Bài giảng Signals and Systems - Chapter: Definitions and classifications - Đặng Quang Hiếu

ET2060 - Signals and Systems
Definitions and classifications
Dr. Quang Hieu Dang
Hanoi University of Science and Technology
School of Electronics and Telecommunications
Autumn 2012
Continuous-time / discrete-time signals
x(t)
sampling−−−−−−→
Ts
x [nTs ]
normalize−−−−−−−→ x [n]
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x(t)
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Figure: Continuous signal x(t) and discrete signal x [n]
Signal energy and power
Given current i(t) and voltage v(t) on resistor R , find its energy,
power?
Continuous signal x(t):
◮ Instantaneous power px(t) = x2(t)
◮ Total energy
Ex = lim
T→∞
∫ T
−T
|x(t)|2dt =
∫ ∞
−∞
|x(t)|2dt
◮ Average power
Px = lim
T→∞
1
2T
∫ T
−T
|x(t)|2dt
Signal energy and power (2)
Discrete signal x [n]:
◮ Total energy
Ex =
∞∑
n=−∞
|x [n]|2
◮ Average power
Px = lim
N→∞
1
2N + 1
N∑
n=−N
|x [n]|2
◮ When 0 < Ex <∞ → x(t), x [n] - energy signal.
◮ When 0 < Px <∞ → x(t), x [n] - power signal.
Operations performed on independent variable(s) (1)
◮ Shift x(t)→ x(t − T )
◮ Reflection x(t)→ x(−t)
◮ Scaling x(t)→ x(kt)
t
x(t)
t
x(t − T )
t
x(−t)
t
x(kt)
Operations performed on independent variable(s) (2)
◮ Draw signals x(kt + T ) and x(k(t + T ))?
◮ Discrete signal case?
Example: Given signals x(t) and x [n] as in the following figure.
(a) Draw x(2t + 1) and x(2(t + 1)).
(b) Draw x [2n + 1] and x [2(n + 1)].
t
x(t)
2 3 4
1
1 2 3 4 5 6 7-1
b b b b b
n
x [n]
1
Periodic signals
◮ Continuous-time
x(t) = x(t + T ), ∀t
◮ Discrete-time
x [n] = x [n + N], ∀n
where N is a positive integer.
◮ Minimum values of T , N are called fundamental periods.
Example: Determine if following signals are periodic? If so, find
the fundamental periods.
(a) cos2(2pit + pi/4)
(b) sin(2n)
Even / odd signals. Deterministic / random signals
◮ Even: x(t) = x(−t); x [n] = x [−n]
◮ Odd: x(t) = −x(−t); x [n] = −x [−n]
◮ Deterministic signals: Signal value at any given time is
deterministic, signal can be represented as a function of time.
◮ Random signals: Random values → random variables,
probability density functions (pdf) and random processes.
Example: Any given signal x(t) can be decomposed into two
components – even and odd signals: x(t) = xe(t) + xo(t). Find
xe(t) and xo(t).
Real exponential signals
x(t) = Ceat , x [n] = Cean, C , a ∈ R
0
1
2
3
4
0 1 2 3 4
x(t) = 3e−2t
decaying
0
20
40
60
80
0 1 2 3 4
x(t) = et
growing
0
1
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4
0 10 20 30 40
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x [n] = 3e−n/10
decaying
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x [n] = en/10
growing
Example: Given an electronic circuit with capacitor C and resistor
R connected in series. Draw voltage v(t) developed across
capacitor C , if initially (t = 0) the capacitor is charged with V0 [v].
Sinusoidal signals
x(t) = sin(ω0t + φ)
Periodic, T = 2piω0→ Discrete-time sinusoidal signal?
1
-1
1 2 3 4 5 t
x(t)
Example: Given an electronic circuit with capacitor C and
inductor L connected in series. Draw voltage v(t) developed across
capacitor C , if initially (t = 0) it is charged with V0 [v].
Complex exponential signals (continuous-time)
If C and a are complex numbers: C = |C |e jθ and a = r + jω0, we
have:
x(t) = |C |erte j(ω0t+θ)
= |C |ert cos(ω0t + θ) + j |C |ert sin(ω0t + θ)
1
-1
1 2 3 4 5 t
Re{x(t)}
envelope |C |ert
Example in electronic circuit?
