Chapter 2 Discrete-Time Signals and Systems
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Transcript Chapter 2 Discrete-Time Signals and Systems
Biomedical Signal processing
Chapter 2 Discrete-Time Signals and
Systems
Zhongguo Liu
Biomedical Engineering
School of Control Science and Engineering, Shandong University
2015/4/13
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Chapter 2 Discrete-Time Signals and Systems
2.0 Introduction
2.1 Discrete-Time Signals: Sequences
2.2 Discrete-Time Systems
2.3 Linear Time-Invariant (LTI) Systems
2.4 Properties of LTI Systems
2.5 Linear Constant-Coefficient
Difference Equations
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Chapter 2 Discrete-Time Signals and Systems
2.6 Frequency-Domain Representation
of Discrete-Time Signals and systems
2.7 Representation of Sequences by
Fourier Transforms
2.8 Symmetry Properties of the
Fourier Transform
2.9 Fourier Transform Theorems
2.10 Discrete-Time Random Signals
2.11 Summary
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2.0 Introduction
Signal: something conveys information
Signals are represented mathematically as
functions of one or more independent variables.
Continuous-time (analog) signals, discretetime signals, digital signals
Signal-processing systems are classified along the
same lines as signals: Continuous-time (analog)
systems, discrete-time systems, digital systems
Discrete-time signal
Sampling a continuous-time signal
Generated directly by some discrete-time process
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2.1 Discrete-Time Signals: Sequences
Discrete-Time signals are represented as
x xn, n , n : integer
Cumbersome, so just use x n
In sampling,
xn xa nT , T : sampling period
1/T (reciprocal of T) : sampling frequency
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Figure 2.1 Graphical representation
of a discrete-time signal
Abscissa: continuous line
x n : is defined only at discrete instants
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x[n] xa (t ) |t nT xa (nT )
EXAMPLE
7
Sampling the analog waveform
Figure 2.2
Basic Sequence Operations
Sum of two sequences
x[n] y[n]
Product of two sequences
x[n] y[n]
Multiplication of a sequence by a numberα
x[n]
Delay (shift) of a sequence
y[n] x[n n0 ]
n0 : integer
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Basic sequences
Unit sample sequence
(discrete-time impulse,
impulse)
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0 n 0
n
1 n 0
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Basic sequences
A sum of scaled, delayed impulses
pn a3 n 3 a1 n 1 a2 n 2 a7 n 7
arbitrary
sequence
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x[n]
x[k ] [n k ]
k
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Basic sequences
Unit step sequence
u[n]
n
k
k
1 n 0
u[n]
0 n 0
n
0,
when
n
0
k 1, when n 0 ,
k
since k 0 k 0
1 k 0
u[n] [n] [n 1] [n 2] [n k ]
[n] u[n] u[n 1]
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k 0
First backward difference
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Basic Sequences
Exponential sequences
x[n] A
n
A and α are real: x[n] is real
A is positive and 0<α<1, x[n] is positive and
decrease with increasing n
-1<α<0, x[n] alternate in sign, but decrease
in magnitude with increasing n
1: x[n] grows in magnitude as n increases
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EX. 2.1 Combining Basic sequences
If we want an exponential sequences that is
zero for n <0, then
A n n 0
x[n]
n0
0
x[n] A u[n]
n
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Cumbersome
simpler
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Basic sequences
Sinusoidal sequence
x[n] A cosw0n
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for all n
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Exponential Sequences
A Ae
j
e
j
x[n] A A e e
n
n
jw0 n
jw0
A e
n
j w0 n
A cosw0 n j A sin w0 n
n
n
Exponentially weighted sinusoids
1
1
1
Exponentially growing envelope
Exponentially decreasing envelope
x[n] Ae
jw0 n
is refered to
Complex Exponential Sequences
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Frequency difference between
continuous-time and discrete-time
complex exponentials or sinusoids
x[n] Ae
j w0 2 n
Ae
jw0 n
e
j 2n
Ae
jw0 n
x[n] A cos w0 2 r n A cos w0 n
w0 : frequency of the complex sinusoid
or complex exponential
: phase
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Periodic Sequences
A periodic sequence with integer period N
x[n] x[n N ]
for all n
A cosw0n A cosw0n w0 N
w0 N 2 k , where k is integer
N 2 k / w0 , where k is integer
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EX. 2.2 Examples of Periodic Sequences
x1[n] cos n / 4
Suppose it is periodic sequence with period N
x1[n] x1[n N ]
cos n / 4 cos n N / 4
n / 4 2 k n / 4 N / 4, k : integer
N 2 k / ( / 4) 8 k
k 1, N 8 2 / w0
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EX. 2.2 Examples of Periodic Sequences
2
3
x
[
n
]
cos
3
n
/
8
1
8
8
Suppose it is periodic sequence with period N
x1[n] x1[n N ]
cos3 n / 8 cos3 n N / 8
3 n / 8 2 k 3 n / 8 3N / 8, k : integer
N 2 k / w0 2 k / (3 / 8) k 3, N 16
N 2 3/ w0 2 / w0 ( for continuous signal)
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EX. 2.2 Non-Periodic Sequences
x2 [n] cos n
Suppose it is periodic sequence with period N
x2 [n] x2 [n N ]
cos n cos(n N )
for n 2 k n N , k : integer,
there is no integer N
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High and Low Frequencies in Discrete-time signal
x[n] A cos(w0n)
(a) w0 = 0 or 2
(b) w0 = /8 or 15/8
(c) w0 = /4 or 7/4
(d) w0 =
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2.2 Discrete-Time System
Discrete-Time System is a trasformation
or operator that maps input sequence
x[n] into a unique y[n]
y[n]=T{x[n]}, x[n], y[n]: discrete-time
signal
x[n]
T{‧}
y[n]
Discrete-Time System
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EX. 2.3 The Ideal Delay System
y[n] x[n nd ], n
If nd is a positive integer: the delay of the
system. Shift the input sequence to the
right by nd samples to form the output .
