Chapter 1: Classification of Signal and System
Houshou Chen
Dept. of Electrical Engineering,
National Chung Hsing University
E-mail: houshou@nchu.edu.tw
H.S. Chen
Chapter1: Classification of signals and systems
• Siganls:
1. What is a signal
2. Classification of signals
3. Basic operations on signals
4. Elementary signals
• Systems:
1. What is a system
2. Classification of systems
3. LTI systems: circuit example
4. More example and motivation
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Chapter1: Classification of signals and systems
Signals
• Signal: a continuous-time signal x(t) (discrete-time signal x[n]) is
a function of an independent continuous variable t (discrete
variable n).
• Elementary continuous-time signals:
1. x(t) = es0 t , s0 = σ0 + jω0 (complex exponential)
2. x(t) = ejω0 t , s0 = jω0 (periodic complex exponential)
3. x(t) = eσ0 t , s0 = σ0 (real exponential)
4. x(t) = cos ω0 t = Re{ejω0 t } (sinusoidal signals)
5. impulse function: δ(t)
6. unit function: u(t)
7. ramp function: r(t)
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Chapter1: Classification of signals and systems
• Elementary discrete-time signals:
1. x[n] = z0n , z0 = r0 ejω0 (complex exponential)
2. x[n] = ejΩ0 n , z0 = ejΩ0 (periodic complex exponential)
3. x[n] = r0n , z0 = r0 (real exponential)
4. x[n] = cos Ω0 n = Re{ejΩ0 n } (sinusoidal signals)
5. impulse function: δ[n]
6. unit function: u[n]
7. ramp function: r[n]
• We will treat continuous-time and discrete-time signals
separately but in parallel.
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Classification of signals
1. continuous-time x(t) vs. discrete-time x[n]
• Usually a discrete-time signal x[n] is obtained from a
continuous time signal x(t) by sampling:
x[n] = x(nT ), n = 0, ±1, ±2... for some fixed T.
2. even vs. odd signals
• even (real): x(−t) = x(t)
• odd (real): x(−t) = −x(t)
• symmetric (complex): x(−t) = x∗ (t)
• anti-symmetric (complex): x(−t) = −x∗ (t)
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Chapter1: Classification of signals and systems
Any signal x(t) can be decompose into the even part xe (t) and
the odd part xo (t) by:
1
1
x(t) = [x(t) + x(−t)] + [x(t) − x(−t)],
2
2
where
xe (t) =
1
1
[x(t) + x(−t)] and xo (t) = [x(t) − x(−t)]
2
2
• It is easy to check that xe (t) = xe (−t) , xo (t) = −xo (t).
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Chapter1: Classification of signals and systems
3. periodic vs. aperiodic signals
• A signal x(t) (x[n]) is called a periodic signal if there exist real
number T (integer N ) such that:
x(t + T ) = x(t) (x[n + N ] = x[n]).
• The smallest T0 (N0 ) such that :
x(t + T0 ) = x(t) (x[n + N0 ) = x[n])
is called the (fundamental) period of x(t) (x[n]).
•
2π 2π
T0 ( N0 )
is called the fundamental frequency ( rad
sec ) of x(t) (x[n]).
• x(t) (x[n]) is called aperiodic if it is not periodic.
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Chapter1: Classification of signals and systems
4. deterministic vs. random
• deterministic signal x(t)
⇒ x(t0 ) is a number, no uncertainity
• random signal x(t)
⇒ x(t0 )is a random variable (with some probability specification)
x(t) = random signal = random process = stochastic process
5. energy signal vs. power signal
• for a continuous signal x(t):
∞ 2
E = −∞ x (t)dt : energy
T2 2
1
P=limT →∞ T −T x (t)dt : power
2
T
2
= T1 −T
x2 (t)dt if x(t) is periodic with period T
2
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Chapter1: Classification of signals and systems
• for a discrete signal x[n]
∞
E = n=−∞ x[n]: energy
N −1 2
1
P = limn→∞ 2N
n=−N x [n]: power
N −1 2
1
= N n=0 x [n] periodic with period N
• x(t)(x[n]) is an energy signal
if 0 < E < ∞
or is a power signal
if 0 < P < ∞
• A signal x(t) (x[n]) can not be an energy signal and a power
signal simultaneously.
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Difference between x(t) and x[n]
• There are many similarities between x(t) and x[n] , but there is
one important difference.
• For a continuous time x(t) = ejw0 t we have:
1. ejw1 t = ejw2 t if w1 = w2 , i.e., any two signals with two
different frequencies are distinct.
2. w1 > w2 ⇒ ejw1 t oscillates faster than ejw2 t .
3. ejw0 t is periodic for any w0 , T0 =
2π
w0 .
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Chapter1: Classification of signals and systems
• The above three properties are not true for a discrete-time signal
x[n] = ejΩ0 n .
