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ANALYSIS IN THE
FREQUENCY DOMAIN
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SPECTRAL DENSITY
Definition
The spectral density of a S.S.P. y (t ) (also called the spectrum of y (t ) )
is defined as:
y ( ) 

  y ( )  e  j
 
 F  y ( ).
In other words, y ( ) is defined as the Fourier transform F  y ( )
of the covariance function.
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Properties of  y ( ) :
1.  y ( ) is a real function of the real variable  ,
Imy ( )   0
  
2.  y ( ) is a positive function,
y ( )  0
  
3.  y ( ) is a even function,
y ( )  y ( )
  
4.  y ( ) is a periodic function with period equal to 2 π
y ( )  y (  k  2π)   , k  Z
Observation: as a consequence of 4, we will plot the spectral density
in the interval [ ,  ] .
y ( )


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Observation
y ( )
π

   is the largest frequency for sinusoidal discrete time signals.
Indeed, the minimum period is T  2
which corresponds to  
2

2
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Inverse transform of  y ( ) :
y ( )  F  y ( ) 

  y ( )  e  j
  

 y ( )  F
bi-univocal relationship
1
1 π
y ( )   y ( )  e j d
2π π
 y ( ) and y ( ) carry the same information on the properties of the
process y (t ) (the spectral density is an alternative representation of
the process 2nd order properties)
Observation:
1 π
1 π
j 0
 y (0) 
y ( )  e d 
y ( )d


2π π
2π π
i.e. the process variance is the rescaled area underlying y ( ) .
y ( )
π
π

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Spectral density of a white noise (WN)
Let us consider e(t ) ~ WN (  , 2 ) . Covariance function:
 y ( )
2
 e ( )  
0
2
if   0
if   0

Spectral density:
e ( ) 

  e ( )  e  j
  
  e (0)e  j 0   e (1)e  j   e (1)e j   
  e ( 0 )  2
2
y ( )
-π

White Noises have constant and equal to 2 spectral density.
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SPECTRAL DENSITY OF S.S.P. GENERATED AS THE
OUTPUT OF DIGITAL FILTERS
Let the process y (t ) the steady-state output of an asymptotically stable
digital filter fed by an S.S.P., i.e. y(t )  F ( z )v(t )
v(t )
y (t )
F (z )
Then, the following formula for the spectral density of y (t ) holds:
y ( )  F e
 y ( )

j 2
 v ( )
output spectral density
 F e j  square absolute value of the filter transfer function
2
evaluate for z  e j (filter frequency response)
 v ( )
input spectral density
If the input v(t ) is a White Noise with variance 2 , then:
y ( )  F e j  2
2
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Example (MA(1) process)
y(t )  e(t )  c  e(t  1) , c   (real coefficient)
e(t ) ~ WN ( 0 ,1 )
 y (0)  12  c 2   1  1  c 2
 y (1)  1  c   1  c
 y ( )  0 when   ...,  3,  4, ... .
Covariance function plot
 y ( )
1 c2
c0
-5
-4
-3
-2
-1
1
2
3
4
5

c0
1 c2
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Spectral density (via the definition)
y ( ) 

  y ( )  e  j

  
(only   y (0),  y (1)e  j  are not null)
  y (0)   y (1)e  j    y (1)e  j   0 
 1  c 2  ce  j  e j 
Euler representation of the exponential
e  j  e  j  cos( )  j sin( )  cos( )  j sin( )  2 cos( )
We have
y ( )  1  c 2  2c cos( )
which is:

real

 0 c
even

periodic with period 2π

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Spectral density (computed through the main theorem)
y (t )  (1  cz 1 )e(t ) ,
e(t ) ~ WN ( 0 ,1 )
2
y ( )  1  ce j  1
Recalling that:
a  jb  (a  jb)(a  jb)  a 2  b 2
2
1.
1  ce  j  1  c cos( )  j sin(  )  1  c cos( )  j sin( )
2.
1  ce  j  1  c cos( )  j sin( )
y ( )  1  ce j 1  ce j   1  c 2  c(e j  e  j )  1  c 2  2c cos( )
Plot of  ( )
y
y ( )
1  c 2
c0
1 c2
1  c 2
-π
π

2
c0
π
2
π
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y (0)  1  c 2  2c  1  c 
2
y ( 2 )  1  c 2
y ( π)  1  c 2  2c  (1  c) 2
Let us compute  y (0) based on y ( )
1 π
2

 y (0) 
1

c
 2c cos( ) d 

2π π
π
1  π

2
   1  c d  2c  cos( )d  
2π  π

π



1
1  c 2    2csin( )   1 2π1  c 2   1  c 2
2π
2π
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An alternative interpretation of the spectral density
(Kinchine-Wiener)
Suppose that an S.S.P. y (t ) is filtered through an (ideal) pass-band
filter:
FPB frequency response
y (t )
FP B
~
y (t )
1
0
FPB  pass - band filter

