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Chapter4
Response Properties of Physical Systems
Up to now we have discussed in great detail the mathematical tools to perform temporal analysis.
In this Chapter we will reviews the theoretical formulas for describing the dynamic behavior (in
other words the response characteristics) of ideal physical systems.
4.1 Basic Dynamic Characteristics
The dynamic characteristics of a constant parameter linear system can be described by a weighting
function h(τ), which is defined as the output of the system at any time to a unit impulse
input applied a time τ before. The usefulness of the weighting function as a description of the
system is due to the following fact. For any arbitrary input x(t), the system output y(t) is given
by the convolution integral
y(t) =
∫ +∞
−∞
h(τ) x(t − τ)dτ (4.1)
That is, the value of the output y(t) is given as a weighted linear (infinite) sum over the entire
history of the input x(t).
In order for a constant parameter linear system to be physically realizable, it is necessary that
the system responds only to past inputs. This implies that
h(τ) = 0 for τ < 0 (4.2)
Hence, for physical systems, the effective lower limit of integration in (4.1) is zero rather that
−∞.
A constant parameter linear system is said to be stable if every possible bounded input function
produces a bounded output function. From (4.1) we have
|y(t)| =
∣
∫ +∞
−∞
∫ +∞
h(τ) x(t − τ)dτ
∣ ≤ |h(τ)||x(t − τ)|dτ (4.3)
When the input x(t) is bounded, there exists some finite constant A such that
63
−∞