1. For a discrete-time signal, we have
x[n] = ej(Ω0 +2π)n = ejΩ0 n × ej2πn = ejωo n
i.e., the signal x[n] at frequency (Ω0 + 2π) is the same as that at
frequency Ω0 , that is unlike the continuous case:
ejw1 t = ejw2 t if w1 = w2
2. I.e., for continuous-time signal, ejw0 t are all distinct for distinct
w0 . On the other hand, in discrete-time, the signal
x[n] = ejΩ0 n = ej(Ω0 +2mπ)n for any m ∈ Z.
=⇒ we only need to consider a frequency interval of length 2π,
usually −π ≤ Ω < π or 0 ≤ Ω < 2π.
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Chapter1: Classification of signals and systems
3. Ω0 is larger ⇒ ejΩ0 n oscillate faster is not true in discrete-time
case
• In discrete-time, since we only need to consider a frequency
interval of length 2π, say −π ≤ Ω < π or 0 ≤ Ω < 2π. We
have: frequencies close to 0, 2π are termed as low frequencies
and frequencies close to π, or −π are termed as high
frequencies.
• I.e., As Ω → 0, 2π, ejΩ0 n oscillates slower, and as Ω → π, −π,
ejΩ0 n oscillates faster.
πn
πn
πn
• cos(0n) = 1, cos( πn
),
cos(
),
cos(
),
cos(
8
4
2
1 ), Ω from 0 to
π, ejΩn oscillates slower to faster
3πn
8πn
• cos( 3πn
2 ), cos( 4 ), cos( 7 )cos(2πn) = 1, Ω from π to 2π,
ejΩn oscillates faster to slower
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Chapter1: Classification of signals and systems
4. The period of discrete-time signal ejΩ0 n
• ejΩ0 (n+N ) = ejΩ0 n ∗ ejΩ0 N = ejΩ0 n ( need ejΩ0 N = 1)
m
0
⇒ Ω0 N = 2πm ⇒ Ω
=
2π
N
i.e., a discrete-time signal ejΩ0 n is not necessary periodic for any
Ω0 . For a periodic ejΩ0 n , we must have Ω0 = s2π, where s ∈ Q.
• ej
nπ
4
(Ωo =
π
4
= 18 2π, N = 8) periodic
• ej3n (Ωo = 3 =
m
N 2π)
not periodic
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Chapter1: Classification of signals and systems
Figure 1: (−1)n
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Chapter1: Classification of signals and systems
Operations on signals
⎧
⎨ t − axis
• operation on
of x(t)
⎩ x − axis
• On dependent variable x(t)
i.e.,
⎧ given x(t), =⇒ want to find y(t) = Ax(t) + B
⎨ y (t) = Ax(t) scaling first
1
⎩ y2 (t) = y1 (t) + B shift next
⇒ y2 (t) = y(t) = Ax(t) + B.
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Chapter1: Classification of signals and systems
• y(t) = Ax(t) + B
⎧
⎪
|A| > 1 expand(A < 0 reverse)
⎪
⎪
⎪
⎪
⎨ |A| < 1 compress
– Remark:
⎪
B > 0 shift up
⎪
⎪
⎪
⎪
⎩ B < 0 shift down
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Chapter1: Classification of signals and systems
• y(t) = 3x(t) + 4
Figure 2:
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Chapter1: Classification of signals and systems
If
⎧ we do
⎨ y (t) = x(t) + B shift next
1
⎩ y2 (t) = Ay1 (t) scaling first
=⇒ y2 (t) = A(x(t) + B)
• Conclusion:
– y(t) = Ax(t) + B then A first ⇒ B next
– y(t) = A(x(t) + B) then B first ⇒ A next
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Chapter1: Classification of signals and systems
• On independent variable t
i.e.,
⎧ given x(t) ⇒ y(t) = x(at + b)
⎨ y (t) = x(t + b) shift first
1
⎩ y2 (t) = y1 (at) scaling next
⇒ y2 (t) = y(t) = y1 (at + b)
⎧
⎪
⎪
⎪
⎪
⎪
⎨
• Remark:
⎪
⎪
⎪
⎪
⎪
⎩
|a| > 1
compress(a < 0 reverse)
|a| < 1 expand
b > 0 shift left (advance version)
b < 0 shift right (delayed version)
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Chapter1: Classification of signals and systems
Figure 3:
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Chapter1: Classification of signals and systems
⎧
⎨ y (t) = x(at) scaling first
1
If we do
⎩ y2 (t) = y1 (t + b) shift next
⇒ y2 (t) = y1 (t + b) = x(a(t + b)) = x(at + ab)
Conclusion:
y(t) = x(at + b) b first⇒ a next
y(t) = x(a(t + b)) a first⇒ b next
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Chapter1: Classification of signals and systems
Why need this: convolutional sum, integral
• x(t) ⇒ Ax(t) + B A first,B next
• x(t) ⇒ x(at + b) b > 0 shift left
b first, a next b < 0 shift right
or equivalent x(t) ⇒ x(at − b) b > 0 shift right
x(t) ⇒ x(at − b) b < 0 shift left
by changing variable
∞
h(t − τ )x(τ )dτ ⇒ t − τ = λ ⇒ τ = t − λ ⇒ dτ = −dλ
−∞
−∞
∞
= ∞ h(λ)x(t − λ)(−dλ) = −∞ x(λ)h(t − λ)(dλ) = x(t) h(t)
−∞
x[n] h[n] = k=∞ h[n − k]x[k]
−∞
= m=∞ h[m]x[n − m] = x[n] h[n]
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Chapter1: Classification of signals and systems
Recall h(τ ) ⇒ h(t − τ ) = h(−τ + t)(h(−τ − (−t)) = h(−(τ − t)))
∞
1. y(t) = −∞ h(t − τ )x(τ ) = h(t) x(t)
2. y[n] = k h[n − k]x[k] = h[n] x[n]
3. X(D)h(D)
Figure 4:
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Chapter1: Classification of signals and systems
Figure 5:
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Chapter1: Classification of signals and systems
Other Elementary signals
1. ramp function:
⎧
⎨ 0 t≤0
r(t) =
⎩ t t≥0
⎧
⎨ 0 n≤0
r[n] =
⎩ n n≥0
Figure 6:
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Chapter1: Classification of signals and systems
2. unit function
⎧
⎨ 0
u(t) =
⎩ 1
⎧
⎨ 0
u[n] =
⎩ 1
t≤0
t≥1
step function
n = −1, −2, . . .
