102_1_lecture4

102_1_lecture4 - UCLA Fall 2010 Systems and Signals Lecture...

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Unformatted text preview: UCLA Fall 2010 Systems and Signals Lecture 4: Systems Characteristics October 6, 2010 EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 1 System Characteristics Today’s topics: • System interconnections • Characteristics (memory, causality, stability) • Differential equations EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 2 Review • A system transforms input signals into output signals . • A system is a function mapping input signals into output signals. • Systems often denoted by block diagrams x y S EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 3 Linearity • A system is linear if it satisfies homogeneity and additivity properties. • Homogeneity: F ( ax ) = aF ( x ) • Additivity: F ( x + ˜ x ) = F ( x ) + F (˜ x ) • Another useful test: zero input to a linear system must produce zero output (ZIZO property). If x ( t ) = 0 , and F ( x ) 6 = 0 , the system F is not linear! EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 4 Combined Homogeneity and Superposition: • If y 1 = Sx 1 and y 2 = Sx 2 , and a and b are constants, ay 1 + by 2 = S ( ax 1 + bx 2 ) Extended Linearity Given system S , input signals x n , output signals y n , scalars (constants) a n . • Summation: If y n = Sx n for all n , an integer from (-∞ < n < ∞ ) X n a n y n = S X n a n x n ! Summation and the system operator commute, and can be interchanged. EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 5 + _ V(t) I R (t) Consider the circuit (a system) above. Input signal is voltage V ( t ) . System output is current through the resistor I R ( t ) . Ohm’s Law: I R ( t ) = V ( t ) /R . Is the system linear? EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 6 + _ V(t) I D (t) Consider a circuit with the resistor replaced by the diode. Input signal is voltage V ( t ) . System output is current through the diode I D ( t ) . Current through the diode is given by I D ( t ) = I s ( e V ( t ) /V T- 1) where I s and V T are physical parameters not relevant to the discussion. Is the system linear? EE102: Systems and Signals; Fall 2010, Jin Hyung Lee 7 Time Invariance • Conceptually: system response is independent of time. • A system is time-invariant if a time shift in the input only produces the same time shift in the output....
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This note was uploaded on 04/20/2011 for the course EE 102 taught by Professor Levan during the Fall '08 term at UCLA.

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102_1_lecture4 - UCLA Fall 2010 Systems and Signals Lecture...

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