 

~
y (t ) is the filtered output process
Theorem (Kinchine-Wiener):
y ( )  lim  ~y (0)
 0
Spectral density = mean energy of process realizations frequency by
frequency.
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Example (White Noise)
A WN process realization is erratic and unpredictable (complete
uncorrelation at different time instants)
8
7
6
5
4
3
2
1
0
-1
-2
0
10
20
30
40
50
60
70
80
90
100
The WN spectral density is constant...
2
-π
y ( )

... i.e. WN energy is equally-distributed all over the frequency domain
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Example (general case)
y (t ) 
1
e(t ), e  WN (0,1)
1  0.9 z 1
4
120
3
100
2
80
1
0
60
-1
40
-2
20
-3
-4
0
20
40
60
80
100
5
0
-4
-2
0
2
4
-2
0
2
4
100
80
60
0
40
20
-5
0
y (t ) 
20
40
60
80
100
0
-4
1
e(t ), e  WN (0,1)
1  0.9 z 1
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Observation
Many different representations for an ARMA process
1. “Time-domain” representation: difference equation
y(t )  a1 y(t  1)  ...  am y(t  m)  c0 e(t )  ...  cn e(t  n)
e(t )  WN (  , 2 )
2. “Operatorial” representation: transfer function
y (t ) 
C ( z)
e(t )
A( z )
e(t )  WN (  , 2 )
3. “Probabilistic” characterization: mean & covariance function
- my
-  y ( )
  0,1,2,3,
4. “Frequency domain” characterization: mean & spectral density
- my
- y ( )
 
(N.B.: neither y ( ) nor  y ( ) carry any kind of information about
the mean value of the process).
Are all the four representations equivalent for a wide-sense process
characterization? Yes!
Clearly 1  2 ,
3  4 , 1,2  3,4
What about 3,4  1,2 ?
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Let us consider an ARMA process
y (t ) 
C ( z)
e(t ) ,
A( z )
e(t )  WN (  , 2 ) .
y (t ) has a rational spectral density:
y ( ) 
C (e j )
2
A(e j )
2
2 ,
where a spectral density is called “rational” if it is a rational function
of the variable e j , that is
p0  p1e j  p2 e j 2  
y ( ) 
q0  q1e j  q2 e j 2  
Is the reverse true? Yes
Theorem. Let y (t ) be a S.S.P. with rational spectral density.
Then, there exists a white noise process  (t ) with suitable mean and
variance and a rational transfer function W (z ) such that:
y(t )  W ( z ) (t )
i.e. y (t ) is an ARMA process.
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Question: is the ARMA representation (i.e. the choice of  (t ) and of
W (z ) ) unique? NO
The same process y(t )  W ( z ) (t ) ,  (t )  WN (  , 2 ) can be
generated according to an infinite number of different ARMA models.
~ ~
~
I.e. y (t ) can be generated also as y (t )  W ( z ) (t ) , where W ( z ) is a
~
rational transfer function different from W (z ) , and  (t ) is a white
noise different from  (t ) .
Case 1. Let  be any real number
y (t )  W ( z )  (t )  W ( z ) 
Here, y(t) does not change,
we have multiplied W(z)
by the identity1!!!

1
~
~
 (t )    W ( z )  (t )  W ( z )  (t )



Thanks to the rules for the
composition of transfer
functions
where:
1
~
 W ( z )   W ( z ) (it’s still a rational transfer function)

~
  (t )   (t )
~
(it’s still a WN,  (t )  WN ( ,  2 2 ) )*
~ ~
Hence, y (t )  W ( z ) (t ) is a new ARMA representation of the process
y (t ) (note that y (t ) is always the same, it never changed).
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~
* Let us verify that  (t ) is actually a white noise
~
~
~
E[ (t )]  E[ (t )]  E[ (t )]  
 ~ ( )  E[( (t )   )( (t   )   )] 
 E[( (t )   )( (t   )   )]    ( )
~
(clearly,  (t ) and  (t ) are two different white noises, although strictly
correlated)
Case 2. Let n be any integer number.
Here, y(t) does not change,
we have multiplied W(z) by
the identity1!!!
zn
~
~
y (t )  W ( z )  (t )  W ( z )  n  (t )  z n  W ( z ) (t  n)  W ( z )  (t )
z
where:
~
 W ( z)  z n  W ( z)
~
  (t )   (t  n)
Thanks to the rules for the
composition of transfer
functions and recalling the
meaning of the shift operator
(it’s still a rational transfer function)
(it’s still a white noise)*
~ ~
Hence, y (t )  W ( z ) (t ) is a new ARMA representation of the process
y (t ) (note that y (t ) is always the same, it never changed).
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~
* Let us verify that  (t ) is actually a white noise
~
E[ (t )]  E[ (t  n)]  
(stationarity)
 ~ ( )  E[( (t  n)   )( (t  n   )   )]    ( )
(stationarity)
~
(note that  (t ) and  (t ) are not the same process although they are
wide-sense equivalent)
Case 3. Let p be any complex number such that p  1.