n = 0, 1, . . .
Figure 7:
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Chapter1: Classification of signals and systems
Remark: Many functions x(t) can be written in term of step
function. This will be very useful since we can deal with the
transform of x(t) by the transform of u(t), e.g., r(t) = tu(t).
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Chapter1: Classification of signals and systems
• u(t) − u(t − 1)
• u(t − a) − u(t − b)
Figure 8:
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Chapter1: Classification of signals and systems
• t · (u(t) − u(t − 1))
• t · (u(t) − u(t − 1)) + (u(t − 1) − u(t − 2))
Figure 9:
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Chapter1: Classification of signals and systems
In general, if we have x(t) in the form as follows.
Figure 10:
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Chapter1: Classification of signals and systems
We can always partition x(t) into:
x(t) = g1 (t)[u(t − a1 ) − u(t − a2 )]
+ g2 (t)[u(t − a2 ) − u(t − a3 )]
..
+ .
+ gn (t)[u(t − an ) − u(t − an+1 )] as follows.
Figure 11:
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Chapter1: Classification of signals and systems
3. impulse function⎧
⎨
δ(t) =
⎩
⎧
⎨
δ[n] =
⎩
0
t = 0
1·∞
t=0
0
n = 0
1
n=0
impulse function
delta function
In general, δ(t) is not a function, it is a generalized function.
(but δ[n] is a function).
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Chapter1: Classification of signals and systems
For example, δ(t) can be defined as the limit of some function.
• We can think of the continuous-time impulse function with the
property
∞
δ(t)dt = 1
−∞
⎧
⎨ 0 (t = 0)
and δ(t) =
⎩ ∞ (t = 0)
• In other words, continuous-time impulse δ(t) has the property:
δ(t) = 0 for all t except at t = 0 and the total area under δ(t) is 1.
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Chapter1: Classification of signals and systems
Figure 12:
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Chapter1: Classification of signals and systems
Properties of impulse function
There are many property of δ(t)
1. sampling property:
x(t) ∗ δ(t − t0 ) = x(t0 ) ∗ δ(t − t0 )
2. sifting property:
∞
x(t)δ(t − t0 )dt = x(t0 )
−∞
b
a
x(t)δ(t − t0 )dt =
⎧
⎪
⎪
⎨ x(t0 ) if t0 ∈ [a, b]
⎪
⎪
⎩
0 else
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Chapter1: Classification of signals and systems
sampling and sifting property
Figure 13:
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Chapter1: Classification of signals and systems
3. δ(at) =
1
|a| δ(t)
Figure 14:
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Chapter1: Classification of signals and systems
4. δ(at + b) = δ(a(t + ab )) =
1
|a| δ(t
+ ab )
Figure 15:
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Chapter1: Classification of signals and systems
• All of these properties can be proved by thinking δ(t) as a
generalized function.
• From the above properties, we have
∞
x(t0 ) = −∞ x(t)δ(t − t0 )dt
∞
= −∞ x(τ )δ(τ − t0 )dτ
∞
= −∞ x(τ )δ(t0 − τ )dτ
( by 1)
( replace t by τ )
( by 3)
Since this is true for ∀t0 ∈ (−∞, ∞), we can replace t0 by t.
