PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Analog and Digital Filters 573

R.6.54 Practical fi lters often used in applications (appliances) are named after the research-

ers, often mathematicians who studied the polynomials that make up the transfer

function that best matches the ideal fi lter characteristics.

The most common practical fi lters are referred to as

a. Butterworth

b. Chebyshev

c. Cauer (elliptic)

d. Bessel

These fi lters return characteristics based on the facts that the system transfer

function presents regions with smooth or ripple (oscillations) behavior.

R.6.55 The Butterworth fi lter is probably the most popular practical fi lter. The fi lter char-

acteristics consist of maximum fl at response in the pass-band region, with no rip-

ples either in the pass band or stop band.

The basic design approach consists of choosing an appropriate model of the

transfer function H(jw). From this transfer function, the complex conjugate H(−jw)

is then obtained, and from the product H(jw) * H(−jw) = H(jw)^2  the poles of the

transfer function can be evaluated.

The Butterworth magnitude-squared response of an n-order LPF is given by the

following equation:

H(jw)*H( jw) H (jw)a



^2

1

1

2

w

wp

n





























(The subindex a in H stands for analog.)

The normalized Butterworth magnitude is given by

FIGURE 6.12

Second-order BRF employing simple fi rst-order LP and HP fi lters.

HLP(w)

HHP(w)

HBR(w)

Vi

Vo

w 1

w 2 w

w

∑