2.0 Linear Time-invariant Systems
2.1 Discrete-time Systems: the Convolution Sum

Representing an arbitrary signal as a sequence of
unit impulses
x[n] 

 x[k ] [n  k ]
k  
an unit impulse located at n = k on the index n
See Fig. 2.1, p.76 of text

u[n]    [n  k ]
k 0
a special case
Vector Space Representation of Discretetime Signals (P.47 of 1.0)

n-dim
𝑎 = (𝑎1 , 𝑎2 , ⋯ 𝑎𝑛 )
𝑥 = (𝑥1 , 𝑥2 , ⋯ 𝑥𝑛 )
𝑏 = (𝑏1 , 𝑏2 , ⋯ 𝑏𝑛 )
𝑥𝑖 𝑣𝑖 合成
𝑥=
𝑣1 = (1, 0, 0, ⋯ 0)
𝑣2 = (0,1, 0, ⋯ 0)
⋮
𝑎⋅𝑏 =
𝑎𝑖 𝑏𝑖
𝑖
𝑖
𝑥𝑗 = 𝑥 ⋅ 𝑣𝑗
分析
Vector Space Representation of Discretetime Signals (P.48 of 1.0)

n extended to ± ∞
𝑥 = (⋯ , 𝑥−2 , 𝑥−1 , 𝑥0 , 𝑥1 , 𝑥2 , 𝑥3 , ⋯ )
𝑣0 = (⋯,
0,
0, 1, 0, 0, 0, ⋯ )=δ [ 𝑛 ]
𝑣1 = (⋯,
0,
0, 0, 1, 0, 0, ⋯ )=δ [ 𝑛 – 1 ]
𝑣𝑘 = (⋯,
0,
0, 0, ⋯, 0, 1, ⋯ )=δ [ 𝑛 – k ]
{𝛿 𝑛 − 𝑘 , 𝑘: inter}
𝑥𝑖 𝑣𝑖 合成
𝑥=
𝑖
𝑥𝑗 = 𝑥 ⋅ 𝑣𝑗
分析
Vector Space Interpretation
Vector Space (P.30 of 1.0)
3-dim Vector Space
N-dim Vector Space (P.31 of 1.0)
𝑁
𝑎𝑘
𝐴=
𝑣𝑘
(合成)
𝑘=1
𝑎𝑗 = 𝐴 ∙ 𝑣𝑗
𝑣𝑖 ∙ 𝑣𝑗 = 𝛿𝑖𝑗
(分析)
Vector Space Interpretation
∞
𝑥𝑛 =
𝑥 𝑘 𝛿 [𝑛 − 𝑘] (是合成也是分析)
𝑘=−∞
𝑥[𝑘]
𝑥[𝑘]
𝑥[𝑛]
𝑘
𝑘
𝐴=
𝑛
𝑛
𝛿[𝑘 − 𝑛]
𝛿[𝑛 − 𝑘]
𝑛
𝑎𝑘 𝑣𝑘 (合成)
𝑘
𝑘
𝑛
𝑘
𝑎𝑗 = 𝐴 ⋅ 𝑣𝑗 分析
𝑎𝑛 = 𝐴 ⋅ 𝑣𝑛 (𝐴 ⋅ 𝐵 =
𝑎𝑘 𝑏𝑘 )
𝑘

Representing an arbitrary signal as a sequence of
unit impulses
– Sifting property of the unit impulse: looked at on the index
k, δ[n-k] is nonzero only at k = n, which “sifts” the value
x[n] out of the function x[k].

Defining the output for an unit impulse input as the
Unit Impulse Response
x[n]=[n]
y[n]=h[n]:unit impulse response
S
0
n
0
n

By Linearity (Superposition Property)
– The output for an arbitrary input signal is the
superposition of a series of “shifted, scaled unit impulse
response”
a
k
x k [ n] 
k
a
k
y k [ n]
k
x[k] [n-k]
y[ n] 
x[k]

h[n-k]
 x[k ]h[n  k ]  x[n]  h[n]
k  
Convolution Sum
See Fig2.3,p.80 of text
Input/Output Relation in Every Dimension
𝛿[𝑛]
ℎ[𝑛]
𝑦[𝑛]
𝑥[𝑛]
𝑥 0 ℎ[𝑛]
𝑥[0]
𝑥[1]
𝑥 1 ℎ[𝑛 − 1]
𝑥[2]
𝑥 2 ℎ[𝑛 − 2]

