Continuous-time Signalscarmenere.ucsd.edu/jorge/teaching/mae143a/lectures/1analogsignals.pdfThe Unit...
Transcript of Continuous-time Signalscarmenere.ucsd.edu/jorge/teaching/mae143a/lectures/1analogsignals.pdfThe Unit...
![Page 1: Continuous-time Signalscarmenere.ucsd.edu/jorge/teaching/mae143a/lectures/1analogsignals.pdfThe Unit Step function u(t)= 1 ,t>0 1/2,t=0 0 ,t](https://reader033.fdocuments.in/reader033/viewer/2022042403/5f164d00959e6b7545104453/html5/thumbnails/1.jpg)
Continuous-time Signals
(AKA analog signals)
![Page 2: Continuous-time Signalscarmenere.ucsd.edu/jorge/teaching/mae143a/lectures/1analogsignals.pdfThe Unit Step function u(t)= 1 ,t>0 1/2,t=0 0 ,t](https://reader033.fdocuments.in/reader033/viewer/2022042403/5f164d00959e6b7545104453/html5/thumbnails/2.jpg)
I. Analog* Signals review
Goals:
*Analog = the argument (time) of the signal iscontinuous (CT). Also known as CT signals.
-Common test signals used in system analysis
- Signal operations and properties: scaling, shifting, periodicity, energy and power
!
t
!
g(t)
![Page 3: Continuous-time Signalscarmenere.ucsd.edu/jorge/teaching/mae143a/lectures/1analogsignals.pdfThe Unit Step function u(t)= 1 ,t>0 1/2,t=0 0 ,t](https://reader033.fdocuments.in/reader033/viewer/2022042403/5f164d00959e6b7545104453/html5/thumbnails/3.jpg)
The Unit Step function
u t( ) =1 , t > 01 / 2 , t = 00 , t < 0
!
"#
$#
Precise Graph Commonly-Used Graph
Note: The signal is discontinuous at zero but is an analog signal
Note: The product signal g(t)u(t) for any g(t) can be thought of asthe signal g(t) “turned on” at time t = 0.
Used to check how a system responds to a “sudden” input
![Page 4: Continuous-time Signalscarmenere.ucsd.edu/jorge/teaching/mae143a/lectures/1analogsignals.pdfThe Unit Step function u(t)= 1 ,t>0 1/2,t=0 0 ,t](https://reader033.fdocuments.in/reader033/viewer/2022042403/5f164d00959e6b7545104453/html5/thumbnails/4.jpg)
The Signum Function
sgn t( ) = 1 , t > 0 0 , t = 0!1 , t < 0
"
#$
%$
&
'$
($= 2u t( )!1
Precise Graph Commonly-Used Graph
![Page 5: Continuous-time Signalscarmenere.ucsd.edu/jorge/teaching/mae143a/lectures/1analogsignals.pdfThe Unit Step function u(t)= 1 ,t>0 1/2,t=0 0 ,t](https://reader033.fdocuments.in/reader033/viewer/2022042403/5f164d00959e6b7545104453/html5/thumbnails/5.jpg)
The Unit Rectangle function
The product can be understood as the signalturned on at and turned off at
5
rect t( ) = 1 , t <1 / 21 / 2 , t = 1 / 2 0 , t >1 / 2
!
"#
$#
%
&#
'#= u t +1 / 2( )( u t (1 / 2( )
!
g(t)rect(t)
!
t = "12
!
