Signals and Systems
Electronics and Telecommunications
Faculty
Communications Department
Instructor: Lecturer Dr. Eng. Corina Nafornita
1
COURSE OBJECTIVES
This 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 discretetime systems.
2
1
COURSE TOPICS
Signals 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 ConstantCoefficient 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 Linear
Constant-Coefficient Difference Equations
3
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
WEBPAGE
http://shannon.etc.upt.ro/teaching/ss-pi/
CONTACT
corina.nafornita [at] gmail [dot] com
4
2
Signals
Signal - A time-variable phenomenon that carries
an information.
Signal types:
Biological, acoustical, chemical, optical,
electronic,…
5
a)
b)
An electrocardiogram.
A voice signal.
6
3
Images.
7
Mathematical model
function having as independent variable the time
x ( t ) = 10 ⋅ sin 2π ⋅103 ⋅ t
[V]
8
4
Discrete-time signals
Sampling x(t) with step Ts=0,05 ms
x̂ ( t ) = x ( nTs ) = 10 ⋅ sin 2 ⋅ π ⋅103 ⋅ 0 ,05 ⋅10−3 ⋅ n =
= 10 ⋅ sin 0 ,1 ⋅ π ⋅ n
[V]
n∈]
n=t/Ts – discrete time
x [ n ] = x ( nTs ) n ∈ ]
9
Sampling.
10
5
Some important signals used in
electrical engineering
i)
Sinusoidal signal
x ( t ) = Acos ( ω0t + ϕ ) ; A, ω0 = 2πf 0 , T0 , ϕ
The sinusoidal signal is periodic
x ( t + T0 ) = x ( t ) , x ( t + nT0 ) = x ( t ) , ∀t and ∀n ∈ ]
Acos ⎡⎣ω0 ( t + T0 ) + ϕ⎤⎦ = Acos ( ω0t + ϕ ) , ∀t
cos [ ω0t + ϕ + ω0T0 ] = cos ( ω0t + ϕ ) , ∀t
ω0T0 = 2π; T0 =
1 2π
=
f0 ω0
11
ii) Sinusoidal discrete-time signal
x [ n] = A cos ( ω0Ts n + ϕ )
[ω0Ts ] = [ω0 ][Ts ] =
rad
⋅ s = rad
s
f
Ω0 = ω0Ts = 2π 0 - discrete frequency
fs
x [ n] = A cos ( Ω0 n + ϕ )
cos ⎡⎣( Ω0 + 2π ) n + ϕ⎤⎦ = cos ( Ω0 n + ϕ )
12
6
Discrete frequency for x [ n] = cos Ω0n
13
“Confusion” due to sampling
Ω0 = 0;
xk ( t ) = Acos k
2π
t; k = 0 ,1,...
Ts
14
7
Ω0 = π;
xk ( t ) = A cos ( 2k + 1)
π
t ; k = 0,1,...
Ts
15
Periodicity of the d.t. sine wave,
period N
Acos ⎡⎣Ω0 ( n + N ) + ϕ⎤⎦ = Acos ( Ω0 n + ϕ ) ,
N =k
Example
∀n, Ω0 N = k 2π
2π
π
∈` ⇒
∈ _ (rational number)
Ω0
Ω0
Ω0 =
4π
7
2⋅7
7
π
⇒
= ⇒ N =k⋅
=k⋅
7
Ω0 4
4
2
minimum k for which N is integer : k=2 ⇒ N=7
⎛ 4π
⎞
x [ n ] = Acos ⎜ n + ϕ ⎟
⎝ 7
⎠
The signal x [ n] = Acos ( 2n + ϕ ) is not periodic.
16
8
iii) Continuous-time unit step
signal
⎧1, t ≥ 0
σ (t ) = ⎨
⎩0 , t < 0
This is only a model. It can not be generated in
practice.
17
iv)Discrete-time unit step signal
⎧1, n ≥ 0
σ [ n ] = σ ( nTs ) = ⎨
⎩0 , n < 0
18
9
v)Continuous-time unit impulse.
