Linear Time-Invariant Systems Discrete-Time LTI Systems: Convolution Sum Continuous-Time LTI Systems: Convolution Integral Properties of LTI Systems Causal.
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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, , can be used to construct any discrete-time signal
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• 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 First-Order 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
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