ on the slide while I was trying to focus the image.

Pete noted that there seemed to be a parabolic curve fit – better NA, numerical aperture, better light collection, and the more expensive the objective lens gets. Here’s the curve, and the supporting data.

### Fourier Transform of 1/f Noise

Posted by Michael Beach in math, technical articles on September 29th, 2009

Here’s a curious article by Steve Smith, author of *The Scientist and Engineer’s Guide to Digital Signal Processing*, where he shows that **1/f noise is its own Fourier transform!**

link to article \”An Interesting Fourier Transform – 1/f Noise\”

I find it much easier to remember complicated ideas when there’s a clear graph. Smith shows a range of graphs, showing that “there is an inverse relationship; if the time domain decays faster, then the frequency domain decays slower, and vice-versa. This means that there must be a certain decay rate that is unique, where both domains are equal.”

Read the article – the mystery of 1/f noise continues. Perhaps the observation of this mathematical property will point toward learning the physical underpinnings that cause 1/f noise.

### Quantization noise

Posted by Michael Beach in books we like, low noise design, math, metrology, tech notes on August 27th, 2009

We notice the assertion that A/D converter quantization noise is equal to ADU/SQRT(12), where ADU is the quantization unit or LSB. We saw this in Hobbs’ excellent book *Building Electro-Optical Systems, Making It All Work*.

So, we decided to derive this. Took us a while to get the ‘trick’, and to remember how to perform calculus, to get that pesky root-mean-squared function.

Think of the quatization error as a sawtooth function that repeats. Then work out the RMS noise of that sawtooth wave (it happens to be the same as a triangle wave). And, yes, it does work out to that value.

Now the next part is Hobbs’ assertion that this quantization noise is not a Gaussian distribution. Get to work.

### CCD Cameras, eyes, and physics

Posted by Michael Beach in application notes, how-to, math, metrology, optics, tech notes, technical articles on March 14th, 2009

This tech note was motivated by the question – how does the response of our eyes

differ from the response of a CCD camera sensor.

Using the data of a particular Hammamatsu CCD camera as an example,

we compared how silicon ‘sees’ to the photopic eye response

and compared both to a Planck black-body curve of a light at a particular

color temperature.

We don’t know what those lumps are in that CCD response curve – maybe some

strange reflection interference??

If you know – tell us!

### Vision response vs. Planck’s Black Body Curve

Posted by Michael Beach in application notes, how-to, math, metrology, optics, tech notes, technical articles on March 14th, 2009

Color temperature is based upon the idea of a Planck black-body radiator.

Here’s a Tech Note that shows how our eyes respond to the Planck Black-Body radiator.

For a lamp filament at a certain ‘color temperature’ there’s a curve of how our eyes

respond to the lamp. Pete put this into a MathCAD model, and there’s a pdf here

that shows off a few nice graphs.

### Actinica Book List

Posted by Michael Beach in application notes, books we like, how-to, math, metrology, optics, research papers, tech notes on May 31st, 2008

Ok, we have a book problem.

Both of us waay like good engineering books. A good explanation, or a great

graph that sums up why that camera ‘sees’ differently than my eyes, etc.

Since we’re always stumbling on more good books, this list will grow.

Drop by later see what’s new.

Here’s some of the books we like, as a pdf file here,

and here’s some more books we like:

- the
, a 3 volume set. Here’s a guy who can explain anything well. Like how sine, cosine and the magic number*Feynman Lectures on Physics**e*all relate to the imaginary number*i*(square root of -1). He also has a great description of how a ’50 Ohm’ transmission line acts like ’50 Ohms’ no matter how long it is. For a really great puzzle – read his description of how charging a capacitor really involves*magnetic fields*the cap’s plates.**outside**