Bài giảng Digital signal processing - Chapter 3: Discrete-time systems - Nguyễn Thanh Tuấn
Chapter 3
Discrete-Time Systems
Nguyen Thanh Tuan,M.Eng.
Department of Telecommunications (113B3)
Ho Chi Minh City University of Technology
Email: [email protected]om
Content
Input/output relationship of the systems
Linear time-invariant (LTI) systems
convolution
FIR and IIR filters
Causality and stability of the systems
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Discrete-Time Systems
1. Discrete-time signal
The discrete-time signal x(n) is obtained from sampling an analog
signal x(t), i.e., x(n)=x(nT) where T is the sampling period.
There are some representations of the discrete-time signal x(n):
x(n)
Graphical representation:
4
1 for n 1,3
Function:
1
1
1
x(n)
4 for n 2
0 elsewhere
-1
0
2 3
4
n
n
…
-2 -1 0
1
2
3
4
5
…
Table:
x(n) …
0
0
0
1
4
1
0
0
…
Sequence: x(n)=[… 0, 0, 1, 4, 1, 0, …]=[0, 1, 4, 1]
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Discrete-Time Systems
Some elementary discrete-time signals
Unit sample sequence (unit impulse):
1 for n 0
0 for n 0
(n)
Unit step signal
1 for n 0
0 for n 0
u(n)
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Discrete-Time Systems
2. Input/output rules
A discrete-time system is a processor that transform an input
sequence x(n) into an output sequence y(n).
Fig: Discrete-time system
Sample-by-sample processing:
that is,
and so on.
Block processing:
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Discrete-Time Systems
Basic building blocks of DSP systems
Constant multiplier
x(n)
y(n) ax(n)
y(n) x(n D)
(amplifier, scale)
x(n)
Delay
x2 (n)
Adder
x1(n)
y(n) x1(n) x2 (n)
y(n) x1(n)x2 (n)
(sum)
x2 (n)
x1(n)
Signal multiplier
(product)
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Discrete-Time Systems
Example 1
Let x(n)={1, 3, 2, 5}. Find the output and plot the graph for the
systems with input/out rules as follows:
a) y(n)=2x(n)
b) y(n)=x(n-4)
c) y(n)=x(n+4)
d) y(n)=x(n)+x(n-1)
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Discrete-Time Systems
Example 2
A weighted average system y(n)=2x(n)+4x(n-1)+5x(n-2). Given the
input signal x(n)=[x0,x1, x2, x3 ]
a) Find the output y(n) by sample-sample processing method?
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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Discrete-Time Systems
3. Linearity and time invariance
A linear system has the property that the output signal due to a
linear combination of two input signals can be obtained by forming
the same linear combination of the individual outputs.
Fig: Testing linearity
If y(n)=a1y1(n)+a2y2(n) a1, a2 linear system. Otherwise, the
system is nonlinear.
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Discrete-Time Systems
Example 3
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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Discrete-Time Systems
3. Linearity and time invariance
A time-invariant system is a system that its input-output
characteristics do not change with time.
Fig: Testing time invariance
If yD(n)=y(n-D) D time-invariant system. Otherwise, the
system is time-variant.
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Discrete-Time Systems
Example 4
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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Discrete-Time Systems
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 (n).
Fig: Impulse response of an LTI system
Fig: Delayed impulse responses of an LTI system
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Discrete-Time Systems
5. Convolution of LTI systems
Fig: Response to linear combination of inputs
Convolution:
y(n) x(m)h(nm) x(n)h(n)
(LTI form)
m
y(n) h(m)x(nm) h(n)x(n)
(direct form)
m
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Discrete-Time Systems
6. FIR versus IIR filters
A finite impulse response (FIR) filter has impulse response h(n)
that extend only over a finite time interval, say 0 n M.
Fig: FIR impulse response
M: filter order; Lh=M+1: the length of impulse response
h={h0, h1, …, hM} is referred by various name such as filter
coefficients, filter weights, or filter taps.
M
FIR filtering equation:
y(n) h(n) x(n) h(m)x(n m)
m0
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Discrete-Time Systems
Example 5
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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Discrete-Time Systems
6. FIR versus IIR filters
A infinite impulse response (IIR) filter has impulse response h(n)
of infinite duration, say 0 n .
Fig: IIR impulse response
IIR filtering equation:
y(n) h(n) x(n) h(m)x(n m)
m0
The I/O equation of IIR filters are expressed as the recursive
difference equation.
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Discrete-Time Systems
Example 6
Determine the output of the LTI system which has the impulse
response h(n)=anu(n), |a| 1 when the input is the unit step signal
x(n)=u(n) ?
rm rn1
n
rk
Remark:
1 r
km
rm
rk
When n= and|r| 1
1 r
km
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Discrete-Time Systems
Example 7
Assume the IIR filter has a casual h(n) defined by
2
for n 0
h(n)
4(0.5)n1 for n 1
a) Find the I/O difference equation ?
b) Find the difference equation for h(n)?
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Discrete-Time Systems
7. 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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Discrete-Time Systems
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