Discrete-time singnal system lecture.ppt

1
Discrete-Time signals
and systems

2
Introduction
 Signal: A signal can be defined as a function that conveys
information, generally about the state or behavior of a physical
system.
 Continuous-time signal: Continuous-time signals are defined
along a continuum of times and thus are represented by a
continuous independent variable. Continuous-time signals are often
referred to as analog signals.
 Discrete time signal: Discrete-time signals are defined at discrete
times and thus the independent variable has discrete values; i.e.,
discrete-time signals are represented as sequences of number.

3
Analog: Linear time invariant
system
f(t) g(t)
input impulse response output
How to retrieve h(t):
δ(t)
h(t)
h(t) h(t)
Impulse Out put will be same as h(t)

4
How to define δ(t) ?
 δ(t)=
And


 
Others
t
0
0








1
)
( dt
t

5
How to define System Output
g(t)?
 g(t)=
 Or g(t)=
 And there is another a
form like
g(t)=h(t)*f(t)
Here * is the commutative
property of convolution




 

 d
t
f
h )
(
).
(




 

 d
t
h
f )
(
).
(

6
Frequency domain:
 In frequency domain system output
G(ω)=
G(ω)=H(ω) x F(ω)
dt
e
t
g t
j




 )
(

7
Definition: Important functions
in Discrete Time Signal
 Unit impulse function:
δ(n) =
Unit step function:
U(n)=
Here “n” is an integer





0
,
0
0
,
1
n
for
n
for





0
,
0
0
,
1
n
for
n
for

8
Relationship between δ(n) and
U(n)
 U(n)=
 U(n)=
 δ(n) = U(n)-U(n-1)




n
k
k)
(





0
)
(
k
k
n

9
Discrete time invariant system
x(n) y(n)
input impulse response output
Output y(n) =
Or y(n) =
How to retrieve h(n)?
δ(n)






i
i
i
n
x
i
h )
(
)
(
h(n)






i
i
i
n
h
i
x )
(
)
(
h(n) h(n)
Impulse Out put will be same as h(n)

10
Definitions:
 FIR (Finite Impulse Response): A finite impulse response
(FIR) filter is a type of a digital filter. The impulse response, the
filter's response to a Kronecker delta input, is 'finite' because it
settles to zero in a finite number of sample intervals.
 IIR(Infinite Impulse Response): They have an impulse
response function which is non-zero over an infinite length of time.
 Casual System: If h(n)=0 for n<0; then this system is called
casual system.

11
Follow graph(FIR digital
filter,IIR digital filter):
 x(n)

D D D D
x
x
a0
X0
a1
X(n-1)
x
a2
X(n-2)
x
a3
X(n-3)
x
an
X(n-N)
FIR Digital Filter:
It is also known as difference
equation.
IIR Digital Filter:
+
)
(
1 n
y n
y
D
D
D
x
x
x b1
b(m-1)
bm


 





m
j
j
N
i
i
n j
n
y
b
i
n
x
a
y
1
0
)
(
)
(




N
i
i
n i
n
x
a
y
0
)
(

12
The System Output h(n)
δ(n)
)
1
(
)
1
(
)
( 1
1
0 




 n
h
b
n
a
n
a
D
x
x
x
+
D
h(n)
b1
a0 a1
h(n)=
n=0 then h(0)=
n=1 then h(1)=
n=2 then h(2)=
n=3 then h(3)=
h(n)= it is general
solution for the system output h(n)
0
a
0
1
1 a
b
a 
)
( 0
1
1
1 a
b
a
b 
)
( 0
1
1
2
1 a
b
a
b 
)
( 0
1
1
1
1 a
b
a
bn



13
The energy of a sequence:
E= 



n
n
x 2
|
)
(
|
Any discrete sequence can be shown by δ(n) :
General equation: x(n)= 





k
k
n
k
x )
(
)
(
Example:
X(n)=0.5δ(n+3)+ δ(n+2)-0.5δ(n+1)-0.5 δ(n)+0.5 δ(n-1)-0.3δ(n-2)
0
-1
-2
-3 1
2
0.5
0.5
1
-0.5 -0.5
-0.3

14
Definition: Different System
The ideal delay system:
X(n)
Moving Average: The moving average system output
y(n)=
Accumulator (Acc) : y(n)=






