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Lecture #1Review of DT Signal

& System Concepts

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Notational Conventions (P-1.2)

Sets of Numbers: R = real numbersC = complex numbersZ = integers ( , -3, -2, -1, 0, 1, 2, 3, )

Signals: x(t ), y(t ) = continuous-time (CT) signals x[n], y[n] = discrete-time (DT) signals

Modulo Operator Clock Math: n mod N

n: -4 -3 -2 -1 0 1 2 3 4 5 6 7 n mod N: 2 0 1 2 0 1 2 0 1 2 0 1 15 oclock = 15 mod 12= 3 oclock

DT Convolution:

==

m

mn ym xn y x ][][]}[*{

DT Circular Convolution: 10]mod)[(][]}[{1

0

=

= N n N mn ym xn y x

N

m

Recall: We dont want Circular convolution it is what we get if we try to implementConvolution in the frequency domain by multiplying DFTs (but not DTFTs)But we can fix it so that it works. We use proper zero-padding!!!

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Transforms (P-2)Fourier Transform for CT Signals

{ }

=

=

=

d e X t x

dt et x

Fx X

t j F

t j

F

)(21

)(

)(

)()( { }

=

=

=

=

d e X n x

en x

Fx X

n j

n

n j

)(21

][

][

)()(

f

f DiscreteT imeFourierT ransform

Discrete Fourier Transform for DT Signals

{ }

1,...,2,1,0][1

][

1,...,2,1,0][

][][

1

0

/ 2d

1

0

/ 2

d

==

==

=

=

=

N nek X N

n x

N k en x

k Dxk X

N

k

N kn j

N

n

N kn j

Discrete Fourier Transform

Z Transform for DT Signals

{ }

=

=

=

n

n z n x

z Zx z X

][

)()(z

Set z = e j

Fourier Transform for DT

Signals

Inverse ZT done using partialfractions & a ZT table

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Fourier Transforms for DT Signals (P-2.7,P-4)DTFT = In Head/On Paper tool for analysis & design

DFT = In Computer tool for Implementation/Analysis&Design(In both cases you generally strive to make the DFT behave like DTFT)

DFT for Implementation:! Compute DFT of signal to be processed and manipulate DFT values! May or may not need zero-padding (depends on application)

DFT for Analysis & Design:!

Compute DFT of given filters TF to see its frequency response! Generally use zero-padding to get fine grid to get approximate DTFT

DFT/DTFT Properties you Must Know: Periodicity. (DFT&DTFT both periodic in FD)

. (IDFT gives periodic TD signal) Time Shift.. (Time Shift in TD " Linear Phase Shift in FD) Frequency Shift. (Mult. by complex sine in TD " Freq Shift in FD) Convolution Theorem (Conv. in Time Domain " Mult. in Freq Domain)

Multiplication Theorem.. (Mult. in Time Domain " Conv. in Freq Domain) Parsevals Theorem. (Sum of Squares in TD = Sum of Squares in FD)

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Summation Rules (P-1.3)Make sure you are familiar with rules for dealing withsummations, as shown in Section 1.3 of Porat.

In addition, a particular summation formula that shows up

often is the Geometric Summation :

=

=

= 1 if ,

1 if ,1

12

1

21

2

1

N N

N N

N

N n

n

Special Case: N 1 = 0

=

=

= 1 if ,

1 if ,1

1

2

1

0

2

2

N

N

N

n

n

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Eulers Formulas

j

ee

ee

j j

j j

2

)sin(

2)cos(

=

+=

Im

je

je

)cos(2

je 2 j s i n

( )

)sin()cos(

)sin()cos(

je

je j

j

=+=

Im

Re

je

je

)cos(

)sin( j

)sin( j Re

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Discrete-Time Processing (P-2, P-7)h[n]

H f () H z( z )

x[n] X f () X z( z )

y[n] = h[n]* x[n]Y f () = H f () X f ()Y z( z ) = H z( z ) X z( z )

h[n] = systems Impulse ResponseIt is response of system to input of [n]

H f () = systems Frequency ResponseIt is DTFT of impulse response

H z( z ) = systems Transfer FunctionIt is ZT of impulse response

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Discrete-Time System Relationships

Time Domain Z / Freq Domain

DTFT

ZT

ZT (Theory)

Inspect (Practice)

Inspect Inspect Roots

Unit Circle

z = e j

p p

qq z

z a z a

z b z bb z H

+++

+++=

L

L1

1

110

1)(

==

+=q

ii

p

ii in xbin yak y

01

][][][

Pole/ZeroDiagramBlock

Diagram

Difference

Equation

Transfer

Function

FrequencyResponseImpulse

Response je z

z z H H == |)()(f

h[k ]

