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
BGL/SNU
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
BGL/SNU
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]
BGL/SNU
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]]
BGL/SNU
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]
BGL/SNU
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=
BGL/SNU
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!
BGL/SNU
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
BGL/SNU
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)
BGL/SNU
~
~
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]
BGL/SNU
h[n]
x[n]
y[n] = x[n]*h[n]
LTI
i) parallel connection
h[n] = h1[n] + h2[n]
BGL/SNU
h[n] = h1[n]*h2[n]
=h2[n]* h1[n]
[note] distinctive feature of digital LTI system (*2)
BGL/SNU
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
br 0 n M finite impulse response!
BGL/SNU
= 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!
BGL/SNU
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
BGL/SNU
^
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 = 
BGL/SNU
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)
BGL/SNU
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
BGL/SNU
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>
BGL/SNU
= ( 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
BGL/SNU
(e.g.) ideal lowpass filter (LPF)
x[n]
y[n]
1
wc
wc
Hl(ejw) = 1. ejwndw
0. elsewhere
wc
periodic with period 2p
Input x[n] = Acos(w0n + j)
output y[n] = Acos(w0(nnd )+ j) , if
0 , otherwise
wo
< wc
BGL/SNU
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 <
BGL/SNU
but not in Chebyshev sense.
Limitation of rectangular windowing
BGL/SNU
2.8 Symmetry Properties (table 2.1)
i) even / odd
e : conjugate symmetric even
o : conjugate antisymmetric odd
BGL/SNU
2.9 Fourier Transform Theorems (table 2.2)
i) linearity
ii) time shifting
iii) frequency shifting
iv) time reversal
BGL/SNU
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.
BGL/SNU