Signals and Systems - tc.etc.upt.ro · 2. ... Barry Van Veen, Signals and Systems, 2nd edition,...
Transcript of Signals and Systems - tc.etc.upt.ro · 2. ... Barry Van Veen, Signals and Systems, 2nd edition,...
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Signals and Systems
Electronics and Telecommunications Faculty
Communications Department
Instructor: Lecturer Dr. Eng. Corina Nafornita
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COURSE OBJECTIVESThis course is frequently found in electrical engineering curricula, the concepts and techniques that form the core of the subject are of fundamental importance in all engineering disciplines. Our approach has been guided by the continuing developments in technologies for signal and system design and implementation, which made it increasingly important for a student to have equal familiarity with techniques suitable for analyzing and synthesizing both continuous-time and discrete-time systems.
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COURSE TOPICSSignals and systems: Continuous-Time and Discrete-Time Signals;
Exponential and Sinusoidal Signals; Continuous-Time and Discrete-Time Systems; Basic System Properties.
Linear time-invariant systems: Discrete-Time LTI Systems: The Convolution Sum; Continuous-Time LTI Systems: The Convolution Integral; Properties of Linear Time-Invariant Systems; Singularity Functions.
Fourier Series Representation: The Response of LTI Systems to Complex Exponentials; Fourier Series Representation of Continuous-Time and Discrete-Time Periodic Signals.
The Continuous-Time Fourier Transform: Representation of Aperiodic Signals: The Continuous-Time Fourier Transform; Properties of the Continuous-Time Fourier Transform, Systems Characterized by Linear Constant-Coefficient Differential Equations.
The Discrete-Time Fourier Transform: Representation of Discrete-Time Aperiodic Signals: The Discrete-Time Fourier Transform; Properties of the Discrete-Time Fourier Transform; Duality; Systems Characterized by LinearConstant-Coefficient Difference Equations
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TEXTBOOKS/REFERENCES 1. Corina Nafornita, Signals and Systems, vol. 1, Politehnica Publishing House, 2009, ISBN 978-606-554-013-2 (978-606-554-014-9 vol I), Table of contents
http://shannon.etc.upt.ro/teaching/ss-pi/Signals_Systems_TOC.pdf
2. Alan V. Oppenheim, Alan S. Willsky with S. Hamid Nawab, Signals & Systems, Second Edition, Prentice Hall, Upper Saddle River, New Jersey, 1997.
3. Simon Haykin, Barry Van Veen, Signals and Systems, 2nd edition, John Wiley& Sons, 2003
WEBPAGEhttp://shannon.etc.upt.ro/teaching/ss-pi/
CONTACTcorina.nafornita [at] gmail [dot] com
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Signals
Signal - A time-variable phenomenon that carriesan information.
Signal types:Biological, acoustical, chemical, optical,
electronic,…
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An electrocardiogram.
a)
b)
A voice signal.
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Images.
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Mathematical model
function having as independent variable the time
( ) [ ]310 2 10 Vx t sin t= ⋅ π⋅ ⋅
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Discrete-time signals
Sampling x(t) with step Ts=0,05 ms
n=t/Ts – discrete time
( ) ( )[ ]
3 310 2 10 0 05 10
10 0 1 V sx̂ t x nT sin , n
sin , n n
−= = ⋅ ⋅π ⋅ ⋅ ⋅ ⋅ =
= ⋅ ⋅ π ⋅ ∈
[ ] ( ) sx n x nT n= ∈
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Sampling.
