review for exam i ece460 spring, 2012. dirichlet conditions fourier series 1. 2. x ( t ) has a...
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![Page 1: Review for Exam I ECE460 Spring, 2012. Dirichlet Conditions Fourier Series 1. 2. x ( t ) has a finite number of minima and maxima over one period 3. x](https://reader035.vdocuments.mx/reader035/viewer/2022062718/56649e875503460f94b8bbb3/html5/thumbnails/1.jpg)
Review for Exam I
ECE460Spring, 2012
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Dirichlet Conditions
Fourier Series
1.
2. x(t) has a finite number of minima and maxima over one period
3. x(t) has a finite number of discontinuities over one period
0T
x t dt
Fourier Transform
1.
2. x(t) has a finite number of minima and maxima in any interval on the real line
3. x(t) has a finite number of discontinuities over any interval on the real line
x t dt
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Fourier Series(Periodic Functions)
Exponential Form
Real Coefficient Trigonometric Form
Complex Coefficient Trigonometric Form
0
1 0 0
( ) cos 2 sin 22 n n
n
a n nx t a t b t
T T
0
0
0 0
0 0
2cos 2
2sin 2
n T
n T
na x t t dt
T T
nb x t t dt
T T
01 0
( ) cos 2n nn
nx t x x t x
Tp
¥
=
æ ö÷ç ÷= + +ç ÷ç ÷çè øå R
2 21
2
arctan
n n n
nn
n
x a b
bx
a
= +
æ ö÷ç ÷=- ç ÷ç ÷çè øR
0
2
( )n
j tT
nn
x t x e
0
0
2
0
1n
j tT
n Tx x t e dt
T
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4
Common Fourier Transform Pairs
Time Frequency 1 t 1
2 1 f
3 0t t 02j f te
4 02j f te 0f f
5
cos 2 of t 0 0
1
2f f f f
6 0sin 2 f t 0 02
jf f f f
7 t sinc f
8 t 2sinc f
9 2sinc t f
10 , 0te u t 1
2j f
11 , 0tt e u t 2
1
2j f
12 , 0te 22
2
2 f
13 2te 2fe
14 sgn t 1
j f
15 u t 1 1
2 2f
j f
16 ( )n t 2n
j f
17 1
t sgnj f
18 0j n tn
n
X e
02 n
n
X n
19 0n
t nT
0 0
1
n
nf
T T
( ) ( ) 2j f tx t X f e dfp¥
- ¥ò@ ( ) 2( ) j f tX f x t e dtp
¥-
- ¥ò@
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5
Fourier Transform Properties
Property Time Frequency Linearity 1 1 2 2( ) ( )x t x t 1 1 2 2( ) ( )X f X f
Time Shift 0x t t 02j f te X f
Duality X t x f
Time Scaling x at 1 f
Xa a
Convolution x t y t X f Y f
Multiplication ( ) ( )x t y t ( ) ( )X f Y f
Parseval’s Theorem
*( )x t y t dt
*( )X f Y f df
Differentiation n
n
dx t
dt 2
nj f X f
Integration t
x d
( ) 1(0) ( )
2 2
X fX t
j f
Rayleigh’s 2x d
2
X f df
Autocorrelation *( )xR t x t x t d
2[ ( )]xR t X fF
Moments nt x d
02
n n
n f
j dX f
df
Modulation 0( ) cos(2 )x t f t 0 0
1 1( ) ( )
2 2X f f X f f
( ) ( ) 2j f tx t X f e dfp¥
- ¥ò@ ( ) 2( ) j f tX f x t e dtp
¥-
- ¥ò@
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6
Sampling TheoremAble to reconstruct any bandlimited signal from its samples if we sample fast enough.
