Bài giảng Signals and Systems - Chapter: Linear time invariant systems - Đặng Quang Hiếu

ET2060 - Signals and Systems
Linear Time Invariant Systems
Dr. Quang Hieu Dang
Hanoi University of Science and Technology
School of Electronics and Telecommunications
Autumn 2012
Outline
Impulse response and convolution
Properties of convolution and systems’ interconnection
Representation of LTI systems
Convolution (1)
Consider a discrete-time LTI system T :
x [n]
T−→ y [n]; y [n] = T{x [n]}
Any input signal x [n] can be represented as
x [n] =
∞∑
k=−∞
x [k]δ[n − k]
By applying the linearity property, we have
y [n] =
∞∑
k=−∞
x [k]T{δ[n − k]}
Convolution (2)
Impulse response of the system: h[n] = T{δ[n]}
δ[n] h[n]
T
Applying time invariance property,
y [n] =
∞∑
k=−∞
x [k]h[n − k] := x [n] ∗ h[n]
Output signal y [n] is the convolution sum of input signal x [n] and
the system’s impulse response h[n].
Convolution calculation steps
y [n0] =
∞∑
k=−∞
x [k]h[n0 − k]
Use graph!
1. Folding (reflection): h[k]→ h[−k]
2. Time shifting: Shift h[−k] by n0 samples to obtain h[n0 − k],
when left / right?
3. Multiplication: vn0 [k] = x [k]h[n0 − k]
4. Summation: Sum all (non-zero) values of the sequence vn0 [k]
to obtain y [n0]
Convolution calculation illustration (1)
0 1 2 3 4 5 6-1-2-3-4
b b b
b b b b b
b b b k
x [k]
0 1 2 3 4 5 6-1-2-3-4
b b b b
b
b
b
b
b b b k
h[k]
0 1 2 3 4 5 6-1-2-3-4
b
b
b
b
b
b b b b b b k
h[−k]
0 1 2 3 4 5 6-1-2-3-4
b b b
b
b
b b b b b b k
v0[k]
y [0] = 0.75 + 1
0 1 2 3 4 5 6-1-2-3-4
b
b
b
b
b b b b b b b k
h[−1− k]
0 1 2 3 4 5 6-1-2-3-4
b b b
b
b b b b b b b k
v−1[k]
y [−1] = 1
0 1 2 3 4 5 6-1-2-3-4
b b
b
b
b
b
b b b b b k
h[1− k]
0 1 2 3 4 5 6-1-2-3-4
b b b
b
b
b
b b b b b k
v1[k]
y [1] = 0.5 + 0.75 + 1
Convolution calculation illustration (2)
0 1 2 3 4 5 6 7 8-1-2-3-4
b b b
b b b b b
b b b b b
n
x [n]
0 1 2 3 4 5 6 7 8-1-2-3-4
b b b b
b
b
b
b
b b b b b
n
h[n]
0 1 2 3 4 5 6 7 8-1-2-3-4
b b b
b
b
b
b b
b
b
b
b b
n
y [n]
Example
Given a system with impulse response
h[n] = rectN [n] := u[n]− u[n − N]
find output y [n] when input is
x [n] =
{
n+3
4 , −3 ≤ n ≤ 1
0, otherwise
Remarks:
◮ If x [n] is a finite length sequence with length L: x [n] = 0,
∀n /∈ [N1,N1 + L− 1], and h[n] is a finite length sequence
with length M: h[n] = 0, ∀n /∈ [N2,N2 +M − 1]. Determine
the length of y [n]?
◮ If x [n] or h[n] is shifted N samples, how does y(n) change?
◮ When h[n] = δ[n]?
◮ Matlab calculation of convolution?
Convolution for continuous-time signals (1)
Any input signal x(t) can be expressed as:
x(t) =
∫ ∞
−∞
x(τ)δ(t − τ)dτ
Let h(t) be the system’s impulse response, apply the linearity and
time invariance properties, we have:
y(t) =
∫ ∞
−∞
x(τ)h(t − τ)dτ := x(t) ∗ h(t)
Example: Given an electronic circuit with R and C connected in
series, where RC = 1[s]. Find voltage y(t) over capacitor C when
the circuit is charge with:
x(t) = u(t)− u(t − 2)
Hint: System’s impulse response is h(t) = e−tu(t)
Convolution for continuous-time signals (2)
1
2
τ
x(τ)
1
τ
h(τ)
1
τ
h(t0 − τ)
1
τ
vt0(τ)
y(t0)
1
2
t
y(t)
Outline
Impulse response and convolution
Properties of convolution and systems’ interconnection
Representation of LTI systems
Commutative
x [n] ∗ h[n] = h[n] ∗ x [n]
LTI systems:
x [n] y [n]
h[n]
h[n] y [n]
x [n]
Associative
(x [n] ∗ h1[n]) ∗ h2[n] = x [n] ∗ (h1[n] ∗ h2[n])
Cascade interconnection of LTI systems:
x [n] y [n]
h1[n] h2[n]
x [n] y [n]
h1[n] ∗ h2[n]
Distributive
x [n] ∗ (h1[n] + h2[n]) = (x [n] ∗ h1[n]) + (x [n] ∗ h2[n])
Parallel interconnection of LTI systems:
+
x [n] y [n]
h1[n]
h2[n]
x [n] y [n]
h1[n] + h2[n]
Memoryless LTI systems
y [n] = x [n] ∗ h[n]
Applying commutative property:
y [n] = h[n] ∗ x [n] =
∞∑
−∞
h[k]x [n − k]
Memoryless system: y [n] depends only on x [n], therefore:
h[k] = 0, ∀k 6= 0
or h[n] = Cδ[n], where C is a constant. The system will be:
y [n] = x [n] ∗ Cδ[n] = Cx [n]
Inversion of an LTI system
x [n] x [n]
h[n] h1[n]
Condition:
h[n] ∗ h1[n] = δ[n]
Example: Determine the inverted systems of following LTI
systems:
(a) h[n] = δ[n − n0]
(b) h[n] = u[n]
Causal LTI systems
Applying commutative property,
y [n] = · · ·+h[−2]x [n+2]+h[−1]x [n+1]+h[0]x [n]+h[1]x [n−1]+· · ·
Therefore, the system is causal if and only if
h[k] = 0, ∀k < 0
Causal signals: x [n] = 0, ∀n < 0.