Dirac impulse
Δk → 0
∞
∫ f k ( t ) dt = 1
−∞
⎧∞ , t = 0
lim f k ( t ) = ⎨
k →∞
⎩ 0, t ≠ 0
Δ k →0
⎧∞ , t = 0
δ (t ) = ⎨
⎩ 0, t ≠ 0
∞
∫ δ ( t ) dt = 1
−∞
19
A remarkable property
ϕ ( t ) fk ( t ) ≅ ϕ ( 0 ) fk ( t )
lim ϕ ( t ) f k ( t ) = ϕ ( 0 ) lim f k ( t )
Δ k →0
Δ k →0
ϕ ( t ) δ ( t ) = ϕ ( 0) δ (t )
∞
∞
∫ ϕ ( t ) δ ( t ) dt = ∫ ϕ ( 0 ) δ ( t ) dt =
−∞
−∞
∞
= ϕ ( 0 ) ∫ δ ( t ) dt =ϕ ( 0 ) ⋅1 = ϕ ( 0 )
−∞
The filtering property of the Dirac impulse
∞
∫ ϕ ( t ) δ ( t ) dt =ϕ ( 0 )
−∞
20
10
Unit impulse and unit step
connection
lim g k ( t ) = σ ( t )
Δ k →0
g'k ( t ) = f k ( t )
lim g'k ( t ) = lim f k ( t ) = δ ( t )
Δ k →0
Δ k →0
'
⎛
⎞
⎜ lim g k ( t ) ⎟ = δ ( t )
⎝ Δ k →0
⎠
σ' ( t ) = δ ( t )
21
σ' ( t ) = δ ( t )
t
⎧1, t > 0
∫ δ ( τ )d τ = ⎨0, t < 0
⎩
−∞
t
∫ δ ( τ )d τ = σ ( t )
−∞
22
11
vi) Discrete-time unit impulse
⎧1, n = 0
δ [ n] = ⎨
⎩0 , n ≠ 0
23
Discrete-time unit impulse and unit
step connection
n
∑ δ [ k ] = σ [ n]
k =−∞
σ [ n ] − σ [ n − 1] = δ [ n ]
n −1
n
n −1
∑ δ [ k ] = σ [ n − 1] ; ∑ δ [ k ] − ∑ δ [ k ] = σ [ n] - σ [ n − 1]
k =−∞
n −1
k =−∞
n −1
k =−∞
∑ δ [ k ] + δ [ n] − ∑ δ [ k ] = σ [ n ] - σ [ n − 1]
k =−∞
k =−∞
24
12
Other properties of the discretetime unit impulse
x [ n ] δ [ n ] = x [ 0] δ [ n ]
∞
∑ x [ k ] δ [ n − k ] = ... + x [ −2] δ [ n + 2] + x [ −1] δ [ n + 1] + x [ 0] δ [ n ] +
k =−∞
+ x [1] δ [ n − 1] + ... + x [ n − 1] δ ⎡⎣ n − ( n − 1) ⎤⎦ + x [ n ] δ [ n − n ] + x [ n + 1] δ ⎡⎣ n − ( n + 1) ⎤⎦ + ...
∞
x [ n] = ∑ x [ k ] δ [ n − k ]
k =−∞
25
26
13
vii) Continuous-time ramp signal
⎧t
⎪ ∫ d τ = t, t ≥ 0
r ( t ) = ∫ σ ( τ )d τ = ⎨0
⎪
−∞
t<0
⎩ 0,
t
⎧ t, t ≥ 0
= tσ ( t )
r (t ) = ⎨
⎩0 , t < 0
27
viii) Discrete-time ramp signal
⎧ n −1
⎪ ∑ 1 = n, n ≥ 1
r [ n] = ∑ σ [ k ] = ⎨k = 0
k =−∞
⎪ 0,
n <1
⎩
n −1
⎧n, n ≥ 0
r [ n] = ⎨
= nσ [ n ]
⎩0, n < 0
28
14
ix) Continuous-time exponential
signal
x ( t ) = eat , a ∈ \, e ~ 2.7182
a > 0 ; lim eat = 0 ; lim eat = ∞ ; e0 = 1
t →−∞
a<0;
lim e
t →−∞
t →∞
at
= ∞ ; lim eat = 0 ; e0 = 1
t →∞
29
Causal exponential
⎧⎪eat , t ≥ 0
x ( t ) = eat σ ( t ) = ⎨
;a<0
⎪⎩ 0 , t < 0
30
15
x)Discrete-time exponential signal
( )
x [ n ] = ebnTs = ebTs
n
; ebTs = a ⇒ x [ n ] = a n , a ∈ \
0<a<1
a>1
Homework: sketch
the signal for a<-1
-1<a<0
31
Discrete-time causal exponential
⎧⎪a n , n ≥ 0
x [ n] = a nσ [ n] = ⎨
⎪⎩ 0 , n < 0
32
16
xi) Oscillation with exponential
envelope in continuous-time
x ( t ) = eat sin ω0t
sin ω0tk = 1; tk =
π
2π
+k
;
2ω0
ω0
sin ω0tl = −1; tl = −
π
2π
;
+k
2ω0
ω0
x ( tk ) = e
atk
x ( tl ) = −e
atl
33
Causal case
⎧⎪eat sin ω t, t ≥ 0
0
x ( t ) = eat sin ω0t σ ( t ) = ⎨
t<0
0,
⎪⎩
34
17
xii) Oscillation with exponential
envelope in discrete-time
x [ n ] = a n cos Ω0 n
Exercise:
Draw the waveform
of this signal for the
case a>1.