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Some important signals used in electrical engineering
i) Sinusoidal signal( ) ( )0 0 0 0 2 T , x t Acos t ; A, f ,= ω + ϕ ω = π ϕ
The sinusoidal signal is periodic
( ) ( ) ( ) ( )( ) ( )
[ ] ( )
0 0
0 0 0
0 0 0 0
0 0 00 0
and
1 22
x t T x t , x t nT x t , t n
Acos t T Acos t , t
cos t T cos t , t
T ; Tf
+ = + = ∀ ∀ ∈
⎡ ⎤ω + + ϕ = ω + ϕ ∀⎣ ⎦ω + ϕ + ω = ω + ϕ ∀
πω = π = =ω
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ii) Sinusoidal discrete-time signal[ ] ( )
[ ] [ ][ ]
[ ] ( )( ) ( )
0
0 0
00 0
0
0 0
cosrad s rads
2 - discrete frequency
cos
cos 2 cos
s
s s
ss
x n A T n
T T
fTf
x n A n
n n
= ω + ϕ
ω = ω = ⋅ =
Ω = ω = π
= Ω + ϕ
⎡ ⎤Ω + π + ϕ = Ω + ϕ⎣ ⎦
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Discrete frequency for [ ] 0x n cos n= Ω
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“Confusion” due to sampling( )0
20 0 1ks
; x t Acos k t; k , ,...TπΩ = = =
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( ) ( )0 ; cos 2 1 ; 0,1,...ks
x t A k t kTπΩ = π = + =
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Periodicity of the d.t. sine wave, period N
( ) ( )0 0 0
0 0
2
2 (rational number)
Acos n N Acos n , n, N k
N k
Ω + + ϕ = Ω + ϕ ∀ Ω = π⎡ ⎤⎣ ⎦π π= ∈ ⇒ ∈
Ω Ω
Example
minimum k for which N is integer : k=2 ⇒ N=7
The signal is not periodic.
00
4 7 2 7 77 4 4 2
N k kπ π ⋅Ω = ⇒ = ⇒ = ⋅ = ⋅Ω
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x n Acos nπ⎛ ⎞= + ϕ⎜ ⎟⎝ ⎠
[ ] ( )2x n Acos n= + ϕ
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iii) Continuous-time unit step signal
( ) 1 00 0, t
t, t
≥⎧σ = ⎨ <⎩
This is only a model. It can not be generated in practice.
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iv)Discrete-time unit step signal
[ ] ( ) 1 00 0s, n
n nT, n
≥⎧σ = σ = ⎨ <⎩
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v)Continuous-time unit impulse. Dirac impulse
( )
( )0
0
1
00 0
k
k
k
kk
f t dt
, tlim f t
, t
∞
−∞
→∞Δ →
Δ →
=
∞ =⎧= ⎨ ≠⎩
∫
( )
( )
00 0
1
, tt
, t
t dt∞
−∞
∞ =⎧δ = ⎨ ≠⎩
δ =∫
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A remarkable property( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
0 0
0
0
0
0
0 0 1 0
k k
k k
k k
t f t f t
lim t f t lim f t
t t t
t t dt t dt
t dt
Δ → Δ →
∞ ∞
−∞ −∞∞
−∞
ϕ ≅ ϕ
ϕ = ϕ
ϕ δ = ϕ δ
ϕ δ = ϕ δ =
= ϕ δ =ϕ ⋅ = ϕ
∫ ∫
∫
( ) ( ) ( )0t t dt∞
−∞ϕ δ =ϕ∫
The filtering property of the Dirac impulse
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Unit impulse and unit step connection
( ) ( )
( ) ( )( ) ( ) ( )
( ) ( )
( ) ( )
0
0 0
0
k
k k
k
k
'k k
'k k
'
k
lim g t t
g t f t
lim g t lim f t t
lim g t t
' t t
Δ →
Δ → Δ →
Δ →
= σ
=
= = δ
⎛ ⎞= δ⎜ ⎟
⎝ ⎠σ = δ
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( )
( ) ( )
1 00 0
t
t
, td
, t
d t
−∞
−∞
>⎧δ τ τ = ⎨ <⎩
δ τ τ = σ
∫
∫
( ) ( )' t tσ = δ
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vi) Discrete-time unit impulse
[ ] 1 00 0, n
n, n
=⎧δ = ⎨ ≠⎩
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Discrete-time unit impulse and unit step connection
[ ] [ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ] [ ]
1 1
1 1
1 ; 1
1
n n n
k k kn n
k k
k n k k n - n
k n k n - n
− −
=−∞ =−∞ =−∞− −
=−∞ =−∞
δ = σ − δ − δ = σ σ −
δ + δ − δ = σ σ −
∑ ∑ ∑
∑ ∑
[ ] [ ]
[ ] [ ] [ ] 1
n
kk n
n n n=−∞
δ = σ
σ − σ − = δ
∑
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Other properties of the discrete-time unit impulse
[ ] [ ] [ ] [ ]0x n n x nδ = δ
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]
[ ] [ ] [ ] ( ) [ ] [ ] [ ] ( )
[ ] [ ] [ ]
2 2 1 1 0
1 1 1 1 1 1
k
k
x k n k ... x n x n x n
x n ... x n n n x n n n x n n n ...