If nd is a negative integer: the system will
shift the input to the left by n samples,
d
corresponding to a time advance.
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EX. 2.4 Moving Average
M2
1
y n
x n k
M 1 M 2 1 k M1
1
x n M 1 x n M 1 1 ... x n x n 1 ... x n M 2
M1 M 2 1
for n=7, M1=0, M2=5
dummy index m
x[m]
n-5
n
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m
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Properties of Discrete-time systems
2.2.1 Memoryless (memory) system
Memoryless systems:
the output y[n] at every value of n depends
only on the input x[n] at the same value of n
yn x[n]
2
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Properties of Discrete-time systems
2.2.2 Linear Systems
If
x1 n
T{‧}
y1 n
x2 n
T{‧}
y2 n
and only If:
x1n x2 n
axn
T{‧}
y1n y2 n additivity property
T{‧}
ayn
homogeneity or scaling
同(齐)次性 property
principle of superposition
x3 n ax1n bx2 n
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T{‧}
y3 n ay1n by2 n
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Example of Linear System
Ex. 2.6 Accumulator system
for arbitrary x1n and x2 n
y1 n
n
y2 n
x k
k
1
yn
n
xk
k
n
x k
k
2
when x3 n ax1n bx2 n
y3 n
n
n
x k ax k bx k
k
3
1
k
n
n
k
k
2
a x1 k b x2 k ay1 n by2 n
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Example 2.7 Nonlinear Systems
Method: find one counterexample
yn x[n]
For
2
counterexample
1 1 1 1
2
2
2
yn log10 x[n]
For
counterexample
10 log10 1 log10 101
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Properties of Discrete-time systems
2.2.3 Time-Invariant Systems
Shift-Invariant Systems
x1 n
T{‧}
x2 n x1n n0
y1 n
y2 n y1n n0
T{‧}
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Example of Time-Invariant System
Ex. 2.8 Accumulator system
n
yn
xk
k
x1 xn n0
y1n
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n
n n0
n
x k xk n xk yn n
k
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k
0
k1
1
0
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Example of Time-varying System
Ex. 2.9 The compressor system
x n
T{‧}
n
yn xMn, n
T{‧}
0
x1 n x n n0 n 1
0
2n
0
T{‧}
2n 1
0
y1n x1Mn xMn n0 yn n0 xM n n0
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Properties of Discrete-time systems
2.2.4 Causality
A system is causal if, for every choice
of n 0 , the output sequence value at
the index n n0 depends only on the
input sequence value for n n0
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Ex. 2.10 Example for Causal System
Forward difference system is not Causal
yn xn 1 xn
Backward difference system is Causal
yn xn xn 1
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Properties of Discrete-time systems
2.2.5 Stability
Bounded-Input Bounded-Output (BIBO)
Stability: every bounded input sequence
produces a bounded output sequence.
if
then
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xn Bx ,
for all n
yn By ,
for all n
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Ex. 2.11 Test for Stability or Instability
yn x[n]
2
if
then
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is stable
xn Bx ,
for all n
yn By B ,
2
x
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for all n
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Ex. 2.11 Test for Stability or Instability
Accumulator system
yn
n
xk
k
0 n 0
xn un
:bounded
1 n 0
n0
0
yn xk xk
: notbounded
k
k
n 1 n 0
n
n
Accumulator system is not stable
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2.3 Linear Time-Invariant (LTI)
Systems
Impulse response
n
n n0
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T{‧}
T{‧}
h n
h n n0
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LTI Systems: Convolution
Representation of general sequence as a
linear combination of delayed impulse
xn
xk n k
k
principle of superposition
yn T xk n k xk T n k
k
k
xk hn k xn hn
k
An Illustration Example(interpretation 1)
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Computation of the Convolution
(interpretation 2)
yn
xk hn k
k
hn k h k n
h k
hk
reflecting h[k] about the origion to obtain h[-k]
Shifting the origin of the reflected sequence to
k=n
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Ex. 2.12
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Convolution can be realized by
–Reflecting h[k] about the origin to obtain h[-k].
–Shifting the origin of the reflected sequences to k=n.
–Computing the weighted moving average of x[k] by
using the weights given by h[n-k].
