Instrumentation & Measurement Magazine 26-5 - 19

M x∫ ()= −
N
n
−∞
∞
= ∑()n k[ ]
x −
1
N
n=1
Fig. 1. VDATS Tester: Digital-Analog (DA, left) and Radio Frequency (RF, right),
from [5] and [11], respectively (©2022 and ©2010 IEEE, used with permission).
further consideration. The second type was more tangible as it
compared the received modulated waveforms to be identified
to a predefined set of library waveforms.
Within the feature-based type, eight potential techniques
were identified: 1) Signal Statistics (SS), 2) Higher Order
Statistics-Cumulants (CM), 3) Cyclostationary Features (CS),
4) Multifractal Features (MF), 5) Discrete Wavelet Transform,
6) Constellation Shape, 7) Zero Crossing and 8) Radon Transform.
Examination of these techniques revealed that only the
first four were potentially universally applicable for the identification
of any conceivable modulation format, the goal
that the authors set out to pursue; thus, the remaining techniques
were disregarded due to their limited applicability. In
subsequent research, an additional Fourier Transform of Continuous
Wavelet Transform (FWT) technique was developed
by the authors and added to the acceptable candidate set. After
running numerous simulations on these five techniques,
it was discovered that the best performing ones were the:
1) CM, 2) CS and 3) FWT techniques. The SS had a major problem
of rapid deterioration with increasing noise and the MF
technique required very long simulation times about an order
of magnitude longer that the next most computation intensive
CS and SS techniques. Thus, in the subsequent presentation
the CM, CS and FWT techniques are described in more detail
along with MF technique which has a potential to work great if
the speed of computation can be improved by any means possible
which is the goal of this article.
Higher Order Statistics - Cumulants (CM)
Technique
The first recommended feature-based technique is Higher
Order Statistics whose most common implementation are Cumulants
of any order n [6]. They can be easily calculated from
the following recursive Moment-to-Cumulant formula:
CM ∑(m nm)!
−
nn
= −
m
n
=
−
1
1
− ⋅−)! (
1
(
n 1)!
C Mm nm
−
(1)
where a central moment M of order n (about mean µ) is defined
as:
August 2023
Cyclostationary Features (CS) Technique
The second technique takes advantage of the fact that many
time signal waveforms/processes can be modeled as cyclostationary
rather than stationary due to the underlying
periodicities of the signals [7]. For such processes both their
means and autocorrelations are periodic. A Spectral Correlation
Function (SCF), also known as Spectral Correlation
Density (SCD), can be obtained from the Fourier transform of
the cyclic autocorrelation.
Cyclic spectral analysis deals with second order transformations
of a function and its spectral representation. A time
waveform (process) x(t) is said to exhibit second order periodicityif
spectral components of x(t) exhibit temporal correlation.
A wide sense stationary process x(t) has time invariant autocorrelation
function:
R t E xt x t{}) xx( )
*
x(, )τ =
( ) ( −τ ττR t R t
(, ) = ∀
(3)
The Wiener relationship relates autocorrelation and power
spectral density:
S f FR τ ττπτ2
xx xR e() −jf
( ) { ()} =
=
−∞
∞
∫
d
(4)
A cyclostationary process x(t) (in wide sense) has periodic
mean and autocorrelation function for some period To. The
Wiener relationship can be established for cyclostationary processes
too and is called cyclic Wiener relation:
αα jfα
S f FR τ ττR e() − 2π τ
xx x
( ) { ()} =
=
−∞
∞
∫
where α = n Tno/ ,
(SCF):
N
ˆα
Sf
NT
=
x () = TT
n 0
11∑X nf X nf −
αα



where X n f
T( , ) =
,
−
+
+
2
nT
nT
∫
α
x () =
Sf
α
/
/
2
2



*


,
x u e() −j fu2π
2
du
which is normalized to obtain the Spectral Coherence (SC):
Cf
x ()
 +−

S f Sf
00
xx
IEEE Instrumentation & Measurement Magazine
( /) ( /)
*
αα
22


12
/



(6)
, ,
d
=±± ±12 ,...3 .
This relation leads to discrete spectral correlation function
(5)
N
µµ
µ
n p xdx) () [ ]⋅
n=1
⋅
x(
=∑ x −
n k p xk))
x(
(
(2)
(7)
19
Instrumentation & Measurement Magazine 26-5

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