a b Slide 15 CUHKSZ EIE 3001 Spring 201819 Variable Transformation Slide 16

# A b slide 15 cuhksz eie 3001 spring 201819 variable

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Slide 15 CUHK(SZ) EIE 3001, Spring 2018/19 Variable Transformation
Slide 16 CUHK(SZ) EIE 3001, Spring 2018/19 Transformation of Independent Variable a (1). Reflection x(t) x(-t) x[n] x[-n] x( t ) t x[ n ] n y(n)=x[- n ] n t y(t)=x(- t )

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Slide 17 CUHK(SZ) EIE 3001, Spring 2018/19 Transformation of Independent Variable (cont.) a (2). Scaling x(t) x(ct)
Slide 18 CUHK(SZ) EIE 3001, Spring 2018/19 Transformation of independent variable (cont.) a (3). Time-shift x[n] x[n-n 0 ] x[ n ] n x[ n- 2] n

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Slide 19 CUHK(SZ) EIE 3001, Spring 2018/19 Transformation of independent variable (cont.) x( t ) 1 2 t x( t+ 1) -1 1 2 t -2/3 2/3 x(1.5 t+ 1) 1 2 t To perform transformation x(t) o x( D t+ You have to do time-shifting then scaling .
Slide 20 CUHK(SZ) EIE 3001, Spring 2018/19 Transformation of independent variable (cont.)

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Slide 21 CUHK(SZ) EIE 3001, Spring 2018/19 Even and Odd functions a A signal is called an even signal (function) if a A signal is called an odd signal (function) if x(t) = - x(-t) x[n] = - x[-n] x(t) = x(-t) x[n] = x[-n]
Slide 22 CUHK(SZ) EIE 3001, Spring 2018/19 Even and Odd functions -10 -8 -6 -4 -2 0 2 4 6 8 10 -1 -0.5 0 0.5 1 t sin(t) -10 -8 -6 -4 -2 0 2 4 6 8 10 -1 -0.5 0 0.5 1 t cos(t)

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Slide 23 CUHK(SZ) EIE 3001, Spring 2018/19 More Examples -10 -8 -6 -4 -2 0 2 4 6 8 10 -1 0 1 t sin(t) -10 -8 -6 -4 -2 0 2 4 6 8 10 -1 0 1 t sin(2t) -10 -8 -6 -4 -2 0 2 4 6 8 10 -1 0 1 t sin(0.5t)
Slide 24 CUHK(SZ) EIE 3001, Spring 2018/19 Even and Odd functions (cont.) a Any signal can be broken into sum of one even and one odd signal. x(t)= Ev {x(t)}+ Od {x(t)} Q: How to find the even (odd) part of a signal. A: Ev {x(t)} = (1/2) [x(t) + x(-t)] Od {x(t)} = (1/2) [x(t) - x(-t)]

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Slide 25 CUHK(SZ) EIE 3001, Spring 2018/19 Even and Odd functions (cont.) a Ex: x[ n ] n -3 -2 -1 0 1 2 3 1 1/2 x[ n ] n -3 -2 -1 0 1 2 3 1 Ev {x[n]} = (1/2) [x[n] + x[-n]] Od {x[n]} = (1/2) [x[n] - x[-n]] x[ n ] n 0 1 2 3 1/2 -1/2
Slide 26 CUHK(SZ) EIE 3001, Spring 2018/19 Periodic and Aperiodic Signal a IF x(t) is periodic with period T, then x(t) = x(t+T) a IF x[n] is periodic with period N, then x[n] = x[n+N] a Fundamental period (T 0 or N 0 ) : the smallest positive value of (T or N) for which the above equation holds. a Aperiodic is also called Non-periodic

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Slide 27 CUHK(SZ) EIE 3001, Spring 2018/19 Periodic and Aperiodic Signal a Periodic signals: x ( t )= x ( t + T ) where T >0
Slide 28 CUHK(SZ) EIE 3001, Spring 2018/19 Periodic and Aperiodic Signal a Periodic signals: x ( n )= x ( n + N ) where N>0

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Slide 29 CUHK(SZ) EIE 3001, Spring 2018/19 Exponential Signal a x(t) = c e at (c>0) (i) Real exponential positive a : negative a (ii) Pure imaginary x(t) = e j Z o t
Slide 30 CUHK(SZ) EIE 3001, Spring 2018/19 Periodic and Sinusoidal Signal Is x(t) = e j Z o t periodic? Let x(t)=x(t+T) e j Z o t = e j Z o (t+T) = e j Z o t x e j Z o T ? e j Z o T = 1 o fundamental period T 0 =2 S /| Z o | Sinusoidal signal : x(t) = A cos ( Z o t+ I ) phase unit Z o : radians/sec I : radians Euler s relation : e j Z o t = cos ( Z o t) + j sin( Z o t) Q: the importance of sinusoidal signal?

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Slide 31 CUHK(SZ) EIE 3001, Spring 2018/19 Periodic and Sinusoidal Signal
Slide 32 CUHK(SZ) EIE 3001, Spring 2018/19 General Complex Exponential a x(t) = c e at where a=r+j Z o ; r, Z o R o c e at = |c| e j T e rt+j Z o t = |c| e rt e j( Z o t + = |c| e rt (cos ( Z o t+ T ) + j sin( Z o t+ T )) Damped Sinusoids (e.g. RLC circuit, car suspension systems, …)

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