• Finally, we have
∞
−∞
x(τ )δ(t − τ )dτ = x(t), ∀t
⇒ x(t) = x(t) ⊗ δ(t)
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Chapter1: Classification of signals and systems
From this property, δ(t) (or δ[n]) is the identity of convolutional
integral (convolutional sum)
∞
• x(t) = −∞ x(τ )δ(t − τ )dτ
or =
∞
−∞
x(t − τ )δ(τ )dτ (continuous-time)
• x[n] = Σ∞
k=−∞ x[k]δ[n − k]
or = Σ∞
k=−∞ δ[k]x[n − k] (discret-time)
We see that any signal x(t) (x[n]) can be written as the ”linear
combination” of δ(t) (δ[n]) and it’s shift version δ(t − τ ) (δ[n − k]),
i.e., the linear integral for continuous-time, and linear sum for
discrete-time.
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Chapter1: Classification of signals and systems
• Remark:
⎧
⎪
r
⎪
⎨ (t) = u(t)
⎪
⎪
⎩
⎧
⎪
⎪
⎨
u (t) = δ(t)
t
δ(τ )dτ = u(t)
−∞
⎪
⎪
⎩ t
−∞
u(τ )dτ = r(t)
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Chapter1: Classification of signals and systems
Also
⎧
⎪
⎪
⎨ r[n] − r[n − 1] = u[n]
⎪
⎪
⎩
u[n] − u[n − 1] = δ[n]
⎧
n
⎪
Σ
⎪
⎨ k=−∞ δ[k] = u[n]
⎪
⎪
⎩
Σnk=−∞ u[k] = r[n]
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Chapter1: Classification of signals and systems
• The relationship between u[n] and δ[n]
• From the identity of convolutional sum, we have
∞
u(t) =
u(t − τ )δ(τ )dτ
−∞
t
=
δ(τ )dτ
−∞
• Similarly, we have
∞
u(t) =
−∞
=
∞
u(τ )δ(t − τ )dτ
δ(t − τ )dτ
0
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Chapter1: Classification of signals and systems
• The relationship between u[n] and δ[n]
• From the identity of convolutional sum, we have
∞
u[n] =
u[n − k]δ[k]
k=−∞
=
n
δ[k]
k=−∞
• Similarly, we have
u[n] =
∞
δ[n − k]u[k]
k=∞
=
∞
δ[n − k]
k=0
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Chapter1: Classification of signals and systems
System
A continuous-time (discrete-time) system H is an operator that
transfer the input x(t) (x[n]) into the output y(t) (y[n]). We denote
the process by
Figure 16:
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Chapter1: Classification of signals and systems
Example: the RLC circuit
Figure 17:
How to describe the relationship between the input vi (t) and the
output v0 (t)?
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Chapter1: Classification of signals and systems
Classification of system
1. linear vs. nonlinear
H is called
⎧ linear if H has the superposition property:
⎨ H{x (t) + x (t)} = H{x (t)} + H{x (t)}
1
2
1
2
⎩ H{cx(t)} = cH{x(t)}
⇔ H{c1 x1 (t) + c2 x2 (t)} = c1 H{x1 (t)} + c2 H{x2 (t)}
n
n
⇔ H{ i=1 ci xi (t)} = i=1 ci H{xi (t)}
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Chapter1: Classification of signals and systems
Figure 18:
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Chapter1: Classification of signals and systems
2. time-invariant vs. time-variant
• H is called time-invariant if the following is true
H{x(t)} = y(t) =⇒ H{x(t − t0 )} = y(t − t0 )
• I.e., a time-shift to in the input x(t) results in an identical
time-shift to in the output
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Chapter1: Classification of signals and systems
Figure 19:
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Chapter1: Classification of signals and systems
3. memory vs. memoryless
• A system H is memoryless if the value y(t0 ) (i.e.,y(t = t0 )) only
depends on the value x(t0 ) for any t0 .
• example: y(t) = x2 (t) is memoryless since y(t0 ) = x2 (t0 ) for ∀t0 .
• example: y(t) = x(t − 1) is a system with memory since
y(t0 ) = x(t0 − 1), e.g., y(0) = x(−1). y(t0 ) depends on x(t) at
t = t0 − 1, not at t0 .
• In other words, output y(t) at current time t = t0 is only affected
by input x(t) at current time t = t0
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Chapter1: Classification of signals and systems
4. causal vs. noncausal
• A system H is causal if the value y(t0 ) only depends on
{x(t) : t ≤ t0 }.
• I.e., current output is produce by current input and past input,
not future input.
• the system y[n] = x[n − 1] is causal (y[0] = x[−1])
• the system y[n] = x[n + 1] is noncausal (y[0] = x[1])
• the system y(t) = x(t + a) is causal if a ≤ 0 and is noncausal if
a>0
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Chapter1: Classification of signals and systems
5. stable vs. nonstable
• H is stable if | x(t) |≤ Mx < ∞ ∀t
then | y(t) |≤ My < ∞ ∀t
• I.e., bounded input x(t) produces bounded output y(t)
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Chapter1: Classification of signals and systems
We will focus on a linear time-invariant system (LTI system) H.
