Instrumentation & Measurement Magazine 26-4 - 29

example which is counter intuitive is the following which
seems non-weakly stationary but it actually is.
Example: Consider a random phase cosine y(n) = A cos(2π Tn
+
ϕ) with sampling period T, amplitude A and phase ϕ∈[0,2 π].
Let ϕ be uniformly distributed in the interval. As a result, we
obtain:
E y n( )

 
2
 2  A Tn x dx) 0
 
1
y n y m E y n y m
2
( ) ( )
2
Cov ( ), ( )     A Tn x)cos(2Tm x)dx
A
1
cos(2 
2
2 0
2 cos(2
cos(2 (Tn m x 
2
) 2 ) cos(2 ( Tn m dx
))
 2 cos(2 ( 
A Tn m))
We conclude as a result, that the random phase cosine is
weakly stationary.
Of course, analytical verification is not possible on a measured
time series. A popular way to visually assess weak
stationarity is by means of the periodogram [2]-[5]. Indeed,
one can segment the measured time series in non-overlapping
windows. For a weakly stationary time series the periodogram
computed per window should not vary too much.
The periodogram can be easily computed. Consider a
measured time series y(n) with n∈{0,1,2,···,N-1}. Partition the
time series in M segments of length L by using the notation
y[j]
ˆ 11
() | ()|
Sk Y k yn i
LL L
[]
y
where S
jj j
n0

ˆ []
[]
2   ( )exp 2
L
[] 


nk
2
(8)
j (.) indicates that the periodogram is computed
from the data specifically for segment j. Indeed, one can
define the power spectrum as follows: Consider a weakly stationary
time series y(n) generated by an absolutely summable
impulse response
 | ( )|

defined as:

Sk r k

[]() 

j
(
) 

[]()
j

yy
k
[]() ([]  
[]
j


j
)


j
(
jj()exp 2i
[] 


nk
L
(9)
where r[] k E yn E yn yn k E yn k
y  is the)
autocorrelation at lag k. Note that an absolutely summable
impulse response is commonly known as a Bounded-inputbounded-output
(BIBO) system.
Equations (8) and (9) are connected by virtue of the WienerKhichine
theorem [9]. By imposing that the impulse response
function satisfies


n 0 n hn| ( )| the following holds:
dlim S[] k  

Ly
ˆ 22
()
S ky()

j


June 2023
y
S k( ) if 
k


1
2

0,

N
2
2
2

if k 0,

N
(10)
Fig. 2. Autoregressive example AR(2).
IEEE Instrumentation & Measurement Magazine
29
n 0 hn , then the power spectrum is
(n)=y(L(j-1)+n). The periodogram is the normalized squared
absolute value of the Fourier coefficients:
where  x) denotes the x-th quantile of the chi-squared distri2
2
(
bution
with 2 degrees of freedom. As a consequence, one can
also consider the confidence interval of an estimated periodogram
in the following way:
 S
2 []i
k Sk    
22 ()

 []
22
i
ˆˆ
() () 1

[]i
2 

Sk

As a result, one can assess whether the periodogram corresponding
to different segments remains in the confidence
interval of the first segment for instance.
In order to illustrate this procedure, we retake the previous
example whose excitation and observed time series was
shown in Fig. 2. Considering eight segments of 128 samples
each results in eight periodograms which can be compared to
the first segment for instance. Fig. 3 shows that the first segment
serving as reference with its uncertainty bounds captures
the computed periodograms of the other seven segments. One
can conclude that weak stationarity holds.
By means of contrast, we consider another example of a
time series which violates the stationarity condition due to a
drifting mean. In this example (Fig. 4a) grip strength is measured
by the classical Jamar Dynamometer (kg). Since grip
where dlim implies limit in distribution whereas  denotes
2
k
the chi-squared distribution with k degrees of freedom. Note
that the result in (10) implies that the periodogram is an unbiased
estimator for the power spectrum but it is inconsistent.
The law-of-large numbers does not apply to the periodogram
[10]. Nonetheless the result allows comparing the periodograms
of different segments. The time series is considered to
be weakly stationary if the following holds for 95% of the frequencies
associated to two segments:
[]


22
22[]
  
22ˆ


ˆ ()
k
Sk
S
i
j
()


1



Instrumentation & Measurement Magazine 26-4

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