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 0O<t<+0 1. Real Exponential signals x()=Cea Decaying exponential. when 0<0 Growing Exponential,when a Is rea a>0 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 α<0 Growing Exponential, when α>0 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