2
1
)
(
1
2
1
1 m
m
k
k
n
x
m
m
D D D
X(n-1) X(n-2)
X(n-N)=y(n)
The ideal delay system output would be y(n)=x(n-N)



n
k
n
x )
(
So the impulse response of the accumulator is same as
the u(n)
δ(n) u(n)
h(n)

15
Linear Shift Invariant System
)]
(
)
(
[ 2
2
1
1
/
n
x
a
n
x
a
T
y 

]]
[
]
[ 2
2
1
1
//
x
T
a
x
T
a
y 

+
a2
a1
[ T[.] ]
x
x
X2(n)
x1(n)
X2(n)
x1(n)
T [.]
T [.]
y1(n)
y2(n)
a2
a1
x
x
+
If y’ = y’’
then system is linear

16
Linear Shift Invariant System
(Cont...)
)]
(
[
/
k
n
x
T
y 

)
(
//
k
n
y
y 

X(n)
x(n)
k-sample
T [.]
x(n-k)
y(n)
T [.]
k-sample
)
(
//
k
n
y
y 

If = T[x(n-k)] then the system is shift invariant

17
Discrete Convolution
 In LSI system
x(n)=
So, y(n)=T[ ]
If the system is linear then
y(n) =






k
k
n
k
x )
(
)
(
x(n)
T [.]
y(n)=T[x(n)]






k
k
n
k
x )
(
)
(






k
k
n
k T
x ]
[ )
(
)
(

18
Discrete Convolution (Cont...)
 If the system is LSI
So we can write
y(n)=
Again if the system is SI: and
if LSI: y(n)= ,it is known as discrete convolution
We can also write as y(n)=
)
( k
n

T [.]
)
(n
hk




k
n
k
k h
x )
(
)
(
]
[ )
(
)
( k
n
k
n x
T
y 
  )
(
)
( k
n
n
k h
h 






k
k
n
k
k h
x )
(
)
(





k
k
n
k
k x
h )
(
)
(

19
Compressor: Down Sampling :Decimator
 The compressor output y(n)=x(M.n) where M is a integer
greater than 1
If M=2;
)
(n
x
Compressor
)
(n
y
0 2 3 4
1
x(n)
0 2
1
y(n)
Discarding M-1 in between samples
A compressor is not SI:
x1(n)=x(n-n0)
y1(n)=x1(mN)=x(Mn-n0)
y(n-n0)=x[M(n-n0)]=x(Mn-Mn0)
Here, x(Mn-n0)!=x(Mn-Mn0)
So, a compressor not SI
n
n

20
Expander : Up Sampling :Interpolation
 The expander output y(n)=x(n/L); where L>1
 If L=2;
)
(n
x
Expander
)
(n
y
0 1 2
2
4
3
4
3
1
0
n
y(n)
x(n)
n

21
The system output y(n) in different
cases
 h(n)=
x(n)=U(n) - U(n-N)
y(n)=?
We know that system output for LSI: y(n)=
1. For n<0 then y(n)=0
2. For 0<=n<N then y(n)=
3. For n>=N-1 then y(n)=
)
(n
x
h(n)
)
(n
y


 

0
,
0
,
0
n
for
a
n
for
n





k
k
n
k h
x )
(
)
(
1
)
1
(
0
0 1
1










 
 a
a
a
a
a
a
n
n
n
k
k
n
n
k
k
n
1
1
0 1
1








 a
a
a
a
N
n
N
k
k
n

22
Definition:
 Stability of a system: A stable system produces finite output when the
input is finite.
For LSI system: and |x(n)|<M<∞
So, |y(n)|<
Causality: A system is casual when for N=n0; the output of the system
depending on input x(n) only for n<=n0
If the system is LSI and h(n)=0 for n<0 then it is causal


 



 |
|
|
)
(
| )
(
k
k
n
k x
h
n
y




 






 k
k
k
k h
h
M |
|
|
| )
(
)
(
N=n0
n
x(n) y(n)
N=n0
Example: s= 







 0
|
|
|
)
(
|
n
n
n
a
n
h
1.|a|<1;S=1/1-|a|;Stable
2.a=1 and a>1; s=∞ then
not stable

23
Impulse response for different
delay
Ideal Delay:
Impulse response for ideal delay h(n)=δ(n-m)
Moving average:
h(n)=
Accumulator: h(n)=u(n)=
)
(n
x
M-Sample
)
(
)
( m
n
x
y n 