t =12
![Page 6: Continuous-time Signalscarmenere.ucsd.edu/jorge/teaching/mae143a/lectures/1analogsignals.pdfThe Unit Step function u(t)= 1 ,t>0 1/2,t=0 0 ,t](https://reader033.fdocuments.in/reader033/viewer/2022042403/5f164d00959e6b7545104453/html5/thumbnails/6.jpg)
The Unit Ramp function
ramp t( ) =t , t > 00 , t ! 0"#$
%&'= u (( )d(
)*
t
+ = t u t( )
The Unit Ramp function is unbounded with time
![Page 7: Continuous-time Signalscarmenere.ucsd.edu/jorge/teaching/mae143a/lectures/1analogsignals.pdfThe Unit Step function u(t)= 1 ,t>0 1/2,t=0 0 ,t](https://reader033.fdocuments.in/reader033/viewer/2022042403/5f164d00959e6b7545104453/html5/thumbnails/7.jpg)
Unit Triangle function
The triangle signal is related to the rectanglethrough an operation called convolution
(to be introduced later…)7
tri t( ) =1! t , t <10 , t "1#$%&
'()&
![Page 8: Continuous-time Signalscarmenere.ucsd.edu/jorge/teaching/mae143a/lectures/1analogsignals.pdfThe Unit Step function u(t)= 1 ,t>0 1/2,t=0 0 ,t](https://reader033.fdocuments.in/reader033/viewer/2022042403/5f164d00959e6b7545104453/html5/thumbnails/8.jpg)
The Impulse function
The impulse function is not a function in the ordinarysense because its value at zero is not a real value
It is represented by a vertical arrowThe impulse function is unbounded and discontinuous
!
"(0) =#,
!
"(t) = 0,
!
t " 0
![Page 9: Continuous-time Signalscarmenere.ucsd.edu/jorge/teaching/mae143a/lectures/1analogsignals.pdfThe Unit Step function u(t)= 1 ,t>0 1/2,t=0 0 ,t](https://reader033.fdocuments.in/reader033/viewer/2022042403/5f164d00959e6b7545104453/html5/thumbnails/9.jpg)
Other approximations are possible…for example, we can use a triangle function
Creating an Impulse
An impulse can be defined as the limit of therectangle function with unit area as goes to zero
!a t( ) =1 / a , t < a / 20 , t > a / 2"#$%
!
a
![Page 10: Continuous-time Signalscarmenere.ucsd.edu/jorge/teaching/mae143a/lectures/1analogsignals.pdfThe Unit Step function u(t)= 1 ,t>0 1/2,t=0 0 ,t](https://reader033.fdocuments.in/reader033/viewer/2022042403/5f164d00959e6b7545104453/html5/thumbnails/10.jpg)
The Unit Sinc function
The unit sinc functionis related to the unitrectangle functionthrough theFourier Transform
It is used for noiseremoval in signals
sinc t( ) = sin ! t( )! t
limt!0
sinc t( ) = limt!0
ddtsin " t( )( )ddt
" t( )= lim
t!0
" cos " t( )"
= 1
![Page 11: Continuous-time Signalscarmenere.ucsd.edu/jorge/teaching/mae143a/lectures/1analogsignals.pdfThe Unit Step function u(t)= 1 ,t>0 1/2,t=0 0 ,t](https://reader033.fdocuments.in/reader033/viewer/2022042403/5f164d00959e6b7545104453/html5/thumbnails/11.jpg)
Real exponentials
!
e0.5t
!
et
!
e2t
!
e"0.5t
!
e"t
!
e"2t
red blue black
red blue black
![Page 12: Continuous-time Signalscarmenere.ucsd.edu/jorge/teaching/mae143a/lectures/1analogsignals.pdfThe Unit Step function u(t)= 1 ,t>0 1/2,t=0 0 ,t](https://reader033.fdocuments.in/reader033/viewer/2022042403/5f164d00959e6b7545104453/html5/thumbnails/12.jpg)
Sinusoids (and Cosinusoids)
redblueblack
redblack
for
radian frequency: (rad/s) cyclic frequency: (cycles/s) wavelength or period: amplitude:
phase:
!
sin(5t)
!
sin(3t)
!
sin(7t)
!
sin(5t)
!
cos(5t)
!
cos("t + #) = cos(2$ft + #)
!
"
!
f
!
1
!
"
!