A Different Way to visualize the convolution sum
– looked at on the index k
y[ n ] 

 x[k ]h[n  k ]
k  
input signal
Contribution to the
output signal at time
n
reflected-over version of h[k]
located at k = n
– on the dummy index k, h[k] is reflected over and shifted to
k=n, weighted by x[k] and summed to produce an output
sample y[n] at time n
See Figs 2.5, 2.6, 2.7, pp. 83-85 of text
Fig. 2.6
𝑥[𝑘]
𝑘
ℎ[𝑘]
𝑘
𝑘
𝑘
𝑘
ℎ −𝑘 , 𝑛 = 0
ℎ 𝑛 − 𝑘 ,𝑛 < 0
ℎ 𝑛 − 𝑘 ,𝑛 = 2 > 0

A linear time-invariant system is completely
characterized by its unit impulse response
y[ n ] 

 x[k ]h[n  k ]
k  
Vector Space Concept

Definition of Inner-product
– Example
A  a1, a2 ,........aN , ai  R  V
N
A  B   ai bi
i 1
– Simple Definition
v1, v2 V,
v1  v2 : V V  R (or C)
Inner Product
𝐴 ⋅ 𝐵 = 𝐴 𝐵 cos 𝜃
𝐴
magnitude similarity
𝐴 = (𝐴 ⋅ 𝐴)
1
2:
1
2
𝑎𝑖2
magnitude,
𝑖
cos 𝜃 =
𝐴⋅𝐵
𝐴 𝐵
𝐴⋅𝐵 = 0
𝐴:
𝐵1 :
𝐵2 :
: similarity, −1 ≤ cos 𝜃 ≤1
: orthogonal
𝐴 ⋅ 𝐵2 > 𝐴 ⋅ 𝐵1
𝐵
Linearity in Inner Product
𝑥 ⋅ 𝑎𝑦 = 𝑎 𝑥 ⋅ 𝑦
𝑥 ⋅ 𝑦1 + 𝑦2 = 𝑥 ⋅ 𝑦1 + 𝑥 ⋅ 𝑦2
magnitude
similarity
Commutative, Non-degeneracy
magnitude
similarity
Vector Space Concept

Definition of Inner-product
–
Physical Properties: magnitude, similarity
–
For the vector space of signals defined in (N1, N2)
xn  yn :V V  R
N2
xn   yn   xny  n
n  N1
–
Inner-product Space
–
(N1, N2) extended to (-∞, ∞)
Vector Space Concept

Magnitude/Similarity
xn  xn  xn
1
2
S xn , yn 

xn   yn
xn
yn
Orthonormal bases
– orthogonal
( i [n])  ( j [n]) 
N2
  [ n]  
n  N1
i

j
[ n]  0
Vector Space Concept

Orthonormal bases
– orthonormal
 k n , k  1, 2, ....M
 i n   j n  0, i 
j
 1, i  j
– orthonormal basis
any vector (signal) in the vector space can be expanded by
the set of orthonormal signals
x[n] 
M
x 
k 1
k
k
[ n]
(合成)
System Characterization

Vector Space
x[n], x[n] defined for -   n    V
– Orthonormal bases
  k [n]   [n  k ] ,
k  0,  1,  2, ......
time-shifted unit impulses, dim = ∞
– A signal expanded by this set of orthonormal basis
x[n] 

 x[k ]  [n  k ]
k  
xk
 k [n]
(合成)
System Characterization

A Different View of the Sifting Property
– 3-dim vector space
A  a1iˆ  a 2 ˆj  a3 kˆ
a 2  A  ˆj  (a1iˆ  a 2 ˆj  a3 kˆ)  ˆj
– Sifting property
x[n] 

 x[k ]  [n  k ]
k  
( x[k ])   n k 
(分析)
System Characterization

Transformation of Vectors
– A System is a Transformation which maps one vector x[n]
to another vector y[n] (linear, time-invariant)
x[ n]
 [n]
H[ ]
y[ n]
h[ n]