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Ex. 2.13 Analytical Evaluation
of the Convolution
For system with impulse response
1 0 n N 1
hn un un N
otherwise
0
input
xn a un
h(k)
n
0
Find the output at index n
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1 0 n N 1
hn
otherwise
0
xn a un
n
h(-k)
0
0
h(n-k)
h(k)
x(k)
0
y n 0 n 0
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h(-k)
h(k)
0
0
n 0, n N 1 0 0 n N 1
n 1
1
a
yn xk h k n a k
1 a
k 0
k 0
n
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n
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h(-k)
h(k)
0
0
n N 1 0 n N 1
n
yn
n
xk h k n a
k n N 1
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a
n N 1
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a
1 a
k
k n N 1
n 1
N
n N 1 1 a
a
1 a
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0,
n 1
1
a
yn
,
1 a
N
1
a
a n N 1
1 a
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n0
0 n N 1
,
N 1 n
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2.4 Properties of LTI Systems
Convolution is commutative(可交换的)
xn hn hn xn
x[n]
h[n]
y[n]
h[n]
x[n]
y[n]
Convolution is distributed over addition
xn h1n h2 n xn h1n xn h2 n
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Cascade connection of systems
hn h1n h2 n
x [n]
h1[n]
h2[n]
y [n]
x [n]
h2[n]
h1[n]
y [n]
x [n]
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h1[n] ]h2[n]
y [n]
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Parallel connection of systems
hn h1n h2 n
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Stability of LTI Systems
LTI system is stable if the impulse response
is absolutely summable .
S
hk
k
yn
k
k
hk xn k hk xn k
xn Bx
y n Bx
h k
k
Causality of LTI systems hn 0, n 0
HW: proof, Problem 2.62
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Impulse response of LTI systems
Impulse response of Ideal Delay systems
hn n nd , nd a positive fixed integer
Impulse response of Accumulator
1, n 0
hn k
un
k
0, n 0
n
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Impulse response of Moving
Average systems
M2
1
hn
n k
M 1 M 2 1 k M1
1
, M1 n M 2
M1 M 2 1
0 ,
otherwise
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Impulse response of Forward Difference
hn n 1 n
Impulse response of Backward Difference
hn n n 1
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Finite-duration impulse
response (FIR) systems
The impulse response of the system has
only a finite number of nonzero samples.
M2
1
n k
such as: hn
M 1 M 2 1 k M1
1
, M1 n M 2
M1 M 2 1
0 ,
otherwise
The FIR systems always are stable.
S
55
h n
n
Infinite-duration impulse
response (IIR)
The impulse response of the system is
infinite in duration.
1, n 0
hn k
un
k
0, n 0
n
Stable IIR System:
a 1
hn a un
n
S
h n
n
56
Equivalent systems
h n n 1 n n 1
n 1 n 1 n n n 1
57
Inverse system
hn hi n hi n hn n
hn un n n 1
un un 1 n
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2.5 Linear Constant-Coefficient
Difference Equations
An important subclass of linear timeinvariant systems consist of those
system for which the input x[n] and
output y[n] satisfy an Nth-order linear
constant-coefficient difference equation.
N
M
a yn k b xn m
k 0
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k
m 0
m
Ex. 2.14 Difference Equation
Representation of the Accumulator
y n
n
x k ,
k
yn xn
y n 1
n 1
x k
k
xk xn yn 1
k
yn yn 1 xn
60
n 1
Block diagram of a recursive
difference equation representing an
accumulator
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y n y n 1 x n
Ex. 2.15 Difference Equation
Representation of the MovingAverage System with M 1 0
M2
1
yn
xn k
M 2 1 k 0
representation 1
1
un un M 2 1
hn
M 2 1
1
n n M 2 1 un
hn
M 2 1
another representation 1
62
1
n n M 2 1 un
hn
M 2 1
yn yn 1 x1n
1
xn xn M 2 1
x1 n
M 2 1
63
1
xn xn M 2 1
yn yn 1
M 2 1
Difference Equation
Representation of the System
An unlimited number of distinct
difference equations can be used to
represent a given linear time-invariant
input-output relation.
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Solving the difference equation
Without additional constraints or
information, a linear constantcoefficient difference equation for
discrete-time systems does not provide
a unique specification of the output for
a given input.
N
M
a yn k b xn m
k 0
65
k
m 0
m
Solving the difference equation
N
M
a yn k b xn m
k 0
k
m 0
m
Output:
yn y p n yh n
y p n Particular solution: one output sequence
for the given input x p n
yh n Homogenous solution: solution for
the homogenous equation( x n 0 ):
N
ak yh n k 0
yh n A z
k 0
where zm is the roots of
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N
m 1
N
n
m m
k
a
z
k 0
k 0
Solving the difference equation
recursively
If the input xn and a set of auxiliary value
y 1 , y 2 , y N
are specified.
y(n) can be written in a recurrence formula:
N
M
ak
bk
y n y n k x n k , n 0,1, 2,3,
k 1 a0
k 1 a0
N 1
M
ak
bk
y n N
y n k
x n k ,
k 0 aN
k 1 aN
n N N 1, N 2, N 3,
67
Example 2.16 Recursive Computation of
Difference Equation
y n ay n 1 x n,
x n K n ,
y 1 c
y0 ac K
y1 ay0 0 aac K a c aK
2
y2 ay1 0 a a c aK a c a K
2
3
2
y3 ay2 0 a a c a K a c a K
y n a c a K
n1
68
n
3
2
for n 0
4
3
Example 2.16 Recursive Computation of
Difference Equation
yn ayn 1 xn
yn 1 a
for n 1
y 2 a
xn K n
1
y1 c
yn xn
y1 x1 a c
1
1 1
2
y 3 a y 2 x 2 a a c a c
1
1 2
3
y 4 a y 3 x 3 a a c a c
1
y n a c
n1
for n 1
y n a c Ka u n
n1
69
1
n
for all n
Example for Recursive Computation
of Difference Equation
y n a c Ka u n
n1
n
for all n
The system is noncausal.
yn 1 a 1 yn xn
The system is not linear.