1/ f
![Page 13: Continuous-time Signalscarmenere.ucsd.edu/jorge/teaching/mae143a/lectures/1analogsignals.pdfThe Unit Step function u(t)= 1 ,t>0 1/2,t=0 0 ,t](https://reader033.fdocuments.in/reader033/viewer/2022042403/5f164d00959e6b7545104453/html5/thumbnails/13.jpg)
Real and complex sinusoids
Recall that a complex number is defined as(here and )
A complex sinusoid is expressed asA graphical example for :
Relation between complex and realsinusoids:
!
z = a + jb
!
Re(z) = a
!
Im(z) = b
!
j = "1
!
e j"t = cos("t) + j sin("t)
!
" = 2#
!
cos("t) =e j"t + e# j"t
2
!
sin("t) =e j"t # e# j"t
2 j
![Page 14: Continuous-time Signalscarmenere.ucsd.edu/jorge/teaching/mae143a/lectures/1analogsignals.pdfThe Unit Step function u(t)= 1 ,t>0 1/2,t=0 0 ,t](https://reader033.fdocuments.in/reader033/viewer/2022042403/5f164d00959e6b7545104453/html5/thumbnails/14.jpg)
Analog signals review
Goals: - Common test signals used in system analysis
- Signal operations and properties: shifting, scaling, periodicity, energy and power
14
![Page 15: Continuous-time Signalscarmenere.ucsd.edu/jorge/teaching/mae143a/lectures/1analogsignals.pdfThe Unit Step function u(t)= 1 ,t>0 1/2,t=0 0 ,t](https://reader033.fdocuments.in/reader033/viewer/2022042403/5f164d00959e6b7545104453/html5/thumbnails/15.jpg)
Signal operations: shifting and scaling
Amplitude scaling
!
g t( )" Ag t( )
Expansion/contraction of signal along Y axisRotation about X axis for negative A
![Page 16: Continuous-time Signalscarmenere.ucsd.edu/jorge/teaching/mae143a/lectures/1analogsignals.pdfThe Unit Step function u(t)= 1 ,t>0 1/2,t=0 0 ,t](https://reader033.fdocuments.in/reader033/viewer/2022042403/5f164d00959e6b7545104453/html5/thumbnails/16.jpg)
Signal operations: shifting and scaling
Time shifting
Time shifting
!
t" t # t0
![Page 17: Continuous-time Signalscarmenere.ucsd.edu/jorge/teaching/mae143a/lectures/1analogsignals.pdfThe Unit Step function u(t)= 1 ,t>0 1/2,t=0 0 ,t](https://reader033.fdocuments.in/reader033/viewer/2022042403/5f164d00959e6b7545104453/html5/thumbnails/17.jpg)
Signal operations: shifting and scaling
Time scaling t! t / a
Expansion/contraction of signal along X axisRotation about Y axis for negative a
![Page 18: Continuous-time Signalscarmenere.ucsd.edu/jorge/teaching/mae143a/lectures/1analogsignals.pdfThe Unit Step function u(t)= 1 ,t>0 1/2,t=0 0 ,t](https://reader033.fdocuments.in/reader033/viewer/2022042403/5f164d00959e6b7545104453/html5/thumbnails/18.jpg)
Example: Doppler effect
Sound heard by firefighters: g(t)Sound we hear when truck comes: A(t) g(at), A increasing, a>1Sound we hear when truck goes: B(t) g(bt), B decreasing, b<1
http://www.lon-capa.org/~mmp/applist/doppler/d.htm
Checkout the website:
![Page 19: Continuous-time Signalscarmenere.ucsd.edu/jorge/teaching/mae143a/lectures/1analogsignals.pdfThe Unit Step function u(t)= 1 ,t>0 1/2,t=0 0 ,t](https://reader033.fdocuments.in/reader033/viewer/2022042403/5f164d00959e6b7545104453/html5/thumbnails/19.jpg)
Signal operations: shifting and scaling
A signal may be expressed by means of basic testsignals that have been time-scaled and amplified
![Page 20: Continuous-time Signalscarmenere.ucsd.edu/jorge/teaching/mae143a/lectures/1analogsignals.pdfThe Unit Step function u(t)= 1 ,t>0 1/2,t=0 0 ,t](https://reader033.fdocuments.in/reader033/viewer/2022042403/5f164d00959e6b7545104453/html5/thumbnails/20.jpg)
Signal operations: shifting and scaling
Caution! The impulse satisfies special properties
Time-scaling property
Sampling property
The impulse functionis not a function in the ordinary sense
g t( )! t " t0( )dt"#
#
$ = g t0( )
! a t " t0( )( ) = 1a ! t " t0( )
![Page 21: Continuous-time Signalscarmenere.ucsd.edu/jorge/teaching/mae143a/lectures/1analogsignals.pdfThe Unit Step function u(t)= 1 ,t>0 1/2,t=0 0 ,t](https://reader033.fdocuments.in/reader033/viewer/2022042403/5f164d00959e6b7545104453/html5/thumbnails/21.jpg)
Signal properties: periodic signals
Periodic signals repeat
Cycle time is the period THere about 1 second 2 seconds is also a period
We only need define the signalover one period and we knoweverything about it
Sinusoids and constant are clearlyperiodic signals
Other examples include periodicpulses (rectangular andtriangular pulses)
!