Defining the output for an unit impulse input as the
Unit Impulse Response
x[n]=[n]
y[n]=h[n]:unit impulse response
S
0
n
0
n
Splitting a single basis vector into a few
more basis vectors
𝑖
a
(a,b,c)
(1,0,0) = 𝑖
b
(𝑎, 𝑏, 𝑐)
(1,0,0)
𝛿 𝑛 →ℎ 𝑛
c
𝑘
𝛿 𝑛−𝑘 →ℎ 𝑛−𝑘
𝑗
𝑥[𝑘] 𝛿[𝑛 − 𝑘] →
𝑘
𝑥[𝑘] ℎ[𝑛 − 𝑘]
𝑘
– transforming a single basis vector enough to describe the
transformation of any vectors

Rotation as a vector transformation
𝑦′
y
(1,0,0)
𝑥′
(𝛼𝑎, 𝛼𝑏, 𝛼𝑐)
(a, b, c)
x
Rotation as a
vector transformation
Rotation plus
vector scaling
(1, 0, 0) → (0.9, 0.3, 0.3) (1, 0, 0, 0,⋯) → (0.9, 0.3, 0.2,⋯)
(0, 1, 0) → (0.3, 0.9, 0.3) (0, 1, 0, 0,⋯) → ( 0, 0.9, 0.3, 0.2,⋯)
(0, 0, 1) → (0.3, 0.3, 0.9) (0, 0, 1, 0,⋯) → ( 0, 0, 0.9, 0.3,02,⋯)
System Characterization

Transformation of Vectors
– unit impulse response is the transformation of δ[n]. A basis
vector is transformed to a set of other bases
– superposition property for convolution sum
a
k
xk n   ak yk n
k
x[k ]
k
 [n  k ]
yn  

x[k ]
h[n  k ]
 xk  hn  k 
k  
System Characterization

Linear, Time-invariant Transformation of vectors can
be expressed by a matrix
x
H  hnk 
y
y  Hx
[ yn ]  hnk xk , yn   hnk xk
hnk  hn  k 
k
Convolution Sum
System Characterization

Matrix Representation of Convolution Sum
example: x[n]=…0,0,x0,x1,x2,0,0,…
h[n]=…0,0,h0,h1,h2,0,0,…
y[n]=…0,0,y0,y1,y2,y3,y4,0,0,…
        
   
  
 0   h 0 0  0 0 0  0 0   0 
 0   h h  0 0 0  0 0   0 
1 0
  
 
 y 0   h 2 h1  h 0 0 0  0 0   x 0 
 y1   0 h 2  h1 h 0 0  0 0   x1 
 y 2    0 0  h 2 h1 h 0  0 0   x 2 
 y   0 0  0 h h  h 0   0 
2 1
0
 3 
 
 y 4   0 0  0 0 h 2  h1 h 0   0 
 0   0 0  0 0 0  h 2 h1   0 
 0   0 0  0 0 0  0 h 2   0 
   
  









 
  
Matrix Representation
h0x2
System Characterization

Matrix Representation of Convolution Sum
– each component of the input vector is transformed to a set
of other components of the output vector, and all these are
superpositioned (superposition property)
– each component of the output vector is contributed
respectively by the transformed component of each input
vector component (reflected, shifted, weighted sum)
2.3 Continuous-time System :
the Convolution Integral

Representing an arbitrary signal as an integral of
impulses

 x(k) (t  k)
x(t )  lim
 0

k  
See Fig 2.12 , p.91 of text
x(t ) 



x( ) (t   )d
an impulse located at t = τ
u (t ) 


0
 (t   ) d
whose value is x(τ) (合成)
a special case
Vector Space Representation of Discretetime Signals (P.48 of 1.0)

