How to Make a 2nd Order Low Pass Filter


A low pass filter is a mathematical system that filters out all but low frequencies from an input signal. Low pass filters are among the most popular and most essential systems used in analog and digital audio signal processing. Simply put, low pass filters work by delaying the input signal, multiplying the delayed signal by a specific value and then adding this signal back to the original input signal. A filter is said to be 2nd order when it uses at most two delays in any part of its system.

  • Determine your cutoff and sampling frequencies. The cutoff frequency (fc) is the highest frequency allowed to pass through your low pass filter, where frequency is measured in cycles per second. Pick this value based on the frequencies you want to pass through your system. The sampling frequency (fs) is how many samples there are per second in your input signal, e.g., digital audio signals typically have 44,100 samples per second.

  • Solve for the angular cutoff frequency (Oc). The angular cutoff frequency is measured in units of radians and is equal to the cutoff frequency multiplied by 2 pi and then divided by the sampling frequency. Mathematically, the equation appears as: Oc= (2pifc) / fs.

  • Calculate the beta value (B), which is a value used in later steps to solve for the coefficients in the final equation. The beta-value equation expressed in mathematical form is: B= 0.5 ((1 - (pi sin[Oc] / (2Oc))) / (1 + (pi sin[Oc] / (2*Oc)))).

  • Obtain the gamma value (G), which is another value used in later steps to solve for the final equation coefficients.

    G= (0.5 B) cos(Oc)

  • Solve for the three feed-forward coefficients (a0, a1 and a2) of the final equation. In signal processing, feed-forward refers to the sections of a filter system that delay the input signal.

    a0= (0.5 + B - G) / 2

    a1= 0.5 + B - G

    a2= a0

  • Compute the two feedback coefficients (b1 and b2) of the final equation. Feedback refers to the sections of a filter system that delay the output signal.

    b1= -2 * G

    b2= 2 * B

  • Plug the coefficients into the final equation. The final equation of a second order low pass filter is:

    y[n]= a0x[n] + a1x[n-1] + a2x[n-2] - b1y[n-1] - b2*y[n-2]

    The output and input signals are represented by the characters y and x respectively. The character n is the index into the signals, i.e., y[n] is equal to the nth sample in the output signal.

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