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.
ANALYSIS IN THE
FREQUENCY DOMAIN
SPECTRAL DENSITY
Definition
The spectral density of a S.S.P. (also called the spectrum of ) is defined as:
.
In other words, is defined as the Fourier transform of the covariance function.
Properties of :
- is arealfunction of the realvariable ,
- is a positive function,
- is a evenfunction,
- is aperiodicfunction with period equal to
Observation: as a consequence of 4,we will plot the spectral density in the interval.
Observation
is the largest frequency for sinusoidal discrete time signals.
Indeed, the minimum period is
which corresponds to
Inverse transform of:
bi-univocal relationship
andcarry the same information on the properties of the process (the spectral density is an alternative representation of the process 2nd order properties)
Observation:
i.e. the process variance is the rescaled area underlying.
Spectral density of a white noise (WN)
Let us consider ~ . Covariance function:
Spectral density:
White Noises have constant and equal to spectral density.
SPECTRAL DENSITY OFS.S.P. GENERATED AS THE OUTPUT OF DIGITAL FILTERS
Letthe process the steady-state output of an asymptotically stable digital filter fed by an S.S.P., i.e.
Then, the following formula for the spectral density ofholds:
- output spectral density
- square absolute value of the filter transfer function evaluate for (filter frequency response)
- input spectral density
If the input is a White Noise with variance , then:
Example (MA(1) process)
, (real coefficient)
~
when.
Covariance function plot
Spectral density (via the definition)
(only are not null)
Euler representation of the exponential
We have
which is:
- real
- even
- periodic with period
Spectral density (computed through the main theorem)
,~
Recalling that:
1.
2.
Plot of
Let us computebased on
An alternative interpretation of the spectral density
(Kinchine-Wiener)
Suppose that an S.S.P. is filtered through an (ideal) pass-band filter:
is the filtered output process
Theorem (Kinchine-Wiener):
Spectral density = mean energy of process realizations frequency by frequency.
Example (White Noise)
A WN process realization is erratic and unpredictable (complete uncorrelation at different time instants)
The WN spectral density is constant...
... i.e. WNenergyisequally-distributedall over the frequency domain
Example (general case)
Observation
Many different representations for an ARMAprocess
1. “Time-domain” representation: difference equation
2. “Operatorial” representation: transfer function
3. “Probabilistic” characterization: mean & covariance function
-
-
4. “Frequency domain” characterization:mean & spectral density
-
-
(N.B.: neither norcarry 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,,
What about ?
Let us consider an ARMA process
,.
has a rationalspectral density:
,
wherea spectral density is called “rational” if it is a rational function of the variable, that is
Is the reverse true? Yes
Theorem. Let be a S.S.P. with rational spectral density.
Then, there exists a white noise process with suitable mean and variance and a rational transfer function such that:
i.e. is an ARMA process.
Question: is the ARMA representation (i.e. the choice of and of ) unique? NO
The same process , can be generated according to an infinite number of different ARMA models.
I.e. can be generated also as , where is a rational transfer function different from , and is a white noise different from .
Case 1. Let be any real number
where:
- (it’s still a rational transfer function)
- (it’sstill a WN, )*
Hence, is a new ARMA representation of the process (note that is always the same, it never changed).
* Let us verify that is actually a white noise
(clearly, and are two different white noises, although strictly correlated)
Case 2. Let be any integer number.
where:
- (it’s still a rational transfer function)
- (it’s still a white noise)*
Hence, is a new ARMA representation of the process (note that is always the same, it never changed).
* Let us verify that is actually a white noise
(stationarity)
(stationarity)
(note that and are not the same process although they are wide-sense equivalent)
Case 3. Let be any complex number such that.
where:
- (it’s still a rational transfer function)
- (plainly, it’s a white noise)
Hence, is a new ARMA representation of the process (note that is always the same, it never changed).
(here and are the same process, but and are different)
Case 4. Let a zero of such that .
That is:
Then,
where:
- (it’s still a rational transfer function)
- is it a white noise?
Let us compute the spectral density of
The spectral density is constant for all values of , so that is a white noise!
Moreover,
Hence, and is a new ARMA representation of the process (note that is always the same, it never changed).
(note instead that and are two different white noises, although strictly correlated)
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 be a S.S.P. with rational spectral density.
Then, there exists anuniquewhite noise process with suitable mean and variance and an unique rational transfer function such that:
,
and, ( and are the numerator and denominator of ):
1. and are monic (i.e. the coefficients of the maximum degree terms of and are equal to 1)
2. and have null relative degree
3. and are coprime (i.e. they have no common factors)
4.the absolute value of the poles and the zeroes of 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 is a canonical representation of
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 is asymptotically stable so that 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
1