The system is not time invariant.
y n ay n 1 x n x n K n n0 y 1 c
y n a c Ka
n1
70
nn0
u n n0 for all n
Difference Equation
Representation of the System
If a system is characterized by a linear
constant-coefficient difference equation and
is further specified to be linear, time
invariant, and causal, the solution is unique.
In this case, the auxiliary conditions are
stated as initial-rest conditions(初始松弛条件).
The auxiliary information is that if the input
xn is zero for n n0 ,then the output, yn
is constrained to be zero for n n0
71
Summary
The system for which the input and
output satisfy a linear constantcoefficient difference equation:
The output for a given input is not
uniquely specified. Auxiliary conditions
are required.
72
Summary
If the auxiliary conditions are in the form
of N sequential values of the output,
y 1, y 2,
, y N
later value can be obtained by
rearranging the difference equation as a
recursive relation running forward in n,
N
M
ak
bk
y n y n k x n k , n 0,1, 2,3,
k 1 a0
k 1 a0
73
Summary
and prior values can be obtained by
rearranging the difference equation as a
recursive relation running backward in n.
N 1
M
ak
bk
y n N
y n k
x n k ,
k 0 aN
k 1 aN
n N N 1, N 2, N 3,
74
Summary
Linearity, time invariance, and
causality of the system will depend on
the auxiliary conditions.
If an additional condition is that the
system is initially at rest, then the
system will be linear, time invariant,
and causal.
75
Example 2.16 with initial-rest conditions
y n ay n 1 x n
since x n 0, n 0
x n K n
y 1 0
y n Ka u n
n
If the input is K n n0 , again with initialrest conditions, then the recursive solution
is carried out using the initial condition
yn 0, n n0
y n Ka
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nn0
u n n0
Discussion
If the input is K n n0 , with initial-rest
conditions,
yn 0, n n0
y n Ka
nn0
u n n0
Note that for n0 0 , initial rest implies
that y 1 0
Initial rest does not always means
y 1
y N 0
It does mean that yn0 1 yn0 N 0
if xn 0, n n0 .
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2.6 Frequency-Domain
Representation of DiscreteTime Signals and systems
2.6.1 Eigenfunction and Eigenvalue for LTI
If
yn T xn H xn xn
xn is called as the eigenfunction of the
, and the associated
system T
eigenvalue is H xn
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Eigenfunction and Eigenvalue
Complex exponentials is the eigenfunction
for discrete-time systems. For LTI systems:
x n e
jwn
, n
y n h n x n
h k e
k
h k x n k
k
jw n k
jw jwn
H e e
eigenvalue
79
jwk jwn
h k e
e
k
eigenfunction
frequency response
Frequency response
jw jwn
jwk jwn
y n h k e
H e e
e
k
H e
is called as frequency response of
the system.
jw
Real part, imagine part
H e
jw
H e jH e
jw
R
I
Magnitude, phase
H e
80
jw
jw
H e e
jw
jH e jw
Example 2.17 Frequency
response of the ideal Delay
jwn
yn xn nd
x n e
jwnnd
jwnd jwn
yn e
e
H e
e
jw
e
jwnd
From defination(2.109):
H e
jw
81
h n e
jwn
n
n n e
n
d
jwn
h n n nd
jwnd
e
Example 2.17 Frequency
response of the ideal Delay
H e
jw
e
jwnd
cos wnd j sin wnd
H R e
1e
82
jwnd
jH e
H e e
jw
jw
I
jw
jH e jw
Linear combination of
complex exponential
xn k e
jwk n
k
e
y n k H e
k
83
jwk
jwk n
Example 2.18 Sinusoidal
response of LTI systems
A j jw0n A j jw0n
x n A cos w0 n e e
e e
2
2
y n H
e
jw0
A j jw0n
e e H
2
if h n is real , H e
jw0
e
jw0
A j jw0n
e e
2
H e
*
jw0
y n A H e jw0 cos w0 n , H e jw0
84
Sinusoidal response of the
ideal Delay
jw
jw
y n A H e cos w0 n , H e
0
0
x n A cos w0n
1,
He
jw
H e
jw
e
w0 nd
y n A cos w0 n w0 nd
A cos w0 n nd
85
jwnd
Periodic Frequency Response
The frequency response of discrete-time
LTI systems is always a periodic function of
the frequency variable w with period 2
H e
e
j w 2
j w 2
H e
86
H e
h n e
j w 2 n
n
e
e
H e
H e ,
j w 2
j w 2 r
jwn j 2 n
e
jwn
jw
jw
for r an integer
Periodic Frequency Response
We need only specify H e jw over 0 w 2
or w
The “low frequencies” are frequencies
close to zero
The “high frequencies” are frequencies
close to
More generally, modify the frequency with
2 r , r is integer.