x n x k n k
∞
=−∞
∞
=−∞
δ − = + − δ + + − δ + + δ +
+ δ − + + − δ − − + δ − + + δ − + +⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦
= δ −
∑
∑
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vii) Continuous-time ramp signal
( ) ( )
( ) ( )
00
0 0
0
0 0
tt d t , t
r t d
, t
t , tr t t t
, t
−∞
⎧τ = ≥⎪= σ τ τ = ⎨
⎪ <⎩≥⎧
= = σ⎨ <⎩
∫∫
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viii) Discrete-time ramp signal
[ ] [ ]
[ ] [ ]
11
01 1
0 1
0
0 0
nn
kk
n, nr n k
, n
n, nr n n n
, n
−−
==−∞
⎧= ≥⎪= σ = ⎨
⎪ <⎩≥⎧
= = σ⎨ <⎩
∑∑
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ix) Continuous-time exponential signal
0
0
0 ; 0 ; ; 1
0 ; ; 0 ; 1
at att t
at att t
a lim e lim e e
a lim e lim e e→−∞ →∞
→−∞ →∞
> = = ∞ =
< = ∞ = =
( ) ~ 2.7182, , at ex t e a= ∈
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Causal exponential( ) ( ) 0 ; 0
0 0
atat e , tx t e t a
, t
⎧⎪ ≥= σ = <⎨<⎪⎩
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x)Discrete-time exponential signal[ ] ( ) [ ]; , ns s s
nbnT bT bTx n e e e a x n a a= = = ⇒ = ∈
Homework: sketch the signal for a<-1
a>10<a<1
-1<a<0
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Discrete-time causal exponential
[ ] [ ] 00 0
nn a , nx n a n
, n
⎧⎪ ≥= σ = ⎨<⎪⎩
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xi) Oscillation with exponentialenvelope in continuous-time
( ) 0sinatx t e t= ω ( )00 0
2sin 1; ; k k kkatt t k x t eπ πω = = + =
2ω ω
( )00 0
sin 1; ; l l llatt t k x t eπ 2πω = − = − + = −
2ω ω
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Causal case
( ) ( ) 00
00 0
atat e sin t , tx t e sin t t
, t
⎧⎪ ω ≥= ω σ = ⎨<⎪⎩
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xii) Oscillation with exponentialenvelope in discrete-time
[ ] 0nx n a cos n= Ω
Exercise:
Draw the waveform of this signal for the case a>1.
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1.3. Complex signals. Phasors.
{ } { }
;
; 2 2
;
j j
j j j j
j j
e cos j sin e cos j sin
e e e ecos sinj
cos Re e sin Im e
θ − θ
θ − θ θ − θ
θ θ
= θ + θ = θ − θ
+ −θ = θ =
θ = θ =
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i) Relation between real sinusoidal signal and complex exponential
( ){ } ( )
( ){ } ( )
( ) { }
0
0
0
0
0
0
0
0
0
Re A cos ;
Re A cos ;
- oscillatory part
- complex amplitude
cos Re
- phasor that rotates with the angular velocity
j t
j tj
j t
j
j
j t
j t
e A t A
e e A t A
e
Ae
A Ae
A t Ae
Ae
ω +ϕ +
ωϕ +
ω
ϕ
ϕ
ω
ω
= ω + ϕ ∈
= ω + ϕ ∈
= ∈
ω + ϕ =
ω
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Evolution in time of the phasor for φ=0. The mobile extremity of the phasor describes a cylindrical helix of radius A.
0j tAe ω
0ϕ =
.
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The negative frequency
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ii) Relation between real oscillationwith exponential envelope and
complex exponential
The vector that rotates with the angular velocity describes a spiral.