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Poles and Zeros of Transfer Function

Consider Here Only p>q

)())((

)())((

)(

)1(

)(

1)(

21

21

11

110

11

110

11

110

p

qq p

p p p

qqqq p

p p

p

qq

qq p

p p

qq z

z z z

z z z z

a z a z

b z b z b z

z a z a z

z b z bb z z

z a z a

z b z bb z H

=

+++

+++=

++++++

=

++++++

=

LL

L

L

L

L

L

L

Re{ z }

Im{ z }

Numerator Roots define zerosDenominator Roots define poles

Poles must be insideunit circle for stability

Poles & zeros must occur inComplex Conjugate pairs

to give real-valued TF coefficients

In this case pq zeros at z =0

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IIR and FIR Filters

p p

qq z

z a z a z b z bb z H

++++++=

LL

11

110

1)(

FIR : Finite Impulse Response h[n] has only finite many nonzero valuesFilter is FIR : If a i = 0 for i = 1, 2, , p (It is IIR otherwise)

qq

z z b z bb z H +++= L110)(

==otherwise

qnbnh n,0

,...,1,0,][ TF coefficients are theimpulse response values!!

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Example System


DifferenceEquation

TransferFunction

ZT (Theory)Inspect (Practice)

)(1)( 1 z X z z Y = )()1()( n xn yn y =Input-Output Form

1

1)(

)()(

==

z z X

z Y z H

)1()()( += n yn xn y

Recursion Form

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Example System (cont.)


Difference

Equation

Block Diagram

ZT (Theory)

Inspect (Practice)

Inspect Inspect

11)( =

z z H

z-1

x(n) y(n)

y(n-1)

)1()()( += n yn xn y

Recursion Form

Transfer

Function

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Z / Freq Domain

11)( =

z z H

Transfer

Function

==

j

j

e z e z

1

OnUnitCircle

Unit Circle = je z 1

],(1

)(

= je

H FrequencyResponse

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[ ]

[ ]

[ ] [ ]

=

+=

+=

=

=

)cos(1)sin(

tan)(

)sin()cos(1)(

)sin()cos(1

)sin()cos(11)(

1

22

H

H

j

je H j

EulersEquation

Plotting Transfer Function

Group intoReal & Imag

StandardEqs for

Mag. & Angle

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TransferFunction

Pole/ZeroDiagram

Roots

Z/Freq Domain

=

= z

z z

z H 11)(

UnitCircle

Zero

Num. = 0When z = 0

Re{z}

Im{z}

If < 1: In side UCIf > 1: Out side UC

Pole

Den. = 0When z =

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ImpulseResponse

DTFT

ZTUnit Circle

11)( =

z z H

],(

1)(

= je H

n

nh =)(

Inv. ZT

Table(See Porat)

=

d enh j1)(

Inv. DTFT


TransferFunction

FrequencyResponse

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Review of StandardSampling Theory

(P-3.1-3.2)

P ti l S li S t U

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0 0 .2 0 . 4 0 .6 0 . 8 1- 2

- 1

0

1

2

Time

S i g n a

l V a

l u e

0 0 .2 0 . 4 0 .6 0 . 8 1- 2

- 1

0

1

2

Time

S i g n a

l V a

l u e

PulseGen

CT LPF

==

n pnT t pnT xt x )(~)()(~

)(~ t xr

DAC

HoldSample at

t = nT

x(t )

ADC

T = Sampling Interval F s = 1/ T = Sampling Rate

Practical Sampling Set-Up

x[n] = x(nT )

Sampling Analysis (#1)

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Goal = Determine Under What Conditions We Have:

Reconstructed CT Signal = Original CT Signal )()(~

t xt x =

)()(~ t t p =Simplify to Develop Theory: Use

Why???? 1. Because delta functions are EASY to analyze!!!2. Because it leads to the best possible case

=

=n

p p nT t nT xt xt x )()()()(~

x p(t ) is called the Impulse Sampled signal

Note: x p(t ) shows up during Reconstruction, not Sampling!!


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Sampling Analysis (#2)Now, the impulse sampled signal x p(t ) is filtered to givethe reconstructed signal xr (t )

CT LPF

=

=

=

=

n

n p

nT t t x

nT t nT xt x

)()(

)()()(

)()()(

???)(

f X f H f X

t x

pr

r

=

=b

)( f H

Need to find X p( f ). But First a Trick!!!

