Instrumentation & Measurement Magazine 26-4 - 28

educationI&M continued
in
ht
( ) 
ml
11   2
 2m l m2

2
g
l m

4
2
yn zpzp

which expresses the solution y(t) of the differential equation (2)
as a convolution integral
y t h s F t s ds

() ( ) ( )
t
 
x
The convolution integral is discretized by equidistant sampling
such that t ∈ {0, T, 2T,..., (N - 1)T} leading to
1
y nT h kT F n k T

N
( )  ( ) ((  ) )
k0

x
By virtue of the z-transform a recursive relationship can be derived
which is equal to:

gg 
ml 
y nT( ) 2exp  T cos T y n T

l 
 
  
  

442
exp
mm m
g
22
22



  
ml

2
    (( 1) )

2ml m
4
T sin 2 T Fx n T



As a result, (3) can be reparametrized into:
12 1
( )    
( 1)
( 2)

 
 (( 1) ) exp (( 2) )

T y n T

(3)
Example: Consider a pole pair zp = 0.8 exp(±πi/6) where
such that the time series y(n) consists of 600 samples.
 1 i
The time series is coherently sampled since the resonance frequency
is part of the discrete Fourier grid: π/6 = 2 π/600 × 50.
The resonance frequency therefore corresponds to the 50th
Fouyn
a y n a yn b F nx( 1)

(4)
where we omitted the sampling period T. If the force excitation
Fx
(n) is a Gaussian independently and identically distributed
noise sequence, the measured or observed time series y(n)
solving (4) is an autoregressive moving average process of order
(2,1) or in short ARMA(2,1). Indeed, it holds 2 memory lags
at the measured output y(n) while one memory lag at the force
excitation Fx
(n).
Autoregressive Moving Average Time Series
The toy example can be easily generalized to an arbitrary
ARMA(p,q) time series model given by:
yn a yn a yn
1
( 1)
2
( )        0
(
)
( 2) pq )
(5)
a yn p b u n b1u n( 1)
( )    
b u(n q
such that u(n) is a Gaussian white noise sequence such that
E[u(n)u(m)] = 0 for n ≠ m whereas 22
E u n  . In order to in[
( )]
1 az a z

b bz b z 2

12
1
01 2
  
  
a z p
b z

p
q
q
where the roots of the first polynomial are referred to as the
poles while the second one provides the zeros. Consider one
28
u
terpret the behavior of the time series y(n) solving (5), it suffices
to study the roots of the characteristic polynomials:
12
rier line. Note that an optimal experimental design strives to
accomplish coherently sampled time series. If the system measured
is unknown, coherent sampling is not possible.
One way to explore the dominant periodicities is by means
of the periodogram [2]-which will be handled in the next
section.
Weak Stationarity and Periodogram
The aforementioned paradigm on ARMA modeling assumes
the data to behave weakly stationary [6]-[8]. Weak stationarity
implies that the expected value of the observed time series
is constant and the covariance between two time instants of the
time series only depends on the lagged time between the considered
samples. Formally weak stationarity can be defined as
follows: Consider a time series y(n) such that the time series exists
in the Hilbert space y(n)∈L2
(Ω, A, P) implying that E[|y(n)
y(m)|]<∞. The time series y(n) is weakly stationary if the following
two conditions are satisfied:
Ey [ ( )]n 
Cov ( ), ( )  ( )
y
yn y m E yn y m ( )

y
(6)
   (  m n) (7)
2
where μ denotes the mean-reverting value around which the
time series behaves, 2
 denotes the variance and ρ(.) the autocorrelation
of the time series.
A common misconception is that condition (6) implies that
a time series must remain close to its mean value as a function
of time. The typical mean reversion property simply implies
that the process can drift away from its mean value but must
return and possibly cross this mean value again in the future.
As a result, condition (6) is often over-interpreted. A good
IEEE Instrumentation & Measurement Magazine
June 2023

p
( ) 2| |cos( ) ( 1) | | ( 2) ( )
 yn z y n u n

2
 
which can be solved analytically leading to
 k 
n
y n () 
k


1
sin ( 1)
sin 

zp
zp
Thus, the phase of the pole corresponding to the angular
frequency of the impulse response whereas the magnitude of
the pole corresponds to the damping amplitude of the impulse
response.

| | ( 
z un k)
k
p
 
exp sin
t  the following AR(2) process:
4
g
t

complex conjugate pole pair:
pp | |exp( )pz
zz i


. This leads to
Instrumentation & Measurement Magazine 26-4

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