 













n
m
n
m
n
for
k
n
m
m
m
m
otherwise
m
m
k
2
1
,
)
(
1
2
1
1 1
2
1
1
0
2
1










0
0
1
0
)
(
n
n
n
k
k

24
Forward difference:
 Difference eqation = x(n+1)-x(n)
 Impulse response = δ(n+1)-δ(n)
)
1
( 
n
x
D
)
(n
x -
y(n)
0
-1
-1
1

25
Backward Difference
 Difference equation= y(n)=x(n)-x(n-1)
 Impulse response h(n)=δ(n+1)-δ(n)
)
1
( 
n
x
D
)
(n
x -
y(n)
1
0
-1
1

26
Stability:
 For a LSI
 For Ideal Delay, Moving Average, Backward
Difference and Forward Difference; if s<∞ then
it is stable
But for Accumulator
s= goes to ∞ then it is not stable


 



n
n
h
s |
)
(
|



0
)
(
n
n
U

27
Accumulator
 Difference equation
y(n)=x(n)+y(n-1)
h(n)=U(n)

This IIR digital filter
D
)
(n
x
+
y(n)
D
)
(n
x
+
y(n)
x
a
Difference equation
y(n)=x(n)+ay(n-1)
Impulse response h(n)=an
u(n)
Stability checking
Condition check: if |a|<1
Result: Stable
 




|
|
1
1
|
|
a
a
s n

28
Other properties of LSI
F.D
)
(n
x y(n)
D
1-sample
h(n)=[δ(n+1)-δ(n)]*δ(n-1)
h(n)=δ(n)- δ(n-1) ; In case of
forward delay, if we add a 1-sample
delay then it will convert to B.D

29
Inverse of a system

Example:
h(n)
)
(n
x x(n)
h(n)
h-1
(n)
Acc
)
(n
x x(n)
y(n)
B.D
h(n)=δ(n)
h1(n)=U(n) h2(n)= δ(n)- δ(n-1) h(n)=h1(n)*h2(n)
h(n)=U(n)*[δ(n)- δ(n-1)]=U(n)-
U(n-1)= δ(n)
)
(n
x y(n)=x(n)
B.D
+
D
-
D
F.D
B.D is inverse system of ACC

30
Inverse of system: Engineering
application
 1. T.V.ghost canceling
 2. Channel multi-path canceling
 3. Equalization in communications channel

31
Frequency domain representation of
discrete time signals and system
 Frequency response of the
system:
H(ejω
)=
k
j
k
e
k
h 




 )
(
Example:
n=0 N-1
h(n)
n






 1
0
,
0
1
0
)
(
N
n
elsewhere
n
h
h(n)=U(n)-U(n-N)
H(ejω
)=
=



j
N
j
N
k
k
j
e
e
e
k
h 







 1
1
)
(
1
0



2
1
)
2
sin(
)
2
sin( 

N
j
e
N
0
|H(ejω
)|
For linear phase
H(ejω
)

32
Inverse Fourier transform:
h(n)=
Example: In the ideal low pass filter








d
e
e
H n
j
j
)
(
2
1







0
0
|
|
|
|
1
0
)
(
c
c
j
e
H






ω
0
-ωc0 ωc0
-π π
|
)
(
| 
j
e
h
n
n
d
e
n
h co
N
j
c
c 





 )
sin(
2
1
)
(
0
0

 

-5
-4 -3 -2
-1 0 1
2
3 4
-1/5π
-1/3π
1/2
1/3π

33
Proof the inverse Fourier transform









d
e
e
x
x n
j
j
n )
(
2
1







n
n
j
n
j
e
x
e
x 

)
(
)
(
:
ˆ that
such
x 





d
e
e
x
n
x n
j
m
m
j
m
 





 )
(
2
1
)
(
ˆ )
(
interchanging the order of integral and summation
 












m
m
n
j
m d
e
x
n
x





)
(
)
(
2
1
.
)
(
ˆ

34






m
m
n
m
x
n
x )
(
)
(
)
(
ˆ 
)
(
)
(
ˆ n
x
n
x 
 












n
m
for
n
m
n
j
m
n
m
n
d
e
,
1
0
)
(
)
(
)
(
sin
2
1








We have,
=δ(n)
Proof the inverse Fourier transform cont..

