Chapter 2 Discrete-time signals and systems

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Chapter 2 Discrete-time signals
and systems
2.1 Discrete-time signalssequences
2.2 Discrete-time system
2.3 Frequency-domain representation
of discrete-time signal and system
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2.1 Discrete-time signalssequences
2.1.1 Definition
2.1.2 Classification of sequence
2.1.3 Basic sequences
2.1.4 Period of sequence
2.1.5 Symmetry of sequence
2.1.6 Energy of sequence
2.1.7 The basic operations of sequences
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2.1.1 Definition
EXAMPLE
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Generate and plot the sequence in MATLAB
n-15
x1,2,1.2,0,-1,-2,-2.5
stem(n,x, '.')
n09
y0.9.n.cos(0.2pinpi/2)
stem(n,y,'.')
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Display the wav speech signal in ULTRAEDIT
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Display the wav speech signal in COOLEDIT
The whole waveform
Display the wav speech signal in
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2.1.2 Classification of sequence
Right-side
Left-side
Two-side
Finite-length
Causal
Noncausal
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2.1.3 Basic sequences
1. Unit sample sequence
2.The unit step sequence
3.The rectangular sequence
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 4.  Exponential sequence
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5. Sinusoidal sequence
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For convenience, sinusoidal signals are usually
expressed by exponential sequences.
The relationship between ? and O
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2.1.4 Period of sequence
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Three kinds of period of sequence
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2.1.5 Symmetry of sequence
Conjugate-symmetric sequence
Conjugate-antisymmetric sequence
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EXAMPLE
n-55 x0,0,0,0,0,1,2,3,4,5,6 xe(xfliplr
(x))/2 xo(x-fliplr(x))/2 subplot(3,1,1) stem
(n,x) subplot(3,1,2) stem(n,xe) subplot(3,1,3) ste
m(n,xo)
Real sequences can be decomposed into two
symmetrical sequences.
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EXAMPLE
Complex sequences can be decomposed into two
symmetrical sequences.
n-55 xzeros(1,11) x((ngt0)(nlt5))(1j).
05 xe(xconj(fliplr(x)))/2 xo(x-conj(fliplr
(x)))/2 subplot(3,2,1) stem(n,real(x)) subplot(3
,2,2) stem(n,imag(x)) subplot(3,2,3) stem(n,
real(xe)) subplot(3,2,4) stem(n,imag(xe)) subpl
ot(3,2,5) stem(n,real(xo)) subplot(3,2,6) st
em(n,imag(xo))
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2.1.6 Energy of sequence
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2.1.7 The basic operations of sequences
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Basic operations of sequences
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Original music sequence
Original speech sequences
sequences after scalar multiplication
sequences after vector addition
sequences after vector multiplication
echo
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The matlab codes on the processions
xwavread('test1.wav',36000) ywavread('test2.wav
',36000) z(xy)/2.0 wavwrite(z,22050,'test3.wa
v') y1y0.5 wavwrite(y1,22050,'test4.wav') y2
zeros(36000,1) for i200036000
y2(i)y(i-20001) end y30.6y0.4y2 wavwrite(
y3,22050,'test5.wav') w01/360001-1/36000' y
4y.w wavwrite(y4,22050,'test6.wav')
Vector addition realizes composition.
scalar multiplication changes the volume.
Delay, scalar multiplication and vector addition
produce echo.
vector multiplication realizes fade-in.
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The matlab codes on the addition of two sequences
EXAMPLE
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n-42 x1,-2,4,6,-5,8,10
x1nxn2 n1n-2 x1x x2nxn-4 n2n
4 x2x yn mmin(min(n1),min(n2))
max(max(n1),max(n2)) y1zeros(1,length(m))
y2y1 y1((mgtmin(n1))(mltmax(n1)))x1y2((mgt
min(n2))(mltmax(n2)))x2 y3y1y2 stem(m,y)
Outputy 3 -6 12 18 -15 24 31 -2 4 6
-5 8 10
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7.convolution sum
stepsturnover, shift, vector multiplication,
addition
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EXAMPLE
nx010 x0.5.nx nh-14 hones(1,length(nh))
yconv(x,h) stem(min(nx)min(nh)max(nx)max(nh
),y)
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8.crosscorrelation
aotocorrelation
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examplecorrelation detection in digital audio
watermark
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2.1 summary

• 2.1.1 Definition
• 2.1.2 Classification of sequence
• 2.1.3 Basic sequences
• 2.1.4 Period of sequence
• 2.1.5 Symmetry of sequence
• 2.1.6 Energy of sequence
• 2.1.7 The basic operations of sequences

