信号与系统（PPT讲稿）Signals and System（Complex Exponentials、Unit Impulse and Unit Step Signal、Singular Functions）

Signals and systems Lecture 2 o Complex exponentials uNit Impulse and unit step signal o Singular Functions 2006 Fall

2006 Fall Signals and Systems Lecture 2 ⚫Complex Exponentials ⚫Unit Impulse and Unit Step Signal ⚫Singular Functions

Chapter 1 Signals and systems 5 1.3 Exponential and Sinusoidal Signals 复指数信号和正弦信号 5 1.3. 1 Continuous-Time Complex Exponential and sinusoidal signals e 0O0 Figure 1.19 2006 Fall

2006 Fall Chapter 1 Signals and Systems § 1.3 Exponential and Sinusoidal Signals 复指数信号和正弦信号 § 1.3.1 Continuous-Time Complex Exponential and Sinusoidal Signals ( ) st x t = Ce −   t  + 1. Real Exponential Signals ( ) at x t = Ce a is real Decaying Exponential, when α0 Figure 1.19

Chapter 1 Signals and systems 2. Periodic Complex Exponential and Sinusoidal signals ④ Period 2丌 ② Euler's relation(欧拉关系) ot e sin ot Jot coS at t sin at 2006 Fall

2006 Fall Chapter 1 Signals and Systems 2. Periodic Complex Exponential and Sinusoidal Signals ① Period ② Euler’ s relation( 欧拉关系) 2 cos j t j t e e t    − + = j e e t j t j t 2 sin    − − = e t j t j t    = cos + sin 0 0 2   T =

Chapter 1 Signals and systems Sinusoids and complex exponential o Probably the most important elemental signal that we will deal with is the real-valued sinusoid. In its continuous-time form, we write the general form as x(t)=Acos(@t +o) o Maybe as important as the general sinusoid, the ca exponential function will become a critical part of our study of signals and systems. Its general form is also written as x()=C 2006 Fall a Is complex

2006 Fall Sinusoids and Complex Exponential ⚫ Probably the most important elemental signal that we will deal with is the real-valued sinusoid. In its continuous-time form, we write the general form as ( ) cos( ) x t = A 0 t + 0 0 2   T = Chapter 1 Signals and Systems ⚫ Maybe as important as the general sinusoid, the complex exponential function will become a critical part of our study of signals and systems. Its general form is also written as t x t Ce ( ) = a is complex

Chapter 1 Signals and systems Sinusoids and complex exponential Decomposition: The complex exponential signal can thus be written in terms of its real and imaginary parts AcoS(O,t+o)=AReelobtg Asin(@t+)=AIme( i+9) This decomposition of the sinusoid can be traced to Euler's relation Oo: Fundamental Frequency ①: Phase A: Amplitude 2006 Fall

2006 Fall Sinusoids and Complex Exponential ⚫ Decomposition: The complex exponential signal can thus be written in terms of its real and imaginary parts cos( ) Re{ } ( ) 0 0    + + = j t A t A e sin( ) Im{ } ( ) 0 0    + + = j t A t A e Chapter 1 Signals and Systems (This decomposition of the sinusoid can be traced to Euler's relation) ω0 :Fundamental Frequency Φ :Phase A :Amplitude

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 k0 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 xn= cos0 n  0,2 ) 0  ω0=2 kπ, low frequency ω0=(2 k+1)π, high frequency Frequency Properties

Chapter 1 Signals and systems Table 1.1 Comparison of the ejoot and ejon J@ot O不同信号不同 o相差2k,信号相同 n∈(0,2兀 ω越大频率越高 o0=2kπ时频率低 00=(2k+1)m时频率高 对任意的om /27为有理数时, 信号均为周期的 信号为周期的 2丌 N 2006 Fall

2006 Fall ω0不同,信号不同. ω0相差2 kπ,信号相同. ω0越大,频率越高. ω0 =2 k π时,频率低; ω0 =(2 k+1)π时,频率高. 对任意的ω0 , 信号均为周期的. 为有理数时, 信号为周期的. Chapter 1 Signals and Systems j t e 0 j n e 0  (0,2 ) 0  0 0 2   T = N m 0 2   = 0 / 2 Table 1.1 Comparison of the and j t e 0 j n e 0