DSP Lec 1.ppt

1
Digital Signal
Processing (DSP)
Lecture no 1
Discrete-Time Signals and Systems

Introduction to Signals and
Systems

14
Sum of two sequences
Product of two sequences
Multiplication of a sequence by a numberα
Delay (shift) of a sequence
Basic Sequence Operations
]
[
]
[ n
y
n
x 
integer
:
]
[
]
[ 0
0 n
n
n
x
n
y 

14
]
[
]
[ n
y
n
x 
]
[n
x



15
Basic sequences
Unit sample sequence
(discrete-time impulse,
impulse)
 






0
1
0
0
n
n
n

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15

16
Basic sequences
Unit step sequence






0
0
0
1
]
[
n
n
n
u
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16
 




n
k
k
n
u 
]
[











0
]
[
]
2
[
]
1
[
]
[
]
[
k
k
n
n
n
n
n
u 


 
]
1
[
]
[
]
[ 

 n
u
n
u
n
 First backward difference
  
  
0, 0 ,
1, 0
0 0
1 0
since
n
k
when n
k
when n
k
k
k



 


 

 

 


 


17
Basic Sequences
Exponential sequences
n
A
n
x 

]
[
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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
 : x[n] grows in magnitude as n increases
1



18
EX. 1 Combining Basic sequences






0
0
0
]
[
n
n
A
n
x
n

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18
If we want an exponential sequences that is
zero for n <0, then
]
[
]
[ n
u
A
n
x n


Cumbersome
simpler

19
Periodic Sequences
A periodic sequence with integer period N
n
all
for
N
n
x
n
x ]
[
]
[ 

   

 


 N
w
n
w
A
n
w
A 0
0
0 cos
cos
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integer
,
2
0 is
k
where
k
N
w 

0
2 / , integer
N k w where k is



21
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
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21
T{‧}
x[n] y[n]
Discrete-Time System

22
EX. 5 The Ideal Delay System






 n
n
n
x
n
y d ],
[
]
[
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22
If is a positive integer: the delay of the
system. Shift the input sequence to the
right by samples to form the output .
d
n
d
n
If is a negative integer: the system will
shift the input to the left by samples,
corresponding to a time advance.
d
n
d
n

23
Properties of Discrete-time systems
Memoryless (memory) system

24
Linear Systems
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24
 If  
n
y1
T{‧}
 
n
x1
 
n
y2
 
n
x2
T{‧}
 
n
ay
 
n
ax T{‧}
     
n
bx
n
ax
n
x 2
1
3 
      
n
by
n
ay
n
y 2
1
3 

T{‧}
   
n
y
n
y 2
1 
   
n
x
n
x 2
1  T{‧} additivity property
homogeneity or scaling
property
 principle of superposition
 and only If:

25
Example Nonlinear Systems
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25
Method: find one counterexample
 2
2
2
1
1
1
1 


 counterexample
   2
]
[n
x
n
y 
 For
   
]
[
log10 n
x
n
y 
   
1
10
log
1
log
10 10
10 


 counterexample
 For

26
Time-Invariant Systems
Shift-Invariant Systems
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26
   
0
1
2 n
n
x
n
x 
    
0
1
2 n
n
y
n
y 

 
n
y1
T{‧}
 
n
x1
T{‧}

29
Causality
A system is causal if, for every choice
of , the output sequence value at
the index depends only on the
input sequence value for
0
n
0
n
n 
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29
0
n
n 

30
Ex:9 Example for Causal System
Forward difference system is not Causal
Backward difference system is Causal
     
n
x
n
x
n
y 

 1
     
1


 n
x
n
x
n
y
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30

32
Properties of Discrete-time system
Stability
Bounded-Input Bounded-Output (BIBO)
Stability: every bounded input sequence
produces a bounded output sequence.
  n
all
for
B
n
x x ,



  n
all
for
B
n
y y ,



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32
if
then

33
Ex:10 Test for Stability or Instability
   2
]
[n
x
n
y 
  n
all
for
B
n
x x ,



  n
all
for
B
B
n
y x
y ,
2




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33
if
then
is stable

34
Accumulator system    




n
k
k
x
n
y
    bounded
n
n
n
u
n
x :
0
1
0
0







Ex:11 Test for Stability or Instability
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     








 
 



bounded
not
n
n
n
k
x
k
x
n
y
n
k
n
k
:
0
1
0
0
Accumulator system is not stable

35
Linear Time-Invariant (LTI)
Systems
Impulse response
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35
 
0
n
n 

 
n
h
 
n

 
0
n
n
h 
T{‧}
T{‧}

36
LTI Systems: Convolution
     






k
k
n
k
x
n
x 
         
 
       


























k
k
k
n
h
n
x
k
n
h
k
x
k
n
T
k
x
k
n
k
x
T
n
y 

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36
Representation of general sequence as a
linear combination of delayed impulse
principle of superposition
An Illustration Example（interpretation 1）

38
Properties of LTI Systems
Convolution is commutative
       
n
x
n
h
n
h
n
x 


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38
h[n]
x[n] y[n]
x[n]
h[n] y[n]
     
         
n
h
n
x
n
h
n
x
n
h
n
h
n
x 2
1
2
1 





Convolution is distributed over addition

39
Cascade connection of systems
     
n
h
n
h
n
h 2
1 

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39
x [n] h1[n] h2[n] y [n]
x [n] h2[n] h1[n] y [n]
x [n] h1[n] ]h2[n] y [n]

40
Parallel connection of systems
     
n
h
n
h
n
h 2
1 

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40

41
Stability of LTI Systems
LTI system is stable if the impulse response
is absolutely summable .
  

 



k
k
h
S
41
Causality of LTI systems   0
,
0 
 n
n
h
HW: proof, Problem 2.62

End chapter problems
1.1, 1.2,
 Example 2.33,
2.6a, 2. 7, 2.17 a b,
42

DSP Lec 1.ppt

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