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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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7

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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9

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= ∈

10

Sampling.

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11

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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13

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

[ ] 47

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

14

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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59

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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61

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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63

Systems

Their mathematical model is the operator.

( ) ( )

( ) ( )t

-

:

:

d x t x' tdt

x t x d∞

→

→ τ τ∫ ∫

64

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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65

Digital system

moving average filter (running averager).

66

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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67

Mathematical model

( ) ( ){ } ( ) ( ) or Sy t S x t x t y t= ⎯⎯→

[ ] [ ]{ } [ ] [ ] or ddSy n S x n x n y n= ⎯⎯→

68

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

35

69

Homogeneity( ){ } ( ){ }S ax t aS x t= [ ]{ } [ ]{ }d dS ax n aS x n=

70

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.

36

71

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+ ⎯⎯→ +

72

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

= ⇒

− = −

37

73

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.

74

Causality•Between the output and input of the system : relation of the type “cause-effect”

•The effect does not appear before the cause.

38

75

76

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.

39

77

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≡ <

78

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

40

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

41

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= =

− = − ≠∑ ∑

82

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.

42

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

86

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

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

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91

Final solution

[ ]( ) ( )

( )[ ]

[ ] ( ) [ ]

00

00

0.5 cos , 01.25 cos

0.5 , 0

linear system null initial condition: 1 0

cos1.25 cos

order system, null

n

n

th

KB n ny n

C n

yKy n n n

N

⎧ + Ω − θ ≥⎪ − Ω= ⎨⎪ <⎩

⇔ − =

= Ω − θ σ− Ω

[ ] [ ] [ ] initial condition is

1 2 ... 0y y y N− = − = = − =