n extended to ± ∞
𝑥 = (⋯ , 𝑥−2 , 𝑥−1 , 𝑥0 , 𝑥1 , 𝑥2 , 𝑥3 , ⋯ )
𝑣0 = (⋯,
0,
0, 1, 0, 0, 0, ⋯ )=δ [ 𝑛 ]
𝑣1 = (⋯,
0,
0, 0, 1, 0, 0, ⋯ )=δ [ 𝑛 – 1 ]
𝑣𝑘 = (⋯,
0,
0, 0, ⋯, 0, 1, ⋯ )=δ [ 𝑛 – k ]
{𝛿 𝑛 − 𝑘 , 𝑘: inter}
𝑥𝑖 𝑣𝑖 合成
𝑥=
𝑖
𝑥𝑗 = 𝑥 ⋅ 𝑣𝑗
分析
Vector Space Interpretation (P.8 of 2.0)
∞
𝑥𝑛 =
𝑥 𝑘 𝛿 [𝑛 − 𝑘] (是合成也是分析)
𝑘=−∞
𝑥[𝑘]
𝑥[𝑘]
𝑥[𝑛]
𝑘
𝑘
𝐴=
𝑛
𝑛
𝛿[𝑘 − 𝑛]
𝛿[𝑘 − 𝑛]
𝑛
𝑎𝑘 𝑣𝑘 (合成)
𝑘
𝑘
𝑛
𝑘
𝑎𝑗 = 𝐴 ⋅ 𝑣𝑗 分析
𝑎𝑛 = 𝐴 ⋅ 𝑣𝑛 (𝐴 ⋅ 𝐵 =
𝑎𝑘 𝑏𝑘 )
𝑘
Vector Space Representation
∞
𝑥 𝑡 =
𝑥 𝜏 𝛿 𝑡 − 𝜏 𝑑𝜏(是合成也是分析)
−∞
𝑥 𝜏
𝑥 𝜏
𝑥 𝑡
𝜏
𝑡
𝑡
𝛿 𝑡−𝜏
𝜏
𝐴=
𝑡
𝑎𝑘 𝑣𝑘 (合成)
𝑡
𝜏
𝜏
𝛿 𝜏−𝑡
𝑎𝑗 = 𝐴 ⋅ 𝑣𝑗 (分析)
𝑘
discrete-time: 𝐴 ⋅ 𝐵 =
continuous-time: 𝐴 ⋅ 𝐵 =
𝑛𝑎 𝑛
∞
𝑎 𝑡
−∞
𝑏∗ 𝑛
𝑏 ∗ 𝑡 𝑑𝑡
Continuous-time Vector Space
𝑥(𝑡) 𝜏2 ≤ 𝑡 ≤ 𝜏1
𝑥(𝑡) −∞ < 𝑡 < ∞
or
𝑎𝑥(𝑡), 𝑥1 𝑡 + 𝑥2 𝑡
{ 𝛿(𝑡 − 𝜏), −∞ < 𝜏 < ∞ } is a set of basis
𝑡
𝑡
𝑡
𝛿(𝑡 − 𝜏1 )
𝛿(𝑡 − 𝜏2 )
𝛿(𝑡 − 𝜏3 )
∞
𝑥 𝑡
⋅ 𝑦 𝑡
=
𝑥 𝑡 𝑦 ∗ 𝑡 𝑑𝑡
−∞
∞
𝑥(𝑡) 2 𝑑𝑡
𝑥(𝑡) =
−∞
1
2
Unit Step
1
∞
𝑢 𝑡 =
𝛿 𝑡 − 𝜏 𝑑𝜏
0
0
𝛿 𝜏−𝑡
0
𝛿 𝜏−𝑡
t
t
𝜏
𝑡>0
𝜏
𝑡<0
0
– running integral
𝑡
𝑢 𝑡 =
𝛿 𝜏 𝑑𝜏, 𝜏 = 𝑡 − 𝜏 ′
−∞
t

Representing an arbitrary signal as an integral of
impulses
– Sifting Property :
(分析)
Looked at on the index τ, δ(t-τ) located at τ = t shifts the
value x(t) out of the function x(τ)
See Fig 2.14 ,p.94 of text

Defining the output for an unit impulse input as the
unit impulse response
x(t )   (t )
y (t )  h (t )
Unit Impulse Response
S
0
t
0
t

By Linearity (Superposition Property)
– The output for an arbitrary input signal is the
superposition of a series of “shifted, scaled unit impulse
response”
a
k
xk (t )   ak yk (t )
k
x(t ) 



k
x( ) (t   ) d
y (t ) 
y (t ) 






x( )h(t   ) d
x( )h(t   )d  x(t )  h(t )
Convolution Integral

Defining the output for an unit impulse input as the
Unit Impulse Response
x[n]=[n]
y[n]=h[n]:unit impulse response
S
0
n
0
n

