Chapter 1 Signals and systems ③ Average power E E perlow period period ④ Harmonic relation 办()=ek,k=0,+1,+2, 3. General Complex Exponential Signals i0(r+joo)t e Cle"cos(ant+0)+jCe"sin(@t+6) 2006 Fall
2006 Fall Chapter 1 Signals and Systems ③ Average Power ④ Harmonic relation 3. General Complex Exponential Signals 0 0 0 2 0 E e dt T T j t period = = 1 1 0 period = Eperiod = T P k (t) = e j k0 t , k = 0,1,2, cos( ) sin( ) 0 0 ( ) 0 = + + + = + C e t jC e t C e C e e r t r t t j r j t
Chapter 1 Signals and systems 5 1.3.2 Discrete-Time Complex Exponential and Sinusoidal Signals x|n=Can-0<n<+∞0 xIn=ce where a=e B 1. Real exponential signals a=a rea rin=a 2006 Fall
2006 Fall Chapter 1 Signals and Systems § 1.3.2 Discrete-Time Complex Exponential and Sinusoidal Signals n x n = C − n + n x n Ce = where = e 1. Real Exponential Signals = a real n x n = a
Chapter 1 Signals and systems 2. Complex Exponential Signals and Sinusoidal Signals ④ Average power ② Euler’ s relation 3 Frequency Properties o Periodicity Properties N=m ⑤ Harmonic relation 3. General Complex exponential signals 2006 Fall a"=Ca cos(@n+0)+jCa" sin(aon+0)
2006 Fall Chapter 1 Signals and Systems 2. Complex Exponential Signals and Sinusoidal Signals ① Average Power ② Euler’ s relation ④ Periodicity Properties ⑤ Harmonic relation ③ Frequency Properties 3. General Complex Exponential Signals 0 2 N = m cos( ) sin( ) C = C 0 n + + jC 0 n + n n n
Chapter 1 Signals and systems Frequency Properties Figure 1.27 x|n|= cOS Oon N∈0,2兀 Low Frequency (a)00=0 (j)h=2 (b)00=丌8N=16 (h)00=15/8N=16 (c)0=π/4N=8 (g)00=7m/4 8 (d)o=丌/2N=4 (f)0033π/2N=4 (e)0o=π N=2 High Frequency 2 kT, low frequency 2006 Fall Oo=(2 k+I)I, high frequency
2006 Fall Chapter 1 Signals and Systems ( a) ω0=0 N=1 ( b) ω0= π /8 N=16 ( c) ω0= π /4 N=8 ( d) ω0 = π /2 N=4 ( e) ω0 = π N=2 ( f) ω0 =3 π/2 N=4 ( g) ω0 =7 π/4 N=8 ( h) ω0 =15 π/8 N=16 ( i) ω0 =2 π N=1 Low Frequency High Frequency Figure 1.27 xn= cos0 n 0,2 ) 0 ω0=2 kπ, low frequency ω0=(2 k+1)π, high frequency Frequency Properties