1. 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

2. Discrete-Time LTI Systems
Representation of Discrete-Time Signals in Terms of Impulses
Discrete-Time Unit Impulse Response and the Convolution-Sum Representation

3. 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]

4. 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

5. Often called the Sifting Property of Discrete-Time unit impulse
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

6. 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

7. 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

8. 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

9. 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

10. 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

11. x[n]
system response to δ[n+1], δ[n], δ[n-1]

12. 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

13. 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

14. 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

15. Singularity Functions
Unit Impulse as an Idealized Short Pulse
Defining the Unit Impulse through Convolution
Unit Doublets and other Singularity Functions