Chap01 - Signal and Systems Chapter 1 Introduction H F...

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Signal and Systems Chapter 1: Introduction H. F. Francis Lu 2011 Spring: Signal and Systems Ver. 2011.02.20 H. F. Francis Lu 1 / 67
Signal and Systems Signal A signal is formally defined as a function of one or more variables that conveys information on the nature of a physical phenomenon. System A system is formally defined as an entity that manipulates one or more signals to accomplish a function, thereby yielding new signals. 2011 Spring: Signal and Systems Ver. 2011.02.20 H. F. Francis Lu 2 / 67
Sec. 1.4 Classification of Signals 2011 Spring: Signal and Systems Ver. 2011.02.20 H. F. Francis Lu 3 / 67
Continuous Time Signals Definition 1 A signal x ( t ) is said to be a continuous time signal if it is defined for all time t. 2011 Spring: Signal and Systems Ver. 2011.02.20 H. F. Francis Lu 4 / 67
Discrete Time Signals Definition 2 A discrete-time signal is defined only at discrete instants of time. A discrete-time signal is often derived from a continuous time signal by sampling it at a uniform rate. For example, given a continuous time signal x ( t ) , let T s be the sampling period. Then the discrete time signal is given by x [ n ] = x ( nT s ) , n = 0 , ± 1 , ± 2 , 2011 Spring: Signal and Systems Ver. 2011.02.20 H. F. Francis Lu 5 / 67
Even and Odd Signals Definition 3 A signal is said to be an even signal if x ( t ) = x ( - t ) , if continuous time x [ n ] = x [ - n ] , if discrete time and is said to be an odd signal if x ( t ) = - x ( - t ) , if continuous time x [ n ] = - x [ - n ] , if discrete time 2011 Spring: Signal and Systems Ver. 2011.02.20 H. F. Francis Lu 6 / 67
Example 1 (p.18) Consider the signal x ( t ) = sin π t T , - T t T 0 , otherwise Is the signal x ( t ) even or odd?
2011 Spring: Signal and Systems Ver. 2011.02.20 H. F. Francis Lu 7 / 67
Even-Odd Decomposition of Signals Given an arbitrary signal x ( t ) , we can decompose x ( t ) into x ( t ) = x e ( t ) + x o ( t ) where x e ( t ) is even and x o ( t ) is odd, according to x e ( t ) = 1 2 [ x ( t ) + x ( - t )] x o ( t ) = 1 2 [ x ( t ) - x ( - t )] The decomposition is similar for discrete time signals x [ n ] . Proof. Straightforward. 2011 Spring: Signal and Systems Ver. 2011.02.20 H. F. Francis Lu 8 / 67
Example 2 (p.19) Decompose the signal x ( t ) = e - 2 t cos ( t ) into even and odd. Sol. x e ( t ) = 1 2 [ x ( t ) + x ( - t )] = cosh ( 2 t ) cos t x o ( t ) = 1 2 [ x ( t ) - x ( - t )] = - sinh ( 2 t ) cos t 2011 Spring: Signal and Systems Ver. 2011.02.20 H. F. Francis Lu 9 / 67
Hermitian Symmetric or Complex Symmetric Definition 4 Let x ( t ) (and the same for x [ n ] ) be a complex-valued signal. We say x ( t ) is Hermitian symmetric if x ( t ) = x * ( - t ) “Hermitian symmetric” is called “conjugate symmetric” in the book, but the latter name is less typical. Proposition 1 x ( t ) = a ( t ) + ı b ( t ) , where a ( t ) and b ( t ) are real-valued functions, is Hermitian symmetric iff a ( t ) is even and b ( t ) is odd. 2011 Spring: Signal and Systems Ver. 2011.02.20 H. F. Francis Lu 10 / 67
Example 3 Show that x ( t ) = exp ( ı 2 π t ) is Hermitian symmetric. Sol. x ( t ) = cos ( 2 π t ) + ı sin ( 2 π t ) x * ( - t ) = cos ( - 2 π t ) - ı sin ( - 2 π t ) = cos ( 2 π t ) + ı sin ( 2 π t ) = x ( t ) 2011 Spring: Signal and Systems Ver. 2011.02.20 H. F. Francis Lu 11 / 67
Periodic Signals Definition 5 A signal x ( t ) (and the same for x [ n ] ) is called a periodic signal if there exists a positive constant T such that x ( t ) = x ( t + T ) for all t R . The smallest such value, say T min is called

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