By Linearity (Superposition Property)
– The output for an arbitrary input signal is the
superposition of a series of “shifted, scaled unit impulse
response”
a
k
x k [ n] 
k
a
k
y k [ n]
k
x[k] [n-k]
y[ n] 
x[k]

h[n-k]
 x[k ]h[n  k ]  x[n]  h[n]
k  
Convolution Sum
See Fig2.3,p.80 of text
Input/Output Relation in Every Dimension
(P.12 of 2.0)
ℎ[𝑛]
𝛿[𝑛]
𝑦[𝑛]
𝑥[𝑛]
𝑥 0 ℎ[𝑛]
𝑥[0]
𝑥[1]
𝑥 1 ℎ[𝑛 − 1]
𝑥[2]
𝑥 2 ℎ[𝑛 − 2]
Input/Output Relation in Every Dimension
𝛿(𝑡)
ℎ(𝑡)
𝑡
0
𝑡
0
𝑥(𝑡)
𝑡
0
𝜏1
𝜏2
𝑡
𝑡
𝑡
0 𝜏1 𝜏2
𝑡
𝑦(𝑡)
𝑡
0
𝜏1
𝜏2
𝑡
𝑡

A Different Way to visualize the convolution integral
– Look on the index τ

y(t )   x( )h(t   )d

input signal
output signal at time t
reflected-over version of h(t)located at  =t
– On the dummy index τ, h(t) is reflected over and shifted to
τ = t, weighted by x(t) and integrated to produce the output
value at time t, y(t)
see Figs.2.19,2.20,2.21,pp.100-101 of text

A linear time-invariant system is completely
characterized by its unit impulse response
−2𝑇
0
ℎ(−𝑡)
2.4 Properties of Linear
Time-invariant Systems

Commutative Property
x[ n]  h[ n]  h[ n]  x[ n]
x(t )  h(t )  h(t )  x(t )
– the role of input signal and unit impulse response is
interchangeable, giving the same output signal
– In evaluating the convolution sum or integral, the input
signal can be reflected over and weighted by the unit
impulse response

Distributive Property
x[n]  (h1 [n]  h2 [n])  x[n]  h1 [n]  x[n]  h2 [n]
x(t )  [h1 (t )  h2 (t )]  x(t )  h1 (t )  x(t )  h2 (t )
x[n ]
h1
y[n ]
h2
x[n ]
h1  h2
y[n ]
( x1[n]  x2 [n])  h[n]  x1[n]  h[n]  x2 [n]  h[n]
additive (linear) property

Associative Property
x[n]  (h1[n]  h2 [n])  ( x[n]  h1[n])  h2 [n]
h1
h2  h1
h2
h1  h2
h2
h1
– Cascade of two systems gives an unit impulse response
which is the convolution of the unit impulse responses of
the two individual systems
– The behavior of a cascade of two systems is independent
of the order in which the two systems are cascaded

Causality
– causal if y[n] dose not depend on x[k] for k > n
y[ n ] 

 x[k ]h[n  k ]
k  
– Causal iff h[n]=0, n < 0
y[n] 
n

k  
k 0
 x[k ]h[n  k ]   h[k ]x[n  k ]

Causality
– continuous-time

y(t )   x( )h(t   )d

– Causal iff h(t)=0,t<0
t

y (t )   x( )h(t   )d   h( ) x(t   )d

0
Causality (P.56 of 1.0)
Causality & Memory
∞
𝑦𝑛 =
𝑥 𝑘 ℎ[𝑛 − 𝑘]
(for future)
non-causal
𝑘=−∞
ℎ[𝑛]
𝑛
memory
ℎ[𝑛 − 𝑘]
𝑛
𝑥[𝑘]
memory
(for past)
𝑛
non-causal
𝑘
𝑘
( ℎ 𝑛 = 0, 𝑛 < 0 ) ⇔ ( ℎ 𝑛 − 𝑘 = 0, 𝑘 > 𝑛 ): Causality

Memoryless / with Memory
– A linear, time-invariant, causal system is memoryless only
if
h[n ]  K [n ]
ht   K t 
y[n ]  Kx[n ]
yt   Kxt 
if k=1 further, they are identity systems
y[n]  x[n] 

 x[k ] [n  k ]  xn  n
k  

y (t )  x(t ) 
 x( ) (t   )d  xt    t 

– sifting property, i.e., convolution sum (or integral) with an
unit impulse function gives the original signal

