# Find minimum sampling rate of non-ideal notch filter

Old test problem:

"You have been asked to design a real-time digital filtering system that eliminates a band of frequencies between 20 and 30 MHz but preserves everything else. The system uses an analog-to-digital converter (ADC) to digitize the analog input, a digital signal processing chip (DSP) to filter the digitized signal, and a digital-to-analog (DAC) converter to convert the signal back to analog. Assume that ideal sampling is used as well as ideal anti-aliasing and reconstruction filters. The maximum frequency that needs to be preserved in the reconstructed analog signal is 90 MHz.

What is the lowest sampling rate allowable in this system given the above constraints and giving that the anti-aliasing and reconstruction filters are not quite ideal: they have 6 MHz transition bands?"

If it was non-ideal it would be the Nyquist equation:

fs > 2*fmax

where fmax = 90 MHz, so,

fs > 2*90MHz

fs > 180 MHz

In the non-ideal situation, my thought would be to add the 6 MHz transition bands to fmax. Since it is a notch filter there are two transition bands. Meaning it would be

90 MHz + 6 MHz + 6 MHz = 102 MHz

I was told that the correct set-up to solve this problem is as follows:

fs > 2*(90MHz + (6MHz/2))

fs > 2*(93)

fs > 186

He mentioned something about not caring whether the transition bands overlap.

My question is why is the 6 MHz transition band divided by 2? Is it because there are two transition bands? And if it were a highpass or lowpass filter, where there is only one transition band, would it not be necessary to divide by two?

## 1 Answer

If you sample at 2*93 = 186 MHz, then the frequencies from 93 to 96 MHz will alias into the range from 90 to 93 MHz. However, since the problem requires that only frequencies from 0 to 90 MHz are preserved, you are not concerned with aliasing occuring after 90 MHz.

• Ok, so in the case of a high pass filter, say the frequency is passed from 30 to 90 MHz with the same 6 MHz transition, it would also be 186 MHz. Correct? – Dr. Derty Dec 6 '15 at 22:59
• I believe so, yes. A high pass filter at 30 MHz will not introduce any aliasing or change your sampling frequency in any way (unless you do bandpass sampling, but that's a different problem). – MBaz Dec 6 '15 at 23:17