Bài giảng Digital signal processing - Chapter 3: Discrete-Time Systems - Nguyen Thanh Tuan

Click to edit Master subtitle style Nguyen Thanh Tuan, M.Eng. 
Department of Telecommunications (113B3) 
Ho Chi Minh City University of Technology 
Email: nttbk97@yahoo.com 
 Discrete-Time Systems 
 Chapter 3 
Digital Signal Processing 
Content 
2 Discrete-Time Systems 
 Input/output relationship of the systems 
 Linear time-invariant (LTI) systems 
 FIR and IIR filters 
 Causality and stability of the systems 
 convolution 
Digital Signal Processing 
1. Discrete-time signal 
3 
 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): 
 Graphical representation: 
 Function: 
 Table: 
 Sequence: x(n)=[ 0, 0, 1, 4, 1, 0, ]=[0, 1, 4, 1] 
1 1,3
( ) 4 2
0
for n
x n for n
elsewhere
n  -2 -1 0 1 2 3 4 5  
x(n)  0 0 0 1 4 1 0 0  
Discrete-Time Systems 
n 
x(n) 
1 2 3 -1 0 4 
4 
1 1 
Digital Signal Processing 
Some elementary discrete-time signals 
4 
 Unit sample sequence (unit impulse): 
 Unit step signal 
1 0
( )
0 0
for n
n
for n

1 0
( )
0 0
for n
u n
for n
Discrete-Time Systems 
Digital Signal Processing 
2. Input/output rules 
5 
 A discrete-time system is a processor that transform an input 
sequence x(n) into an output sequence y(n). 
 Sample-by-sample processing: 
Discrete-Time Systems 
Fig: Discrete-time system 
that is, and so on. 
 Block processing: 
Digital Signal Processing 
Basic building blocks of DSP systems 
6 
 Adder 
 (sum) 
Discrete-Time Systems 
 Constant multiplier 
 (amplifier, scale) 
)(nx )()( naxny 
)(1 nx
)(2 nx
)()()( 21 nxnxny 
)(nx )()( Dnxny  Delay 
 Signal multiplier 
 (product) 
)(1 nx
)(2 nx
)()()( 21 nxnxny 
Digital Signal Processing 
Example 1 
7 Discrete-Time Systems 
 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) 
Digital Signal Processing 
Example 2 
8 Discrete-Time Systems 
 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 ? 
Digital Signal Processing 
3. Linearity and time invariance 
9 
 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. 
Discrete-Time Systems 
Fig: Testing linearity 
 If y(n)=a1y1(n)+a2y2(n)  a1, a2 linear system. Otherwise, the 
system is nonlinear. 
Digital Signal Processing 
Example 3 
10 
 Test the linearity of the following discrete-time systems: 
Discrete-Time Systems 
 a) y(n)=nx(n) 
 b) y(n)=x(n2) 
 c) y(n)=x2(n) 
 d) y(n)=Ax(n)+B 
Digital Signal Processing 
3. Linearity and time invariance 
11 
 A time-invariant system is a system that its input-output 
characteristics do not change with time. 
Discrete-Time Systems 
Fig: Testing time invariance 
 If yD(n)=y(n-D)  D time-invariant system. Otherwise, the 
system is time-variant. 
Digital Signal Processing 
Example 4 
12 
 Test the time-invariance of the following discrete-time systems: 
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) 
Digital Signal Processing 
4. Impulse response 
13 
 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). 
Discrete-Time Systems 
Fig: Impulse response of an LTI system 
Fig: Delayed impulse responses of an LTI system 
Digital Signal Processing 
5. Convolution of LTI systems 
14 Discrete-Time Systems 
Fig: Response to linear combination of inputs 
 (LTI form) )()()()()( nhnxmnhmxny
m
 
)()()()()( nxnhmnxmhny
m
  (direct form) 
 Convolution: 
Digital Signal Processing 
6. FIR versus IIR filters 
15 
 A finite impulse response (FIR) filter has impulse response h(n) 
that extend only over a finite time interval, say 0 n M. 
Discrete-Time Systems 
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
m
mnxmhnxnhny
0
)()()()()( FIR filtering equation: 
Digital Signal Processing 
Example 5 
16 
 The third-order FIR filter has the impulse response h=[1, 2, 1, -1] 
Discrete-Time Systems 
 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) ? 
Digital Signal Processing 
6. FIR versus IIR filters 
17 
 A infinite impulse response (IIR) filter has impulse response h(n) 
of infinite duration, say 0 n . 
