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I am writing the theoretical background in my thesis but my thesis is more centered on deep learning and the signals that I am using for classification were filtered through the [b, a] = butter(n,[Wl, Wh]) and after with the function Y = filter (b, a, X) made by another author, I saw by typing: help butter at the command line from Octave that there is a book called Proakis & Manolakis (1992). Digital Signal Processing. New York: Macmillan Publishing Company. listed in the references from the butter function, and that when the butter function related from the Butterworth low-pass filter is used through a two-column vector it imposes that it is a band-pass filter with edges pi*Wl and pi*Wh radians.

Is there any theoretical approach of how it is applied, or any pseudo algorithm where is possible to see which formulas are used? I have tried to debug these functions using step into but they are very extensive until his finish, and at the book, I only saw an example to determine the order and the poles of a lowpass Butterworth filter, and I wanna know if after determining them, so am I supposed to convert the analog low-pass filter to a digital low-pass filter and after to a digital band-pass filter to arrive in the theory behind the application of these functions?

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A Butterworth filter is a a maximally flat, continuous time (analog) filter. That is it has the flattest passband possible, for a given cutoff frequency and filter order.

Frequency Response

The frequency response of a low-pass Butterworth filter is given by: $$G^2(\omega) = |H(j\omega)|^2 = \frac{G_0^2}{1+\left(\frac{j\omega}{j\omega_c}\right)^{2n}}$$ where $G(\omega)$ is the gain as a function of the radian frequency ($\omega = 2\pi f$), $H(j\omega)$ the transfer function of the filter, $H(s)$ evaluated at $s = j\omega$, $G_0$ the DC gain (gain at zero frequency), $n$ the filter order, $\omega_c$ the cutoff frequency, and $j$ the imaginary unit.

Often filter prototypes are defined with $G_0 = 1$ and that will be assumed here.

The cutoff frequency is the frequency at which the gain of the filter has been reduced by a factor of 2, approximately -3 dB ($10\log_{10}(1/2) \approx 3.01$).

The figure below is a plot of the frequency response for a lowpass Butterworth filter with order varying from 1 to 5 with $\omega_c = 1$.

Plot of the gain of Butterworth low-pass filters of orders 1 through 5, with cutoff frequency ω0=1.
Image from Wikipedia.

Transfer Function

The transfer function can be expressed as a ratio of a numerator polynomial and a denominator polynomial. For the low pass Butterworth filter the numerator is always $1$ and the denominator is a function of the filter order. For normalized cutoff, $\omega_c = 1$: $$H(s) = \frac{1}{B_n(s)}$$ where for even $n$: $$B_n(s) = \prod_{k=1}^{n/2} \left[s^2 - 2s\cos\left(\frac{2k+n-1}{2n}\pi\right)+1\right]$$ and for odd $n$: $$B_n(s) = (s+1)\prod_{k=1}^{(n-1)/2} \left[s^2 - 2s\cos\left(\frac{2k+n-1}{2n}\pi\right)+1\right]$$

The first few value are |n|$B_n$(s)| |-|--------| |1|$(s+1)$ | |2|$(s^2 +\sqrt{2}s+1)$| |3|$(s+1)(s^2+s+1)$|

Low-Pass to Band-Pass Transformation

Next, the prototype filter needs to be converted from low pass to band pass. Assuming a normalized prototype, $\omega_c = 1$, this can be achieve with a substitution of variables: $$H(j\omega) \rightarrow H\left(Q\left[\frac{j\omega}{\omega_0} + \frac{\omega_0}{j\omega}\right]\right)$$ where $Q=\frac{\omega_0}{\Delta\omega}$, $\omega_0$ is the center of the bandpass passband, and $\Delta\omega$ is the bandwidth of the passband.

Conversion to Discrete Time (Digital)

According to the MATLAB documentation for the butter function, it uses the bilinear transformation with frequency prewarping to convert from continuous to discrete domain.

To provide a more faithful digital response, prewarping is applied to the cutoff frequencies of the band-pass filter (as it is bandpass there is a low and an upper cutoff). This would be done in the previous section when computing the frequency mapping to convert from low pass to bandpass.

The prewarping transformation is given by: $$\omega_p=\frac{2}{T}\tan\frac{\omega_1 T}{2}$$ where $\omega_p$ is the prewarped cutoff frequency, $\omega_1$ is the corresponding cutoff frequency prior to prewarping, and $T$ is the sample period of the desired discrete time filter.

Finally, the bilinear transformation is applied to convert from continuous to discrete time: $$s \rightarrow \frac{2(1-z^{-1})}{T(1+z^{-1})}$$

With this substitution made, the desired discrete time transfer function, $H(z)$, is arrived at.

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  • $\begingroup$ "it has the flattest passband possible, for a given cutoff frequency and filter order" -- among the all-pole filters, otherwise the inverse Chebyshev has the flattest passband. $\endgroup$ Mar 14 at 11:01
  • $\begingroup$ Thank you for the help @GrapefruitIsAwesome!!! It's exactly it what I was needing!! By the way, do you have the references from these equations and the explanation is possible?? I would like to reference them in my defense master dissertation!! $\endgroup$
    – Victor
    Apr 6 at 19:56

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