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8/6/2019 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 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
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
Time Domain Z / Freq Domain
)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
Example System (cont.)
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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
Example System (cont.)
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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
Example System (cont.)
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TransferFunction
FrequencyResponse
ImpulseResponse
DTFT
ZTUnit Circle
Time Domain Z / Freq Domain
Example System (cont.)
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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Sampling Analysis (#1)
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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Sampling Analysis (#3)
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Sampling Analysis (#3)
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
Sampling Analysis (#4)
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Sampling Analysis (#4)
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
Sampling Analysis (#5)
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Sampling Analysis (#5)
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 =
Sampling Analysis (#6)
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Sampling Analysis (#6)
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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d
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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
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. )