35
Sampling theorem:
)
1
....(
..........
)
(
)
( dt
e
t
x
x t
j
c
c








Analog signal xc(t)
Xc(t)
T(sampling period)
x(nt)
FT:
IFT: )
2
........(
..........
)
(
2
1
)
( 

 



 d
e
x
t
x t
j
c
c

Fourier transform for discrete time signal:
Digital freq: ω=Ω.T……………………………………..(3)

36
)
4
..(
..........
)
(
)
(
)
( T
jn
n
n
jn
j
e
nT
x
e
nT
x
e
x 









 
 

)
5
....(
..........
..........
)
(
2
1
)
( 

 



d
e
x
nT
x nT
j
c


From eq.(2) for t=n.T
On the other hand we had before:
)
6
.....(
..........
..........
)
(
2
1
)
( 









d
e
e
x
nT
x n
j
j

37
)
7
.....(
..........
]
)
(
2
1
)
(  












n
jn
nT
j
c
j
e
d
e
x
e
x 


Putting (5) into (4)
Interchanging ∑ with integral
)
8
.....(
..........
)
(
2
1
)
( )
(








  








d
e
x
e
x
n
T
jn
c
j 



38
)
9
....(
..........
)
2
(
2
)
(















k
n
T
jn
T
k
T
T
e




But we have:
Then from (8) and (9) :












  







d
T
k
T
T
x
e
x
k
c
j
)
2
(
2
).
(
2
1
)
(





)
10
.........(
)
2
(
)
(
1
)
(  














k
c
j
d
T
k
T
x
T
e
x




39






k
c
j
T
k
T
x
T
e
x )
2
(
1
)
(



T



)
11
......(
..........
)
2
(
1
)
( 






k
c
j
T
k
x
T
e
x


Then
Since,
Then is a periodic function of Ω=ω/T with period of 2π/T
)
( 
j
e
x
Ω
0
-ω/T
-2π/T
1/T
ω/T 2π/T
)
( 
j
e
x

40
spectrum
Ana
x log
:
)
(
s
s
f
T







2
0
-Ω0 Ω0
-π/T π/T
Ω
Nyquist rate for maximum frequency Ω0 is sampling rate in order
not to have aliasing effect
if


 



41








k
c
c
kT
t
T
kT
t
T
kt
x
t
x
)]
(
[
)]
(
sin[
)
(
)
(








T
T
t
j
c
c d
e
x
t
x
/
/
)
(
2
1
)
(



In this case we can recover the analog signal from its sample as follows:
This formal is obtained as follows:
T
T







42
)
(
1
)
(
)
( 

 c
T
j
j
x
T
e
x
e
x 


 


 d
e
e
x
T
t
x t
j
T
T
j
c
/
/
)
(
.
2
1
)
(













k
Tk
j
T
j
j
e
kT
x
e
x
e
x )
(
)
(
)
( 







 






  d
e
e
kT
x
T
t
x t
j
T
T
k
Tk
j
c
/
/
)
(
2
)
(




43














k
c
kT
t
T
kT
t
T
kT
x
t
x
)
)(
(
)
)(
(
sin
)
(
)
(


 














k
T
T
kT
t
j
c d
e
T
kT
x
t
x
/
/
)
(
2
)
(
)
(



To recover analog signal from its sample

44
Fourier transform properties
x(n) X(ejω
)
1. Time shift: x(n-M)
2. Frequency shift X(ej(ω-ω0)
)
3. Time reversal: x(-n) X(e-jω
)
if x(n) is a real sequence then:
x(-n) X.(ejω
)
ƒ
ƒ
e-jωM
X(ejω
)
ejω
0
n
x(n)
ƒ
ƒ
ƒ

45
contd..
4. Differentiations in frequency domain:
n.x(n)
5. Convolution theorem:
y(n)=
Y(ejω
)=
X(ejω
) . H (ejω
)


d
e
dx
j
j
)
(





k
k
n
h
k
x )
(
)
(

46
contd..
6. Parseval’s Theorem (Energy)
E=
7. The modulation or windowing
theorem:
 





n
j
d
e
x
n
x 




2
2
|
)
(
|
2
1
|
)
(
|
X(n)
y(n)=w(n) . x(n)








d
e
w
e
X
e
Y j
j
j
)
(
.
)
(
2
1
)
( )
( 




47
Z-Transform
 X(z)=
z: complex variable in z-plane
Similar to Laplace transform
Convergency of Z-transfer should be check
Example 1. x(n)=an
U(n)
X(z)=





n
n
z
n
x ).
(









0
1
0
)
(
n
n
n
n
n
az
z
a From the geometric series if we have |az-1
|<1=>|z|>|a|
x(z)=1/1-az-1