Invertibility / Inverse system
x[n ]
h[n ]
h[n]  h1[n]   [n]
y[n]
h1[n]
h(t )  h1 (t )   (t )
x[n ]

Stability
stable if bounded input gives bounded output
|x[n]| < B, all n



k  
k  
k  
 h[k ]x[n  k ]   h[k ] x[n  k ]  B   h[k ]
y[n] 
– Stable iff the impulse response is absolutely summable,

 | h[k ] | 
k  
or absolutely integrable,



| h(t ) | dt  
the necessary condition can be proved

Unit step response
output for an unit step function input
s[n]  u[n] * h[n] 
n
 h[k ]
Running sum
k  
h[n]  s[n]  s[n  1]
First difference
similarly
s (t ) 

t

h ( ) d
ds (t )
h (t ) 
dt
Running integral
First derivative
2.5 Systems Described by
Differential/Difference Equations
Continuous-time

Differential Equation Specification for Input/Output
Relationships
N

k 0
k
M
k
d y (t )
d x(t )
ak
  bk
k
k
dt
d
t
k 0
– derived by physical phenomena and relationships
– very often auxiliary conditions are needed to completely
specify the system
Continuous-time

Differential Equation Specification for Input/Output
Relationships
– the response y(t) to an input x(t) in general consists of two
parts:
(1) homogeneous solution (natural response): a solution for
N

k 0
d k y (t )
ak
0
k
dt
(2) particular solution: to the complete differential equation
Continuous-time

Initial Rest Condition for Causal Systems
x(t )  0, t  t 0 
y(t )  0, t  t 0
– initial conditions
dy (t 0 ) d 2 y (t 0 )
d N y (t 0 )
y (t 0 ) 

 ....... 
0
2
N
dt
dt
dt
Discrete-time

Difference Equation Specification
N
a
k 0
M
k
y[n  k ]   bk x[n  k ]
k 0
– derived by sequential behavior of different processes
– auxiliary conditions needed
– response y(t) consists of two parts
(1) homogeneous solution (natural response) for
N
a
k 0
k
y[ n  k ]  0
(2) particular solution
Discrete-time

Initial Rest Condition for Causal Systems
x[n]  0, n  n0 

y[n]  0, n  n0
Recursive Equation
N
1 M

y[n] 
 bk x[n  k ]   a k y[n  k ]
a 0  k 0
k 1

– output at time n expressed in terms of previous values of
input/output
N
1 M

y[n] 
 bk x[n  k ]   a k y[n  k ]
a 0  k 0
k 1

M
bk
y ( n)   (
) x[n  k ]
k 0 a0
Block Diagram Representation

Elementary Operations
x2[n]
x2(t)
x1[n]+ x2[n]
x1(t)+ x2(t)
x1[n]
x1(t)
x[n]
x(t)
a
ax[n]
ax(t)
Block Diagram Representation

Elementary Operations
x[n]
D
x(t)
d
dt
x(t)

x[n-1]
dx (t )
dt

t

x( )d
Block Diagram Representation

An Example
y[n]  ay[n  1]  bx[n]
x[n]
b
y[n]
D
-a
y[n-1]
– Feedback, with memory, initial value of the memory
element as the initial condition
– Initial rest condition: initial value in the memory element
is zero
Block Diagram Representation

Continuous-time Example
dy (t )
 ay (t )  bx(t )
dt
x(t)
b
a
y(t)
d
dt

1
a
dy (t )
dt
Block Diagram Representation

Continuous-time Example
dy (t )
 ay (t )  bx(t )
dt
– Expressed by integrator, assuming initially at rest
t
y(t )   [bx( )  ay( )]d

t
y (t )  y (t0 )   [bx( )  ay ( )]d
t0
x(t)
b

y(t)
-a
– The integrator represents the memory element
2.6 The Unit Impulse for
Continuous-time Cases
Many Different Functions
Approaching δ(t) in the Limit