35
1.3. Complex signals. Phasors.
e jθ = cos θ + j sin θ ; e− jθ = cos θ − j sin θ
e jθ + e − jθ
cos θ =
2
;
e jθ − e− jθ
sin θ =
2j
cos θ = Re e jθ
;
sin θ = Im e jθ
{ }
{ }
36
18
i) Relation between real sinusoidal
signal and complex exponential
{
}
j ω t +ϕ
Re Ae ( 0 ) = A cos ( ω0t + ϕ ) ;
{(
)
}
Re Ae jϕ e jω0t = A cos ( ω0t + ϕ ) ;
A ∈ \+
A ∈ \+
e jω0t - oscillatory part
Ae jϕ - complex amplitude
A = Ae jϕ ∈ ^
{
jω0t
A cos ( ω0t + ϕ ) = Re Ae
}
jω0t - phasor that rotates with the angular velocity ω
Ae
0
37
ϕ=0
.
jω0t for φ=0. The mobile extremity
Evolution in time of the phasor Ae
of the phasor describes a cylindrical helix of radius A.
38
19
The negative frequency
39
ii) Relation between real oscillation
with exponential envelope and
complex exponential
x ( t ) = Aeσ0t cos ( ω0t + ϕ ) ,
A ( t ) = Ae jϕeσ0t
{
}
{
σ0 ∈ \
}
Re A ( t ) e jω0t = Re Aeσ0t e jϕe jω0t = Aeσ0t cos ( ω0t + ϕ )
A ( t )
complex envelope of the signal x ( t )
The vector that rotates with the angular velocity ω0
describes a spiral.
40
20
iii) Sampling (discrete-time case)
x [ n ] = Aeσ0Ts n cos ( ω0Ts n + ϕ ) = Aa n cos ( Ω0 n + ϕ )
j Ω n +ϕ )
Aa n e ( 0
associated phasor; A [ n ] = Aa n e jϕ complex envelope
{
} {
j Ω n +ϕ )
x [ n ] = Re Aa n e ( 0
= Re A [ n ] e jΩ0 n
}
σ0 = 0 : complex envelope constant A [ n ] = Ae jϕ
the vector rotating with angular velocity Ω0 - constant magnitude
ϕ = 0 : A cos Ω0 n =
A j Ω0 n A − j Ω 0 n
e
+ e
→ "negative frequency"
2
2
41
1.4. Simple signal transformations
i)Weighting- amplification or attenuation of signal
42
21
ii) Time shifting
x ( t − t0 ) shifted to the right if t0 > 0
x [ n − n0 ] shifted to the right if n0 > 0
to the left if t0 < 0
to the left if n0 < 0
43
x ( t ) = x ( −t )
iii) Time reversal
x [ n] = x [ −n]
44
22
iv) Time scaling
compresses or dilates the signal by multiplying the time variable by a constant
y ( t ) = x ( at ) , a ∈ \
45
v) Discrete-time scaling
⎧ ⎡n⎤
, if k divides n
⎪x
x( k ) [ n ] = ⎨ ⎢⎣ k ⎥⎦
⎪ 0,
otherwise
⎩
46
23
vi) Simple
transformations
x ( t ) ⎯⎯
→ 2 x ( −2t − 2 )
47
Even and odd parts of a real signal
x ( t ) = xe ( t ) + xo ( t ) ; xe ( t ) =
xe ( −t ) = xe ( t )
x [ n ] = xe [ n ] + xo [ n ]
;
x ( t ) + x ( −t )
x ( t ) − x ( −t )
; xo ( t ) =
2
2
xo ( −t ) = − xo ( t )
;
xe [ n ] =
x [ n] + x [ −n ]
2
; xo [ n ] =
x [ n ] − x [ −n]
2
48
24
Energy and Power
∞
Energy W =
∫ x (t )
2
dt < ∞ for complex signal
−∞
∞
Discrete-time: W =
∑ x [n]
2
<∞
n =−∞