87
Example 2.19 Ideal
Frequency-Selective Filters
Frequency Response of Ideal Low-pass Filter
88
Frequency Response of Ideal
High-pass Filter
89
Frequency Response of Ideal
Band-stop Filter
90
Frequency Response of Ideal
Band-pass Filter
91
Example 2.20 Frequency Response of
the Moving-Average System
1
, M1 n M 2
hn M 1 M 2 1
0,
otherwise
M2
1
H e
M 1 M 2 1 n M
jw
e
jwn
1
1
e
M1 M 2 1
92
jwM1
jw M 2 1
e
jw
1 e
H e
jw
M1M 2 1
1
e
1
jw M1 M 2 1
2
M1 M 2 1
M1 M 2 1
e
jw M1 M 2 1
2
e
jw M 2 1
e
jw
1 e
jw M1 M 2 1
1 e
e
1
e
jwM1
jw 2
2
jw
e
e
e
jw M1 M 2 1
2
jw 2
jw M 2 M11
2
e
jw M 2 M1
2
sin w M 1 M 2 1 2 jw M 2 M1
e
M1 M 2 1
sin w 2
93
1
2
Frequency Response of the MovingAverage System
sin w M 1 M 2 1 2 jw M 2 M1
H e
e
M1 M 2 1
sin w 2
jw
1
相位也取决于符号,不仅与指数相关
M1 = 0 and M2 = 4
4
5
94
2
4
5
2.6.2 Suddenly applied Complex
Exponential Inputs
In practice, we may not apply the complex
exponential inputs ejwn to a system, but the
more practical-appearing inputs of the form
x[n] = ejwn u[n]
i.e., x[n] suddenly applied at an arbitrary time,
which for convenience we choose n=0.
For causal LTI system:
y n 0, n 0
95
for x n 0, n 0
2.6.2 Suddenly applied Complex
Exponential Inputs
jwn
For causal LTI system
xn e un
n0
0,
n
yn xn hn
jwk jwn
hk e
e , n 0
k 0
For n≥0
jwk jwn
jwk jwn
y n h k e
e h k e
e
k 0
k n1
H e
96
jw
e
jwk jwn
h k e
e yss n yt n
k n1
jwn
2.6.2 Suddenly applied Complex
Exponential Inputs
Steady-state Response
yss n H e
jw
e
jwn
h k e
k 0
Transient response
yt n hk e
k n 1
97
jwk
e
jwn
jwk
e
jwn
2.6.2 Suddenly Applied Complex
Exponential Inputs (continue)
For infinite-duration impulse response (IIR)
yt n
h k e
k n 1
jwk
e
jwn
k n 1
k 0
h k h k
For stable system, transient response must
become increasingly smaller as n ,
Illustration of a real part of suddenly applied complex
exponential Input with IIR
98
2.6.2 Suddenly Applied Complex
Exponential Inputs (continue)
If h[n] = 0 except for 0 n M (FIR), then the
transient response yt[n] = 0 for n+1 > M.
For n M, only the steady-state response
exists
Illustration of a real part of suddenly applied complex
exponential Input with FIR
99
2.7 Representation of Sequences
by Fourier Transforms
(Discrete-Time) Fourier Transform, DTFT,
analyzing
jwn
jw
x n e
n
X e
If xn is absolutely summable, i.e. x n
n
then X e jw exists. (Stability)
Inverse Fourier Transform, synthesis
1
xn
2
100
X e e
jw
jwn
dw
Fourier Transform
X e jX e X e e
X e
jw
jw
R
jw
jX e jw
I
rectangular form
: Fouriertransform,
Xe
jw
polar form
jw
Fourier spectrum, spectrum
X e
jw
: magnitude, magnitude spectrum,
amplitude spectrum
: phase, phasespectrum
X e
101
jw
Principal Value(主值)
X e is not unique because any 2 r
jw
may be added to X e
without affecting
the result of the complex exponentiation.
jw
e
jX e jw
jw
X
e
Principle value:
is restricted to the
range of values between and . It is
jw
denoted as ARG X e
jw
arg X e : phase function is referred as a
continuous function of
102
w
for 0 w
Impulse response and
Frequency response
The frequency response of a LTI
system is the Fourier transform of the
impulse response.
H e
jw
h n e
1
hn
2
103
jwn
n
e
He
jw
jwn
dw
Example 2.21: Absolute Summability
xn a un
Let
n
The Fourier transform
X e
jw
a e
n jwn
n 0
jw n+1
ae
n 0
1 (ae )
1
lim
jw
jw
n
1 ae
1 ae
104
jw
if ae
1
or if a 1
1
a
, if a 1
1 a
n 0
n
jw n
Discussion of convergence
Absolute summability is a sufficient
condition for the existence of a Fourier
transform representation, and it also
guarantees uniform convergence.
