Lecture 4 Systems Characteristics

Lecture 4 Systems Characteristics - UCLA Fall 2011 Systems...

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Unformatted text preview: UCLA Fall 2011 Systems and Signals Lecture 4: Systems Characteristics October 5, 2011 EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 1 Agenda • My office hours is now T 2-4pm. • Zhongnan’s office hour is now T 11-12pm. EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 2 System Characteristics Today’s topics: • System interconnections • Characteristics (memory, causality, stability) • Differential equations EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 3 Review A signal can be of one of the following types: • Energy signal, < E x = lim T →∞ Z T- T | x ( t ) | 2 dt < ∞ • Power signal, < P x = lim T →∞ 1 2 T Z T- T | x ( t ) | 2 dt < ∞ • Neither energy nor power signal. For instance, E x = ∞ and P x = 0 . A signal CANNOT be both an energy and power signal. Why? EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 4 • An energy signal x ( t ) has zero power P x = lim T →∞ 1 2 T Z T- T | x ( t ) | 2 dt | {z } → E x < ∞ = 0 • A power signal has infinite energy E x = lim T →∞ Z T- T | x ( t ) | 2 dt = lim T →∞ 2 T 1 2 T Z T- T | x ( t ) | 2 dt | {z } → P x > = ∞ . • But, a signal with zero power may or may not be an energy signal. • Similarly, a signal with infinite energy may or may not be a power signal. EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 5 System Characteristics • 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 2011, Jin Hyung Lee 6 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 2011, Jin Hyung Lee 7 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 2011, Jin Hyung Lee 8 + _ 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?...
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This note was uploaded on 01/30/2012 for the course ELEC ENGR 102 taught by Professor Jinhyunglee during the Fall '11 term at UCLA.

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Lecture 4 Systems Characteristics - UCLA Fall 2011 Systems...

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