z
y (t )  W ( z )  (t )  W ( z ) 
z

where:
z p
~
 W ( z)  W ( z) 
z p
~
  (t )   (t )
p
~
~

(
t
)

W
(
z
)

(t )
p 
Here, y(t) does not change,
we have multiplied W(z)
by the identity1!!!
(it’s still a rational transfer function)
(plainly, it’s a white noise)
~ ~
Hence, y (t )  W ( z ) (t ) is a new ARMA representation of the process
y (t ) (note that y (t ) is always the same, it never changed).
~
~
(here  (t ) and  (t ) are the same process, but W (z ) and W ( z ) are
different)
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Case 4. Let q a zero of W (z ) such that q  1.
That is:
W ( z)  W1 ( z)  ( z  q)
Here, y(t) does not change,
we have multiplied W(z)
by the identity1!!!
Then,

z  1q 
y (t )  W ( z )  (t )  W1 ( z )  ( z  q)  (t )  W1 ( z )  ( z  q) 
 (t ) 
1
z

q

z  q
 ~
~
 W1 ( z )  ( z  ) 

(
t
)

W
(
z
)

(t )

1
z

q


1
q
where:
~
 W ( z )  W1 ( z )  ( z  1q )
Thanks to the rules for the
composition of transfer
functions
(it’s still a rational transfer function)
zq
~
  (t ) 
 (t )
z  1q
is it a white noise?
~
Let us compute the spectral density of  (t )
~ ( ) 
e j  q
e
j

1
q
2

2
2
e

e
 q e  j  q  2
 
j
 1q e  j  1q 
j
1  q 2  qe j  e  j  2 1  q 2  2q cos( ) 2

 
 
1 1 j
1 2
 j
1  2  e  e 
1  2  cos( )
q
q
q
q
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1
2

1

cos( )
2
q
q
 q2
 2  q 2  2
1 2
1  2  cos( )
q
q
~
The spectral density is constant for all values of  , so that  (t ) is a
white noise!
Moreover,
1  1q
1 q
E[ (t )] 
 E[ (t )]  q 
  q
1
1
1 q
1 q
~
~
~ ~
Hence,  (t )  WN (q   , q  2 ) and y (t )  W ( z ) (t ) is a new ARMA
representation of the process y (t ) (note that y (t ) is always the same,
it never changed).
~
(note instead that  (t ) and  (t ) are two different white noises,
although strictly correlated)
Model Identification and Data Analysis (MIDA) – ANALYSIS IN FREQUENCY DOMAIN
21
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exam requires integrating this material with teacher explanations and textbooks.
Apart from the four examined ones, there are no other sources of
ambiguity in defining an ARMA process. This is expressed by the
following theorem which better clarifies the previous one.
Theorem (Spectral Factorization)
Let y (t ) be a S.S.P. with rational spectral density.
Then, there exists an unique white noise process  (t ) with suitable
mean and variance and an unique rational transfer function W (z ) such
that:
y(t )  W ( z ) (t ) ,
and, ( C (z ) and A(z ) are the numerator and denominator of W (z ) ):
1.
C (z ) and A(z ) are monic (i.e. the coefficients of the maximum
degree terms of C (z ) and A(z ) are equal to 1)
2.
C (z ) and A(z ) have null relative degree
3.
C (z ) and A(z ) are coprime (i.e. they have no common factors)
4.
the absolute value of the poles and the zeroes of W (z ) is less than
or equal to 1 (i.e. poles and zeroes are inside the unit circle)
When all the four conditions above are satisfied, we will say that
y(t )  W ( z ) (t ) is a canonical representation of y (t )
Model Identification and Data Analysis (MIDA) – ANALYSIS IN FREQUENCY DOMAIN
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This material is protected by copyright and is intended for students’ use only. Sell and distribution are strictly forbidden. Final
exam requires integrating this material with teacher explanations and textbooks.
Observation.
Conditions 1, 2, and 3 remove any ambiguity as due to the process
described in Case 1, Case 2, and Case 3, respectively.
The first part of Condition 4 assure that W (z ) is asymptotically stable
so that y (t ) is well defined, while the second part removes any
ambiguity as due to the process described in Case 4.
Model Identification and Data Analysis (MIDA) – ANALYSIS IN FREQUENCY DOMAIN
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