Continuous-time finite energy signals - square integrable functions (space L2 )
∞
∫ x (t )
−∞
2
dt < ∞ ⇒ x ( t ) ∈ L2
Discrete-time finite energy signals - square summable functions (space l 2 )
∞
∑
n =−∞
x[n] < ∞ ⇒ x [ n ] ∈ l 2
2
49
1 – Causal decreasing exponential
x ( t ) = e −t σ ( t )
∞
W = ∫e
0
−2t
−2t ∞
e
dt =
−2
0
1 − e−∞ 1
=
=
2
2
50
25
2 – Causal oscillation with exponential
envelope
x ( t ) = e−t sin ω0tσ ( t )
∞
∞
0
0
W = ∫ e−2t sin 2 ω0tdt = ∫ e−2t
W=
ω02
1
1
−
=
4 4 1 + ω2
4 1 + ω02
0
(
) (
1 − cos 2ω0t
1∞
1∞
dt = ∫ e−2t dt − ∫ e−2t cos 2ω0tdt
2
20
20
)
“fast” oscillations (ω0 large), approximation
1
W≅
4
half of the energy for the case when there are no oscillations
51
3- Discrete-time causal exponential
signal
x [ n] = a n σ [ n] , a < 1
∞
W =∑ a
n =0
2n
=
1
1− a
2
1 + a + a 2 + ... + a n + ... =
1
,
1− a
a <1
52
26
4- Sine wave
Periodic signal x ( t ) = A sin ω0t
average energy computed over one period T0
T0
WT0 = ∫ A2 sin 2 ω0tdt =
0
A2
2
T0
∫ (1 − cos 2ω t ) dt =
0
0
A2
T0
2
53
5 – Unit step signal
x [ n] = σ [ n]
∞
N
W = ∑ 12 = lim ∑ 1 = lim ( N + 1) = ∞
n=0
N →∞
n=0
N →∞
54
27
Power
Average power of the signal P : average flux of
energy, ratio of signal energy and time interval when
that energy was developed.
infinite duration signals:
1 τ
2
x ( t ) dt
∫
τ→∞ 2τ −τ
P = lim
N
1
2
∑ x [ n]
N →∞ 2 N + 1 n =− N
P = lim
55
Energy and average power, finite duration signals
Discrete-time signals
Continuous-time signals,
support [t1,t2]
t2
2
support {N1, N1+1,…,N2}
N2
W = ∫ x ( t ) dt
W = ∑ x [ n]
2
n = N1
t1
t
1 2
W
2
P=
=
x ( t ) dt
∫
t2 − t1 t2 − t1 t
1
P=
N2
1
W
2
=
∑ x [ n]
N 2 − N1 + 1 N 2 − N1 + 1 n = N1
56
28
Periodic signals
average power is computed over one period:
P=
1
2
x
t
dt
(
)
∫
T0 T0
P=
1
2
∑ x [ n]
N n∈ N
57
6-Sine wave, average power
Sinusoidal Signal
1 τ 2 2
A2 τ 1 − cos 2ω0t
P = lim
dt =
∫ A sin ω0tdt = lim 2τ ∫
2
τ→∞ 2τ −τ
τ→∞
−τ
τ ⎤
⎡ 2
2
2
2
2
2
sin
ω
t
A
A
0
⎥ = A − A lim sin 2ω0 τ = A
= lim ⎢
⋅ 2τ −
⋅
⎥ 2
4τ
2ω0
4 τ→∞ 2ω0 τ
2
τ→∞ ⎢ 4τ
−τ ⎥⎦
⎢⎣
58
29
1.7. Distributions
function
distribution
operator
59
Example of Distribution: The Dirac Impulse
δ (t ) : ϕ (t ) → ϕ ( 0)
δ ( t − t0 ) : ϕ ( t ) → ϕ ( t0 )
δ(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
∞
ϕ ( t ) ⎯⎯
→ ∫ ϕ ( t ) f ( t ) dt
f
−∞
f : ϕ ( t ) ⎯⎯
→ f (t ) , ϕ (t ) =
∞
∫ ϕ ( t ) f ( t ) dt
60
−∞
30
The Derivative of a Distribution
∞
f ' ( t ) , ϕ ( t ) = − ∫ ϕ' ( t ) f ( t ) dt = f ( t ) , −ϕ' ( t )
−∞