Discrete-Time Systems 
Fig: IIR impulse response 

0
)()()()()(
m
mnxmhnxnhny IIR filtering equation: 
 The I/O equation of IIR filters are expressed as the recursive 
difference equation. 
Digital Signal Processing 
Example 6 
18 
 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) ? 
Discrete-Time Systems 
 Remark: 
r
rr
r
nmn
mk
k

1
1
When n= and|r| 1 
r
r
r
m
mk
k
 
 1
Digital Signal Processing 
Example 7 
19 
 Assume the IIR filter has a casual h(n) defined by 
Discrete-Time Systems 
 a) Find the I/O difference equation ? 
 1)5.0(4
02
)(
1 nfor
nfor
nh
n
 b) Find the difference equation for h(n)? 
Digital Signal Processing 
7. Causality and Stability 
20 
 LTI systems can also classified in terms of causality depending on 
whether h(n) is casual, anticausal or mixed. 
Discrete-Time Systems 
Fig: Causal, anticausal, and mixed signals 
 A system is stable (BIBO) if bounded inputs (|x(n)| A) always 
generate bounded outputs (|y(n)| B). 
 A LTI system is stable 
 n
nh |)(|
Digital Signal Processing 
Example 8 
21 
 Consider the causality and stability of the following systems: 
Discrete-Time Systems 
 a) h(n)=(0.5)nu(n) 
 b) h(n)=(-0.5)nu(-n-1) 
Digital Signal Processing 
8. Static versus Dynamic systems 
22 
 Static (memoryless): output at any instant depends at most on the 
input sample at the same time, but not on past or future samples of 
the inputs. 
Otherwise, the system is dynamic. 
 Finite memory 
 Infinite memory 
Discrete-Time Systems 
Digital Signal Processing 
9. Interconnection of discrete time systems 
23 
 Cascade (series): 
 LTI systems: 
 Parallel: 
Discrete-Time Systems 
Digital Signal Processing 
10. Energy versus Power signals 
24 
 Energy: 
 Power: 
Discrete-Time Systems 
Digital Signal Processing 
11. Periodic versus Aperiodic signals 
25 
 Periodic: 
Otherwise, the signal is nonperiodic or aperiodic. 
Discrete-Time Systems 
Digital Signal Processing 
12. Symmetric versus Antisymmetric signals 
26 
 Symmetric (even): 
 Antisymmetric (odd): 
Discrete-Time Systems 
Digital Signal Processing 
13. Crosscorrelation and Autocorrelation 
27 
 Crosscorrelation: 
 Autocorrelation: 
Discrete-Time Systems 
Digital Signal Processing 
Example 9 
28 Discrete-Time Systems 
Digital Signal Processing 
Homework 1 
29 Discrete-Time Systems 
Digital Signal Processing 
Homework 2 
30 Discrete-Time Systems 
Digital Signal Processing 
Homework 3 
31 Discrete-Time Systems 
Digital Signal Processing 
Homework 4 
32 Discrete-Time Systems 
Digital Signal Processing 
Homework 5 
33 Discrete-Time Systems 
Digital Signal Processing 
Homework 6 
34 Discrete-Time Systems 
Digital Signal Processing 
Homework 7 
35 Discrete-Time Systems 
Digital Signal Processing 
Homework 8 
36 Discrete-Time Systems 
Digital Signal Processing 
Homework 9 
37 Discrete-Time Systems 
Digital Signal Processing 
Homework 10 
38 Discrete-Time Systems 
Digital Signal Processing 
Homework 11 
39 
Cho hệ thống rời rạc tuyến tính bất biến có đáp ứng xung h(n)={0↑, 
@, -1}. 
a) Xác định phương trình sai phân vào-ra của hệ thống trên. 
b) Vẽ 1 sơ đồ khối thực hiện hệ thống trên. 
c) Tìm giá trị của mẫu tín hiệu ngõ ra y(n = 1) khi tín hiệu ngõ vào 
x(n) = {1, 0↑, -1}. 
d) Tìm giá trị của mẫu tín hiệu ngõ ra y(n = 2) khi tín hiệu ngõ vào 
x(n) = δ(n) – δ(n–2). 
e) Tìm giá trị của mẫu tín hiệu ngõ ra y(n = 3) khi tín hiệu ngõ vào 
x(n) = u(n) – u(n–3). 
f) Tìm giá trị của mẫu tín hiệu ngõ ra y(n = 4) khi tín hiệu ngõ vào 
x(n) = u(n+4) – u(n–4). 
g) Tìm giá trị của mẫu tín hiệu ngõ ra y(n = 5) khi tín hiệu ngõ vào 
x(n) = u(–n) – u(–n–5). 
