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# Bài giảng Xử lý tín hiệu số: Chapter 3 - Hà Hoàng Kha

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Chapter
p 3
Discrete-Time Systems
Ha Hoang Kha, Ph.D.Click to edit Master subtitle style
Ho Chi Minh City University of Technology
@
Email: hhkha@hcmut.edu.vn

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Content

 Input/output
I
t/ t t relationship
l ti hi off the
th systems
t
 Linear time-invariant
time invariant (LTI) systems
 convolution

 FIR andd IIR filters
fil
 Causality
C
li and
d stability

bili off the
h systems

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1. Discrete-time signal
 The discrete-time signal x(n) is obtained from sampling an analog
signal x(t),
(t) i.e.,
i e x(n)=x(nT)
(n)= (nT) where
here T is the sampling period.
period
 There are some representations of the discrete-time signal x(n):
x(n)

 Graphical representation:
 Function:

 Table:
T bl

⎧1

x ( n) = ⎨ 4
⎪0

n

x(n) …

for n = 1,3
for n = 2

4

1
‐1
1

elsewhere
l
h

1

0 1 2 3 4

n

‐2

‐1

0

1

2

3

4

5

0

0

0

1

4

1

0

0

 Sequence:
q
x(n)=[…
( ) [ 0,, 0,, 1,, 4,, 1,, 0,, …]=[0,
] [ , 1,, 4,, 1]]
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Some elementary of discrete-time signals
 Unit sample sequence (unit impulse):
⎧1
δ ( n) = ⎨
⎩0

for n = 0
for n ≠ 0

 Unit step signal
⎧1

u ( n) = ⎨
⎩0

f n≥0
for
for n < 0

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2. Input/output rules
 A discrete-time system is a processor that transform an input
seq ence x(n)
sequence
(n) into an output
o tp t sequence
seq ence y(n).
(n)

Fig: Discrete-time system

 Sample-by-sample
Sample by sample processing:
that is,

and so on.

 Block processing:

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Basic building blocks of DSP systems
 Constant multiplier
p
 Delay
D l

y (n) = ax(n)

x((n)

y ( n) = x ( n − D )

x(n
( )
x2 ((n
n)

dde

y (n) = x1 (n) + x2 (n)

x1 (n)
x2 ( n )

 Signal multiplier

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x1 (n)

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y (n) = x1 (n) x2 (n)

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Example
 Let x(n)={1,
( ) { , 3,, 2,, 5}.
} Find the output
p and plot
p the graph

g p for the
systems with input/out rules as follows:
y( )
( )
a)) y(n)=2x(n)
b) y(n)=x(n-4)
c) y(n)=x(n)+x(n-1)
y(n)=x(n)+x(n 1)

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Example
 A weighted
g
average
g system
y
y(n)=2x(n)+4x(n-1)+5x(n-2).
y( )
( )
( )
( ) Given the
input signal x(n)=[x0,x1, x2, x4 ]

p y(n)
y( ) byy sample-sample
p
p p
processingg method?
a)) Find the output
b) Find the output y(n) by block processing method.
c) Plot the block diagram to implement this system from basic
building blocks ?

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3. Linearity and time invariance
 A linear system has the property that the output signal due to a
linear combination of ttwo
o inp
inputt signals can be obtained b
by forming
the same linear combination of the individual outputs.

Fig: Testing linearity
 If y(n)=a1y1(n)+a2y2(n) ∀ a1, a2 Æ linear system. Otherwise, the
system is nonlinear.

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Example
 Test the linearity of the following discrete-time systems:
a) y(n)=nx(n)
b) y(n)=x(n2)
c) y(n)=x2(n)
d) y(n)=Ax(n)+B

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3. Linearity and time invariance
 A time-invariant system is a system that its input-output
characteristics do not change with
ith time.
time

Fig:
g Testingg time invariance
 If yD(n)=y(n-D) ∀ DÆ time-invariant system. Otherwise, the
system is time-variant.

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Example
 Test the time-invariance of the following discrete-time systems:
a) y(n)=x(n)-x(n-1)
b) y(n)=nx(n)
c) y(n)=x(-n)
d) y(n)=x(2n)

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4. Impulse response
 Linear time-invariant (LTI) systems are characterized uniquely by
their impulse response sequence h(n), which is defined as the
response of the systems to a unit impulse δ(n).

Fig: Impulse response of an LTI system

Fig:
i Delayed
D l d impulse
i
l responses off an LTI
T system
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5. Convolution of LTI systems

Fig: Response to linear combination of inputs
 Convolution:

y (n) = ∑ x(m)h(n − m) = x(n) ∗ h(n) (LTI form)
m

y (n) = ∑ h(m) x(n − m) = h(n) ∗ x(n) (direct form)
m

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5. FIR and IIR filters
 A finite impulse response (FIR) filter has impulse response h(n)
that extend only over a finite time interval,
interval say 0 ≤n ≤ M.
M

Fi FIR impulse
Fig:
i
l response
 M: filter order; Lh=M+1: the length
g of impulse
p
response
p
 h={h0, h1, …, hM} is referred by various name such as filter
coefficients, filter weights, or filter taps.

 FIR filtering equation: y (n) = h(n) ∗ x(n) =

M

∑ h ( m) x ( n − m)

m =0
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Example
 The third-order FIR filter has the impulse response h=[1, 2, 1, -1]
a) Find the I/O equation, i.e., the relationship of the input x(n) and the
output y(n) ?
b) Given x=[1, 2, 3, 1], find the output y(n) ?

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5. FIR and IIR filters
 A infinite impulse response (IIR) filter has impulse response h(n)
of infinite duration,
duration say 0 ≤n ≤ ∞.

Fi IIR impulse
Fig:
i
l response
 IIR filtering equation: y (n) = h( n) ∗ x(n) =

∑ h ( m) x ( n − m)

m =0

 The I/O equation of IIR filters are expressed as the recursive
difference equation.
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Example
 Determine the output of the LTI system which has the impulse
r p n h(n)=
response
h(n)=anu(n),
(n) |a|≤
| |≤ 1 when
h n th
the inp
inputt is
i the
th unit
nit step
t p signal
i n l
x(n)=u(n) ?

 Remark:

m
n+
+1
1
r

r
k
r
=

1− r
k =m
n

 When n= ∞ and|r|≤ 1
Ha H. Kha

m
r
k
r
=

1− r
k =m
18

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Example
 Assume the IIR filter has a casual h(n) defined by

for n = 0
for n ≥ 1

⎧ 2
h( n) = ⎨
n −1
4
(
0
.
5
)

a)) Find
Fi d the
h I/O difference
diff
equation
i ?
b) Find the difference equation for h(n)?

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6. Causality and Stability

Fig: Causal, anticausal, and mixed signals
 LTI systems can also classified in terms of causality depending on
whether h(n) is casual, anticausal or mixed.
 A system is stable (BIBO) if bounded inputs (|x(n)| ≤A) always
generate bounded outputs (|y(n)| ≤B).
 A LTI system is stable ⇔

∑ | h( n) | < ∞

n = −∞
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Example
 Consider the causality and stability of the following systems:
a) h(n)=(0.5)nu(n)
b)) h(n)=-(0.5)
( ) ( )nu(-n-1)
(
)

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Homework
 Problems: 3.1, 3.2, 3.3, 3.4, 3.5, 3.6

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