0ω
( ) ( )( )
( ){ } { } ( )( ) ( )
0
0
0 0 0 0
0 0
0
cos ,
Re Re cos
complex envelope of the signal
t
tj
j t t j t tj
x t Ae t
A t Ae e
A t e Ae e e Ae t
A t x t
σ
σϕ
ω σ ω σϕ
= ω + ϕ σ ∈
=
= = ω + ϕ
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iii) Sampling (discrete-time case)[ ] ( ) ( )
( ) [ ][ ] ( ){ } [ ]{ }
[ ]
0
0
0 0
0 0cos cos
A associated phasor; complex envelope
Re Re
0 : complex envelope constant 0the vector rotating with angular velocit
sT n ns
j nn n j
j n j nn
j
x n Ae T n Aa n
a e A n Aa e
x n Aa e A n e
A n Ae
σ
Ω +ϕ ϕ
Ω +ϕ Ω
ϕ
= ω + ϕ = Ω + ϕ
=
= =
σ = =
0 0
0
0
y - constant magnitude
0 : cos "negative frequency"2 2
j n j nA AA n e eΩ − Ω
Ω
ϕ = Ω = + →
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1.4. Simple signal transformationsi)Weighting- amplification or attenuation of signal
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ii) Time shifting( )0 0
0
shifted to the right if 0 to the left if 0x t t t
t− >
<[ ]0 0
0
shifted to the right if 0 to the left if 0x n n n
n− >
<
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iii) Time reversal( ) ( )x t x t= − [ ] [ ]x n x n= −
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iv) Time scalingcompresses or dilates the signal by multiplying the time variable by a constant
( ) ( ) , y t x at a= ∈
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v) Discrete-time scaling
( ) [ ] , if divides
0 , otherwisek
nx k nx n k
⎧ ⎡ ⎤⎪ ⎢ ⎥= ⎣ ⎦⎨⎪⎩
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vi) Simple transformations
( ) ( )2 2 2x t x t⎯⎯→ − −
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Even and odd parts of a real signal( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]
; ; 2 2
;
; ; 2 2
e o e o
e e o o
e o e o
x t x t x t x tx t x t x t x t x t
x t x t x t x t
x n x n x n x nx n x n x n x n x n
+ − − −= + = =
− = − = −
+ − − −= + = =
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Energy and Power( )
[ ]
( ) ( )
2
2
2
2 2
Energy for complex signal
Discrete-time:
Continuous-time finite energy signals - square integrable functions (space )
Discrete-time finite
n
W x t dt
W x n
L
x t dt x t L
∞
−∞∞
=−∞
∞
−∞
= < ∞
= < ∞
< ∞ ⇒ ∈
∫
∑
∫
[ ]
2
2 2
energy signals - square summable functions (space )
[ ]n
l
x n x n l∞
=−∞
< ∞ ⇒ ∈∑
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1 – Causal decreasing exponential( ) ( )
22
0 0
1 12 2 2
t
tt
x t e t
e eW e dt
−
∞− −∞∞−
= σ
−= = = =−∫
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2 – Causal oscillation with exponentialenvelope
( ) ( )
( ) ( )
0
2 2 2 2 200 0
0 0 0 020
2 20 0
sin
1 cos 2 1 1sin cos 22 2 2
1 14 4 1 4 1
t
t t t t
x t e t t
tW e tdt e dt e dt e tdt
W
−
∞ ∞ ∞ ∞− − − −
= ω σ
− ω= ω = = − ω
ω= − =
+ ω + ω
∫ ∫ ∫ ∫
“fast” oscillations (ω0 large), approximation
half of the energy for the case when there are no oscillations
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W ≅
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3- Discrete-time causal exponential signal
[ ] [ ]2
20
2
, 1
11
11 ... ... , 11
n
n
n
n
x n a n a
W aa
a a a aa
∞
=
= σ <
= =−
+ + + + + = <−
∑
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4- Sine wave( ) 0sinx t A t= ω
0TPeriodic signal
average energy computed over one period
( )0 0
0
2 22 2
0 0 00 0
sin 1 cos 22 2
T T
TA AW A tdt t dt T= ω = − ω =∫ ∫
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5 – Unit step signal
( )2
0 01 lim 1 lim 1
N
N Nn nW N
∞
→∞ →∞= =
= = = + = ∞∑ ∑
[ ] [ ]x n n= σ
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PowerAverage power of the signal P : average flux of energy, ratio of signal energy and time interval when that energy was developed.