Some sequences are not absolutely
summable, but are square summable,
i.e.,
2
x n
n
105
Discussion of convergence
Sequences which are square
summable, can be represented by a
Fourier transform, if we are willing to
relax the condition of uniform
convergence of the infinite sum
defining X e jw .
Is called Mean-square Cconvergence
106
Discussion of convergence
Mean-square convergence
xne
X e
jw
jwn
n
lim
M
xne
M
XM e
jw
jwn
n M
X e dw 0
Xe
jw 2
jw
M
The error X e X M e
may not approach
zero at each value of w as M , but
total “energy” in the error does.
jw
107
jw
Example 2.22 : Square-summability
for the ideal Lowpass Filter
1,
w wc
H lp e
0, wc w
1 wc jwn
1 jwn wc
hlp n
e
dw
e
wc
2 wc
2 jn
jw
e
2 jn
1
jwc n
jwc n
e
sin wc n
, n
n
Since hlp n is nonzero for n 0 , the ideal
lowpass filter is noncausal.
108
Example 2.22 Square-summability
for the ideal Lowpass Filter
sin wc n
hlp n
approaches zero as n ,
n
but only as 1 n .
hlp n is not absolutely summable.
sin wc n jwn does not converge
e
uniformly for all w.
n
n
Define H M e
109
M
jw
n M
sin wc n jwn
e
n
Gibbs Phenomenon
HM e
110
jw
1
2
sin[( 2M+1) ( w ) / 2]
wc sin[( w ) / 2] d
wc
M=1
M=3
M=7
M=19
Example 2.22 continued
As M increases, oscillatory behavior at
w wc is more rapid, but the size of
the ripple does not decrease. (Gibbs
Phenomenon)
As M , the maximum amplitude of
the oscillation does not approach zero,
but the oscillations converge in location
toward the point w w .
c
111
Example 2.22 continued
sin wc n jwn
e
does not converge uniformly
n
n
jw
to the discontinuous function Hlp e
.
However, hlp n is square summable,
and H M e jw converges in the meansquare sense to Hlp e jw
H e dw 0
lim Hlp e
M
112
jw
jw
M
2
Example 2.23 Fourier Transform
of a constant
xn 1 for all n
The sequence is neither absolutely summable
nor square summable.
The Fourier transform of x n
is
2 w 2 r
X e
jw
r
The impulses are functions of a continuous
variable and therefore are of “infinite height,
zero width, and unit area.”
113
Example 2.23 Fourier Transform
of a constant: proof
1
jw
jwn
x n
X e e dw
2
1
jwn
2 w 2 r e dw
2 r
jwn
w 2 r e dw
r
w e
114
jwn
dw e
j 0n
w dw 1
Example 2.24 Fourier Transform of
Complex Exponential Sequences
xn e
jw0 n
2 w w
X e
jw
0
r
1
x n
2
1
2
X e jw e jwn dw
jwn
2 w w0 2 r e dw
r
w w0 e
115
2 r
jwn
dw e
jw0 n
Example: Fourier Transform of
Complex Exponential Sequences
xn ak e
jwk n
, n
k
2 a w w
X e
jw
r k
116
k
k
2 r
Example: Fourier Transform
of unit step sequence
xn un
Ue
117
jw
1
w
2
r
jw
1 e
r
2.8 Symmetry Properties of the
Fourier Transform
Conjugate-symmetric sequence
xe n x n
e
Conjugate-antisymmetric sequence
xo n x n
o
xn xe n xo n
118
1
xe n x n x n xe n
2
1
xo n xn x n xo n
2
Symmetry Properties of real sequence
even sequence: a real sequence that is
Conjugate-symmetric x n x n
e
e
odd sequence: real, Conjugate-antisymmetric
xo n xo n
real sequence: xn xe n xo n
119
1
xe n xn x n xe n
2
1
xo n xn x n xo n
2
Decomposition of a Fourier
transform
X e X e
Xe
jw
jw
e
Conjugate-symmetric
o
Conjugate-antisymmetric
1
jw
jw
jw
X e X e
Xe e
2
1
jw
jw
jw
X e X e
Xo e
2
Xe e
Xo e
120
jw
jw
jw
xn X e
x[n] is complex
x n X e
jw
x n X e
jw
jw
j Imxn X e
1
1
xe n x n x n X e X e X e
2
2
1
jw 1
jw
jw
Re x n x n x n X e e X e X e
2
2
jw
o
jw
jw
R
j ImX e
xo n jX I e
121
jw
jw
jw
x[n] is real
x n x n X e
X R e jX I e
jw
X R e
jw
X e
jw
R
X e
X e
jw
xe n X R
122
jw
X
X e
e jX e
jw
jw
jw
R
jw
I
X e
XI e
jw
jw
I
X e
e x n jX e
jw
X e
jw
jw
jw
jw
o
I
Ex. 2.25 illustration of Symmetry Properties
1
jw
n
X e
if a 1
xn a un
jw
1 ae
1
jw
X e
X e
jw
1 ae
1 a cos w
jw
jw
XR e
X
e
R
2
1 a 2a cos w
jw
XI e
jw
a sin w
jw
X
e
I
2
1 a 2a cos w
Xe
jw
X e
123
jw
1 a
1
2
2a cos w
12
X e jw
a sin w
X e jw
tan
1 a cos w
1
Ex. 2.25 illustration of Symmetry Properties
Real part
1 a cos w
1 a 2 2a cos w
Imaginary part
a sin w
1 a 2 2a cos w
a=0.75(solid curve) and a=0.5(dashed curve)
124
Ex. 2.25 illustration of Symmetry Properties
Its magnitude is an even function, and phase is
odd.
