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Impulse Function Discrete impulse 1, n 0 0, n 0 [ n] Continuous time impulse (t ) 0 for t 0 (t )dt 1 (t ) d u (t ) dt t u (t ) ( )d Ramp Function t, t 0 r (t ) 0, t 0 Discrete Time Function n, n 0 r[n] 0, n 0 x (t ) /2 /2 /2 /2 Systems viewed as interconnections of Operations is the collection of all real-valued continuous functions defined on some interval . is the collection of all functions with continuous th derivatives. A function space is a topological vector space whose "points" are functions. http://mathworld.wolfram.com/FunctionSpace.html A vector space with a Hausdorff topology such that the operations of vector addition and scalar multiplication are continuous. An operator assigns to every function a function . It is therefore a mapping between two function spaces. If the range is on the real line or in the complex plane, the mapping is usually called a functional instead. http://mathworld.wolfram.com/Operator.html 1. A vector space is a set that is closed under finite vector addition and scalar multiplication. 2. A topological space fulfilling the -axiom. In the terminology of Alexandroff and Hopf (1972), a space is called a Hausdorff space. a. Given any two distinct points , there exist neighborhoods and of and , respectively, with . y (t ) H {x(t )}, y[n] H {x[n]} Moving Average System (a) cascade form of implementation and (b) parallel form of implementation. 1 y[n] ( x[n] x[n 1] x[n 2]) 3 1 H (1 S S 2 ) 3 Properties of Systems Stability BIBO y (t ) M y for x(t ) M x Memory A system is said to possess memory if its output depends on past or future values of the input signal. Causality A system is said to be causal if the present value of the output signal depends only on the present or past values of the input signal Invertibility A system is said to be invertible if the input of the system can be recovered from the output. H inv{ y(t )} H inv{H {x(t )}} H inv H {x(t )} H inv H I Time Invariance A system is said to be time invariant if a time delay or time advance of the input signal leads to an identical time shift of the output signal. Linearity A system is said to be linear in terms of the system input (excitation) and the system output (response) if it satisfies the properties of: 1. Superposition 2. Homogenity