infinite duration signals: ( )
[ ]
2
2
12
12 1
N
N n N
P lim x t dt
P lim x nN
τ
τ→∞ −τ
→∞ =−
=τ
=+
∫
∑
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Energy and average power, finite duration signals
( )
( )
2
1
2
1
2
2
2 1 2 1
1
t
tt
t
W x t dt
WP x t dtt t t t
=
= =− −
∫
∫
Continuous-time signals, support [t1,t2]
Discrete-time signals
support {N1, N1+1,…,N2}
[ ]
[ ]
2
1
2
1
2
2
2 1 2 1
11 1
N
n NN
n N
W x n
WP x nN N N N
=
=
=
= =− + − +
∑
∑
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average power is computed over one period:
( )
[ ]
0
2
02
1
1
T
n N
P x t dtT
P x nN ∈
=
=
∫
∑
Periodic signals
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6-Sine wave, average power
Sinusoidal Signal2
2 2 00
2 2 2 2 20 0
0 0
1 212 2 2
2 224 4 2 2 4 2 2
cos tAP lim A sin tdt lim dt
sin t sinA A A A Alim lim
τ τ
τ→∞ τ→∞−τ −ττ
τ→∞ τ→∞−τ
− ω= ω = =τ τ
⎡ ⎤ω ω τ⎢ ⎥= ⋅ τ − ⋅ = − =⎢ ⎥τ τ ω ω τ⎢ ⎥⎣ ⎦
∫ ∫
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1.7. Distributions
function
distribution
operator
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Example of Distribution: The Dirac Impulse
( ) ( ) ( )( ) ( ) ( )0 0
: 0
:
t t
t t t t
δ ϕ → ϕ
δ − ϕ → ϕδ(t) associates to any test function ϕ(t), its value from origin, ϕ(0)
δ(t-t0) associates to any test function ϕ(t), its value from t0, ϕ(t0)
f – distribution. The test function φ and a number (scalar product between f and φ) are associated
( ) ( ) ( )ft t f t dt∞
−∞
ϕ ⎯⎯→ ϕ∫
( ) ( ) ( ) ( ) ( ): ,f t f t t t f t dt∞
−∞
ϕ ⎯⎯→ ϕ = ϕ∫
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The Derivative of a Distribution( ) ( ) ( ) ( ) ( ) ( )f ' t , t ' t f t dt f t , ' t
∞
−∞ϕ = − ϕ = −ϕ∫
( ) ( ) ( ) ( ) ( )0' t , t t , ' t 'δ ϕ = δ −ϕ = −ϕ
( ) ( )0'
t 'δ
ϕ →− ϕ ( )' tδ ( )tϕassociates to the test function thenegative value of its first derivative computed in zero
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Unit Step Distribution
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )( ) ( )
0
00
0
0
t
t
t
t t dt
t t , t t dt t
t
t t
∞σ
∞′σ ∞
′σ
ϕ ⎯⎯⎯→ ϕ
′ ′ϕ ⎯⎯⎯→ σ −ϕ = − ϕ = −ϕ = ϕ − ϕ ∞
ϕ ⎯⎯⎯→ϕ′⇒ σ = δ
∫
∫
i) Functions are useful for modeling signals,ii) Distributions are useful for modeling some signals and processes
like sampling,iii) Operators are useful for modeling signal processing systems.
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Systems
Their mathematical model is the operator.
( ) ( )
( ) ( )t
-
:
:
d x t x' tdt
x t x d∞
→
→ τ τ∫ ∫
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2.1. Systems2
;
2 2
2
S
S
10000
100005 ;
V 500 ; V 0
;
100 ; 500 5 100 k
u
uS
Su
Si
in
in
i
Au AV
u uVA
u V
V
ViR
R kVi nA
>
=
= <
<
< μ=
=
= Ωμ< =Ω
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Digital system
moving average filter (running averager).
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Simulated analog system
( ) [ ]converter on 10 bits (1024 quantization levels)domain of the input voltage 10V
10V 10mV1024
Max quantization error 5mV
sx nT qx n
q
≅
≅ ≅
±
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Mathematical model
( ) ( ){ } ( ) ( ) or Sy t S x t x t y t= ⎯⎯→
[ ] [ ]{ } [ ] [ ] or ddSy n S x n x n y n= ⎯⎯→
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2.2. Linear systems
( ) ( ){ } ( ){ } ( ){ }[ ] [ ]{ } [ ]{ } [ ]{ }
1 1 2 2 1 1 2 2
1 1 2 2 1 1 2 2d d d
S a x t a x t a S x t a S x t
S a x n a x n a S x n a S x n
+ = +
+ = +
Superposition principle
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Homogeneity( ){ } ( ){ }S ax t aS x t= [ ]{ } [ ]{ }d dS ax n aS x n=
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Incremental linear systems( ){ } ( ){ }Homogeneity, 0: 0 0 0a S x t S x t= = =
Systems with increments of the output, proportional with the increments of the input, not homogeneous ⇒ linear system; at the output the zero-input response y0 is added.