If H is a LTI system, x(t) and y(t) are usually described by
• impulse response h(t)
• transfer function H(s)
• differential equation
• block diagram
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Chapter1: Classification of signals and systems
Preview and Review: t and s domain
1. t-domain: impulse response h(t)
x(t) =
∞
−∞
x(τ )δ(t − τ )dτ
∞
⇒ y(t) = H{x(t)} = H{ −∞ x(τ )δ(t − τ )dτ }
=
∞
−∞
x(τ )H{δ(t − τ )}dτ =
∞
−∞
x(τ )h(t − τ )dτ
2. s-domain: transfer function H(s)
x(t) =
=
∞
∞
jwt
X(w)e
dw ⇒ y(t) = H{x(t)}
−∞
jwt
X(w)H{e
−∞
}dw =
∞
jwt
X(w)H(w)e
dw
−∞
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• est is an eigenfunction of a continuous-time LTL system
∞
∞
y(t) = −∞ h(τ )x(t − τ )dτ = −∞ h(τ )es(t−τ ) dτ
∞
= ( −∞ h(τ )e−sτ dτ )est = H(s)est (= H(s)x(t))
• z n is an eigenfunction of a discrete-time LTL system
∞
∞
y[n] = −∞ h[k]x[n − k] = −∞ h[k]z n−k
∞
= ( −∞ h[k]z −k )z n = H(z)z n (= H(z)x[n])
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Chapter1: Classification of signals and systems
Change of basis: two domains
A vector x in terms of one basis {e1 , e2 · · · , en }
x = (x1 , x2 , · · · xn ) = x1 (100 · · · 0) + x2 (010 · · · 0) + · · · + xn (000 · · · 1)
= x1 e1 + x2 e2 + · · · xn en (∈ e1 , e2 · · · en )
The same vector x in terms of another basis {v1 , v2 · · · vn }
x = x1 e1 + x2 e2 + · · · xn en = x1 v1 + x2 v2 + · · · + xn vn = x
• A vector x has two representations in terms of two bases
x = (x1 , x2 , · · · , xn ) = (x1 , x2 , · · · , xn )
• We can change from {ei }ni=1 to {vi }ni=1 and vice versa; if {vi }ni=1
are eigenvectors, we can simplify operation y = Ax in {ei }ni=1
domain to y = Dx in {vi }ni=1 domain.
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• The reason is as follows. In {ei }ni=1 domain, we have y = Ax.
• If Avi = λvi for all i
{v1 , v2 · · · vn } = eigenvectors with eigenvalues {λ1 , λ2 · · · λn }
x = x1 e1 + x2 e2 + · · · xn en = x1 v1 + x2 v2 + · · · + xn vn = x
x = (x1 x2 · · · xn ) = x1 v1 + x2 v2 + · · · + xn vn
y = Ax = A(x1 v1 + x2 v2 + · · · + xn vn )
= x1 λ1 v1 + x2 λ2 v2 + · · · + xn λn vn =y1 v1 + · · · yn vn
where yi = λi xi
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Or equivalently,
A(v1 v2 · · · vn ) = (Av1 , Av2 , · · · Avn ) = (λ1 v1 , λ2 v2 , · · · λn vn )
⎤
⎡
λ
⎢ 1
⎢
= (v1 v2 · · · vn ) ⎢ 0
⎣
0
0
..
0
0
.
0
⎥
⎥
⎥ ⇒ AV = V D ⇒ A = V DV −1
⎦
λn
y = V DV −1 x ⇒ y = Dx ⇒ V −1 y = DV −1 x
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Motivation of LTI system
• Motivation I: O.D.E and Circuit ⇔ signal and system
A RLC circuit
Figure 20:
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Chapter1: Classification of signals and systems
Or block diagram
Figure 21:
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Chapter1: Classification of signals and systems
From
the circuit theory, we have
⎧
⎪
VR (t) = R · i(t)
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
VL (t) = L di(t)
dt
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩ i(t) = C dVC (t) ⇒ di(t) = C d2 VC (t)
dt
dt
dt2
Therefore, by KVL, we have :VC (t) + VL (t) + VR (t) = Vs (t)
C (t)
⇒ VR (t) = R · i(t) = RC dVdt
VL (t) = L ·
d2 VC (t)
C dt2
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Finally, we have
L·
d2 VC (t)
C dt2
⇒ Vc (t) +
C (t)
+ RC dVdt
+ VC (t) = Vs (t)
R L Vc (t)
+
1
Lc Vc (t)
=
1
LC Vs (t)
Input signal: x(t) = Vs (t)
output signal: y(t) = Vc (t)
⇒ The differential equation describing the relationship between
input x(t) & output y(t) is as follows.
y (t) +
R L y (t)
+
1
LC y(t)
=
1
LC x(t)
This is a 2nd order constant coefficient linear ODE.
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A complete solution y(t) is given by:
y(t) = yh (t) + yp (t) (O.D.E.)