Stable LTI systems
Necessary and sufficient condition:
∞∑
n=−∞
|h[n]| <∞
Proof of sufficient condition: Easy!
Proof of necessary condition: a→ b ≡ b¯ → a¯
◮ Show that if
∑∞
n=−∞ |h[n]| =∞, there exists at least one
bounded input that produces unbounded output.
◮ Choose a bounded input:
x [n] =
{
h∗[−n]
|h[−n]| h[n] 6= 0
0, h[n] = 0
◮ Output at n = 0?
Example: Check the stability of the system: h[n] = anu[n].
Step response of an LTI system
If input of an LTI system is a unit step sequence, then its output is
called the step response of the system
s[n] = u[n] ∗ h[n]
u[n] s[n]
h[n]
Applying commutative law,
s[n] = h[n] ∗ u[n] =
n∑
k=−∞
h[k]
Reversly, we have: h[n] = s[n]− s[n − 1]
Homework (1)
1. Derive properties of continuous-time LTI systems.
2. Chapter 2 exercises.
3. Write a Matlab program myconv to calculation convolution of
two discrete-time signals. Compare implementation speed
with Matlab built-in function conv using profile command.
4. Use Matlab to draw step response s[n] of an LTI system given
its impulse response h[n].
5. Write a Matlab program to calculate convolution of two
continuous-time signals. Is it possible to use myconv function?
Compare results in a same figure.
Outline
Impulse response and convolution
Properties of convolution and systems’ interconnection
Representation of LTI systems
Constant-coefficient differential equation
N∑
k=0
ak
dk
dtk
y(t) =
M∑
k=0
bk
dk
dtk
x(t)
◮ Find solution yh(t) of the homogeneous differential equation
N∑
k=0
ak
dk
dtk
y(t) = 0 =⇒ yh(t) =
N∑
i=0
cie
ri t
Characteristic equation:
∑N
k=0 akr
k = 0
◮ Find particular solution yp(t) (similar to x(t)).
◮ Find unknown coefficients by using initial conditions on
y(t) = yh(t) + yp(t).
Example: Consider an RC electronic circuit:
y(t) + RC ddt y(t) = x(t). Find y(t) (t > 0) when
x(t) = cos(ω0t)u(t) and y(0) = 2 [V], R = 1 [Ω], C = 1 [F].
Constant-coefficient difference equation
N∑
k=0
aky [n − k] =
M∑
k=0
bkx [n − k]
◮ Finite Impulse Response (FIR) systems: N = 0
◮ Infinite Impulse Response (IIR) systems: N > 0
Similarly, but:
◮ Solution of the homogeneous equation
yh[n] =
N∑
i=1
ci r
n
i
◮ Characteristic equation:
N∑
k=0
ak r
N−k = 0
LTI system implementation: Elementary operators
x [n]
a
ax [n] x(t)
a
ax(t)
x1[n]
x2[n]
x1[n] + x2[n]+ x1(t)
x2(t)
x1(t) + x2(t)+
x [n] x [n − 1]D x(t) dx(t)
dt
D
LTI system implemetation: Direct form I
y [n] = −
N∑
k=1
aky [n − k] +
M∑
r=0
brx [n − r ]
x [n] y [n]
b0
b1
b2
+
+
+
+
D
D
D
D
−a1
−a2
LTI system implemetation: Direct form II
b
b
b
x [n] y [n]
b0
b1
b2
+
+
+
+
D
D
−a1
−a2
Correlation
Similarity between two signals.
Cross-correlation:
rxy [n] =
∞∑
m=−∞
x [m]y [m − n]
Auto-correlation:
rxx [n] =
∞∑
m=−∞
x [m]x [m − n]
How to calculate?
Matlab?
Application: Radar (1)
Transmit a signal via Gaussian channel (white noise only) with an
unknown delay τ .
0 1 2
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
Time [sec]
Am
pl
itu
de
Transmitted waveform
0 1 2
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
Time [sec]
Am
pl
itu
de
Received waveform
Application: Radar (2)
Find the maximum value of the cross-correlation function and the
corresponding time instant τˆ = 0.88 [s] (SNR = 20 dB). Compare
with the true delay τ .
−3 −2 −1 0 1 2 3
−0.5
0
0.5
1
delay [sec]
xc
o
rr
Cross correlation
Cross−correlation
True delay
Homework (2)
1. Solve constant-coefficient difference equation in Matlab.
2. Write the illustrated radar application in Matlab.