Calculating the 3dB frequency of an nth order low pass filter and nth order high pass filter

Question

I am using the 6th order high pass filter in the C code. And the cut-off frequency selected for it is 150 Hz and the sampling frequency being used is 32000Hz. Now, I want to calculate the 3 dB frequency for this implementation (theoretically). I have read that the 3 dB frequency is the frequency at which the filter magnitude response is $1 / \sqrt{2}$ of the peak amplitude response. But I am not sure about how to calculate it? And also, I want to know how to calculate 3 dB frequency for an nth order low pass filter.

Note:

I am using the Chebyshev type II design in my code to implement a 6th order high pass filter.

Answer 1

In general there is no straightforward analytical solution. As you know, you need to solve

$$\left|H(e^{j\omega_c})\right|=\frac{1}{\sqrt{2}}\tag{1}$$

for $\omega_c$, where it is assumed that the maximum filter gain equals $1$.

For Butterworth filters, the specified cut-off frequency always equals the $3\textrm{ dB}$ frequency. This is not the case for other filter types.

In general, you have to either estimate $\omega_{c}$ from the magnitude plot, or you solve $(1)$ numerically.

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EDIT: For Chebyshev filters it's relatively straightforward to compute an analytical expression for the $3\textrm{ dB}$ cut-off frequency.

As an example I'll use a Chebyshev II highpass filter, as mentioned in the question. The magnitude of the normalized frequency response of the analog prototype filter is given by

$$\left|H(\Omega)\right|=\frac{1}{\sqrt{1+\frac{1}{\epsilon^2T_n^2(\Omega)}}}\tag{2}$$

where $T_n(\Omega)$ is the $n^{th}$-order Chebyshev polynomial, and $\epsilon$ is a parameter determining the ripple size in the stop band. The normalized $3\textrm{ dB}$ cut-off frequency $\Omega_c$ can be determined from the equation $\left|H(\Omega_c)\right|=1/\sqrt{2}$, which is equivalent to

$$T_n(\Omega_c)=\frac{1}{\epsilon}\tag{3}$$

From the trigonometric definition of the Chebyshev polynomials

$$T_n(\Omega)=\cosh\big[n\; \textrm{arcosh}(\Omega)\big],\qquad\Omega>1\tag{4}$$

we can solve $(3)$ for $\Omega_c$:

$$\Omega_c=\cosh\left[\frac{1}{n}\; \textrm{arcosh}\left(\frac{1}{\epsilon}\right)\right]\tag{5}$$

Now we just need to transform $\Omega_c$ to the discrete-time domain according to the warping introduced by the bilinear transform:

$$\omega_{c}=2\arctan\left[\Omega_c\tan\left(\frac{\omega_s}{2}\right)\right]\tag{6}$$

where $\omega_s$ is the specified stop band edge of the discrete-time Chebyshev II high pass filter.

Note that we can also rewrite $(6)$ to determine the required value of $\omega_s$ given a specified $3\textrm{ dB}$ cut-off frequency $\omega_c$:

$$\omega_s=2\arctan\left[\frac{1}{\Omega_c}\tan\left(\frac{\omega_c}{2}\right)\right]\tag{7}$$

The following Matlab/Octave script illustrates the procedure of computing the $3\textrm{ dB}$ cut-off frequency of a Chebyshev II high pass filter:

% example filter specs of Chebyshev II high pass filter
n = 6;          % filter order
Rs = 60;        % stop band ripple in dB
ws = .5 * pi;   % stop band edge of Chebyshev II filter

% Cheby II: stopband ripple = ep / sqrt( 1 + ep^2 )
ep = 1 ./ sqrt( 10^( .1 * Rs ) - 1 );

% compute normalized 3dB cut-off frequency of analog prototype
Wc = cosh( acosh( 1 / ep ) / n );

% transform 3dB cut-off frequency to discrete-time domain
wc = 2 * atan( tan( .5 * ws ) * Wc );       % bilinear transform

I designed a discrete-time $6^{th}$-order Chebyshev II high pass filter according to above specifications. The resulting $3\textrm{ dB}$ cut-off frequency obtained from above script is $\omega_{c}=0.69\pi$. This is the correct value as shown in the plot below:

Answer 2

I've used this method in Octave:

function [c] = Hz_at_3dB(b, a, fs, samples)
     [H,W] = freqz(b,a,samples);
      magresp = 20*log10(abs(H));
      maxresp = max(magresp);
      [I,~] = find(magresp < maxresp-3.0103,3,'first');
      c=(fs/2 * W(I(1)))/pi;

EDIT: SECTION 8.4: STANDARD RESPONSES - https://www.analog.com/media/en/training-seminars/design-handbooks/Basic-Linear-Design/Chapter8.pdf