##### From Signals and Systems with MATLAB Applications, Second Edition

## 11.4 Design of Butterworth Analog Low-Pass Filters

We will consider the *Butterworth low-pass filter* whose amplitude-squared function is

where *k* is a positive integer, and ? _{ C} is the cutoff ( *3 dB*) frequency. Figure 11.10 shows relation (11.20) for *k* = *1, 2, 4*, and *8*. The plot was created with the following MATLAB code.

<span class="serif">w_w0=0:0.02:3; Aw2k1=sqrt(1./(w_w0.^2+1)); Aw2k2=sqrt(1./(w_w0.^4+1));...Aw2k4=sqrt(1./(w_w0.^8+1)); Aw2k8=sqrt(1./(w_w0.^16+1));...plot(w_w0,Aw2k1,w_w0,Aw2k2,w_w0,Aw2k4,w_w0,Aw2k8); grid</span>

Figure 11.10: Butterworth low-pass filter amplitude characteristics

All Butterworth filters have the property that all poles of the transfer functions that describes them, lie on a circumference of a circle of radius ? _{ C}, and they are *2* ?/ *2k* radians apart. Thus, if *k* = *odd*, the poles start at zero radians, and if *k* = *even*, they start at *2* ?/ *2k*. But regardless whether *k* is odd or even, the poles are distributed in symmetry with respect to the *j* ? axis. For stability, we choose the left half-plane poles to form *G*( *s*).

We can find the *nth* roots of a the complex number *s* by *DeMoivre's theorem*. It states that

Derive the transfer function *G*( *s*) for the third order ( *k* = *3*) Butterworth low-pass filter with *normalized cutoff frequency* ? _{ C} = *1 rad*/ *s*.

**Solution:**

With *k* = *3* and ? _{ c} = *1 rad*

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