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I am referring to the calculations in https://courses.cs.washington.edu/courses/cse474/18wi/labs/l8/QRSdetection.pdf for the second order Lynn's low pass filter.

On the third page, the author mentions the transfer function used as:

$$ H(z)=\frac{(1−z^{-6})^2}{ (1−z^{-1})^2 } $$

The amplitude frequency response for this is given as: $$ |H(e^{j\omega T})|=\frac{\sin^2(3\omega T)}{\sin^2(\omega T/2)} $$ Now, he has calculated the 3dB cutoff frequency to be around 11 Hz for the sampling frequency of 200 Hz (i.e. sampling time = 0.005 seconds). Substituting these values in the above equation for amplitude response, I get: $$ |H(e^{j\omega_c T})|=\frac{\sin^2(3 \cdot 11 \cdot 0.005)}{\sin^2(11 \cdot 0.005/2)} = \frac{0.026979}{0.000756} = 35.69 $$ At the 3dB cutoff frequency indicated, shouldn't the amplitude response work out to around 0.707 and not 35.69? Where is my understanding incorrect?

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    $\begingroup$ Did you forget that $\omega = 2\pi f$ ? $\endgroup$
    – Ben
    Mar 22, 2021 at 13:03
  • $\begingroup$ @Ben , yes that was it! Now |H(ωcT)| works out to 25 which is ~ 0.7 of the peak amplitude (gain) of 36. Hope this understanding is correct $\endgroup$
    – WebDev
    Mar 22, 2021 at 13:13
  • $\begingroup$ That's one weird filter with pole cancellation. You can just do an FIR with b = [1 2 3 4 5 6 5 4 3 2 1] and get the same result. $\endgroup$
    – Hilmar
    Mar 22, 2021 at 13:21
  • $\begingroup$ @Hilmar yes that is possible, but it is expressed in a recursive form here so that it has a low computational cost. For details, you can have a look at section 7.1.4 of fdocuments.in/document/… $\endgroup$
    – WebDev
    Mar 22, 2021 at 13:39
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    $\begingroup$ I see the intent but I don't think it's a correct assumption, at least not these days. You can break this down into a simple FIR filter with b = [1 0 2 0 3 0 2 0 1] followed by two first order integrators. This will on most platforms be significantly faster than a recursive implementation. It's about the same number of arithmetic operations but much easier to pipeline efficiently and it can be made parallel and make use of SIMD instructions. $\endgroup$
    – Hilmar
    Mar 22, 2021 at 14:42

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Here's your mistake, you forgot that $ \omega = 2\pi f$

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