  t    t  as   0

Sifting Property
x(t )  xt    t , xt    t 
 t    t    t 
 t    t    t    t 
Unit Impulse
𝛿(𝑡)
𝑥(𝑡)
ℎ(𝑡)
ℎ(𝑡)
𝑦 𝑡 =𝑥 𝑡 ∗ℎ 𝑡
∞
𝑥 𝑡 =
𝑥 𝜏 𝛿 𝑡 − 𝜏 𝑑𝜏 = 𝑥 𝑡 ∗ 𝛿(𝑡)
−∞
(是分析, 是合成, 是單位元素)
𝑥(𝑡)
𝑥(𝑡)
𝛿(𝑡)
𝑦 𝑡 = 𝑥 𝑡 ∗ 𝛿 𝑡 = 𝑥(𝑡)
𝛿(𝑡)
𝑥(𝑡)
𝑥(𝑡)
𝛿(𝑡)
𝛿(𝑡)
𝛿(𝑡)
𝛿 𝑡 = 𝛿 𝑡 ∗ 𝛿 𝑡 ∗ 𝛿(𝑡)
Many Different Functions
Approaching δ(t) in the Limit

Let r (t )    t     t 
See Fig. 2.33, p. 128 of text
then
lim r (t )   (t )
 0
lim [r (t )    (t )]   (t )
 0
lim [r (t )  r (t )]   (t )
 0
.
.
.
.
Many Different Functions
Approaching δ(t) in the Limit

Let r (t )    t     t 
Operational Definition of Unit Impulse

A Function can be defined by
– what it is at each value of the independent variable
– what it does under some mathematical operation such as
an integral or a convolution, or how it behaves with a
system, or some mathematical constraints: Singularity
Function
x t   x t    t  for any x t , 𝑥 𝑡 =
∞
𝑥 𝜏 𝛿 𝑡 − 𝜏 𝑑𝜏
−∞
ht   ht    t 




 ( )d  1


g ( ) ( )d  g (0)
Operational Definition of δ(t)

{𝛿 𝑡 − 𝜏 , −∞ < 𝜏 < ∞} is a set of basis
∞

𝑥 𝑡 =
𝑥 𝜏 𝛿 𝑡 − 𝜏 𝑑𝜏 = 𝑥 𝑡 ∗ 𝛿(𝑡)
−∞
𝑡
𝑡
𝑡

是合成也是分析
是單位元素
All are operational definitions
Differentiator, Integrator and
Singularity Functions

Differentiator
x(t)
d
dt
unit impulse response
d (t )
h(t ) 
 u1 (t )
dt
dx(t )
 x(t )  u1 (t )
dt
dx (t )
y t  
dt
d  (t ) 1
 [ (t )   (t  )]
dt

d  (t ) 1
dxt 
xt  
 [ x(t )  x(t  )] 
dt

dt
See Fig. 2.36, p. 134 of text
Differentiator, Integrator and
Singularity Functions

Cascade of Differentiators
d 2 xt 
 xt   u1 t   u1 t   xt   u 2 t 
2
dt
d 2 t 
u 2 t  
dt 2
u 2 t   u1 t   u1 t 
d k  t 
u k t  u 1 t  u 1 t   ........ u 1 t , all k  0
k
dt
k times
Differentiator, Integrator and
Singularity Functions

Integrator
xt 

yt  

t

unit impulse response
ht  

t

 ( )d  u (t ) unit step function
  x d  xt   u t  for any xt 
t

x d
Differentiator, Integrator and
Singularity Functions

Cascade of Integrators
u 2 t   ut   ut    u d  tut 
t

unit ramp function
t



xt   u 2 t   xt   ut   ut    ( x( )d )d
Differentiator, Integrator and
Singularity Functions

Cascade of Integrators
u k t   ut   ut   ........  ut    u ( k 1) ( )d
t

k times
t k 1

u (t )
(k  1)!
Differentiator, Integrator and
Singularity Functions

Unified Definition
 (t )  u 0 (t )
u (t )  u 1 (t )
, u  k (t ), , u  2 (t ), u 1 (t ), u 0 (t ), u1 (t ), u 2 (t ), , u k (t ), 
||
integrators
u1 (t ) * u 1 (t )   (t )
u 2 (t ) * u  2 (t )   (t )
u k (t ) * u r (t )  u k  r (t )
δ(t)
inverse systems
differentiators
Differentiator, Integrator and
Singularity Functions

