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.

Review for Exam I

ECE460Spring, 2012

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

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

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@

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@

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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