Complex exponential signals (discrete-time)
If C and a are complex number: C = |C |e jθ and a = r + jω0, we
have:
x [n] = |C |erne j(ω0n+θ)
= |C |ern cos(ω0n + θ) + j |C |ern sin(ω0n + θ)
Remarks about e j(ω0n+θ):
◮ Not always periodic (depends on ω0), period?
◮ Only need to consider ω0 within [0, 2pi], when high / low
frequency?
Illustration of x [n] = e j(ω0n)
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Im{x [n]}
ω0 = 0.8pi
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Im{x [n]}
ω0 = 1.8pi
Unit step function
u(t) =
{
1, t ≥ 0
0, otherwise
u[n] =
{
1, n ≥ 0
0, otherwise
1
t
u(t)
1
b
b b b b b b b b b b
n
u[n]
Example in electronic circuit?
Unit impulse function (discrete-time)
δ[n] =
{
1, n = 0
0, otherwise
1
b b b
b
b b b b b
n
δ[n]
Given unit step function?
δ[n] = u[n]− u[n − 1]
u[n] =
∞∑
k=0
δ[n − k]
Given arbitrary signal x [n]?
x [n] =
∞∑
k=−∞
x [k]δ[n − k]
Dirac delta function (continuous-time)
δ(t) = 0, ∀t 6= 0∫ ∞
−∞
δ(t)dt = 1
t
x(t)
1
t
δ(t)
Some properties:
δ(t) =
d
dt
u(t), u(t) =
∫ t
−∞
δ(τ)dτ
x(t0) =
∫ ∞
−∞
x(t)δ(t − t0)dt
δ(at) =
1
a
δ(t)
Ramp function
r(t) =
{
t, t ≥ 0
0, otherwise
r [n] =
{
n, n ≥ 0
0, otherwise
t
u(t)
b b b
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n
u[n]
Systems
x [n]
T−→ y [n]
x(t) y(t)
continuous system
x [n] y [n]
discrete system
System interconnections: cascade, parallel and feedback
input output
system 1 system 2
+
input output
system 1
system 2
+
input output
system 1
system 2
Stability
A system T is called BIBO stable if and only if every bounded input
|x(t)| <∞, ∀t
produces bounded output
|y(t)| <∞, ∀t
Example: Check the stability of the system
y [n] = rnx [n]
for |r | > 1.
Memory
◮ A system is called memoryless if its output depends only on
present value of the input.
◮ A system is said to possess memory if its output depends on
past or future values of the input.
Example: Determine if following systems are memoryless
(a) y [n] = x [n]− x [n − 1] + 2x [n + 2]
(b) i(t) = 1R v(t)
Causality
A system is said to be causal if the output at any time n depends
only on the present and past inputs.
y(n) = F [x(n), x(n − 1), x(n − 2), . . . ]
Example: Determine if the following systems are causal or
non-causal
(a) y [n] = x [n]− x [n − 1] + 2x [n + 2]
(b) i(t) = 1L
∫ t
−∞ v(τ)dτ
Time invariance
A system T is said to be time invariant if and only if
x [n]
T−→ y [n] then x [n − n0] T−→ y [n − n0] ∀n, n0
for every input signal x(n) and every time shift n0.
Example: Determine if the system is time invariant
y [n] = nx [n]
Linearity
A system T is linear if and only if
T{a1x1[n] + a2x2[n]} = a1T{x1[n]}+ a2T{x2[n]}
for any input sequences x1[n], x2[n] and any constants a1, a2.
Example: Determine if the systems are linear
(a) y(t) = tx(t)
(b) y(t) = x2(t)
Invertibility
A system is invertible if the input can be recovered from its output
(diferrent inputs produce different output).
x(t) x(t)y(t)
T T−1
Example: Determine if the systems are invertible, if so, find the
inverted system
(a) y [n] =
∑n
k=−∞ x [k]
(b) y(t) = x2(t)
Homework
◮ Exercises in Chapter 1.
◮ Use Matlab to draw elementary signals.