δ' ( t ) , ϕ ( t ) = δ ( t ) , −ϕ' ( t ) = −ϕ' ( 0 )
δ'
ϕ ( t ) →− ϕ' ( 0 )
ϕ (t )
the
negative value of its first derivative computed in zero
δ' ( t ) associates to the test function
61
Unit Step Distribution
σ( t )
∞
ϕ ( t ) ⎯⎯⎯
→ ∫ ϕ ( t ) dt
0
σ ′( t )
∞
∞
ϕ ( t ) ⎯⎯⎯→ σ ( t ) , −ϕ′ ( t ) = − ∫ ϕ′ ( t ) dt = −ϕ ( t ) = ϕ ( 0 ) − ϕ ( ∞ )
0
0
σ ′( t )
ϕ ( t ) ⎯⎯⎯→ ϕ ( 0 )
⇒ σ′ ( t ) = δ ( t )
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.
62
31
Systems
Their mathematical model is the operator.
d
dt
∫
: x ( t ) → x' ( t )
t
: x ( t ) → ∫ x ( τ )d τ
-∞
63
2.1. Systems
Au > 10000
u2
= Au ;
VS
u
u
VS = 2 < 2
Au 10000
u2 < 5V ;
VS < 500 μV ;
VS = 0
V
ii = S
Rin
;
Rin = 100 k Ω ;
ii <
500 μV
= 5 nA
100 kΩ 64
32
Digital system
moving average filter (running averager).
65
Simulated analog system
x ( nTs ) ≅ qx [ n]
converter on 10 bits (1024
quantization levels)
domain of the input voltage 10V
10V
≅ 10mV
1024
Max quantization error ± 5mV
q≅
66
33
Mathematical model
y ( t ) = S { x ( t )}
S
or x ( t ) ⎯⎯
→ y (t )
y [ n ] = S d { x [ n ]}
d
or x [ n ] ⎯⎯→
y [ n]
S
67
2.2. Linear systems
S {a1x1 ( t ) + a2 x2 ( t )} = a1S { x1 ( t )} + a2 S { x2 ( t )}
Sd {a1x1 [ n ] + a2 x2 [ n ]} = a1Sd { x1 [ n ]} + a2 Sd { x2 [ n ]}
Superposition principle
68
34
Homogeneity
S d {ax [ n ]} = aS d { x [ n ]}
S {ax ( t )} = aS { x ( t )}
69
Incremental linear systems
Homogeneity, a = 0: S {0 x ( t )} = 0 S { x ( t )} = 0
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.
70
35
Additivity
The linear system response
at the sum of two input
signals equals the sum of
responses at each signal.
x1 ( t ) + x2 ( t ) ⎯⎯
→ y1 ( t ) + y2 ( t )
71
2.3. Time invariant systems
S { x ( t )} = y ( t )
⇒
S { x ( t − t0 )} = y ( t − t0 )
S d { x [ n ]} = y [ n ] ⇒
S d { x [ n − n0 ]} = y [ n − n0 ]
72
36
Stability
a) Stable equilibrium: the
impulse applied to the
ball creates attenuated
oscillations of its
position.
b) Neutrally stable
equilibrium: the
impulse applied to
the ball modifies the
equilibrium position.
c) Unstable
equilibrium: the
impulse applied to
the ball produces
loss of equilibrium
73
Causality
•Between the output and input of the system : relation of the type “cause-effect”
•The effect does not appear before the cause.
74
37
75
2.6. Systems described by linear constant-coefficients
differential equations and difference equations
First order linear system.
Second order linear system.
Homework: Prove the linearity of these systems.
