The poles are plotted as the red x’s on the left side of Figure 4. ![]() = butter(6,2*fc/fs) % Matlab function for Butterworth LP IIR (Note butter is a function in the Matlab signal processing toolbox that synthesizes IIR Butterworth filters). The following code finds the unquantized poles of the 6 th order Butterworth filter with -3 dB frequency f c = 5 Hz. To illustrate this, we’ll first look at how quantizing coefficients effects z-plane pole locations of a 6 th order IIR filter. Top: response of each biquad section (blue= h1, green= h2, red= h3).Īs I stated at the beginning, the cascaded-biquad design is less sensitive to coefficient quantization than a single high-order IIR, particularly for lower cut-off frequencies. 6 th order lowpass Butterworth cascaded-biquad response. With the peaking response (h3) last minimizes the chance of clipping.įigure 2. Sequence of the biquads doesn’t matter in theory however, placing the biquad The magnitude response of each biquad and the overall We can compute the frequency response of each biquad. The gains for each biquad are, from equation 4: As we already determined, the numerator coefficients b are the same for all three biquads: Here is the function call and the function output:Įach row of the matrix a contains the denominator coefficients of a biquad. Note biquad_synth contains code developed in an earlier post on IIR Butterworth filter synthesis. biquad_synth computes the denominator (feedback) coefficients a of each biquad. The filter will consist of three biquads, as shown in Figure 2. Note biquad_synth uses the bilinear transform with prewarping to transform H(s) to H(z). In this example, we’ll use biquad_synth to design a 6 th order Butterworth lowpass filter with -3 dB frequency of 15 Hz and f s= 100 Hz. This is evident from Equation 3.įigure 1 Biquad (second-order) lowpass all-pole filter Will cascade N/2 biquads to implement an NĬoefficients b have the same value for all N/2 biquads in a filter. ![]() Summarizing the coefficient values, we have:Ī biquad lowpass block diagram using the Direct form II Where a = are the denominator coefficients of the biquad section. If we stipulate that N is even, then we can write H(z) as: Our goal is to convert H(z) into a cascade of second-order sections. The N zeros at z = 1 (ω= π or f = f s/2) occur when we transform the lowpass analog zeros from the s-domain to z-domain using the bilinear transform. This article is available in PDF format for easy printing
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