= yZ.I.R (t) + yZ.S.R. (t) (circuit)
= ynatural (t) + yforced (t) (circuit)
In general, y(t) for t t0 depends on both the initial state s(t0 ) and
the input function x(τ ), t t0 we write:
y(t) = F (s(t0 ); x(τ ), τ t0 )
then ZIR(t) = f (s(t0 ); 0); ZSR(t) = f (0; x(τ ), τ t0 )
1. For a linear-system, Complete system response=ZIR+ZSR
2. We will assume s(t0 ) = 0 from now on and turn attention
to ZIR when we discuss the Laplace Transform.
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1. Solving 2nd oreder O.D.E.:
1
yh (t): solving λ2 + R
λ
+
L
LC = 0 (two roots λ1 & λ2 )
⇒ yh (t) = c1 eλ1 t + c2 eλ2 t (λ1 = λ2 distinct roots )
or yh (t) = c1 eλ1 t + c2 teλ2 t (λ1 = λ2 repeat roots )
or yn (t) = eα1 t (c1 cos β1 t + c2 sin β1 t)
where λ1 = α1 + ıβ1 ,λ2 = λ1 = α1 − ıβ1
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65
2. Solving 1st order differential system:
states:i(t) & vC (t)
⇒ VC (t) = 1c i(t)
i (t) = L1 VL (t)= L1 (Vs (t) − VC (t) − VR (t))
= L1 (−VC (t) − Ri(t) + Vs (t))
⎡
⇒⎣
⎡
⇒⎣
⎤
VC (t)
i(t)
x1 (t)
x2 (t)
⎡
⎦ =⎣
⎤
⎤⎡
0
−1
L
⎡
⎦ = A⎣
1
C
−R
L
x1 (t)
⎤
⎦⎣
⎤
VC (t)
i(t)
⎡
⎦+⎣
⎤
0
1
L Vs (t)
⎦ + F (t)
x2 (t)
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Chapter1: Classification of signals and systems
66
in general,we have X (t) = Ax(t) + F (t)
and we have the solution of the first order differential system:
X(t) = Xh (t) + Xp (t) where Xh (t) is obtained by
diagonalizing the matrix A
A
V1
V2
=
λ1 V 1
λ2 V2
=
V1
V2
⎤
⎡
⎣
λ1
0
0
λ2
⇒ A = V DV −1
x (t) = V DV −1 x(t)
⇒ V −1 x−1 (t) = D V x(t)
Y (t)
Y (t)
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H.S. Chen
Chapter1: Classification of signals and systems
⎧
⎨ y (t) = c eλ1 t
1
1
⇒
⎩ y2 (t) = c2 eλ2 t
⎡
⇒⎣
⎤
x1 (t)
x2 (t)
⎦=
v1
v2
⎡
⎣
⎤
y1 (t)
⎦
y2 (t)
⎡
⎤
⎡
⎤
c eλ1 t
x1 (t)
⎦
⎦ = v1 (t) v2 (t) ⎣ 1
⇒⎣
c2 eλ2 t
x2 (t)
= c1 eλ1 t v1 + c2 eλ2 t v2
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Summary
From the above example, we can see that there are several ways to
describe the relationship between the input x(t) and the output y(t)
for a LIT sytem x(t) ↔ y(t). These are:
1. Block diagram
Figure 22:
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2. differential equation (SISO system)
y +
R L y (t)
+
1
LC y(t)
=
1
LC x(t)
(λ2 +
R
Lλ
+
1
LC
= 0 two roots)
⇒ y(t) = yh (t) + yp (t) = yZIR (t) + yZSR (t)
where ⎧
λt
λt
⎪
e
+
c
e
(λ1 = λ2 real)
c
1
2
⎪
⎨
yh (t) =
c1 eλt + c2 teλt (λ1 = λ2 real)
⎪
⎪
⎩
yh (t) = c1 eα1 t (cos βt + sin βt) (λ1 = λ2 = α + ıβ)
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3. differential system (MIMO system)
⎡
⎡
⎤
⎤
y1 (t)
y1 (t)
⎣
⎣
⎦
⎦ + F (t)
y (t) =
=A
y2 (t)
y2 (t)
⇒ y(t) = yh (t) + yp (t)
& yh (t) =
v1
v2
⎡
⎣
⎤
c1 eλ1 t
c2 eλ2 t
⎦
where Av1 = λ1 v1 , Av2 = λ2 v2 (λ1 = λ2 )
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4. our focus
⎧
⎨ time domain h(t): impulse response
⎩ frequency domain H(s): transfer function
We can find the transfer function H(s), or the frequency
response H(jw) (H(ejΩ )) directly from the circuit diagram or
from the differential equation (system). After that, we can get
the impulse response h(t) from H(s). The idea is connecting
with phasors in circuit theory.
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Phasors
R: VR (t) = Ri(t)
ZR
i(t) = ejwt
⇒ VR (t) = R ejwt ⇒ ZR = R( independce)
L: VL (t) = L di(t)
dt
ZL
i(t) = ejwt
⇒ VL (t) = L · jw ejwt ⇒ ZL = jwL (SL)
C (t)
C: i(t) = C dVdt
ZC
VC (t) = ejwt
⇒ i(t) = jwC ejwt ⇒ ZC =
1
jwC
1
( sC
)
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Figure 23:
We can replace R by R, C by 1/jwC, and L by jwL; then by KVL
or KCL we can solve the transfer function H(s) directly from the
circuit diagram.