X e
125
jw
1 a
1
2
2a cos w
12
a sin w
X e tan
1 a cos w
jw
1
a=0.75(solid curve) and a=0.5(dashed curve)
2.9 Fourier Transform Theorems
X e
jw
F {x[n]}
x[n] F { X e
1
x[n]
X e
F
jw
jw
}
2.9.1 Linearity
x1 n
F
X 1
e
ax1 n bx2 n
126
jw
x2 n
F
aX 1
F
X 2
e
jw
e bX e
jw
jw
2
Fourier Transform Theorems
2.9.2 Time shifting and frequency shifting
xn X e
x n nd e
e
127
jw0 n
jw
jwnd
xn X e
jw
X e
j w w0
Fourier Transform Theorems
2.9.3 Time reversal
x n X e
xn X e
jw
jw
If xn is real,
x n X e
jw
e X e
x n x n X e X e
x n x n X
128
jw
jw
jw
jw
Fourier Transform Theorems
2.9.4 Differentiation in Frequency
jw
dX e
nxn j
dw
x n X e
j
dX e
129
dw
jw
j
jw
jwn
x n e
n
jwn
de
x n dw
n
jwn
nx ne
n
Fourier Transform Theorems
2.9.5 Parseval’s Theorem
E
xn
2
n
X e
1
2
xn X e
X e
jw 2
jw 2
jw
dw
is called the energy density spectrum
1
jw
jwn
E x n x n [ X e e dw]x n
2
n
n
1
jw
jwn
X e x ne dw
jw
X
e
2
n
130
Fourier Transform Theorems
2.9.6 Convolution Theorem
hn H e
xn X e
if y n
jw
jw
x k h n k x n h n
k
X e H e
Ye
131
jw
jw
jw
HW: proof
Fourier Transform Theorems
2.9.7 Modulation or Windowing Theorem
xn X e
jw
wn W e
jw
yn xn wn
Ye
jw
1
2
X e W e
j
j w
d
HW: proof
132
Fourier transform pairs
n 1
n n0 e
jwn0
1
n 2 w 2 k
k
a u n
n
1
a 1 1 ae jw
1
u n
w 2 k
jw
1 e
k
n 1 a u n a
n
133
1
1
1 ae
jw 2
Fourier transform pairs
r sin w p n 1
n
sin w p
un
1
r 1
jw
2 j 2w
1 2r cos w p e r e
jw
jw
1
e
e
(re )
jwp
jwp
jwp jw
jwp jw
r (e e ) 1 re e
1 re e
1,
w wc
sin wc n
jw
X e
n
0 , wc w
jwp n 1
sin w M 1 2 jwM
1, 0 n M
x n
e
0,
otherwise
sin
w
2
134
2
Fourier transform pairs
e
jw0n
2 w w
0
k
1 jw0n j
cos w0 n (e
2
e w w 2 k e
k
135
j
0
j
2 k
e
jw0n j
)
w w0 2 k
Ex. 2.26 Determine the Fourier Transform
of sequence
xn a un 5
n
x1 n a un X 1 e
n
x2 n x1 n 5 a
jw
1
jw
1 ae
un 5
n 5
j 5w
e
X2 e e
X1 e
jw
1 ae
5 j 5w
ae
5
jw
5
jw
xn a x2 n X e a X 2 e
jw
1 ae
jw
136
j 5w
jw
Ex. 2.27 Determine an inverse Fourier
1
Transform of X e jw
jw
jw
1 ae 1 be
a a b b a b
jw
jw
1 ae
1 be
X e
jw
a n
b n
xn
a un
b un
a b
a b
137
Ex. 2.28 Determine the impulse response
from the frequency respone:
0,
w
w
c
jw
H hp e jwn
d , w w
e
c
1,
H lp e
0,
jw
Hhp e
jw
e
jwnd
w wc
wc w
1 H e e
jw
lp
jwnd
jwnd
e
Hlp e
jw
sin wc n nd
hhp n n nd hlp n nd n nd
n nd
138
Ex. 2.29 Determine the impulse response
for a difference equation:
1
1
yn yn 1 xn xn 1
2
4
xn n
Impulse response
1
1
hn hn 1 n n 1
2
4
He
jw
1 jw
1 jw
jw
e H e 1 e
2
4
jw
jw
1
1
1 e
e
1
jw
4
H e
4
jw
jw
jw
1
1
1
1 e
1 e
1 e
2
2
2
139
Ex. 2.29 Determine the impulse response
for a difference equation:
1
H e
jw
1
1 e
2
jw
n
1 u n
2
n
1 e jw
4
jw
1
1 e
2
1
4
1
2
n 1
n 1
u n 1
1
1 1
hn un un 1
2
4 2
140
2.10 Discrete-Time Random Signals
Deterministic: each value of a sequence is
uniquely determined by a mathematically
expression, a table of data, or a rule of
some type.