Discrete-Time Systems 
Digital Signal Processing 
Homework 12 
40 
Cho hệ thống rời rạc có phương trình sai phân vào-ra y(n) = 2x(n) – 
3x(n–3). 
a) Tìm đáp ứng xung của hệ thống trên. 
b) Tìm các giá trị của tín hiệu ngõ ra khi tín hiệu ngõ vào x(n) = 
δ(n+@) + 2δ(n – 2). 
c) Tìm 5 giá trị (n=0,1,2,3,4) của tín hiệu ngõ ra khi tín hiệu ngõ vào 
x(n) = u(n). 
d) Tìm 5 giá trị (n=0,1,2,3,4) của tín hiệu ngõ ra khi tín hiệu ngõ vào 
x(n) = u(– n). 
e) Tìm 5 giá trị (n=0,1,2,3,4) của tín hiệu ngõ ra khi tín hiệu ngõ vào 
x(n) = u(2 – n). 
f) Tìm 5 giá trị (n=0,1,2,3,4) của tín hiệu ngõ ra khi tín hiệu ngõ vào 
x(n) = u(n – 2). 
Discrete-Time Systems 
Digital Signal Processing 
Homework 13 
41 
Cho hệ thống rời rạc tuyến tính bất biến nhân quả có phương trình sai 
phân vào-ra y(n) = 2x(n–2) – y(n–1). 
a) Vẽ 1 sơ đồ khối thực hiện hệ thống trên với số bộ trễ là ít nhất có 
thể. 
b) Tìm giá trị của đáp ứng xung h(n = @). 
c) Tìm giá trị mẫu ngõ ra y(n = @) khi ngõ vào x(n) = 2δ(n). 
d) Tìm giá trị mẫu ngõ ra y(n = @) khi ngõ vào x(n) = δ(n–2). 
e) Tìm giá trị mẫu ngõ ra y(n = @) khi ngõ vào x(n) = δ(n)–δ(n–2). 
f) Tìm giá trị mẫu ngõ ra y(n = @) khi ngõ vào x(n) = u(n)–u(n-2). 
g) Tìm giá trị mẫu ngõ ra y(n = @) khi ngõ vào x(n) = u(n). 
h) Tìm giá trị mẫu ngõ ra y(n = @) khi ngõ vào x(n) = u(–n). 
i) Tìm giá trị mẫu ngõ ra y(n = @) khi ngõ vào x(n) = u(–n–1). 
j) Tìm giá trị mẫu ngõ ra y(n = @) khi ngõ vào x(n) = 1. 
Discrete-Time Systems 
Digital Signal Processing 
Homework 14 
42 
Kiểm tra tính chất tuyến tính, bất biến, nhân quả, ổn định, tĩnh của các 
hệ thống rời rạc sau: 
1) y(n) = x(n) + 2. 
2) y(n) = 2 – x(n). 
3) y(n) = x(2 – n). 
4) y(n) = x2(n). 
5) y(n) = x(n2). 
6) y(n) = x(2n). 
7) y(n) = x(2n + 1). 
8) y(n) = nx(n). 
9) y(n) = x(2|n|). 
10) y(n) = 2x(n). 
11) y(n) = 2nx(n). 
12) y(n) = 2-nx(n). 
Discrete-Time Systems 
Digital Signal Processing 
Homework 15 
43 
Kiểm tra tính chất tuyến tính, bất biến, nhân quả, ổn định, tĩnh của các 
hệ thống rời rạc sau: 
1) y(n) = cos{x(n)}. 
2) y(n) = cos{x(2n)}. 
3) y(n) = cos{x2(n)}. 
4) y(n) = cos2{x(n)}. 
5) y(n) = cos(n)x(n). 
6) y(n) = cos{nx(n)}. 
7) y(n) = cos(n) + x(n). 
8) y(n) = x(n) + 2x(n – 3) – 3x(n + 2). 
9) y(n) = 2x(n) + y(n – 1). 
10) y(n) = x(n) + 2y(n – 1). 
11) y(n) = x(n) + y(n – 1)/2. 
12) y(n) = y(n – 1) – y(n – 2). 
Discrete-Time Systems 
Digital Signal Processing 
Homework 16 
44 
Xác định và vẽ tín hiệu ngõ ra tương ứng với tín hiệu ngõ vào x(n) = 
{– @, 0, 1, 2, 3} của các hệ thống rời rạc sau: 
1) y(n) = nx(n). 
2) y(n) = x(n – 2). 
