## 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

where we made use of the fact that the impedance of a capacitor is , as derived above. The driving point impedance of the whole RC filter is thus

Alternatively, we could simply note that impedances always sum in series and write down this result directly.

### 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,

The parameter is called the 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.,

where denotes the Heaviside unit step function

This result is most easily checked by taking the Laplace transform of an exponential decay with time-constant :

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

 (E.5)

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,

 (E.6)

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

An impulse is not a function in the usual sense, so it is called instead a 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

Thus, there is a single 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.
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