48
Example
 x(n)=a|n|
|a|<1
X(z)=
=
 








1 0
n
n
n
n
n
n
z
a
z
a








0
1 n
n
n
n
n
n
z
a
z
a
1 2
1- convergent for |az|<1
2-convergent for |az-1
|<1
where |a|<|z|<1/|a|
a>1
a<1
Don’t shape

49
Example contd…
So,
X(z)=
)
1
)(
1
(
1
)
(
1
1
1
1
1









az
az
a
z
Z
az
az
az
z

50
Example
 Example 2:
x(n)=δ(n)
X(z)=1
x(n)=δ(n-m)
X(z)=z-m
Example 3: x(n)=an
sin(ω0n)U(n)
x(z)= 2
2
1
0
1
0
cos
2
1
)
sin(




 z
a
z
a
z
a



51
Example
 Example 4
x(n)=nan
U(n)
X(z)=



0
n
n
n
z
na
With a little change in above summation














 







 1
1
1
0
1
1
1
1
)
(
az
dz
d
z
z
a
dz
d
z
z
X
n
n
n
 2
1
1
1
)
(




az
az
z
X

52
Properties of z-transform
 1. Linearity: a1x1(n)+a2x2(n)
 2. Shift: x(n k) Z k
X(z)
 3 Convolution: y(n)=
Y(Z)=h(Z).X(Z)





k
k
n
x
k
h )
(
)
(
Z-trans
a1X1(z)+a2X2(z)
 

53
The relation between Z transform to
Fourier transform
 X(z)= 




n
n
z
n
x )
(
If we put: z=ejω
then Z.T F.T
For more general case Z=rejω
So,  







n
n
j
n
j
e
r
n
x
re
X 

)
(
)
(

54
z-transform derived from Laplace
transform
Consider a discrete-time signal x(t) below sampled every T sec
x(t) = x0 δ (t) + x1δ (t −T) + x2 δ (t − 2T) + x3 δ (t − 3T) +.....
The Laplace transform of x(t) is therefore:
X (s) = x0 + x1 e−sT
+ x2 e−s 2T
+ x3 e−s 3T
+.....

55
Range of convergency (ROC)
 x(n)=u(n)
in case z=ejω
not convergent
in case z=rejω
for r>1 then convergent because























0
|
|
|
)
(
|
|
)
(
|
n
n
n
n
n
n
r
r
n
u
r
n
x

56
ROC Contd….
 Step function u(n) has z-transform for ROC: |z|>1
If ROC includes the unit circle then z=ejω
and the sequence has
Fourier Transform
There is possibility that two sequences are different but they may
have a similar algebraic form of their z-transform, however their
ROC’s are different



59
The properties of ROC
 ROC has a ring form or a disc form
 The Fourier transform of x(n) has Fourier transform if and only if that
its z-transform’s ROC includes unit circle
 ROC cannot contain any pole
 If the sequence x(n) has finite length then ROC contains all z-plane
(excluding z=0 or z=∞)
 If x(n) is right-sided, then ROC is located outside of the largest pole.
 If x(n) is left sided then ROC is located inside of the smallest pole.
 If the sequence x(n) is both-sided then the ROC has ring shape
which is limited to inside and outside poles and there is no pole in
ROC.
 ROC must be a connected area.

60
Calculation of inverse Z-transform



c
n
dz
z
z
x
j
n
x 1
)
(
2
1
)
(

1
5
.
0
1
1
)
( 


z
z
X
1
5
.
0
1
1
)
( 


z
z
X
C is a closed curve
Example:
|z|>0.5
=1+0.5z-1
+0.25z-2
+0.125z-3







0
....
..........
..........
..........
..........
125
.
0
,
25
.
0
,
5
.
0
,
1
)
(
n
n
x
0 n<0
=>x(n)=(0.5)n
u(n)

61
Inverse z-transform
 If the Z transform can be expanded out as a series in powers of z, then the coefficients
of each term of the series constitutes the inverse. In the following expression, the
inverse would be the coefficients in blue
 Consider a Z transform which can be expanded as in the expression below
 Then the inverse is the coefficients in blue and can be written as follows.

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