Handbook for Sound Engineers

Filters and Equalizers 797

(23-53)

where,

P(s) and Q(s) are polynomials expressible in the form

P(s)=(s  p 1 )(s  p 2 )...(s  pn),

Q(s)=(s  q 1 )(s  q 2 )...(s qn).

23.3.2.1 Zeros

The zeros of the function are the values of s at which
P(s) is zero and consequently H(s) is zero. These occur
at values p 1 , p2, and so on, and represent frequencies at
which the transfer function exhibits maximum
attenuation.

23.3.2.2 Poles

The poles of the function are the values of s at which
Q(s) is zero and consequently H(s) is infinite. These
occur at values q 1 , q2, and so on, and represent frequen-
cies at which the transfer function exhibits maximum
gain.

23.3.2.3 Stability

A pole or a zero in the right-hand side of the s-plane

means that for that value of s, V is greater than zero. In

the time domain representation, the signal is given as

(23-54)

The term est may be expanded to eVt × ejZt. If the

value of V is greater than zero, the expression represents

an exponentially increasing factor, meaning that the

filter is unstable. This situation cannot arise in a passive

filter so they are inherently stable.

23.4 Switched Capacitor Filters

Any active filter based on resistive and capacitive

components may be reconfigured as a switched capac-

itor filter. The resistive elements are replaced by an

equivalent switched capacitive element. The advan-

tages of using switched capacitors in place of resistors

is that they are easier to implement in silicon, since

capacitors take up less space than resistors, and toler-

ances of capacitor-to-capacitor ratios can be more easily

controlled the resistor-capacitor products.

The circuit shown in Fig. 23-21 transfers charge, and

therefore current, between the two voltage sources

under control of the switch. The charge 'Q transferred

every switch period of length ts may be expressed in

terms of current Eq. 23-53 or voltage Eq. 23-54.

(23-55)

(23-56)

Combining these two equations we can find the equiva-

lent resistance.

(23-57)

The equivalent resistor value in Eq. 23-55 has a

fixed capacitive term and a frequency term. Its value

may be controlled by varying the switching frequency.

This makes switched capacitor filters ideal for filters

that need to be tuned.

23.5 Digital Filters

Filters may be implemented using entirely mathematical

means from their transfer function representations in the

time domain. The time and frequency domains are

Figure 23-20. Pole-zero plot with poles represented as X

and zeros as 0.

0 0

X

X

W

S

2.0 1.5 1.0 0.5 0 0.5 1.0 1.5 2.0

2.0

1.5

1.0

0.5

0

0.5

1.0

1.5

2.0

Imaginary

Real

Hs Ps^

Qs

=-----------

ft e

st

Fs sd

0

f

=³

'QIt= s

I

fs

= ---

'QCv= 1 – v 2

I

fs

--- =Cv 1 – v 2

R

v 1 – v 2

I

---------------------=

I

Cfs

=-------