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therefore by voltage divider, we have
Vc =
1
jwc
1
jwc
+ R + jwL
Vs , (∗jw L1 on top and bottom)
H(jw)
i.e. H(jw) =
or H(s) =
1
LC
1
(jw)2 + R
L jw+ LC
1
LC
1
S2 + R
L S+ LC
Note: O.D.E. Vc (t) +
R 1
L Vc (t) LC
+ Vc (t) =
1
LC Vs (t)
• It seems that we can find H(s) aslo from the ODE.
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Figure 24:
• x(t) = ejwt (or in general x(t) = est ) is an eigenfunction of a
continuous-time LTI system.
• x[n] = ejΩn (or in general x[n] = z n ) is an eigenfunction of a
discrete-time LTI system.
• Let x(t) = ejwt then y(t) = H(jw)ejwt
• Let x[n] = ejΩn then y[n] = H(ejΩn )ejωn
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Figure 25:
I.e., mathematically, for a LTI system H, we have
1. h(t) = H{δ(t)}, h[n] = H{δ[n]}
2. H(jw) =
H{ejwt }
ejwt ,
jΩ
H(e ) =
H{ejΩn }
ejΩn
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77
e.g. the ODE for RLC circuit is:
1
Let x(t) = ejwt , then y(t) = H(jw)ejwt y (t)+ R
y
+
L
LC y(t) =
then y (t) = (jw)H(jw)ejwt , y (t) = (jw)2 H(jw)ejwt
⇒ ((jw)2 +
⇒ H(jw) =
R
L jw
+
1
jwt
)H(jw)e
RL
=
1 jwt
RL e
1
LC
1
(jw)2 + R
L jw+ LC
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LC x(t)
H.S. Chen
Chapter1: Classification of signals and systems
This is always true for any nth order linear constant coefficient ODE.
That is, given a differential equation for a LTI system
an y (n) (t) + an−1 y (n−1) (t) + · · · + a1 y (t) + a0 y(t)
= bm x(m) (t) + bm−1 x(m−1) (t) + · · · + b1 x1 (t) + b0 x(t)
i.e.,
n
(i)
i=1 ai y (t) =
m
(j)
b
x
(t)
j
j=1
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Substitute: x(t) = ejwt & y(t) = H(jw)ejwt
into the ODE & use the fact
di jw
dti e
= (jwt)i ejwt we have
(an (jw)n + an−1 (jw)n−1 + · · · + a1 (jw) + a0 )H(jw)ejwt
= (bm (jw)m + bm−1 (jw)m−1 + · · · + b1 (jw) + b0 )ejwt
↔ H(jw) =
H(s) =
bm (jw)m +···+b1 (jw)+b0
an (jw)n +···+a1 (jw)+a0
bm sm +···+b1 sm +b0
an sn +···+a1 s+a0
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Figure 26:
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Chapter1: Classification of signals and systems
Usually H(s) =
=
N (s)
D(s)
(degD(s) = n)
An
+ · · · + s+p
(assume D(s) has n distinct toots)
n
(by P.E.F. partial fraction Expansion)
⇒ h(t) = L−1 {H(s)}
⇒ h(t) = A1 e−p1 t u(t) + A2 e−p2 t u(t) + · · · + An e−pn t u(t)
A1
s+p1
+
A2
s+p2
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In general,
block diagram > differential system
> O.D.E
> h(t)(H(S))
where > means providing more information.
In signal & system,we study the zero-state response
Figure 27:
in particular,the system H will be a L.I.T. system.(linear & time
invariant)
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Chapter1: Classification of signals and systems
Motivation II: (linear algebra⇔ signal & system)
⎡
a d
⎢
⎢ b
⎢
A=⎢
⎢ c
⎣
d
c
b
⎤
⎥
a c d ⎥
⎥
⎥ (circulant matrix)
b a d ⎥
⎦
c b a
• How to find the eigenvectors and eigenvalues for the circulant
matrix A?
• We can use the fact that A represents a discrete-time LTI system
to find the eigenvectors and eigenvalues.
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The matrix A represents a LTI system for a discrete-time with
periodic input x[n]. That is, if x is a periodic input, then y = Ax is
the periodic output with the fact that y[n] is obtained by the circular
convolution between x[n] and h[n]:
y[n] =
N
x[k]h[n − k]
k=1
In this example, we have h[n] = (a, b, c, d).
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Find the eigenvalues and eigenvectors for A. First,we can find
eigenvalues of A by
Av = λv ⇒ (λI − A)v = 0
with v = 0
Therefore we must have λI − A is a singular matrix, i.e.
det(λI − A) = 0 (characteristic polynomial).