76
38
General form of the linear constantcoefficients differential equation that
describes an Nth order system
d k y (t ) M
d k x (t )
=
, aN ≠ 0 (at least)
b
∑
k
dt k
dt k
k =0
k =0
The initial conditions should be null if the system is linear:
N
∑ ak
y ( t0 ) =
dy ( t )
d 2 y (t )
d N −1 y ( t )
=
=
...
=
=0
dt t =t
dt 2 t =t
dt N −1 t =t
0
0
0
if the input signal is applied at the moment of time t0
x ( t ) ≡ 0 for t < t0
77
Digital case - first order system
dy ( t )
+ y ( t ) = x ( t ) - equivalent digital system?
dt
dy ( t )
+ y ( nTs ) = x ( nTs )
RC
dt t =nT
RC
s
dy ( t )
y ( nTs ) − y ( nTs − Ts ) y [ n ] − y [ n − 1]
≅
=
dt t =nT
Ts
Ts
s
⎛ RC ⎞
RC
+ 1⎟ y [ n ] −
y [ n − 1] = x [ n ]
⎜
T
T
s
⎝ s
⎠
linear constant-coefficients difference equation, obtained by the
approximation of the differential equation
78
39
• the slope of the secant line is a good
approximation for the slope of the tangent
line for a small sampling step - Ts
79
Digital case - second order system
LC
d 2 y (t )
dt
d 2 y (t )
dt
=
2
t = nTs
+ RC
2
t = nTs
d ⎛ dy ( t ) ⎞
⎜
⎟
dt ⎝ dt ⎠
t = nTs
dy ( t )
+ y ( nTs ) = x ( nTs )
dt t = nT
s
dy ( t )
dy ( t )
−
dt t = nT
dt t = nT −T
s
s
s
≅
Ts
y [ n ] − y [ n − 1] y [ n − 1] − y [ n − 2]
−
y [ n ] − 2 y [ n − 1] + y [ n − 2]
Ts
Ts
=
=
Ts
Ts2
⎛ LC RC ⎞
⎛ 2 LC RC ⎞
LC
+ 1⎟ y [ n ] − ⎜
+
⎜ 2 +
⎟ y [ n − 1] + 2 y [ n − 2] = x [ n ]
2
⎜T
⎟
⎜ T
Ts
Ts ⎟⎠
Ts
⎝ s
⎠
⎝ s
linear constant-coefficients difference equation, obtained by the
approximation of the differential equation
80
40
General form of the linear constantcoefficients difference equation that
describes an Nth order system
N
M
∑ A y [n − k ] = ∑ B x [n − k ],
k
k =0
k
AN ≠ 0 (at least)
k =0
81
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.
82
41
ii) Ideal differentiator system
y (t ) =
dx ( t )
dt
y [ n] =
1
( x [ n] − x [ n − 1])
Ts
system that implements the approximation of the derivative for the digital case83
iii) Ideal integrator system
t
y ( t ) = ∫ x ( τ )d τ
−∞
n
n −1
k =−∞
k =−∞
y [ n] = ∑ x [ k ] = ∑ x [ k ] + x [ n]
y [ n ] = y [ n − 1] + x [ n ]
Continuous time
systems with memory :
Discrete time:
adder
•
digital adder
•
digital differentiator
84
42
2.8. Examples
1.Linear analog system with time-variable parameters
d 2 y (t )
dy ( t ) 2
+ 2t
+ t y (t ) = x (t )
2
dt
dt
a) Linearity
Additivity
→ y1 ( t ) ⇒
x1 ( t ) ⎯⎯
d 2 y1 ( t )
dt
x2 ( t ) ⎯⎯
→ y2 ( t ) ⇒
d
2
2
2
+ 2t
d 2 y2 ( t )
dt
2
dy1 ( t ) 2
+ t y1 ( t ) = x1 ( t )