If X(f) is band limited with bandwidth W
then it is possible to reconstruct x(t) from samples i.e., 0 for X f f W
s nx nT
1if
2sTW
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7
Example
Properties of a System:
– Linear
– Time-Invariant
– Causality
– Stability
y t x t h t x h t d
Filter( )x t
( )h t
( )y t
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8
Narrowband SignalsGiven:
0
0
:bandpass signal with center frequency
: impulse response of LTI system
- narrowband
- centered on frequency f
x t f
h t
020
Definitions:
2
which gives
X which gives
(in-phase and quadrature components)
Final
j f tl l
l c s
Z f u f X f
jz t t X f
t
z t x t j x t
f Z f f x t z t e
x t x t x t
02
0 0
0 0
0 0
0 0
ly,
z
cos 2 sin 2
+ sin 2 cos 2
cos 2 sin 2
= cos 2 sin 2
j f tl
c s
c s
c
s
t x t e
x t f t x t f t
j x t f t x t f t
x t x t f t x t f t
x t x t f t x t f t
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9
Bandpass Signals & SystemsFrequency Domain:
Low-pass Equivalents:
Let
Giving
To solve, work with low-pass parameters (easier mathematically), then switch back to bandpass via
Y f X f H f
0 0 02lY f u f f X f f H f f
0 0
0 0
2
2
l
l
X f u f f X f f
H f u f f H f f
1
21
2
l l l
l l l
Y f X f H f
y t x t h t
2Re oj f tly t y t e
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10
Analog ModulationAmplitude Modulation (AM)
Message Signal:Sinusoidal Carrier:
• AM (DSB)
• DSB – SC
• SSB• Started with DSB-SC signal and filtered to one sideband• Used ideal filter:
•
• Vestigial
( )m t
( ) cos(2 )c cc t A f t
( ) 1 ( ) cos(2 )
( ) ( ) ( ) ( ) ( )2 2
c a c
c a cc c c c
s t A k m t f t
A k AS f f f f f M f f M f f
( ) cos(2 ) ( )
( ) ( ) ( )2
c c
c c
s t A f t m t
AcS f M f f M f f
1,( )
0, otherwisecf f
H f
ˆcos 2 sin 2
where
1ˆ
c c c cs t A m t f t A m t f t
m t m tt
4
cc c
AV f M f H f f H f f
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11
Analog ModulationAngle Modulation
Definitions:
FM (sinusoidal signal)
2
2
cos 2 cos 2 2
t
p f
p f
t
c c p c c f
PM FM
t k m t k m d
d dt k m t k m t
dt dt
s t A f t k m t A f t k m d
Deviation constants ,
Modulation Index ( ) max
max max
f p
p p
f f fm
k k
k m t
m t m tk k
f W
( ) cos 2 sin 2
Re cos 2
2
c c m
c n c mn
cn c m c m
n
s t A f t f t
A J f n f t
AS f J f f n f f f n f
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12
Combinatorics1. Sampling with replacement and ordering
2. Sampling without replacement and with ordering
3. Sampling without replacement and without ordering
4. Sampling with replacement and without ordering
Bernouli Trials
Conditional Probabilities
where = population size and = subpopulation sizern n r
!
!
n
n r
!
! !
n n
r n r r
1n r
r
probability of success and 1 probability of failure
A event of k-success in n-trialsk
p p
1
; , Binomial Law
n kkk
nP A p p
k
b k n p
1 2
21 2 2
, 0( | )
0, Otherwise
P E EP E
P E E P E
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13
Random Variables• Cumulative Distribution Function (CDF)
• Probability Distribution Function (PDF)
• Probability Mass Function (PMF)
• Key Distributions• Bernoulli Random Variable
• Uniform Random Variable
• Gaussian (Normal) Random Variable
XF x P X x
X X
df x F x
dx
i ip P X x
1
where = center line and width
X
xf x
1 , 0
, 1
0, otherwise
p x
P X x p x
2
2 221 or : ,
2
1
xx m
X x
xX X
X
f x e X N m
x mF x
Q x x
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14
Functions of a Random VariableGeneral:
Statistical Averages• Mean
• Variance
:
where number of g x equal to y
Y
X iY i
i i
Y g X
F y P g X y
f xf y i
g x
x Xm E X x f x dx
22x xE X m
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15
Multiple Random VariablesJoint CDF of X and Y
Joint PDF of X and Y
Conditional PDF of X
Expected Values
Correlation of X and Y
Covariance of X and Y - what is ρX,Y
Correlation of X and Y
, , ,X YF x y P X x Y y
2
, , ,X Y XYf x y F x yx y
,, , ,X YE g X Y g x y f x y dx dy
,( , ) ( ( , )) ,XY X YR x y E g X Y x y f x y dx dy
,COV( , ) ( )( ) ,x y X YX Y x m y m f x y dx dy
,
|
,, 0
|
0, otherwise
X YX
Y X X
f x yf x
f y x f x
, ,X Y X Yf x y f x f y
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Jointly Gaussian R.V.’sX and Y are jointly Gaussian if
Matrix Form:
Function:
16
, 221 2
2 2
1 2 1 22 21 2 1 2
1 1, exp
2 12 1
2
X Yf x y
x m y m x m y m
1 1 2 1
2 1 2 2
1
Var Cov , Cov ,
Cov , Var Cov ,
Cov , Var
covariance matrix of .
