Transcript Linearity - The University of Texas at Austin

EE313 Linear Systems and Signals
Fall 2010
Signal Energy
Prof. Brian L. Evans
Dept. of Electrical and Computer Engineering
The University of Texas at Austin
Initial conversion of content to PowerPoint
by Dr. Wade C. Schwartzkopf
Time/Frequency Domains
• Signal
Time domain: waveform, signal duration, waveform decay,
causality, periodicity
Frequency domain: sinusoidal components, relative
amplitudes and phases
• LTI system
Impulse response: width indicates system time constant,
a.k.a. response time, and amount of dispersion (spreading)
Frequency response: bandwidth filter selectively transmits
certain frequency components and suppresses the other,
transfer functions, stability
15 - 2
Time/Frequency Domains
• Time-Frequency
Fourier transform gives a global average of the frequency
content of a signal
Does not indicate when Fourier components are present in
time
In speech and audio processing, it is important to know
when certain frequencies occurred
15 - 3
Signal Energy
• Instantaneous power: |f(t)|2
Complex signal: |f(t)|2 = f(t) f *(t)
Real-valued signal: |f(t)|2 = f 2(t)
Example: f(t) is a voltage across a 1- resistor
• Energy of signal, f(t)
Ef  
Energy dissipated when voltage
f(t) is applied to 1 resistor
• Parseval’s Theorem

2
E f   f t  dt   

1
2





f t  dt  
2
F   d
2
Choose domain in which it is easier to compute energy
15 - 4
Example #1
• f(t) = e- a t u(t) is real-valued and causal


E f   f t dt   e  2 at dt 
2

F   
0
1
2a
1
j  a
F    F  F    
2
1
Ef 
2



1
1
1
 2
j   a  j  a   a 2
1  1
2
F   d   2
d
2
0
  a

1
1 
 
 1

tan 1   

0


a
 a 0  a 2
 2a
15 - 5
Example #2
• Determine the frequency W (rad/s) such that
energy contributed by frequencies   [0, W] is
95% of the total energy Ef = 1 / (2 a)
1
1
0.95
d 
2
2

0
  a
2a
1
0.95
1  W 
tan   
a
 a  2a
W
 tan 0.475    W  12.706 a rad/s
a
W
• W is known as the effective bandwidth
15 - 6
Average Signal Power
• Time average of signals with infinite energy
Signal does not go to zero as time goes to infinity
T /2
1
2
Pf  lim
f
(t ) dt
T  T 
T / 2
Time average of amplitude squared (mean-squared value)
Square root is root mean squared (RMS) value of signal
• Power signals must have infinite duration
Periodic signals are power signals if finite energy in period
Random signals are power signals
• Power of energy signal is zero
• Energy of power signal is infinite
15 - 7
Examples
• Lathi, example 1.1
Finite Energy
Ef =8
Infinite energy
but finite power
Pf = 1/3
15 - 8