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# Transfer Functions

The design of filters involves a detailed consideration of input/output
relationships because a filter may be required to pass or attenuate input
signals so that the output amplitude-versus-frequency curve has some
desired shape. The purpose of this section is to demonstrate how
the equations that describe output-versus-input
relationships of filters can be written directly in a mathematical
form called a **transfer function**.

When a voltage is applied to the filter of figure below, the transfer function
is *H*(*s*) and it is equivalent to
(*v*_{out}/*v*_{in}) (*s*), *s* being in
general a complex mathematical operator having both real and imaginary parts
(**Laplace operator**); in this text, however, *s* will be equated
to its steady-state sinusoidal equivalent (*jω*).
The magnitude of the complex transfer function |*H*(*jω*)|
is then referred to as **amplitude response**, while the phase of the complex
transfer function is called **phase response**.

The transfer function can be expressed as the
ratio of two polynomials, *N*(*s*) in the numerator and
*D*(*s*) in the denominator, such as

The roots of the polynomial in the denominator *D*(*s*) are referred
to as poles, and the roots of *N*(*s*), which are located in the
numerator, are referred to as zeros. The order of the filter is the
largest exponent of *s* in the polynomials.

It is possible to study quite generalized expressions for transfer functions,
but in order to understand certain essential points, attention will be
focused on three specific transfer functions: those for low-pass,
high-pass, and band pass filters of the second order. The term "second order"
refers to the fact that the transfer equations involve terms no higher than
*s*^{2}. Second-order transfer functions describe 2-pole filters;
third-order functions involve *s*^{3} and describe 3-pole filters,
and so on. Second-order transfer functions are a useful starting point
in the study of filters because they are foundations upon which more complex
filters can be built.

Typical second-order transfer functions for low-pass, high-pass, and bandpass filters are:

Note that all three transfer functions have the same denominator and that the
numerators have increasing powers of *s*. That the equations do represent
low-pass, high-pass, and bandpass filters can be seen by noting that *s*
is proportional to frequency and by studying the behavior of the equations at
such points as *ω* = 0 and *ω* = ∞. For example,
putting *ω* = 0 into the above equation for low-pass filter
*K*/(*s*^{2}+*bs*+*a*) gives *H* =
*K/a*, while *ω* = ∞ gives *H* = 0, which
indicates that this equation describes a filter that passes DC with a gain
of *K/a* and attenuates infinite and high frequencies; in other words,
it describes a low-pass filter.

The numerator and the denominator of the above equation for bandpass filter
*Ks*/(*s*^{2}+*bs*+*a*)
can be divided by *s* and then *s* can be replaced with
*jω* to give

At *ω* = 0 and *ω* = ∞, the above equation reduces
to zero; when *ω* = *a/ω* or
*ω*^{2} = *a*, the *H* is equal to *K/b*,
which indicates that this equation describes a filter that attenuates low and
high frequencies and passes midband frequencies; in other words, the equation
describes a bandpass filter.

The numerator and denominator of the above equation for high-pass filter
*Ks*^{2}/(*s*^{2}+*bs*+*a*)
can be divided by *s*^{2} and *s* can be replaced with
*jω* to give

This equation reduces to 0 at *ω* = 0 and to *K* at
*ω* = ∞, indicating that the equation describes a filter
that passes high frequencies and attenuates low frequencies; in other words,
the equation describes a high-pass filter.

Although the above equation
*K*/(*s*^{2}+*bs*+*a*)
describes a low-pass filter, there are many different types of low-pass filters;
for example, the next chapter contains information about Butterworth, Chebyshev,
and Bessel low-pass filters. The type of filter, that is, the shape of its
response curve, is determined by the constants *a* and *b* in the
equation for low-pass filter; the constant *K* is a multiplier that
affects the filter gain. Constants *a* and *b* in the above equations
for bandpass and high-pass filters also determine the type of filter and *K*
is again a multiplying factor affecting the gain.

There are available tables listing constants such as *a* and *b* for
various types of filters. (*K* is a scaling factor chosen by the designer
to give a specific gain.) Typically, *a* and *b* are given as
coefficients in polynomial functions for various types of filters.