1 Signal and System

1 Signal and System
C. General Complex Exponential Signals x(t) = C e jat , in which C = |C| ej , a = r + j 0 So x(t) = |C| ej eat ej0t = |C| eat ej(0t+ ) = |C| eat cos(0t+ ) + j |C| eat sin(0t+ )
(3) A Speech Signal
1 Signal and System
(4) A Picture
1 Signal and System
(5) Vertical Wind Profile
1 Signal and System
B. Types of Signals
(1) Continuous-time Signal
1 Signal and System
Unit Step Function:
0, u[n] 1,
n0 n0
(2) Relation Between Unit Sample and Unit Step
[ n] u[ n] u[ n 1]
u[ n]
m
[ m]
1 Signal and System Signal waves
1 Signal and System
1.3.2 Discrete-time Complex Exponential and Sinusoidal Signals Complex Exponential Signal (sequence) : x[n] = C n or x[n] = C en
1 Signal and System
1. Signals and Systems
1.1 Continuous-time and discrete-time signals
1.1.1 Examples and Mathematical Representation A. Examples (1) A simple RC circuit
Calculate period:
By definition: e j0n = e j0(n+N) thus So e j0N = 1 or 0N = 2 m N = 2m/0
Condition of periodicity: 2/0 is rational
1 Signal and System Periodicity Properties
1 Signal and System
(2) Discrete-time Signal
1 Signal and System
C. Representation
(1) Function Representation Example: x(t) = cos0t x(t) = ej 0t (2) Graphical Representation Example: ( See page before )
1 Signal and System
A. Real Exponential Signal
Real Exponential Signal x[n] = C n (a) >1 (b) 0<<1 (c) -1<<0 (d) <-1
1 Signal and System
B. Sinusoidal Signals
x 2 [ n]
x 2 [ n]

n2
E
P lim
N
n
Average Power:
N 1 x 2 [ n] 2 N 1 n N
1 Signal and System
C. Finite Energy and Finite Power Signal Finite Energy Signal : (P0)
Complex exponential: x[n] = e j0n = cos 0n + jsin0n Sinusoidal signal: x[n] = cos(0n+)
1 Signal and System
C. General Complex Exponential Signals Complex Exponential Signal: x[n] = C n in which C = |C| ej , = ||ej0 (polar form) then x[n]=|C| ||ncos(0n + )+j|C| ||nsin(0n + )
1 Ev{x(t )} xe (t ) [ x(t ) x(t )] 2 1 Od {x(t )} xo (t ) [ x(t ) x(t )] 2
or:
1 Ev{x[n]} xe [n] {x[n] x[n]} 2
Od {x[ n]} xo [ n] 1 {x[ n] x[ n]} 2
E
E

n
x 2 [ n]

x 2 (t )dt
Finite Power Signal : ( E ) 1 T 2 P lim T x (t )dt T 2T
P lim
N N 1 x 2 [ n] 2 N 1 n N
Source voltage Vs and Capacitor voltage Vc
1 Signal and System
(2) An automobile
Force f from engine Retarding frictional force ρV Velocity V
1 Signal and System
A. Real Exponential Signals x(t)= C eat ( C, a are real value)
1 Signal and System
B. Periodic Complex Exponential and Sinusoidal Signals (1) x(t) = e j0t (2) x(t) = Acos(0t+) (3) x(t) = e jk0t All x(t) satisfy for x(t) = x(t+T) , and T=2/ 0 So x(t) is periodic.
1 Signal and System
Energy over t1 t t2:

t2
t1
p(t )dt v (t )dt x (t )dt
2 2 t1 gy:
E lim x (t )dt
2 T t1
t2
Average Power: P lim
C. Time Scaling
x(at) ( a>0 ) Stretch if a<1 Compressed if a>1
Example 1.1
1 Signal and System
1.2.2 Periodic Signals
Definition: There is a posotive value of T which : x(t)=x(t+T) , for all t x(t) is periodic with period T . T Fundamental Period
1 Signal and System
1.4 The Unit Impulse and Unit Step Functions
1.4.1 The Discrete-time Unit Impulse and Unit Step Sequences
0, n 0 (1) Unit Sample(Impulse): [n] 1, n 0
1 Signal and System Real or Imaginary of Signal
1 Signal and System
1.3.3 Periodicity Properties of Discrete-time Complex Exponentials Continuous-time: e j0t , T=2/0 Discrete-time: e j0n , N=?
1 Signal and System
Euler’s Relation: e j0t = cos0t + sin 0t and cos0t = (e j0t + e -j0t ) / 2 sin0t = (e j0t - e -j0t ) / 2
We also have
A j j 0t A j j 0t A cos( 0t ) e e e e 2 2
1 Signal and System
1.1.2 Signal Energy and Power
A. Energy (Continuous-time)
Instantaneous power: 1 2 p(t ) v(t ) i(t ) v (t ) R i 2 (t ) R Let R=1Ω, so p(t)=i2(t)=v2(t)=x2(t)
1 Signal and System Examples
1 Signal and System
1.3 Exponential and Sinusoidal signal
1.3.1 Continuous-time Complex Exponential and Sinusoidal Signals
n
or
u[ n]
[n k ]
k 0
1 Signal and System
(3) Sampling Property of Unit Sample