Design a filter that allows frequencies below 5 kHz to pass freely, but all frequencies above 5.2 kHz must be undetectable

Question

My biggest challenge with this question is its ridiculously steep roll-off rate.

I am assuming that the signal is undetectable if its gain is -20 dB. This means that, within the transition band of 200 Hz, the signal strength needs to drop by 20 dB.

If my calculations are correct, this filter requires a roll-off rate of 1200 dB/dec. That requires 60 poles, which is obviously not feasible.

I would like to use an analog active filter with minimal ripple in the pass band. A large phase shift is not too important.

One potential solution is to use a notch filter at 5.2 kHz. However, frequencies above the bandwidth of the notch filter are still not sufficiently filtered.

Please point out any flaws in my logic and or propose potential solutions. Thank you.

Answer 1

You have assumed a 20dB/dec per filter order roll-off for your filter. This is not true for all filter types.

Let \$f_0 = 5 \mathrm{kHz}\$ and \$f_{\mathrm{stop}} = 5.2 \mathrm{kHz}\$. Then $$\frac{f_{\mathrm{stop}}}{f_0} = 1.04.$$

Have a look at this fourth order elliptic filter taken from the Wikipedia article.

Although it does not quite meet your requirements you can see it is feasible. A higher order elliptic filter can achieve what you are after.

You should keep in mind that elliptic filters can do disturbing things to the phase of the signal. Since you did not mention anything about your phase constraints, I have assumed that an elliptic filter is suitable.

Answer 2

This kind of sharp rolloff requires a digital filter. Don't even think about analog. You need to convolve the input with a sinc function. The width of the sinc function (the number of kernel points) defines the stop band attenuation.

I haven't done the math, but some very quick (could be off, your job to do properly) calculations says you probably need a few 100 points if sampling at 20 kHz. 200 points at 20 kHz means a MAC rate of 4 MHz. That's doable, in fact well below what modern DSPs can do rather easily. That means your problem is quite tractable. Something like a E series dsPIC can do this, and that's rather low end if you're only looking for DSP capability.

Answer 3

If you allow a substantial delay or process a recorded signal you can simply do FFT , remove unwanted components and invert the transform. You must truncate the fft with proper window function to keep the ringing acceptable.

Answer 4

I'd pick an audio codec chip (ADC+DAC), route the ADC digital output to the DAC input, and set the sample rate to 10kHz.

These chips already include the digital filter engine you need. A quick datasheet check seems to confirm you'll get the filter performance you need.

Answer 5

You already have many nice answers with good traditional solutions, elliptic filters, (short time) FFT, etc., so I was thinking I can add something from the sub-band coding / wavelet transform world.

Sub-band coding means to subdivide the frequency spectrum into "bins", each of these bins has its own associated filter. The tighter bands, the broader filters in time domain (naturally) - but in areas where we don't need very tight bands we can get away with really short & cheap-to-calculate filters.

Wavelets are functions which are the result of a specific type of sub-band filters which are generated by iterated filtering followed by subsampling.

The idea would be to find the sub-bands of interest which would allow us to squeeze the computations the most, but still get good granularity at the band of interest.

Example of a Daubechies 12 tap packet decomposition in three levels (Wikipedia):

We can then selectively sum up these to get the response we want. And the ones which we don't want to add - we don't even have to compute! We will need slimmer ones closer to the 5-5.2 kHz band to be able to get steep enough behaviour. But on the other hand, far away from the 5-5.2 kHz band we can get away with only a few subdivisions.

Answer 6

if you can GUARANTEE a sinusoidal input, then a one-shot monostable (74121) may suffice. Or the re-triggerable 122/123.

Use a comparator prior to the 74121/122/123

Some MCUs include analog comparators as their peripherals; once converted to square wave, you can use timers etc to detect above/below 5,000Hz, if the MCU has a XTAL-stabilized clock. No need for a temperature-sensitive monostable.