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Linear System Theory Introduction

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1 Linear Systems, Superposition, and Convolution
In this section we provide a brief introduction to linear systems theory. It is based on the followingdeﬁnition:Let
L
denote a
time-invariant, linear system
with input
and output
:
L
Such systems are
additive
(the response to the sum of two inputs
and
equals the sum of theresponses to each input taken individually):
and
homogeneous
(the system can be scaled by the magnitude of the input
, where
is a scalar):
Taken together, they enjoy the
Principle of Superposition:
Several idealized input functions are of special importance in analyzing systems, the
Dirac delta function
and the
unit step function
. The unit step is deﬁned as:
if
;
if
;undeﬁned
if
(1)and the Dirac delta function is its derivative:
Of course, since the step function is not continuous (notice the value at 0), one has to be careful indeﬁning exactly what is meant by the above equation. Formally, the
is a
distribution
deﬁned by theintegral equation:
It can be represented as the limit of a sequence of functions such as:
with
, see Fig. 1; During this limiting process we have something that resembles a physicalapproximation to the unit impulse, the ﬁrst test pattern used to evaluate the lateral inhibitory network.The result is called the
impulse response
1
−20 −10 0 10 20−0.500.51t(epislon=1)−20 −10 0 10 20−0.500.511.52t(epislon=1/2)−20 −10 0 10 20−0.500.511.52t(epislon=1/4)−20 −10 0 10 20−1−0.500.511.522.5t(epislon=1/16)
Figure 1:Similarly, since the step function is the integral of the impulse function, the
step response
, or theresponse to a unit step function
is the integral of the impulse response function:
In general, of course, we are interested in the response of a system not to these special functions, butto an arbitrary input. The trick is to approximate this arbitrary input as a sequence of step functions (Fig.2.12 in Oppenheim and Willsky, p91), and then to use superposition to obtain the overall response. It isnecessary to
assume that the input function
is continuous
, so that the approximation is meaningful.Now, suppose the input starts at
, and time is discretized into bins
seconds apart. The ﬁrststep in the approximation has height
, a constant, and the additional step at
has height
. Remembering that
, the signal
can thus be approximated by
(2)
(3)The approximate output is then given by2
(4)
(5)Now, introduce a limiting process in which the time steps become vanishingly close, so that
,
, the sum becomes an integral and we have:
(6)
(7)Repeating this approximation procedure for a continuous input
starting at
and such that
as
yields:
(8)
(9)
(10)Thus the output can be computed as an integral function of the impulse response. Such an integral iscalled a
convolution integral
, and the above expression is often abbreviated:
ReferenceAlan V. Oppenheim and Alan S. Willsky,
Signals and Systems
, second edition, Prentice-Hall Inc.,1997.3

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