I-1 DSP Review_2007

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I-1: Review 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

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 don’t want Circular convolution – it is what we get if we try to implementConvolution in the frequency domain by multiplying DFT’s (but not DTFT’s)But… we can fix it so that it works…. We use proper zero-padding!!!

Proakis & Manolakis

don’t follow this

Proakis & Manolakisdon’t follow this

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Transforms (P-2)Fourier Transform for DT Signals

{ }

∫

∑

−

∞

∞−=

−

=

=

=

π

π

θ

θ

θ θ π

θ θ

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

π

π

Discrete Fourier Transform

Fourier Transform for CT Signals

{ }

∫

∫

∞

∞−

∞

∞−

−

=

=

=

ω ω π

ω ω

ω

ω

d e X t x

dt et x

Fx X

t j F

t j

F

)(21

)(

)(

)()(

Z Transform for DT Signals

{ }

∑∞∞−=

−=

=

n

n z n x

z Zx z X

][

)()(z

Inverse ZT done using partialfractions & a ZT table

Set z = e jθ

Proakis & Manolakis don’t follow this Notation

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Fourier Transforms for DT Signals (P-2.7,P-4, PM-4, PM-7)

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 valuesMay or may not need zero-padding (depends on application)

• DFT for Analysis & Design:

Compute DFT of given filter’s TF to see its frequency responseGenerally 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)• Parseval’s 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 α

α α α α

α

N N

N N

N

N n

n

Special Case: N 1 = 0

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=

≠−

−

=∑−

= 1 if ,

1 if ,1

1

2

1

0

2

2

α

α α

α

α

N

N

N

n

n

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Euler’s Formulas

)sin()cos(

)sin()cos(

θ θ

θ θ θ

θ

je

je j

j

−=+=

− j

ee

ee

j j

j j

2

)sin(

2)cos(

θ θ

θ θ

θ

θ

−

−

−=

+=

θ je−

)cos(2 θ Re

Im

θ jeθ je−− 2 j

s i n ( θ )

Re

Im

θ je

)cos(θ

)sin(θ j

θ je−

)sin(θ j−

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Discrete-Time Processing (P-2, PM-5)h[n]

H f (θ) H z( z )

x[n] X f (θ) X z( z )

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

h[n] = system’s “Impulse Response”It is response of system to input of δ[n]

H f (θ) = system’s “Frequency Response”

It is DTFT of impulse response

H z( z ) = system’s “Transfer Function”It is ZT of impulse response

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Difference

Equation

Transfer

Function

FrequencyResponseImpulse

Response

Block

Diagram

Pole/ZeroDiagram

DTFT

ZT

ZT (Theory)

Inspect (Practice)

Inspect Inspect Roots

Unit Circle

z = e jθ

Time Domain Z / Freq Domain

Discrete-Time System Relationships

h[k ]

p p

qq z

z a z a

z b z bb z H −−

−−

+++

+++=

11

110

1)(

θ je z z z H θ H == |)()(f

∑∑==

−+−−=q

ii

p

ii in xbin yak y

01

][][][

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Poles and Zeros of Transfer Function

)())((

)())((

)(

)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

α α α

β β β

−−−−−−

=

+++

+++=

++++++

=

++++++

=

−

−

−−

−−

−−−

−−

−−

Consider Here Only p>q

Numerator Roots define “zeros”Denominator Roots define “poles”

In this case p–q zeros at z =0

Poles must be insideunit circle for stability

Poles & zeros must occur inComplex Conjugate pairs

to give real-valued TF coefficients

Re{ z }

Im{ z }

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p p

qq z

z a z a z b z bb z H −−

−−++++++=

11

110

1)(

IIR and FIR Filters

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

qq

z z b z bb z H −− +++= 110)(

⎩⎨⎧ ==

otherwiseqnbnh n

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

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DifferenceEquation

Time Domain

)()1()( n xn yn y β α =−−Input-Output Form

TransferFunction

ZT (Theory)Inspect (Practice)

)1()()( −+= n yn xn y α β Recursion Form

Z / Freq Domain

1

1)(

)()( −

−==

z z X

z Y z H

α

β

Example System

)(1)( 1 z X z z Y β α =− −

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Difference

Equation

Transfer

Function

Block Diagram

ZT (Theory)

Inspect (Practice)

Inspect Inspect


)1()()( −+= n yn xn y α β

Recursion Form11

)( −−=

z z H

α β

Σz-1

βα

x(n) y(n)

y(n-1)

Example System (cont.)

