UNIT-1
INTRODUCTION
Introduction
 Signal: something conveys information
 Signals are represented mathematically as functions of one or
more independent variables.
 Continuous-time (analog) signals, discrete-time 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
2.1 Discrete-Time Signals: Sequences
Discrete-Time signals are represented as
x  xn,    n  , n : integer
Cumbersome, so just use x  n 
In sampling,
xn  xa nT , T : sampling period
1/T (reciprocal of T) : sampling frequency
Figure : Graphical representation of
a discrete-time signal
Abscissa: continuous line
x  n  : is defined only at discrete instants
x[n]  xa (t ) |t nT  xa (nT )
EXAMPLE
Sampling the analog waveform
Figure 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
Basic sequences
Unit sample sequence
(discrete-time impulse,
impulse)
0 n  0
 n  
1 n  0
Basic sequences
A sum of scaled, delayed impulses
pn  a3 n  3  a1 n  1  a2 n  2  a7 n  7
arbitrary
sequence
x[n] 

 x[k ] [n  k ]
k  
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]
k 0





First backward difference
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
7/29/2017
EX. Combining Basic sequences
If we want an exponential sequences that is
zero for n <0, then
 A
x[n]  
0
n
n0
n0
x[n]  A u[n]
n
11
Cumbersome
simpler
Basic sequences
Sinusoidal sequence
x[n]  A cosw0 n   
12
for all n
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  cosw0 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
13
Frequency difference between
continuous-time and discrete-time
complex exponentials or sinusoids
x[n]  Ae
j  w0  2 n
 Ae
jw0 n
e
j 2n
 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
Periodic Sequences
A periodic sequence with integer period N
x[n]  x[n  N ]
for all n
A cosw0 n     A cosw0 n  w0 N   
w0 N  2 k , where k is integer
N  2 k / w0 , where k is integer
EX. 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
16
EX. 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 ]
cos3 n / 8  cos3 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)
EX. 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
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
.
Properties of Discrete-time systems
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
yn   x[n]
2
20
Properties of Discrete-time systems
Linear Systems
If
x1 n
T{‧}
y1 n
x2 n
T{‧}
y2 n
and only If:
x1n  x2 n
axn
T{‧}
y1n  y2 n additivity property
T{‧}
ayn
homogeneity or scaling
同(齐)次性 property
principle of superposition
x3 n  ax1 n  bx2 n
21
T{‧}
y3 n  ay1 n  by2 n
Example of Linear System
Ex. Accumulator system
for arbitrary x1n and x2 n
y1 n  
n
y2 n  
 x k 
k  
1
yn  
n
 xk 
k  
n
 x k 
k  
2
when x3 n  ax1 n  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
22
Properties of Discrete-time systems
Time-Invariant Systems
Shift-Invariant Systems
x1 n
T{‧}
x2 n  x1 n  n0 
y2 n  y1 n  n0 
T{‧}
23
y1 n
Example of Time-Invariant System
Ex. Accumulator system
n
yn  
 xk 
k  
x1  xn  n0 
y1 n 
24
n
n  n0
n
 x k    xk  n    xk   yn  n 
k  
1
k  
0
k1  
1
0
Example of Time-varying System
Ex. 2.9 The compressor system
x  n
T{‧}
  n
yn  xMn,    n  
T{‧}
0
x1  n  x  n  n0    n  1
0
  2n
0
T{‧}
  2n 1
0
y1 n  x1 Mn  xMn  n0   yn  n0   xM n  n0 
25
Properties of Discrete-time systems
Causality
A system is causal if, for every choice
of n0 , the output sequence value at
the index n  n0 depends only on the
input sequence value for n  n0
26
Ex. Example for Causal System
Forward difference system is not Causal
yn  xn  1  xn
Backward difference system is Causal
yn  xn  xn 1
27
Properties of Discrete-time systems
Stability
Bounded-Input Bounded-Output (BIBO)
Stability: every bounded input sequence
produces a bounded output sequence.
if
then
28
xn  Bx  ,
for all n
yn  By  ,
for all n
.
Ex. Test for Stability or Instability
yn   x[n]
2
if
then
29
is stable
xn  Bx  ,
for all n
yn  By  B  ,
2
x
for all n
Ex. Test for Stability or Instability
Accumulator system
yn  
n
 xk 
k  
0 n  0
xn  un  
:bounded
1 n  0
n0
0
yn   xk    xk   
: not bounded
k  
k  
n  1 n  0
n
n
Accumulator system is not stable
30
.
Linear Time-Invariant (LTI)
Systems
Impulse response
 n
 n  n0 
31
T{‧}
T{‧}
h n
h n  n0 
LTI Systems: Convolution
Representation of general sequence as a
linear combination of delayed impulse
xn 

