Chapter2. DISCRETETIME SIGNALS AND SYSTEMS. 2.0 Introduction 2.1 DiscreteTime Signals : Sequences 2.2 DiscreteTime Systems 2.3 Linear TimeInvariant Systems 2.4 Properties of Linear TimeInvariant Systems 2.5 Linear ConstantCoefficient Difference Equations
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Chapter2. DISCRETETIME SIGNALS AND SYSTEMS
2.0 Introduction
2.1 DiscreteTime Signals : Sequences
2.2 DiscreteTime Systems
2.3 Linear TimeInvariant Systems
2.4 Properties of Linear TimeInvariant Systems
2.5 Linear ConstantCoefficient Difference Equations
2.6 FrequencyDomain Representation
2.7 Representation of Sequences of the Fourier Transform
2.8 Symmetry Properties of the Fourier Transform
2.9 Fourier Transform Theorems
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2.1.Discrete  Time Signals
x[n]= x(t)t=nT
n : 1,0,1,2,…
T: sampling period
x(t) : analog signal
ii) unit step sequence
u[n] = 1,n0
0,n0
i) unit impulse signal(sequence)
d[n] = 1,n=0
0,n0
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iii) exponential/sinusoidal sequence
x[n]= Aej(won+), Acos(won+)
 not necessarily periodic in n with period 2p/wo
 periodic in nwithperiod N (discrete number)
for woN=2pk or wo= 2pk/N
[note] x(t)= Ae j(wo t+)
is periodic in t with period T= 2p/wo (continuous value)
iv) general expression
x[n] = Sx[k]d[nk]
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x[n]
y[n]
T[ ]
2.2DiscreteTime Systems
System : signal processor
i) memoryless or with memory
y[n] = f(x[n]), y[n]=f(x[nk]) with delay
ii) linearity
x1[n]y1[n]
x2[n]y2[n]
a1x1[n] + a2x2[n] a1y1[n] + a2 y2[n]
 e.g. T[a1x1[n] + a2x2[n]] = T[a1x1[n]] + T[a2x2[n]]
= a1T[x1[n]] + a2T[x2[n]]
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iii) timeinvariance
x[n] y[n] x[nn0] y[nn0]
 e.g., T[x[nn0]] = T[x[n]]  n nno
 e.g., d[n] h[n] d[nk] h[nk]
 counterexample : decimator T[ ] = x[Mn]
iv) causality
y[n] for n=n1, depends on x[n] for nn1 only
 counterexample : y[n] = x[n+1]  x[n]
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v) stability
bounded input yields bounded output(BIBO)
x[n] < for all n y[n] < for all n
 counterexample : y[n] = S u[k] = 0,n<0
n+1,n0
unbounded ( no fixed value By exists that keeps
y[n] By < .)
n
k=
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In general, let x[n] = Sx[k]d[nk]
In general, let x[n] = Sx[k]d[nk]
y[n] = T[ Sx[k]d[nk]]
k=
k
2.3 Linear TimeInvariant Systems
d[n]
h[n]
LTI
T[d[n]] : impulse response
y[n]
x[n]
( by linearity)
= Sx[k]T[d[nk]]
k
( by timeinvariance)
= Sx[k]h[nk]
coefficient
= x[n]*h[n]
= Sx[nr]h[r]
r=
Convolution!
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d[n] = 1,n=0
0,n0
Sh[k]x[nk]
k=0
In summary,
h[n]
x[n]
y[n]
y[n] = x[n]*h[n]
LTI
h[n] : unique characteristic of the LTI system
 causal LTI system
y[n] =
Sh[k]x[nk] =
k=
[note] h[n] = T[d[n]] = 0 n<0. as
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 Stable LTI System
S  x[nk]  •  h[k] 
Sh[k]x[nk] 
y[n] = 
k=
k=
S h[k] 
<
By
Bx
k=
S h[k] 
Therefore,
<
k=
In fact, this is necessary and sufficient condition for stability of a BIBO system.
( You prove it! )(*1)
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~
~
~
x[n]
y[n]
y[n]
 Example of nonLTI system  Decimator
x[n]
Decimator
M
y[n] = x[Mn]
?
= x[n1]
= y[n1] = x[M[n1]]
M=3
y[n] = x[Mn]
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~
y[n]
y[n]
y[n1]
No!
= x[Mn1]
x[M[n1]] = y[n1]
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2.4 Properties of LTI System
h[n]
x[n]
y[n] = x[n]*h[n]
LTI
i) parallel connection
h[n] = h1[n] + h2[n]
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ii) cascade connection
h[n] = h1[n]*h2[n]
=h2[n]* h1[n]
[note] distinctive feature of digital LTI system (*2)
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M
M
= Sbrx[nr]
= Sbrx[nr]
r = 0
r = 0
N
Saky[nk]
k = 0
2.5 Linear Difference Equations
LTI
x[n]
y[n]
i) Case 1 : N=0 FIR System
(set a0 =1, for convenience)
y[n]
For impulse input, x[n]=d[n], the response is
h[n]=0,n<0 or n>M
br0 n M finite impulse response!
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M
= Sbrx[nr]
r = 0
N
Saky[nk]
k = 1
ii) Case 2 : N 0 IIR System
(set a0 =1, for convenience)
y[n]
e.g., set N=1 (lst order), and a1 = a
y[n] = b0 x[n] + ay[n1]

For impulse input x[n] = d[n], the response is
1) If assume a causal system, i.e., y[n]=0 n<0.
y[0] = b0 d[0] + ay[1] = b0
y[1] = b0 d[1] + ay[0] = ab0
• • •
y[n] = b0 d[n] + ay[n1] = anb0
h[n] = anb0u[n]
infinite impulse response!