x(t + kT ) = x(t)
![Page 22: Continuous-time Signalscarmenere.ucsd.edu/jorge/teaching/mae143a/lectures/1analogsignals.pdfThe Unit Step function u(t)= 1 ,t>0 1/2,t=0 0 ,t](https://reader033.fdocuments.in/reader033/viewer/2022042403/5f164d00959e6b7545104453/html5/thumbnails/22.jpg)
Signal properties: periodic signals
The sum of two periodic signals
with periods and is:
!
x(t) = x1(t) + x2(t)
!
T1
!
T2
!
T =LCM(a1,a2)GCD(b1,b2)
!
T1T2
!
T1 =a1b1
!
T2 =a2b2
- periodic when is rational. If , then the period is
- not periodic when is irrational
!
T1T2
![Page 23: Continuous-time Signalscarmenere.ucsd.edu/jorge/teaching/mae143a/lectures/1analogsignals.pdfThe Unit Step function u(t)= 1 ,t>0 1/2,t=0 0 ,t](https://reader033.fdocuments.in/reader033/viewer/2022042403/5f164d00959e6b7545104453/html5/thumbnails/23.jpg)
Signal energy and power
Quantifying the “size” of a signal is important inmany applications: How much electricity can beused in a defibrillator? How much energy should anaudio signal have to be heard?
The energy of the signal is Ex = x t( ) 2 dt!"
"
#
!
x(t)
![Page 24: Continuous-time Signalscarmenere.ucsd.edu/jorge/teaching/mae143a/lectures/1analogsignals.pdfThe Unit Step function u(t)= 1 ,t>0 1/2,t=0 0 ,t](https://reader033.fdocuments.in/reader033/viewer/2022042403/5f164d00959e6b7545104453/html5/thumbnails/24.jpg)
Signal energy and power
Some signals have infinite energy. In that case, wemay use the concept of average signal power
For a periodic signal, , with period T, the averagesignal power is
If the signal is not periodic, then
Px =1T
x t( ) 2 dtT!
Px = limT!"
1T
x t( ) 2 dt#T /2
T /2
$
!
x(t)
![Page 25: Continuous-time Signalscarmenere.ucsd.edu/jorge/teaching/mae143a/lectures/1analogsignals.pdfThe Unit Step function u(t)= 1 ,t>0 1/2,t=0 0 ,t](https://reader033.fdocuments.in/reader033/viewer/2022042403/5f164d00959e6b7545104453/html5/thumbnails/25.jpg)
Analog signals summary
We have seen:
-Signals can be seen as inputs/outputs to systems
-Analog signals can be represented as functions ofcontinuous time
-The unit step, impulse, ramp and rectangle functions areexamples of test signals to systems
-A general signal can be expressed as a combination of somebasic test signals by using scaling/shifting operations
-Properties of signals include periodicity, even/odd,continuity, differentiability, etc
-Power and energy are concepts that measure signal “size”