# Calculating sampling frequency off of estimated stop band attenuation

I'm trying to calculate what I will have to set my microcontroller's ADC sampling rate at in order to sample a band of 500kHz to 1.6065MHz without aliasing. Some constraints here are that I have to work with only 3 high gain-bandwidth product op-amps (conceivably at my level of expertise this means designing a maximum 6th order filter or 3 2nd order stages cascaded) and the maximum sampling rate the ADC can do is 7MSPS in a triple interleaved mode.

I read in a reference book that for a 10 bit ADC the average attenuation at the end of the transition band and beginning of the stop band should be about 62dB down. From the equation for a butterworth filter I work out the sampling rate as follows:

$$\A_{min} = 62dB = 20log_{10}(1+(\frac{f'max}{fc})^6)^\frac{1}{2}\$$ where $$\f'max\$$ is the frequency the stopband begins at and $$\f_c\$$ is the cutoff frequency $$\1.6065MHz\$$. The power 6 comes from the filter order.

$$\(10^{\frac{62}{20}})^2-1\ = (\frac{f'max}{fc})^6\$$

$$\f'_{max} = 17346kHz (17.3MHz) \$$

and according to my reference source $$\F_s=2f'max\$$ meaning Fs needs to be like $$\34MHz\$$

This seems really high. Is there a flaw in my math or my reasoning here? It seems that the higher the filter order the lower the sampling frequency needs to be but due to the 3 opamp constraint I can't implement something upwards of a 10th order (for a 10th order filter I work out Fs = 13MHz) and the ADC does a max of 7MSPS.

• What phase response is required ? – Tony Stewart Sunnyskyguy EE75 Apr 26 '19 at 3:27
• @SunnyskyguyEE75 I don't think phase response is much of a concern. The project aim is to sample a simple sinusoid amplitude modulated to within the band mentioned above. Then to display the FFT of it on an LCD screen, demodulate in software and output the result via DAC. – Blargian Apr 26 '19 at 3:28
• Yes you are right, you need a 10th order filter to pass 1.6MHz at -3dB and stop -60 dB at 3.5MHz for a fs=7MHz – Tony Stewart Sunnyskyguy EE75 Apr 26 '19 at 3:39
• But you wont have enough GBW to do that (2.5GHz) – Tony Stewart Sunnyskyguy EE75 Apr 26 '19 at 3:43
• @Blargian - a)A Butterworth filter may not be your best choice here; an elliptical filter can provide a much much sharper cutoff. b)You should only care about aliasing that folds on top of your signal (0.5~1.6MHz), so if f'max is the frequency where you get the attenuation you want, you only need Fs>1.6MHz + f'max., as opposed to Fs>2*f'max. – joribama Apr 26 '19 at 6:56 