This is a poly of degree n if A is a n × n matrix.
In general, it is not easy to find the eigenvalues for a given n × n
matrix A.
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For this circulant
⎡
⎤ matrix,we can show that
1
⎢
⎥
⎢ 1 ⎥
⎢
⎥
j 2π
v1 = ⎢
⎥ = (e 4 ·0·n )n=0,1,2,3 = (i0 )n=0,1,2,3
⎢ 1 ⎥
⎣
⎦
1
⎡
⎤
1
⎢
⎥
⎢ i ⎥
⎢
⎥
j 2π
v2 = ⎢
⎥ = (e 4 ·1·n )0≤n≤3 = (i1·n )0≤n≤3
⎢ −1 ⎥
⎣
⎦
−i
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⎡
1
⎤
⎢
⎥
⎢ −1 ⎥
⎢
⎥
j 2π
v3 = ⎢
⎥ = (e 4 ·2·n )0≤n≤3 = (i2·n )0≤n≤3
⎢ 1 ⎥
⎣
⎦
−1
⎡
⎤
1
⎢
⎥
⎢ −i ⎥
⎢
⎥
j 2π
v4 = ⎢
⎥ = (e 4 ·3·n )0≤n≤3 = (i3·n )0≤n≤3
⎢ −1 ⎥
⎣
⎦
i
are eigenvectors of A.
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eigenvalue of v1
=a+d+c+b
eigenvalue of v2
(a − c) + i(d − b)
eigenvalue of v3
(a + c) − (d + b)
eigenvalue of v4
(a − c) − i(d − b)
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Moveover, v1 , v2 , v3 , v4 are orthogonal vectors, i.e.,
(vi , vj ) = 0 for any i = j
.
Let ei =
√1 vi
4
⇒ {e1 , e2 , e3 , e4 } are orthonormal eigenvector for A.
In other words,we have A [e1 , e2 , e3 , e4 ] = [e1 , e2 , e3 , e4 ] D
V
V
⎤
⎡
λ1
0
⎥
⎢
⎥
⎢
λ2
⎥
⎢
where D = ⎢
⎥,
⎥
⎢
λ
3
⎦
⎣
0
λ4
and λi is an eigenvalue of ei .
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we can define a = h(0), b = h(1), c = h(2), d = h(3) then
⎡
h(0) h(3) h(2) h(1)
⎤
⎢
⎥
⎢ h(1) h(0) h(3) h(2) ⎥ ⎢
⎥
A=⎢
⎥ = h((n − k))4
⎢ h(2) h(1) h(0) h(3) ⎥
⎣
⎦
h(3) h(2) h(1) h(0)
where h(3)4 = h(−1), h(2)4 = h(−2), · · ·
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Then
Ax =
3
h[n − k]x[k]
=
k=0
3
h[k]x[n − k]
k=0
• This is just the discrete-time convolution sum.
j 2π
4 nk0
• If we let x[n] = e
(k0 = 0, 1, 2, 3)
3
2π
⇒ Ax = k=0 h[k]ej 4 (n−k)k0
3
−j 2π
kk0
j 2π
4
4 nk0 .
=
h[k]e
· e k=0
x[n]
λ
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j 2π
4 nk0
• i.e., e
, (0 ≤ k0 ≤ 3) is an eigenvector of A with eigenvalue
−j 2π
4 kk0 .
h[k]e
k
In matrix language, we have
⇒ y = Ax -time domain
y = V DV −1 x (since V −1 = V t )
= V DV T x
⇒ V T y = DV T x
⇒ y = Dx
-frequency domain
• If V is an orthonormal matrix,then V T = V −1
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In general,we can show
⎡
h(0)
h(N − 1)
⎢
⎢ h(1)
h(0)
⎢
⎢
h(1)
A=⎢
⎢ h(2)
⎢ .
⎢ ..
⎣
h(N − 1) h(N − 2)
h(1)
⎤
⎥
h(2) ⎥
⎥
⎥
h(3) ⎥
⎥
⎥
..
⎥
.
⎦
h(0)
always has eigenvectors
2π
√1 (ej N ·0n )0≤n≤N −1
N
2π
√1 (ej N ·1n )0≤n≤N −1
N
= e1
= e2
..
.
2π
√1 (ej N ·(N −1)n )0≤n≤N −1
N
= eN , (N eigenvectors)
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Such {e1 , . . . , eN } are orthonormal eigenvector for A and
h[n − k]x[k] =
h[k]x[n − k]
Ax =
k
k
j 2π
N nk0
, 0 ≤ k0 ≤ N − 1
Similarly, if we let x[n] = e
j 2π
h[k]e N (n−k)k0
⇒ Ax =
=
k
k
j 2π
N kk0
h[k]e
j 2π
N nk0
· e x[n]
λ
Also y = Ax = V DV −1 x ⇒ V −1 y = DV −1 x
⇒ y = DV T x
⇒ y = Dx
Department of Electrical Engineering, National Chung Hsing University
94