Even and Odd Signals A signal is said to be an even signal if x(-t)=x(t)or x[-n]=x[n].Even signals are symmetric about the vertical axis, or time origin. A signal is said to be an odd signal if x(-t)=-x(t)or x[-n]=-x[n].Odd signals are antisymmetric about the time origin
Even and Odd Signals • A signal is said to be an even signal if x(-t)=x(t) or x[-n]=x[n]. Even signals are symmetric about the vertical axis, or time origin. • A signal is said to be an odd signal if x(-t)=-x(t) or x[-n]=-x[n]. Odd signals are antisymmetric about the time origin
Example Is the signal an even or odd function of time? sin -T≤t≤T otherwise Solution: -a孕7-m受2≤7 0,otherwise 0,otherwise =-x(t) So,x(t)is an odd signal
Example Is the signal an even or odd function of time? otherwise T t T T t x t 0, sin , ( ) Solution: sin( ), sin( ), ( ) 0, otherwise 0, otherwise ( ) So, ( ) is an odd signal. t t t T t T x t T T x t x t
Even-Odd Decomposition Given an arbitrary signal,we may develop an even-odd decomposition of that signal. (t)= x(t)+x(-t),x(t)-x(-t) 2 2 e(t)= x(t)+x(-t) even component 2 x()-x():odd component 2
Even-Odd Decomposition Given an arbitrary signal, we may develop an even-odd decomposition of that signal. : odd component 2 ( ) ( ) ( ) : even component 2 ( ) ( ) ( ) 2 ( ) ( ) 2 ( ) ( ) ( ) x t x t x t x t x t x t x t x t x t x t x t o e
Example Find the even and odd components of the signal x(t)=e2 cost Solution: x(-t)=e2 cost x()=x()+x(-1)ecost+ecost e"e cost 2 2 2 cosh(2t)cost x(t)=x(1)-x(-1)_e2cost-e"cost e2 -e-21 coSt 2 2 2 =-sinh(2t)cost
Example Find the even and odd components of the signal 2 2 2 2 2 2 2 2 2 Solution: ( ) cos ( ) ( ) cos cos ( ) cos 2 2 2 cosh(2 ) cos ( ) ( ) cos cos ( ) cos 2 2 2 sinh(2 ) cos t t t t t e t t t t o x t e t x t x t e t e t e e x t t t t x t x t e t e t e e x t t t t 2 ( ) cos t x t e t
Conjugate Symmetry A complex-valued signal x(t)is said to be conjugate symmetric if x(-t)=x(t) x(t)=a(t)+jb(t)→ x(-t)=a(-t)+jb(-t),x()=a(t)-jb(t) x(-t)=x(t)→a(-t)=a(t),b(-t)=-b(t)
Conjugate Symmetry A complex-valued signal x(t) is said to be conjugate symmetric if * * ( ) ( ) ( ) ( ) ( ) ( ), ( ) ( ) ( ) ( ) ( ) ( ) ( ), ( ) ( ) x t a t jb t x t a t jb t x t a t jb t x t x t a t a t b t b t * x(t) x (t)