final exam review
DESCRIPTION
Final Exam Review for Arizona State University course EEE 203 Signals and Systems. This exam covers Fourier Transforms, Laplace Transofrms, Z-transforms.TRANSCRIPT
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Chapter 4/5 CT Fourier Transform DT Fourier Transform For non-periodic signal
() = ()
= []=
is periodic in with period 2
() = 12 () [] = 12 2 For periodic signal () = 2( 0)
=
0 = 2 = 2( 0)
=
0 = 2 Properties Table 4.1 Table 5.1 Basic Transform Pairs
Table 4.2 Table 5.2
Chapter 9/10 Laplace Transform Z Transform
() = ()
() = []=
= + () = ()|=0
() = {()} = = ()|=1() = {[]} Properties Table 9.1 Table 10.1 Basic Transform Pairs
Table 9.2 Table 10.2
ROC ROC Finite Duration Signal
If x(t) is absolutely integrable, entire s-plane
Entire z-plane except possibly z = 0 and/or z =
Right -sided Signal
If () = ()() (rational), right half
plane to the right of the right-most pole
If () = ()() (rational), outside the circle of
the outmost pole (special care for z = )
Left-sided signal
If () = ()() (rational), left half plane
to the left of the left-most pole
If () = ()() (rational), inside the circle of
the innermost nonzero pole (special care for z = 0)
Two-sided signal
If () = ()() (rational), the strip
parallel to the j axis bounded by poles
If () = ()() (rational), the ring bounded by
poles ()
() , . ., |()|
< ({} = 0)
[] , . ., |[]|
=
< (|| = 1)
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LTI System LTI
h(t)x(t) y(t)
LTIh[n]
x[n] y[n]
Impulse response
h(t)
h[n]
() = () ()= ()( )
[] = [] []= [][ ]=
Frequency response
H(j) H(ej)
() = ()() = () System or Transfer function
H(s) H(z)
() = ()() () = ()() Properties Causal If () = ()
() (rational), ROC of a causal system function is the right half plane to the right of the right-most pole.
If () = ()() (rational), ROC of a causal
system function is the exterior of a circle outside of the outmost pole and the order of the numerator cannot be greater than the order of the denominator.
Stable ROC of a stable system function includes the entire j axis ({} = 0). ROC of a stable system function includes the unit circle (|z|=1).
Causal and Stable
ROC of a causal and stable system function has all poles in the left-half plane, i.e., all of the poles have negative real parts.
ROC of a causal and stable system function has all poles inside the unit circle, i.e., all of the poles have magnitude smaller than 1.
CT LTI system characterized by linear-
constant coefficient differential equation
DT LTI system characterized by linear-constant coefficient difference equation
()
=0
= () =0
[ ]=0
= [ ]=0
Frequency Response () ()
=0
= () ()=0
=0
= =0
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() = ()() = ()=0 [ ()=0 ]
= () = =0 [ =0 ]
System or Transfer Function
() =0
= () =0
() = ()() = =0 [ =0 ]
() =0
= () =0
() = ()() = =0 [ =0 ]
Interconnections of LTI system
Serial h1(t) h2(t)x(t) y(t)
h1[n] h2[n]x[n] y[n]
() = 1() 2() () = 1()2() () = 1()2()
[] = 1[] 2[] = 12 () = 1()2()
Parallel
+x(t) y(t)
h1(t)
h2(t)
+x[n] y[n]
h1[n]
h2[n]
() = 1() + 2() () = 1()+2() () = 1()+2()
[] = 1[] + 2[] = 1+2 () = 1()+2()
Chapter 7 Sampling
Sampling Theorem
If > 2, we can use [] =() to reconstruct () perfectly. : sampling frequency : highest frequency in signal 2: Nyquist rate
If 2, aliasing occurs.
Impulse Train Sampling
() = ()() () = ( )= , = 2. () is a periodic signal with Fourier series coefficient = 1. () = 1
2( )=
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() = 12() () = 1 ( )=
1X(j)
-M
1/Ts
Xp(j)
-s
... ...
M -M M s
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