Lecture 1
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1/27/2015 1 EE110, S15: Circuits & Systems, Lec 01 - Example Signals and Systems. - Signal Transformations Prof. Ping Hsu 1 Examples of Signals & Systems (Lec 00) © 2013 National Technology and Science Press. All rights reserved 2 - We start by describing the nature of the input or output signals & some basic operations on these signals
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EE 110
Transcript of Lecture 1
1/27/2015
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EE110, S15: Circuits & Systems, Lec 01
- Example Signals and Systems. - Signal Transformations
Prof. Ping Hsu
1
Examples of Signals & Systems (Lec 00)
© 2013 National Technology and Science Press. All rights reserved
2
- We start by describing the nature of the input or output signals & some basic operations on these signals
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Expression of Signals
© 2013 National Technology and Science Press. All rights reserved
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Time signals are often expressed as mathematical functions of time, such as
0.1( )( ) sin( )tx t e t
A new signal y(t) can be derived from scaling another signal (x(t)) by multiplying the original function by a number.
0.1( )( ) 3 ( ) 3 sin( )ty t x t e t
0 5 10 15 20
20
0
20
x t( )
t
0 5 10 15 20
20
0
2030
30
y t( )
200 t
Expression of Signals4
/10
( 1) /10
Example: ( ) sin( )
( ) ( 1) sin( 1)
( ) is a '1-second delayed' version of ( ).
t
t
x t e t
y t x t e t
y t x t
0 5 10210
1
2
3
x t( )
y t( )
x(4) = -1 y(5)=-1
1
A delayed version of a signal x(t) can be expressed as
where T is the amount of delay. If T is negative, y(t) is a time advanced version of x(t).
( ) ( )y t x t T
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Signal Transformations© 2013 National Technology and Science Press. All rights reserved
T > 0 : y(t) is the “right shifted” version of x(t)
T < 0 : y(t) is the “left shifted” version of x(t)
( ) ( ) ( )
x t y t x t T
Signal Transformations
1-2.2 Time-Scaling Transformation
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The value of y at t is the value of x at at
x(t) y(t)=x(0.5t)y(t)=x(2t)
|a|<1: expansion from zero, |a|>1: Compression toward zero,
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Signal Transformations
© 2013 National Technology and Science Press. All rights reserved
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y(t) = x(-t), a = -1: time reversal, also called reflection
Signal Transformation Procedure
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Generalized transformation: Standard form: y(t) = x(at-b).
Step 1: Time delay (shift to right) by b.Step 2: Time scale by a.
Example: ( ) ( 2( 3)).
(1) Express ( ) in the standrad form.
( ) ( 2( 3))
= 2 ( 6) ( 2, 6)
(2) Shift x(t) to the lef
( )
t by 6 (since b<0)
x at b
y t x t
y t
y t x t
x t a b
(3) Scale z(t) by 2 (compressed toward zero)
and then 'flip' about zero (since a<0)
-3
-9
0
0
4.50
( )x t
( )z t
( )y t
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Signal Transformation Procedure
© 2013 National Technology and Science Press. All rights reserved
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Alternative method: Standard form: y(t) = x[a(t-b)].
Step 1: Time scale by a.Step 2: Time delay (or shift to right) by b.
Example: ( ) ( 2( 3)).
(1) 2, 3
(2) Scale x(t) by 2 (compress toward zero)
and then 'flip' about zero (since a<0)
(3) Shift z(t) to the right by 3
y t x t
a b
-3
-9
0
1.5
4.50
( )x t
( )z t
( )y t-1.5
© 2013 National Technology and Science Press. All rights reserved
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Signal Symmetries
x(t)
t0-2 -1 1 2
1
The values of x(t) at ±t , for any t, are identical
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© 2013 National Technology and Science Press. All rights reserved
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Signal Symmetries
t0-2 -1 1 2
1
x(t)
The values of x(t) at ±t , for any t, have identical absolute values and are opposite in sign