T
n
n
n n
E
X X X X X
X X X X X
X X X
C X m X m
X
1[ ]
mean vector of .
[ ]n
E X
E
E X
m X X
11
22
1 1exp
22
T
nf
X x x m C x mC
Y AX b
Y XE E m Y A X b Am b
T
Y Y Y
T TX X
TX
E
E
C Y m Y m
A X m X m A
AC A
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17
Random ProcessesNotation:
Understand integration across time or ensemblesMean
Autocorrelation
Auto-covariance
Power Spectral Density
Stationary Processes• Strict Sense Stationary• Wide-Sense Stationary (WSS)• Cyclostationary
Ergodic
x X tm t E X t f d
1 2
1 2 1 2
1 2 1 2 1 2,
,
,
X
X t X t
R t t E X t X t
x x f x x dx dx
1 2 1 1 2 2,X x xC t t E X t m t X t m t
1 2,X XS f R t tF
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18
Transfer Through a Linear System
Mean of Y(t) where X(t) is wss
Cross-correlation function RXY(t1,t2)
Autocorrelation function RY(t1,t2)
Spectral Analysis
h t X t Y t
0Y X Xm t E Y t m h s ds m H
1 2 1 2,XY
X
R t t E X t Y t
R h
1 2 1 2,Y
XY
X
R t t E Y t Y t
R h
R h h
2
X X
XY XY X
Y X
S f R
S f R S f H f
S f S f H f
F
F
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19
Energy & Power ProcessesFor a sample function
For Random Variables we have
Then the energy and power content of the random process is
2 ,i ix t dt
E
X
2
2
X
,
X
X
E E
E t dt
E X t dt
R t t dt
E
2 , ix t
2X X t dt
E
2 2
2
lim ,T
i iTTP x t dt
2
2
21lim
T
TX TX t dt
T P
2
2
2
2
2
2
X
2
2
1lim
1lim
1lim ,
T
T
T
T
T
T
X
T
T
XT
P E
E X t dtT
E X t dtT
R t t dtT
P
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20
Zero-Mean White Gaussian NoiseA zero mean white Gaussian noise, W(t), is a random process with
4. For any n and any sequence t1, t2, …, tn the random variables W(t1), W(t2), …, W(tn), are jointly Gaussian with zero mean
and covariances
1. 0
2.2
3. Watt/Hz2
oW
oW
E W t t
NR E W t W t
NS f
, cov
(since zero mean)
2
X i j i j
i j
W j i
oj i
K t t W t W t
E W t W t
R t t
Nt t
0 for 1,2,...,iE W t i n
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21
Bandpass ProcessesX(t) is a bandpass process
Filter X(t) using a Hilbert Transform:
and define
If X(t) is a zero-mean stationary bandpass process, then Xc(t) and Xs(t) will be zero-mean jointly stationary processes:
Giving
0
is a deterministic bandpass signal
and is non-zero about
X
X X
R
S f R f
F
1; sgnh t H f j f
t
0 0
0 0
cos 2 sin 2
cos 2 sin 2c
s
X t X t f t X t f t
X t X t f t X t f t
0
,
,
,
c c
s s
c s c s
c s
X X
X X
X X X X
E X t E X t
R t t R
R t t R
R t t R
0 0
0 0
cos 2 sin 2
sin 2 cos 2c s
c s
X X X X
X X X X
R R R f R f
R R f R f