=

=

=

=

=

=

k

t kF j

k

T kt j

n p

set xT

eT

t x

nT t t xt x

2

/ 2

FSuseperiodic

)(1

1)(

)()()(

4 4 34 4 21 { }

=

=

=

=

k

s

k

t kF j p

kF f X T

et x F T

f X s

)(1

)(1

)( 2


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ImpulseGen CT LPF x[n] = x(nT ))(t x p )(t xr

DAC

HoldSample at

t = nT

ADC

f

f

f

X ( f )

X p( f )

X r ( f )

F s 2F s F s 2 F s

B B

X r ( f ) = X ( f )If F s 2 B

x(t )

H ( f )

f


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What this says: Samples of a bandlimited signal completely define

it as long as they are taken at F s 2 BImpact: To extract the info from a bandlimited signal we only need tooperate on its (properly taken) samples

# Use computer to process signals

Hold

Sample att = nT

x(t ) x[n] = x(nT ) Extracted

InformationComputer


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FT of Impulse Sampled Signal gives view of Original FT

&FT of Impulse Sampled Signal = DTFT of Samples

#DTFT gives view of FT of original signal

Lets see this

T en xenT x

nT t F nT xnT t nT x F t x F

n

jn

n

T jn

n en p

T jn

===

==

=

=

=

=

][)(

)}({)()()()}({ 4 4 34 4 21

)() / ( f F p X T X =


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f F

T

s=

=

2

X ( f )

DTFT

B B f

f F s F s /2 2F s F s 2 F s F s /2

DT Freq(rad/sample)2 4 2 4

/ Normalized DT Freq2 4 2 4 11

Read notes on web on Concept of Digital Frequency

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Introduction toEE521 Case Study

d

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EE521 Case Study: Emitter Location

Data Link

DataLink

Receiver #3(Rx3)

Processing Tasks Intercept RF Signal @ Rxs Sample Signal Suitably for Processing Detect Presence of Emitters Signal

Estimate Characteristics of Signal Use Estd Chars to Classify Emitter Share Data Between Rxs Cross-Correlate Signals to Locate Tx

Radio Freq Transmitter (Tx) Communications Radar

Receiver #1(Rx1) Receiver #2

(Rx2)

Rx1Overall Processing Set-Up

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RFFront-End

DigitalFront-End

CrossCorrelation

DetectSignal

Estimate

SignalParameters

ClassifyEmitter

Data Link &Decompress

Antenna

Rx2RF Data Link

SamplingSub-System

DigitalAnalog

RFFront-End

DigitalFront-End

DetectSignal

Estimate

SignalParameters

ClassifyEmitter

Antenna

Compress& Data Link

Rx1

SamplingSub-System

DigitalAnalog

Processing Tasks Overview (#1)

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RF Front-End (Analog Circuitry): RF = Radio Frequency Selects reception band; may need to scan bands

Amplifies signal to level suitable for ADC, etc. Frequency-shifts signal spectrum to range suitable for ADC

! Technology trend is toward requiring less shifting Unfortunately: noise introduced by analog electronics ( Random Signals )

Topic Well Cover

RF

Front-End200 MHz

FT of Signal

Before RF-FE

3 GHz1 GHz 4 GHz

Antenna

FT of SignalAfter RF-FE


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Sampling Sub-System (Mixed Analog/Digital Circuitry): Converts analog CT signal into digital DT signal ( Bandpass Sampling )

Converts real-valued RF signal into complex LP signal ( Equiv. LP Signals )Digital Front-End (Digital Processing): Convert to complex LP signal ( Equiv. LP Signals ) Equalizes response of analog front-end ( DFT-Based Filtering )

! Magnitude and Phase Bank of digital filters splits signal into sub-bands

for further processing ( Filter Banks ) Reduce sampling rate after filtering to sub-bands ( Multi-Rate Processing )

TopicsWellCover

rad/sample

DTFT of SignalBefore Digital FE

DigitalFront-End

Signals areOversampled


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Detect Presence of Signal (Digital Processing): Has signal been intercepted or only noise in subband ( DFT-Based Proc. )

Estimate Parameters of Signal (Digital Processing) (see also EE522): Estimate frequency of signal ( DFT-Based Processing)

Classify/Model Signal (Digital Processing):

What type of signal is it? (Spectral Analysis of Random Signals)

je z

qq

p p

x z a z a z a

z b z b z bb P

=

++++

++++=

L

L

2211

22

1102

1)(

PSD Model

Use PSD Model parameters { bi} and { a i} to classify signal type

Compress/De-Compress Signal (Digital Processing) (see also EE523): Exploit signal structure to allow efficient transfer( DFT-Based Proc./ Filter Banks / Spectral Analysis )

Cross-Correlate Signals (Digital Processing): Compute relative delay and Doppler ( DFT-Based Proc.& Multi-Rate Proc. )

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