Linear Time-Invariant Systems
• Discrete-Time LTI Systems: Convolution Sum
• Continuous-Time LTI Systems: Convolution
Integral
• Properties of LTI Systems
• Causal LTI Systems Described by
– Differential and Difference Equations
• Singularity Functions
Discrete-Time LTI Systems
• Representation of Discrete-Time Signals in
Terms of Impulses
• Discrete-Time Unit Impulse Response and
the Convolution-Sum Representation
Representation of
Discrete-Time Signals
in Terms of Impulses
Discrete-time unit impulse,
any discrete-time signal
, can be used to construct
Discrete-time signal is a sequence of individual impulses
Consider x[n]
• 5 time shifted impulses
scaled by x[n]
• Therefore
• x[n] = x[-3]δ[n+3] + x[-2]δ[n+2] +
x[-1]δ[n+2] + x0]δ[n] + x[1]δ[n-1] +
x[2]δ[n-2] + x[3]δ[n-3]
• or
• Represents arbitrary sequence as linear
combination of shifted unit impulses δ[n-k],
where the weights are x[k]
• Often called the Sifting Property of
Discrete-Time unit impulse
– Because δ[n-k] is nonzero only when k = n the
summation “sifts” through the sequence of values x[k]
and preserves only the value corresponding to k = n
Discrete-Time Unit Impulse Response
and the Convolution-Sum Representation
• Sifting property represents x[n] as a
superposition of scaled versions of very simple
functions
– shifted unit impulses, δ[n-k], each of which is
nonzero at a single point in time specified by the
corresponding value of k
• Response of Linear system will be
– Superposition of scaled responses of the system
to each shifted impulse
• Time Invariance tells us that
– Responses of a time-invariant system to
– time-shifted unit impulses are
– time-shifted versions of one another
• Convolution-Sum representation for D-T
LTI systems is based on these two facts
Convolution-sum Representation
of LTI Systems
• Consider response of linear system to x[n]
•
says input can be represented
as linear combination of shifted unit impulses
• let hk[n] denote response of linear system to
shifted unit impulse δ[n-k]
• Superposition property of a linear system
says the response y[n] of the linear system
to x[n] is weighted linear combination of
these responses
• with input x[n] to a linear system the output
y[n] can be expressed as
If x[n]
is applied to a
system
Whose
responses
h-1[n], h0[n], and
h1[n] to the
signals δ[n+1],
δ[n], and δ[n-1]
are
Superposition allows us to write the response to x[n] as a linear
combination of the responses to the individual shifted impulses
x[n]
system response to δ[n+1], δ[n], δ[n-1]
Continuous-Time LTI Systems:
Convolution Integral
• Representation of Continuous-Time Signals
in Terms of Impulses
• Continous-Time Unit Impulse Response and
the Convolution Integral Representation of
LTI Systems
Properties of LTI Systems
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Commutative Property
Distributive Property
Associative Property
LTI Systems with and without Memory
Invertibility of LTI Systems
Causality of LTI Systems
Stability for LTI Systems
Unit Step Response of an LTI System
Causal LTI Systems Described by
Differential and Difference Equations
• Linear Constant-Coefficient Differential
Equations
• Linear Constant-Coefficient Difference
Equations
• Block Diagram Representations of FirstOrder Systems Described by Differential
and Difference Equations
Singularity Functions
• Unit Impulse as an Idealized Short Pulse
• Defining the Unit Impulse through
Convolution
• Unit Doublets and other Singularity
Functions