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Lecture 4 Systems Characteristics

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

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UCLA Fall 2011 Systems and Signals Lecture 4: Systems Characteristics October 5, 2011 EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 1

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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

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Review A signal can be of one of the following types: Energy signal, 0 < E x = lim T →∞ Z T - T | x ( t ) | 2 dt < Power signal, 0 < 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 > 0 = . 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

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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

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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? EE102: Systems and Signals; Fall 2011, Jin Hyung Lee 9

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+ _ V(t) I D (t) Consider a circuit with the resistor replaced by the diode.
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