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The linear system response at the sum of two input signals equals the sum of
responses at each signal.
Additivity
( ) ( ) ( ) ( )1 2 1 2x t x t y t y t+ ⎯⎯→ +
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2.3. Time invariant systems
( ){ } ( )( ){ } ( )0 0
S x t y t
S x t t y t t
= ⇒
− = −[ ]{ } [ ][ ]{ } [ ]0 0
d
d
S x n y n
S x n n y n n
= ⇒
− = −
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Stability
c) Unstable equilibrium: the impulse applied to the ball produces loss of equilibrium
b) Neutrally stable equilibrium: the impulse applied to the ball modifies the equilibrium position.
a) Stable equilibrium: the impulse applied to the ball creates attenuated oscillations of its position.
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Causality•Between the output and input of the system : relation of the type “cause-effect”
•The effect does not appear before the cause.
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2.6. Systems described by linear constant-coefficients
differential equations and difference equations
Homework: Prove the linearity of these systems.
First order linear system. Second order linear system.
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General form of the linear constant-coefficients differential equation that
describes an Nth order system( ) ( )
( ) ( ) ( ) ( )
0 0
2 1
0 2 1
0 0 0
, 0 (at least)
The initial conditions should be null if the system is linear:
... 0
if the input signal is applied at the moment
k kN M
k k Nk kk k
N
Nt t t t t t
d y t d x ta b a
dt dt
dy t d y t d y ty t
dt dt dt
= =
−
−= = =
= ≠
= = = = =
∑ ∑
( )0
0
of time 0 for
tx t t t≡ <
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Digital case - first order system( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) [ ] [ ]
[ ] [ ] [ ]
- equivalent digital system?
1
1 1
s
s
s s
s s s
s st nT
s s
t nT
dy tRC y t x t
dtdy t
RC y nT x nTdt
y n y ndy t y nT y nT Tdt T T
RC RCy n y n x nT T
=
=
+ =
+ =
− −− −≅ =
⎛ ⎞+ − − =⎜ ⎟
⎝ ⎠
linear constant-coefficients difference equation, obtained by the approximation of the differential equation
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79
• the slope of the secant line is a good approximation for the slope of the tangent line for a small sampling step - Ts
80
( ) ( ) ( ) ( )2
2ss
s st nTt nT
d y t dy tLC RC y nT x nT
dtdt ==
+ + =
Digital case - second order system
( ) ( )( ) ( )
[ ] [ ] [ ] [ ][ ] [ ] [ ]
2
2
2
1 1 22 1 2
s s s
ss
t nT t nT T
st nTt nT
s s
s s
dy t dy tdt dtd y t dy td
dt dt Tdt
y n y n y n y ny n y n y nT T
T T
= = −
==
−⎛ ⎞
= ≅⎜ ⎟⎝ ⎠
− − − − −−
− − + −= =
[ ] [ ] [ ] [ ]2 2 221 1 2
s ss s s
LC RC LC RC LCy n y n y n x nT TT T T
⎛ ⎞ ⎛ ⎞+ + − + − + − =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
linear constant-coefficients difference equation, obtained by the approximation of the differential equation
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81
General form of the linear constant-coefficients difference equation that
describes an Nth order system
[ ] [ ]0 0
, 0 (at least)N M
k k Nk k
A y n k B x n k A= =
− = − ≠∑ ∑
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2.7. Some examples of systems i) Proportional ideal system
( ) ( ) , y t ax t a= ∈ [ ] [ ], y n ax n a= ∈
memoryless system: the output signal at each time depends only on the input signal at the same value of the time, and it doesn’t depend on previous values.
