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Properties of LTI Systems
Commutative Property x[n]*h[n]=h[n]*x[n] Distributive Property x[n]*(h1[n]+ h2[n])= x[n]*h1[n]+x[n]* h2[n] Associative Property x[n]*(h1[n]* h2[n])= (x[n]*h1[n])* h2[n] Memoryless If h[n]=0 for n not equal 0. I.e. h[n]=Kd[n].
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Properties of LTI Systems
Invertibility w(t)=x(t) x(t) h1(t) h(t) y(t) x(t) Identity System d(t)
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Properties of LTI Systems
Causality h[n]=0 for n<0 or h(t)=0 for t<0. Stability
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Unit Step Response of LTI System
s[n] h[n] The step response of a discrete-time LTI system is the convolution of the unit step with the impulse response:- s[n]=u[n]*h[n]. Via commutative property of convolution, s[n]=h[n]*u[n]. That means s[n] is the response to the input h[n] of a discrete-time LTI system with unit impulse response u[n]. h[n] s[n] u[n]
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Using the convolution sum:-
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Unit Step Response of Continuous-time LTI System
Similarly, unit step response is the running integral of its impulse response. The unit impulse response is the first derivative of the unit step response:-
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Causal LTI Systems Described By Differential & Difference Equations
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Example 2.14
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Example 2.14 with impulse input (Problem 2.56 (a) Pg 158-159 OWN)
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General Higher N-order DE
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Linear Constant-Coefficient Difference equations
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Linear Constant-Coefficient Difference equations
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Recursive case when N > or = 1.
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Block Diagram Representations of First- Order Systems.
Provides a pictorial representation which can add to our understanding of the behavior and properties of these systems. Simulation or implementation of the systems. Basis for analog computer simulation of systems described by DE. Digital simulation & Digital Hardware implementations
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First-Order Recursive Discrete-time System.
y[n]+ay[n-1]=bx[n] y[n]=-ay[n-1]+bx[n] y[n] b + x[n] D -a y[n-1]
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First-Order Continuous-time System Described By Differential Equation
y(t) b/a + x(t) Difficult to implement, sensitive to errors and noise. D -1/a
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First-Order Continuous-time System Described By Differential Equation Alternative Block Diagram.
y(t) b + x(t) -a
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