## RC Filter Analysis

Referring again to Fig.E.1, let's perform an *impedance
analysis* of the simple RC lowpass filter.

### Driving Point Impedance

Taking the Laplace transform of both sides of Eq.(E.1) gives

### Transfer Function

Since the input and output signals are defined as and
, respectively, the *transfer function* of this analog
filter is given by, using *voltage divider rule*,

*RC time constant*, for reasons we will soon see.

### Impulse Response

In the same way that the impulse response of a digital filter is given
by the inverse *z* transform of its transfer function, the impulse response
of an *analog* filter is given by the inverse *Laplace*
transform of its transfer function, *viz.*,

*Heaviside unit step function*

In more complicated situations, any rational (ratio of
polynomials in ) may be expanded into first-order terms by means of
a *partial fraction expansion* (see §6.8) and each term in
the expansion inverted by inspection as above.

### The Continuous-Time Impulse

The continuous-time impulse response was derived above as the
inverse-Laplace transform of the transfer function. In this section,
we look at how the *impulse* itself must be defined in the
continuous-time case.

An *impulse* in continuous time may be loosely defined as any
``generalized function'' having *``zero width''* and *unit
area* under it. A simple valid definition is

More generally, an impulse can be defined as the limit of

*any*pulse shape which maintains unit area and approaches zero width at time 0. As a result, the impulse under every definition has the so-called

*sifting property*under integration,

provided is continuous at . This is often taken as the

*defining property*of an impulse, allowing it to be defined in terms of non-vanishing function limits such as

*distribution*or

*generalized function*[13,44]. (It is still commonly called a ``delta function'', however, despite the misnomer.)

### Poles and Zeros

In the simple RC-filter example of §E.4.3, the transfer function is

*pole*at , and we can say there is one

*zero at infinity*as well. Since resistors and capacitors always have positive values, the time constant is always non-negative. This means the impulse response is always an exponential

*decay*--never a growth. Since the pole is at , we find that it is

*always in the left-half plane*. This turns out to be the case also for any

*complex*analog one-pole filter. By consideration of the partial fraction expansion of any , it is clear that, for stability of an analog filter,

*all poles must lie in the left half of the complex plane*. This is the analog counterpart of the requirement for digital filters that all poles lie inside the unit circle.

**Next Section:**

RLC Filter Analysis

**Previous Section:**

Inductors