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83
ii) Ideal differentiator system
( ) ( ) [ ] [ ] [ ]( )1 1s
dx ty t y n x n x n
dt T= = − −
system that implements the approximation of the derivative for the digital case
84
iii) Ideal integrator system
( ) ( )t
y t x d−∞
= τ τ∫ [ ] [ ] [ ] [ ]
[ ] [ ] [ ]
1
1
n n
k ky n x k x k x n
y n y n x n
−
=−∞ =−∞= = +
= − +
∑ ∑
systems with memory :
• digital adder
• digital differentiator
Continuous time
Discrete time: adder
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85
2.8. Examples1.Linear analog system with time-variable parameters
( ) ( ) ( ) ( )2
22 2
d y t dy tt t y t x t
dtdt+ + =
a) Linearity
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
21 1 2
1 1 1 12
22 2 2
2 2 2 22
22
1 2 1 2 1 2 1 22
1 2 1 2
2
2
2
d y t dy tx t y t t t y t x t
dtdtd y t dy t
x t y t t t y t x tdtdt
d dy t y t t y t y t t y t y t x t x tdtdt
x t x t y t y t
⎯⎯→ ⇒ + + =
⎯⎯→ ⇒ + + =
+ + + + + = +⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦ ⎣ ⎦
⇒ + ⎯⎯→ +
Additivity
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( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
22
2
22
2
2
2
d y t dy tx t y t t t y t x t
dtdtd day t t ay t t ay t ax t ax t ay t
dtdt
⎯⎯→ ⇒ + + =
+ + = ⇒ ⎯⎯→⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦ ⎣ ⎦
Homogeneity
b) Time shift invariance
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
22
2
223 3
0 3 0 0 3 02
2
2
d y t dy tx t y t t t y t x t
dtdtd y t dy t
x t t y t t t t t y t x t tdtdt
⎯⎯→ + + =
− ⎯⎯→ + − + − = −
( ) ( )3 0 linear system, but not time-invarianty t y t t≠ − ⇒
( ) ( ) ( ) ( )2
0 0 20 02time invariant system: 2
d y t t dy t tt t y t t x t t
dtdt
− −+ + − = −
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87
ii) The influence of null initial conditions on linearity of analog systems
( ) ( ) ( )
( ) ( ) 00
2
00 0
dy ty t x t
dtK cos t , t
x t K cos t t, t
+ =
ω ≥⎧= ω σ = ⎨ <⎩
88
Particular solution - steady state( ) ( )
( ) ( )
0
020
2 cos , 0
cos , 04
ff
f
dy ty t K t t
dtKy t t t
+ = ω ≥
= ω − ≥+
θω
Homogeneous solution – transient state( ) ( )
( ) ( )2 2
2 0
, 0 and , 0
trtr
t ttr tr
dy ty t t
dty t Be t y t Ce t− −
+ = ∈
= ≥ = <
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89
Final Solution
( ) ( ) ( )( )
( )( )
( ) ( )
2 20 02
02
0
2 20 02
0
, 0, 0
cos cos , 04
, 0
cos cos4
tr f
tr
t t
t
t t
y t y t ty t
y t t
Ky e t e ty t
y e tKy t y e t e
− −
−
− −
+ ≥⎧⎪= ⎨ <⎪⎩⎧ ⎡ ⎤+ ω − θ − θ ≥⎣ ⎦⎪= + ω⎨⎪ <⎩
= + ω − θ −+ ω
( ) t t⎡ ⎤θ σ ∈⎣ ⎦
( ) ( ) 200
0
linear system0 : 0 0
0 null initial value for linear system!
tK
K x t y t y e
y
−=
= = ⎯⎯⎯⎯⎯→ = ≡
⇒ =
90
iii) Influence of the null initial conditions on the linearity of digital systems
[ ] [ ] [ ] [ ] [ ] 00
, 00 5 1
0 , 0K cos n n
y n , y n x n x n K cos n nn
Ω ≥⎧− − = ⇒ = Ω σ = ⎨ <⎩
Particular solution - steady state[ ] [ ] { }[ ] ( )
00
0
0
00
0.5 1 cos Re , 0
cos
0.5sin; arctg1 0.5cos1.25 cos
j nf f
f
j
y n y n K n Ke n
y n A n
KA e
Ω
− θ
− − = Ω = ≥
= Ω − θ
Ω= θ =− Ω− Ω
[ ] [ ][ ] ( ) [ ] ( )
0 5 1 0 ,
0 5 , 0 and 0 5 , 0
tr trn n
tr tr
y n , y n n
y n B , n y n C , n
− − = ∈
= ≥ = <
Homogeneous solution – transient state