Linear Time-Invariant Systems Discrete-Time LTI Systems: Convolution Sum Continuous-Time LTI Systems: Convolution Integral Properties of LTI Systems Causal.

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