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requirementsjudge the period of sequence
calculate convolution with graphical
and analytical
evaluation .
key convolution
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2.2 Discrete-time system
2.2.1 Definitioninput-output description of
systems
2.2.2 Classification of discrete-time system
2.2.3 Linear time-invariant system(LTI)
2.2.4 Linear constant-coefficient difference
equation
2.2.5. Direct implementation of discrete-time
system
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2.2.1 definitioninput-output description of
systems
the impulse response
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EXAMPLE
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2.2.2 classification of discrete-time system
1.Memoryless (static) system the output
depends only on the current input.
2.Linear system
3.Time-invariant system
4.Causal system the output does not depend
on the latter input.
5.Stable system
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2.2.3 linear time-invariant system(LTI)
How to get hn from the input and output
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the impulse response in LTI
EXAMPLE
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Properties of LTI
Figure 2.12
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classification of linear time-invariant system
IIR hns length is infinite
the latter input the former
input
FIR must be stable?
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2.2.4 linear constant-coefficient difference
equation
1.relation with input-output description and
convolution
For IIR,the latter two are consistent.
EXAMPLE
input-output description
convolution description infinite
items,unrealizable
difference equation description Finite items,
realizable
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EXAMPLE
For FIR,the followings are consistent
input-output description and difference equation
description (non-recursion)
Convolution description
Another difference equation description,recursion,
lower rank
For FIR and IIR,difference equations are not
exclusive.
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2.Recursive computation of difference equations
For IIR, there needs N initial conditions , then
,the solution is unique. For FIR, there needs
no initial conditions. With initial-rest
conditions (linear, time invariant, and causal),
the solution is unique.
EXAMPLE
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3.computation of difference equations with
homogeneous and particular solution
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2.2.5. Direct implementation of discrete-time
system
EXAMPLE
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EXAMPLE
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EXAMPLE
The matlab codes on the direct realization of LTI
B1 A1,-1 n0100 xngt0
yfilter(B,A,x) stem(n,y) axis(0,20,0,20)
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2.2 summary
2.2.1 Definitioninput-output description of
systems
2.2.2 Classification of discrete-time system
2.2.3 Linear time-invariant system(LTI)
2.2.4 Linear constant-coefficient difference
equation
2.2.5. Direct implementation of discrete-time
system
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keys judge the type of a system(from the
relationship between the input and output, and
from hn for LTI). the physics meaning of
convolution representation for LTI the output
signals are the weighted combination of the input
signals, hn is the weight? the similarities
and differences between linear constant-coefficien
t difference equations and convolution
representation,recursive computation? the
difference between IIR and FIR FIR
IIR hn finite length
infinite length yn?xn??? finite items
infinite items
realization convolution or
difference difference , recursion
stability stable maybe
stable
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2.3 frequency-domain representation of
discrete-time signal and system
2.3.1 definition of fourier transform
2.3.2 frequency response of system
2.3.3 properties of fourier transform
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EXAMPLE
The intuitionistic meaning of frequency-domain
representation of signals
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The intuitionistic meaning of frequency-domain
representation of systems
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The effect of lowpass and highpass filters to
image signals
EXAMPLE
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Frequency-domain analysis of de-noise process
through bandstop filter
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Derivation of Fourier transform
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2.3.1 definition of fourier transform
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EXAMPLE
subplot(2,2,1) fplot('real(1/(1-0.2exp(-1jw)))
',-2pi,2pi) title('??') subplot(2,2,2) fplot
('imag(1/(1-0.2exp(-1jw)))',-2pi
,2pi) title('??') subplot(2,2,3) fplot('abs(1/
(1-0.2exp(-1jw)))',-2pi,2pi) title('??') s
ubplot(2,2,4) fplot('angle(1/(1-0.2exp(-1jw)))
',-2pi,2pi) title('??')
Matlab codes to draw the frequency chart of
signals
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Fourier transforms of non-absolutely summable or
non-square summable signals
EXAMPLE

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2.3.2 frequency response of system
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Ideal filter in frequency and time domain
EXAMPLE
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Matlab codes to draw the frequency response of a
system
EXAMPLE
h1,0,0,0,0,0,0,0,0,0.5 freqz(h,1)
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Eigenfunction and steady-state response
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causal FIR system acts on causal signal
Causal and stable IIR system acts on causal signal
Figure 2.20
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example of steady-state response
Sin(0.1pin)
hn1,1,1,1,1,1,1,1,1,1/4
B1,0,1,0,1A1,0.81,0.81,0.81
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2.3.3 properties of fourier transform
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2.3 summary
2.3.1 definition of fourier transform
2.3.2 frequency response of system
2.3.3 properties of fourier transform
requirementscalculation of fourier
transforms steady-state response linearity
time shifting frequency shifting the
convolution theorem windowing theorem Parsevals
theorem symmetry properties
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Keys and difficulties the convolution
theorem the frequency spectrum of a real
sequence is conjugate symmetric the frequency
spectrum of a conjugate symmetric sequence is a
real function.
exercises 2.35 2.45 2.57