dt
+ 2t
dy2 ( t ) 2
+ t y2 ( t ) = x2 ( t )
dt
d
2
⎣⎡ y1 ( t ) + y2 ( t ) ⎦⎤ + 2t dt ⎣⎡ y1 ( t ) + y2 ( t ) ⎦⎤ + t ⎣⎡ y1 ( t ) + y2 ( t ) ⎦⎤ = x1 ( t ) + x2 ( t )
dt
⇒ x1 ( t ) + x2 ( t ) ⎯⎯
→ y1 ( t ) + y2 ( t )
85
Homogeneity
x ( t ) ⎯⎯
→ y (t ) ⇒
d 2 y (t )
dt
+ 2t
2
dy ( t ) 2
+ t y (t ) = x (t )
dt
d2
d
⎡⎣ ay ( t ) ⎤⎦ + 2t ⎡⎣ ay ( t ) ⎤⎦ + t 2 ⎡⎣ ay ( t ) ⎤⎦ = ax ( t ) ⇒ ax ( t ) ⎯⎯
→ ay ( t )
dt
dt 2
b) Time shift invariance
x ( t ) ⎯⎯
→ y (t )
x ( t − t0 ) ⎯⎯
→ y3 ( t )
d 2 y (t )
dt
2
+ 2t
d 2 y3 ( t )
time invariant system:
dt
2
dy ( t )
dt
+ t 2 y (t ) = x (t )
+ 2 ( t − t0 )
d 2 y ( t − t0 )
2
dt
+ 2t
dy3 ( t )
dt
dy ( t − t0 )
dt
2
+ ( t − t0 ) y3 ( t ) = x ( t − t0 )
+ t 2 y ( t − t0 ) = x ( t − t0 )
y3 ( t ) ≠ y ( t − t0 ) ⇒ linear system, but not time-invariant
86
43
ii) The influence of null initial conditions
on linearity of analog systems
dy ( t )
+ 2 y (t ) = x (t )
dt
⎧ K cos ω0t, t ≥ 0
x ( t ) = K cos ω0t σ ( t ) = ⎨
0,
t<0
⎩
87
Particular solution - steady state
dy f ( t )
dt
+ 2 y f ( t ) = K cos ω0t , t ≥ 0
y f (t ) =
K
4 + ω02
cos ( ω0t − θ ) , t ≥ 0
Homogeneous solution – transient state
dytr ( t )
+ 2 ytr ( t ) = 0 t ∈ \
dt
ytr ( t ) = Be −2t , t ≥ 0 and ytr ( t ) = Ce−2t , t < 0
88
44
Final Solution
⎧⎪ y ( t ) + y f ( t ) , t ≥ 0
y ( t ) = ⎨ tr
t<0
⎪⎩ ytr ( t ) ,
K
⎧
−2 t
⎡ cos ( ω0t − θ ) − e −2t cos θ ⎤⎦ , t ≥ 0
⎪ y0 e +
2 ⎣
y (t ) = ⎨
4 + ω0
⎪ y e −2 t ,
t<0
⎩ 0
y ( t ) = y0 e −2t +
K
2
0
4+ω
⎡⎣cos ( ω0t − θ ) − e −2t cos θ ⎤⎦ σ ( t ) t ∈ \
linear system
K = 0 : x ( t ) = 0 ⎯⎯⎯⎯⎯
→ y ( t ) K =0 = y0e−2t ≡ 0
⇒ y0 = 0 null initial value for linear system!
89
iii) Influence of the null initial conditions
on the linearity of digital systems
⎧ K cos Ω0 n , n ≥ 0
y [ n ] − 0 ,5 y [ n − 1] = x [ n ] ⇒ x [ n ] = K cos Ω0 nσ [ n ] = ⎨
0,
n<0
⎩
Particular solution - steady state
{
}
y f [ n ] − 0.5 y f [ n − 1] = K cos Ω0 n = Re Ke jΩ0 n , n ≥ 0
y f [ n ] = A cos ( Ω0 n − θ )
A=
0.5sin Ω0
K
e − jθ ; θ = arctg
1 − 0.5cos Ω0
1.25 − cos Ω0
Homogeneous solution – transient state
ytr [ n ] − 0,5 ytr [ n − 1] = 0 , n ∈ `
ytr [ n ] = B ( 0,5 )
n
, n ≥ 0 and ytr [ n ] = C ( 0,5 )
n
, n<0
90
45
Final solution
K
⎧
n
⎪ B ( 0.5 ) + 1.25 − cos Ω cos ( Ω0 n − θ ) , n ≥ 0
0
y [ n] = ⎨
⎪
n
n<0
⎩C ( 0.5) ,
linear system ⇔ null initial condition: y [ −1] = 0
y [ n] =
K
cos ( Ω0 n − θ ) σ [ n ]
1.25 − cos Ω0
N th order system, null initial condition is
y [ −1] = y [ −2] = ... = y [ − N ] = 0
91
46