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TransferFunction

FrequencyResponse

Unit Circle

Z / Freq Domain

11)( −−=

z z H

α β

Ω−−

Ω

==

j

j

e z e z

1

OnUnitCircle

],(1

)(

π π α

β

−∈Ω−

=Ω Ω− j

e H

Ω−− = je z 1


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[ ]

[ ]

[ ] [ ]

⎥⎦

⎤⎢⎣

⎡

Ω−Ω−=Ω∠

Ω+Ω−=Ω

Ω+Ω−=

Ω−Ω−=

−=Ω

−

Ω−

)cos(1)sin(

tan)(

)sin()cos(1)(

)sin()cos(1

)sin()cos(11)(

1

22

α α

α α

β

α α β

α α β

α β

H

H

j

je H j

Plotting Transfer Function

Euler’sEquation

Group intoReal & Imag

StandardEqs for

Mag. & Angle


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TransferFunction

Pole/ZeroDiagram

Roots

Z/Freq Domain

α β

α β

−=

−= − z

z z

z H 11)(

Pole

Den. = 0When z = α

Zero

Num. = 0When z = 0

UnitCircle

Re{ z }

Im{ z }

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


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TransferFunction

FrequencyResponse

ImpulseResponse

DTFT

ZTUnit Circle



11)( −−=

z z H

α β

n

nh βα =)(

Inv. ZT

Table(See Porat)

],(

1)(

π π

α β

−∈Ω−

=Ω Ω− je H

∫ − Ω− Ω−=π

π α β

d enh j1)(

Inv. DTFT

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Review of StandardSampling Theory

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

Practical Sampling Set-Up

PulseGen

CT LPF

∑∞

−∞=−=

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

DAC

)(~ t xr

x(t ) x[n] = x(nT )“Hold”

Sample att = nT

ADC

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

Sampling Analysis (#1)

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Goal = Determine Under What Conditions We Have:

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

t xt x =

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

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 )()()()(~ δ

x p(t ) is called the “Impulse Sampled” signal

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

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ImpulseGen CT LPF

)(t x p )(t xr

DAC x(t )

x[n] = x(nT )“Hold”Sample at

t = nT

ADC

f

X ( f )

B –B

f

X p( f )

F s 2F s –F s – 2 F s

f

H ( f )

f

X r ( f )

X r ( f ) = X ( f )If F s ≥ 2 B


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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 B

Impact: To extract the info from a bandlimited signal we only need tooperate on its (properly taken) samples

Use computer to process signals

Computer x[n] = x(nT )

“Hold”

Sample att = nT

x(t )ExtractedInformation


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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 signalLet’s 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

ω θ

δ δ

θ ω

ω

===

−=⎪⎭⎪⎬⎫

⎪⎩⎪⎨⎧ −=

∑∑

∑∑∞

−∞=−

∞

−∞=−

∞

−∞=

∞

−∞= −

][)(

)}({)()()()}({

)() / ( θ θ f F p X T X =


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f

X ( f )

B –B

f

DTFT

F s 2F s –F s – 2 F s F s /2 –F s /2

θ / π Normalized DT Freq2 4 – 2 – 4 1–1

θ DT Freq(rad/sample)2π 4π – 2π – 4π π–π

f F

T

s ⎥⎦

⎤

⎢⎣

⎡=

=π

ω θ

2

Read notes on web on “Concept of Digital Frequency”

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EE521 Case Study: Emitter Location

Radio Freq Transmitter (Tx)• Communications• Radar

Data Link

DataLink

Receiver #1(Rx1) Receiver #2

(Rx2)

Receiver #3(Rx3)

Processing Tasks• Intercept RF Signal @ Rx’s• Sample Signal Suitably for Processing• Detect Presence of Emitter’s Signal

• Estimate Characteristics of Signal• Use Est’d Char’s to Classify Emitter• Share Data Between Rx’s• Cross-Correlate Signals to Locate Tx

Overall Processing Set-Up

Rx1

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RFFront-End

Digital“Front-End”

CrossCorrelation

DetectSignal

Estimate

SignalParameters

ClassifyEmitter

Data Link &Decompress

Antenna

RFFront-End

Digital“Front-End”

DetectSignal

Estimate

SignalParameters

ClassifyEmitter

Antenna

Compress& Data Link

Rx2

Rx1

RF Data Link

SamplingSub-System

SamplingSub-System

→ Digital

→ Digital

Analog ←

Analog ←

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 )

RF

Front-End200 MHz

1 GHz 3 GHz 4 GHz

Antenna

FT of Signal

Before RF-FE

FT of SignalAfter RF-FE

Topic We’ll Cover

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Processing Tasks Overview (#3)

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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)

π θ–π θ

σ θ je z

qq

p p

x z a z a z a

z b z b z bb P

=−−−

−−−

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

++++

++++=

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. )

I-1 DSP Review_2007

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