Stochastic signal: a member of an
ensemble of discrete-time signals that is
characterized by a set of probability density
function.
141
2.10 Discrete-Time Random Signals
For a particular signal at a particular time,
the amplitude of the signal sample at that
time is assumed to have been determined
by an underlying scheme of probability.
That is, xn is an outcome of some
random variable x n
142
2.10 Discrete-Time Random
Signals
xn is an outcome of some random variable
x n ( not distinguished in notation).
The collection of random variables is called a
random process.
The stochastic signals do not directly have
Fourier transform, but the Fourier transform
of the autocorrelation and autocovariance
sequece often exist.
143
Fourier transform in stochastic signals
The Fourier transform of autocovariance
sequence has a useful interpretation in
terms of the frequency distribution of
the power in the signal.
The effect of processing stochastic
signals with a discrete-time LTI system
can be described in terms of the effect
of the system on the autocovariance
sequence.
144
Stochastic signal as input
Let xn be a real-valued sequence that is a
sample sequence of a wide-sense stationary
discrete-time random process.
xn
yn
145
hn
yn
k
k
hn k xk hnxn k
Stochastic signal as input
In our discussion, no necessary to distinguish
between the random variables Xn andYn and
their specific values x[n] and y[n].
mXn = E{xn }, mYn= E(Yn}, can be written as
mx[n] = E{x[n]}, my[n] =E(y[n]}.
The mean of output process
my E y n
mx
146
h k E x n k
k
h k H e m
j0
k
x
Stochastic signal as input
The autocorrelation function of output
yy m E y n y n m
E h k h r x n k x n m r
k r
h k h r m ( r k ) m l c l
k r
l
where
xx
lk
chh l
l
xx
l
h k h l k
k
hh
chh n is called a deterministic autocorrelation
sequence or autocorrelation sequence of hn
147
Stochastic signal as input
yy m E y n y n m xx m l chh l
l
where
c l h k h l k
hh
k
C e e
yy e
jw
jw
jw
hh
xx
H e H e H e
the power
e H e e
spectrum
Chh e
jw
jw
yy
jw
jw
jw 2
jw
2
jw
xx
DTFT of the autocorrelation function of output
148
Total average power in output
yy e
jw
H e
jw
2
xx e jw
provides the motivation for the term
power density spectrum.
1
2
jw
jw0
E y n yy 0
e
dw
e
yy
2
2
能量
1
jw
jw
H
e
e
dw
xx
无限
2
total average power in output
能量有限
Parseval’s E
Theorem
149
xn
n
2
1
2
X e
jw 2
dw
For Ideal bandpass system
jw
m
Since xx
is a real, even, its FT xx e is
e
so is e H e e
1
1
0 e dw e dw
2
2
also real and even, i.e.,
xx e
jw 2
jw
能量
非负
xx
jw
yy
yy
jw
jw
xx
b
a
b
jw
xx
lim yy 0 0
(b a ) 0
xx e
jw
a
0
jw
xx
for all w
the power density function of a real signal is
150 real, even, and nonnegative.
Ex. 2.30 White Noise
A white-noise signal is a signal for which
xx m x2 m
Assume the signal has zero mean. The
power spectrum of a white noise is
xx e
jw
m e
m
xx
jwm
2
x
for all w
The average power of a white noise is
1
xx 0
2
151
1
xx e dw 2
jw
2
x
dw
2
x
Color Noise
A noise signal whose power spectrum is
not constant with frequency.
A noise signal with power spectrum yy e jw
can be assumed to be the output of a LTI
system with white-noise input.
H e
yy e
152
jw
jw 2
2
x
Color Noise
Suppose h n anu n
,
1
H e
jw
1 ae
jw
yy e
jw
H e
jw
153
2
2
1
2
x
jw
1 ae
2
x
2
x
1 a 2a cos w
2
Cross-correlation between the
input and output
xy m E x n y n m
E x n h k x n m k
k
h k m k
k
xx
h k xx k
H e e
xy e
154
jw
jw
jw
xx
Cross-correlation between
the input and output
2
If xx m x m
,
xy m h k xx k
h k m k h m
k
2
x
2
x
That is, for a zero mean white-noise input,
the cross-correlation between input and
output of a LTI system is proportional to
the impulse response of the system.
155
Cross power spectrum between
the input and output
, w
e H e
xx e
jw
jw
xy
2
x
2
x
jw
The cross power spectrum is proportional
to the frequency response of the system.
156
2.11 Summary
Define a set of basic sequence.
Define and represent the LTI systems
in terms of the convolution, stability and
causality.
Introduce the linear constant-coefficient
difference equation with initial rest
conditions for LTI , causal system.
Recursive solution of linear constantcoefficient difference equations.
157
2.11 Summary
Define FIR and IIR systems
Define frequency response of the LTI
system.
Define Fourier transform.
Introduce the properties and theorems
of Fourier transform. (Symmetry)
Introduce the discrete-time random
signals.
158
Chapter 2 HW
2.1, 2.2, 2.4, 2.5,
2.7, 2.11, 2.12,2.15,
2.20, 2.62,
159
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Zhongguo Liu_Biomedical Engineering_Shandong Univ.
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