# 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 ? Apr 26, 2019 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. Apr 26, 2019 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 Apr 26, 2019 at 3:39
• But you wont have enough GBW to do that (2.5GHz) Apr 26, 2019 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. Apr 26, 2019 at 6:56 