# Shannon Sampling Theorem

For the data sampled in digital systems two conditions hold:

• The Signal must have a finite bandwidth: above a cutoff frequency all frequency components must be zero..
• The sampling frequency must be minimum twice the cutoff frequency of the signal.

These rules are called Shannon sampling theorem , or Nyquist Shannon Sampling Theorem.

If the conditions are not met, is for example the sampling frequency not minimum twice the cutoff frquency, there will be components in the spectrum that are not given in the signal.
If the lens resolution is for example higher than the pixel pitch’s frequency. we get Moiré effects. Yes, lenses can be “too good”.
See https://www.optowiki.info/faq/why-can-color-cameras-use-lower-resolution-lenses-than-monochrome-cameras/

This effect is called aliasing and results from mirroring of frequencies above into the frequencies below the cutoff limit..

The cutoff frequency is called „Nyquist-Frequency“.

In Optics the Nyquist-Frequency equals onle line per pixel = half a line pair per pixel.

At the Nyquist-Frequency the MTF reaches Zero. We don’t have contrast then.

The cutoff frequency depends on the FNumber and the wavelength.

cutoff frequency = 2 / Airydisk-Diameter = 1 / Airydiskradius = 1 / (1.22 * Wavelength * FNumber)
One can resolve 70% of the Nyquist frequency = 0.7*Nyquist frequency = Nyquist frequency / 1.41
If you want a contrast of about 20%, you should devide this Nyquist frequency by 1.41:
If you even want a chance to reach 50% contrast, you should even divide by 2.

This contrast is then reachend over a range of 4 pixels.

Lets say we have a sensor with 2.2um pixel pitch.Then the Nyquist frequency is 1/2 line pairs per Pixel.

On 1mm = 1000um fit 1000/2.2u =   454pixel.

So the Nyquist frequency = 0.5lp / pixel = (454/2) lp/mm = 227lp/mm.

there we have 0% Contrast.

To have a chance for 50% Contrast haben, we have to divide this value by 2 and get (227/2) lp/mm = 113.5 lp/mm.