3) y(n) = x(n + 2). 
4) y(n) = x(n) + 2. 
5) y(n) = x(2n). 
6) y(n) = x(2n – 1). 
7) y(n) = x(– n). 
8) y(n) = x(2 – n). 
9) y(n) = x2(n). 
10) y(n) = x(n) + x(n + 2). 
11) y(n) = x(n) – x(n – 2). 
12) y(n) = x(n) + x(– n). 
Discrete-Time Systems 
Digital Signal Processing 
Homework 17 
45 
Xác định và vẽ tín hiệu ngõ ra tương ứng với tín hiệu ngõ vào x(n) = 
{0, 4, 5, @} của các hệ thống rời rạc sau: 
1) y(n) = nx(n). 
2) y(n) = x(n – 2). 
3) y(n) = x(n + 2). 
4) y(n) = x(n) + 2. 
5) y(n) = x(2n). 
6) y(n) = x(2n – 1). 
7) y(n) = x(– n). 
8) y(n) = x(2 – n). 
9) y(n) = x2(n). 
10) y(n) = x(n) + x(n + 2). 
11) y(n) = x(n) – x(n – 2). 
12) y(n) = x(n) + x(–n). 
Discrete-Time Systems 
Digital Signal Processing 
Homework 18 
46 
Xác định và vẽ tín hiệu ngõ ra tương ứng với tín hiệu ngõ vào x(n) = 
{– @, 0, 1, 2, 3, 4, 5, @} của các hệ thống rời rạc sau: 
1) y(n) = nx(n). 
2) y(n) = x(n – 2). 
3) y(n) = x(n + 2). 
4) y(n) = x(n) + 2. 
5) y(n) = x(2n). 
6) y(n) = x(2n – 1). 
7) y(n) = x(– n). 
8) y(n) = x(2 – n). 
9) y(n) = x2(n). 
10) y(n) = x(n) + x(n + 2). 
11) y(n) = x(n) – x(n – 2). 
12) y(n) = x(n) + x(– n). 
Discrete-Time Systems 
Digital Signal Processing 
Homework 19 
47 
Xác định và vẽ tín hiệu ngõ ra tương ứng với tín hiệu ngõ vào x(n) = 
@δ(n) + 2δ(n – 2) – 3δ(n + 3) của các hệ thống rời rạc sau: 
1) y(n) = nx(n). 
2) y(n) = x(n – 2). 
3) y(n) = x(n + 2). 
4) y(n) = x(n) + 2. 
5) y(n) = x(2n). 
6) y(n) = x(2n – 1). 
7) y(n) = x(– n). 
8) y(n) = x(2 – n). 
9) y(n) = x2(n). 
10) y(n) = x(n) + x(n + 2). 
11) y(n) = x(n) – x(n – 2). 
12) y(n) = x(n) + x(–n). 
Discrete-Time Systems 
Digital Signal Processing 
Homework 20 
48 
Vẽ sơ đồ khối thực hiện các hệ thống rời rạc sau: 
1) y(n) = x(n) + 2x(n – 1) – 3x(n – 3). 
2) y(n) = 2x(n – 1) + y(n – 1). 
3) y(n) = x(n – 1) + 2y(n – 1). 
4) y(n) = x(n – 1) + y(n – 1)/2. 
5) y(n) = y(n – 1) – y(n – 2). 
6) y(n) = x(n – 1) – y(n – 2). 
7) y(n) = x(n – 2) – y(n – 2). 
8) y(n) = x(n – 2) – y(n – 1). 
9) y(n) = 2x(n) – y(n – 2). 
10) y(n) = 0.5{2x(n) – y(n – 2)}. 
11) y(n) = x(n) + 2x(n – 1) – 3y(n – 2). 
12) y(n) = x(n) + 2x(n – 2) – 3y(n – 2). 
Discrete-Time Systems 
Digital Signal Processing 
Homework 21 
49 
Vẽ dạng sóng của các tín hiệu rời rạc sau: 
1) x(n) = δ(n) – δ(n – 2). 
2) x(n) = 2δ(n – 2) – δ(n + 2). 
3) x(n) = u(n) – u(n – 2). 
4) x(n) = u(–n). 
5) x(n) = u(2 – n). 
6) x(n) = u(2 + n). 
7) x(n) = u(n) + u(–n). 
8) x(n) = u(– n) – u(–n – 1). 
9) x(n) = nu(n). 
10) x(n) = nu(–n – 1). 
11) x(n) = u(n) – 1. 
12) x(n) = 1 – u(–n – 1). 
Discrete-Time Systems