 xk  n  k 
k  
principle of superposition
 
 
yn  T   xk  n  k    xk T  n  k 
k  
 k  


 xk hn  k   xn hn
k  
An
Illustration Example（interpretation
1）
32
.
33
7/29/2017
Zhongguo Liu_Biomedical Engineering_Shandong Univ.
Computation of the Convolution
（interpretation 2）
yn 

 xk hn  k 
k  
hn  k   h k  n
h k 
hk 
reflecting h[k] about the origion to obtain h[-k]
Shifting the origin of the reflected sequence to
k=n
34
.
Ex.
35
7/29/2017
Zhongguo Liu_Biomedical Engineering_Shandong Univ.
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].
36
Properties of LTI Systems
Convolution is commutative(可交换的)
xn hn  hn xn
x[n]
h[n]
y[n]
h[n]
x[n]
y[n]
Convolution is distributed over addition
xn h1n  h2 n  xn h1n  xn h2 n
37
Cascade connection of systems
hn  h1n h2 n
x [n]
h1[n]
h2[n]
y [n]
x [n]
h2[n]
h1[n]
y [n]
x [n]
38
h1[n] ]h2[n]
y [n]
Parallel connection of systems
hn  h1n  h2 n
39
Stability of LTI Systems
LTI system is stable if the impulse response

is absolutely summable .
S
 hk   
k  
yn 


k  
k  
 hk xn  k    hk  xn  k 
xn  Bx
y  n   Bx
Causality of LTI systems
40

 h k   
k 
hn  0, n  0
Impulse response of LTI systems
Impulse response of Ideal Delay systems
h  n    n  nd  , nd a positive fixed integer
Impulse response of Accumulator
1, n  0
hn    k   
 un
k  
0, n  0
n
41
Impulse response of Forward Difference
hn   n  1   n
Impulse response of Backward Difference
hn   n   n 1
42
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 yn  k    b xn  m
k 0
43
k
m 0
m
Ex. Difference Equation
Representation of the Accumulator
y  n 
n
 x k  ,
k 
yn  xn 
y  n  1 
n 1
 x k 
k 
 xk   xn  yn  1
k  
yn  yn 1  xn
44
n 1
Block diagram of a recursive
difference equation representing an
accumulator
y  n  y  n  1  x  n
45
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.
46
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 yn  k    b xn  m
k 0
47
k
m 0
m
Solving the difference equation
N
M
 a yn  k    b xn  m
k 0
k
m 0
m
Output:
yn  y p n  yh n
 y p n  Particular solution: one output sequence
for the given input x p n 
 y h n  Homogenous solution: solution for
the homogenous equation( x  n  0 ):
N
 ak yh n  k   0
yh n    A z
k 0
 where
48
zm
N
m 1
N
is the roots of
n
m m
k
a
z
 k 0
k 0
Solving the difference equation
recursively
If the input xn 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,
49
Fourier Transform
   X e  jX e   X e e
Xe
jw
jw
R
jw
 
jX e jw
I
rectangular form
  : Fourier transform,
X e
jw
polar form
jw
Fourier spectrum, spectrum
X e
jw
 : magnitude, magnitude spectrum,
amplitude spectrum
  : phase, phase spectrum
X e
50
jw
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
hn 
2
51
 jwn
n 



 e
He
jw
jwn
dw
Fourier Transform Theorems
X e
jw
  F {x[n]}
x[n]  F { X  e
1
x[n] 
 X e
F
jw
jw
}

Linearity
x1  n
F
 X1
e 
ax1  n  bx2  n
52
jw
x2  n
F
 aX1
F
 X 2
e 
jw
e   bX e 
jw
jw
2
Fourier Transform Theorems
Time shifting and frequency shifting
 
xn  X e
jw
x  n  nd   e
 jwnd
e
jw0 n

xn  X e
jw
X e

j  w w0 

Fourier Transform Theorems
Time reversal
 
x n  X e 
xn  X e
jw
 jw
If xn  is real,
 
x n  X e

jw
e   X e 
x  n  x  n  X  e   X  e 
x  n   x  n   X



 jw
jw
jw

 jw
Fourier Transform Theorems
Differentiation in Frequency
 
dX e
nxn  j
dw
x  n  X  e
j
55
dX  e
dw
jw
 j

jw
jw

   x  n e
 jwn
n 
 jwn
de
 x n dw
n


 nx n e
n 
 jwn
Fourier Transform Theorems
 
xn  X e
Parseval’s Theorem
E

 xn
n  
 X e


jw 2
2
1

2
  X e 


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  n e dw

jw

X
e
 
2 
n 

Fourier Transform Theorems
Convolution Theorem
  hn  H e 
xn  X e
if y  n  
jw
jw

 x k  h n  k   x n  h n
k 
   X e H e 
Ye
jw
jw
jw
Fourier Transform Theorems
Modulation or Windowing Theorem
 
xn  X e
jw
 
wn  W e
jw
yn  xn wn
 
Ye
jw
1

2
  X e W e


j
j  w  
d