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2) If assume an anticausal system, i.e., y[n]=0 n>0.
y[n1] = a1(b0 d[n] + y[n])
y[0] = a1(b0 d[1] + y[1]) = 0
y[1] = a1(b0 d[0] + y[0]) = a1b0 h[n] = anb0u[n1]
• • •
y[n] = a1( b0 d[n+1] + y[n+1]) = anb0
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^
^
y = Ax=lx
y[n] = Sh[k] ejw(nk)
^
^
x
x
k = 
2.6 FrequencyDomain Representation
• Linear System
x
y = Ax
A
scalar, eigenvalue
for eigenvector input
• LTI System
x[n]
y[n]=x[n]*h[n]
h[n]
y[n]=ejwn*h[n]
ejwn
= H(ejw)ejwn
Fourier transform
( Sh[k] ejwk )ejwn =H(ejw) ejwn
=
k = 
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Sh[k] ejwk
k = 
p
h[k] = 1/2p H(ejw) ejwkdw
p
• Fourier Transform
H(ejw) =
You prove this! (*3)
• Condition for existence of FT
 X(ejw)  < S  x[n]  <
“ absolutely summable”
(BIBO stable condition)
• Real  imaginary
H(ejw) = HR(ejw) + j HI(ejw)
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• Magnitudephase
H(ejw) =  H(ejw) ej H(ejw)
(e.g.) ideal delay system
x[n]
y[n] = x[nnd]
h[n]
y[n] = ejw(nnd)= H(ejw) ejwn
H(ejw) = ejwnd
ejwn
HR(ejw) = coswnd HI(ejw) =  sinwnd
 H(ejw)  = 1 H(ejw) = wnd
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(e.g.) sinusoidal input
x[n] = Acos(w0n + j) = (A/2) ejjejw0n + (A/2) ejjejw0n
y[n] = H(ejw0) (A/2) ejjejw0n + H(ejw0) (A/2) ejjejw0n
= (A/2)(H(ejw0) ejjejw0n + H(ejw0) ejjejw0n)
y[n] = H(ejw) ejwn
<1>
= (A/2){ (H(ejw0) ejjejw0n ) + (H(ejw0) ejjejw0n)* }
= ARe{H(ejw0) ejjejw0n}
= A  H(ejw0) (cosw0n + j + q)
= A cos (w0(nnd) + j)
<2>
<3>
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= ( Sh*[n] ejw0n)* = H*(ejw0)
n
<1> if h[n] real
h[n] = hR[n]+jhI[n] = hR[n] = h*[n]
H(ejw0) =
Sh[n] e(jw0n)
n
<2> H(ejw) =  H(ejw) ejq
<3> if ideal delay system
with  H(ejw)  = 1, H(ejw) = q = wnd
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(e.g.) ideal lowpass filter (LPF)
x[n]
y[n]
1
wc
wc
Hl(ejw) = 1. ejwndw
0.elsewhere
wc
periodic with period 2p
Inputx[n] = Acos(w0n + j)
output y[n] = Acos(w0(nnd )+ j) , if
0 ,otherwise
wo
< wc
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sinw0[nnd ]
p[nnd ]
wc
hl [k] = ejwnd ejwndw
wc
(e.g.) Fourier transform of anu[n]a<1
X(ejw)
= Sanejwn = S (aejw)n
=
n = 0
n = 0
(e.g.) inverse Fourier transform of ideal LPF
=
 <n <
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 Data length vs. Spectrum Change
 Gibb’s phenomenon (page 52)
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 Error reduces in RMS sense
but not in Chebyshev sense.
Limitation of rectangular windowing
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2.8 Symmetry Properties (table 2.1)
i) even / odd
e : conjugate symmetric even
o : conjugate antisymmetric odd
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ii) real/imaginary
iii) conjugation/reversal
S
= ( S )*
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iv) real/imaginary  even/odd
v) for real x[n]
real even
imaginary odd
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(e.g.) x[n] = anu[n]a<1, real (example 2.25)
even
odd
even
odd
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Dashed line : a = 0.5Solid line : a = 0.9
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2.9 Fourier Transform Theorems (table 2.2)
i) linearity
ii) time shifting
iii) frequency shifting
iv) time reversal
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v) differentiation in frequency
vi) Parserval’s relation
vii) convolution relation
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viii) modulation/windowing relation
ix) fundamental functions
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1.
0.
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H.W. of Chapter 2
Matlab: [1] Consider the following discretetime systems characterized by the difference equations:
y[n]=0.5x[n]+0.27x[n1]+0.77x[n2]
Write a MATLAB program to compute the output of the above systems for an input x[n]=cos(20n/256)+cos(200n/256), with 0n<299 and plot the output.
[2] The Matlab command y=impz(num,den,N) can be used to compute the first N samples of the impulse response of the causal LTI discretetime system.
Compute the impulse response of the system described by
y[n]0.4y[n1]+0.75y[n2]
=2.2403x[n]+2.4908x[n1]+2.2403x[n2]
and plot the output using the stem function.
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H.W. of Chapter 2
Text: [3]215 [4]230 [5]242 [6]256 [7]258
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