Unified Definition
, uk (t ),, u2 (t ), u1 (t ), u0 (t ), u1 (t ), u2 (t ),, uk (t ),
||
integrators
δ(t)
differentiators
– operational definitions with singularity functions
– manipulate operations efficiently and easily
2.7 Vector Space Interpretation for
Continuous-time Systems
Vector Space Concept

The Set of All Continuous-time Signals Defined in
(t1, t2) Forms A Vector Space
{x(t), x(t) is a continuous-time signal defined in
(t1,t2)}=V
– definitions and properties
xt   yt , axt 
Vector Space Concept

Definition of Inner-product
xt   yt : V  V  R or C
xt   yt : t xt y  t dt
t1
2
– similar to
N2
xn   yn :  xny  n
n  N1
N
A  B   ai bi
n 1
Vector Space Concept

Magnitude/Similarity
xt   xt  xt 
1
2

xt   y t 
S xt , y t  
xt  y t 
Inner Product for Continuous-time Signals
∞
𝑥 𝑡
⋅ 𝑦 𝑡
𝑥 𝑡 𝑦 ∗ 𝑡 𝑑𝑡
=
−∞
𝑥 𝑡
𝑦1 𝑡
𝑦2 𝑡
𝑥 𝑡
⋅ 𝑦1 𝑡
> 𝑥 𝑡
⋅ 𝑦2 𝑡
Vector Space Concept

Orthonormal Bases
– orthogonal
i t   j t   t
t2
i t  t dt  0
1
– orthonormal
k t , k  1,2,3,...M
i t   j t   0, i  j
 1, i  j

j
Vector Space Concept
– a typical example of orthonormal signal set
k t   b  t  k , k  k1 , k1  1, ...k 2 
b: scaling factor
k1  t1 , k2  1  t2
– orthonormal basis
any vector in the vector space can be expanded by the set
of orthonormal signals
M
xt    xkk t 
k 1
Vector Space for Continuous-time
Signals

Vector Space
xt , xt  defined in  ,   V

Orthonormal Bases(?)
 t    t  ,    , 
time-shifted unit impulses, dim=∞
– Inner-product
Vector Space for Continuous-time
Signals

Orthonormal Bases(?)
 t    t  ,    , 
– Inner-product
– Not really orthonormal (but are orthogonal), but makes
sense under the operational definition
System Characterization

A Signal expanded by unit impulse bases
(合成)
(分析)
System Characterization

A System is a Transformation
x(t)
(t)
y(t)
H
h(t)
 a x t    a
k
k
k
k
yk t 
k

xt    x   t   d


yt    x  ht   d


Can’t be represented by a matrix due to the discrete
property of the matrices, but the concept is the same
Data Transmission
“0”
“1”
…1101 …
Transmitter
1
1 0 1
ℎ𝑐 𝑡
channel distortion
𝑛 𝑡
noise
…1101 …
ℎ2 𝑡
Decision
equalizer
ℎ2 [𝑡] ≈ ℎ𝑐−1 [𝑡]
ℎ1 𝑡
front-end
filter
ℎ1,2 [𝑛]
ℎ1,2 [𝑡]
Examples
• Example 2.11, p.110 of text
− time shift of signals
x(t)
(t)
h(t)
y(t)=x(t-t0)
h(t)= (t-t0)
x(t-t0 )  x(t) δt-t0 
− convolution of a signal with a shifted impulse is
the signal itself but shifted
Examples
• Example 2.12, p.111 of text
− running sum or accumulation
x[n]
[n]
h[n]
h[n]=u[n]
− first difference is the inverse
y[n]=x[n]-x[n-1]
h 1[ n ] =  [ n ] -  [ n - 1 ]
− u[n]*{
u[n] {δ[n][n] [n-1]}=u[n]-u[n-1]=
δ[n  1]}  u[n]  u[n[n]
1]  δ[n]
…
Examples
Problem 2.64, p.166 of text
• A simplified echo system and its inverse
x(t)
y(t) …… x(t)
T
y(t)
½
(-½)
h(t)
g(t)
T
Problem 2.64, p.166 of text
• A more realistic model
x(t)
y(t)
……
x(t)
α T
y(t)
T
h(t)
(-α)
g(t)
−stability analysis
0<α<1,  h(t ) dt    , stable




k
k 0
α>1,  h(t ) dt not integrable, “NOT” stable