Instrumentation & Measurement Magazine 23-9 - 32

Table 3 - The allowed values, the returned values by the CCD analysis and the rounded values
of the Bw and CR factors, used to carry out the design points
Factor

Allowed Values

Returned Five Values by the CCD Analysis

Rounded Five Values

[125, 250, 500]

[125, 219, 313, 406, 500]

[125, 250, 250, 500, 500]

[1, 2, 3, 4]

[1, 1.75, 2.5, 3.25, 4]

[1, 2, 2, 3, 4]

variable. As an example, in Fig. 3, graphic representation of the
CCD design consisting of two factors, X1 and X2, on five levels,
 
2 1.414 .
is shown. For two factors
Once the experiments indicated by the CCD are carried out,
the obtained measurements can be used to identify the whole
response surface by applying a linear regression technique:
y   0   1X1   2 X 2     k X k
 11X12   22 X22     kk Xk2

(2)

 12 X1X2     k1,k X k1Xk  

where y is the measured response, β are the regressors, X1, X2,
. . ., Xk stand for the experimental variables (k is the number of
factors) and ε indicates the residual error associate with the
model.

Experimental Results
LoRa performance assessment has been carried out by applying the CCD technique and considering 4 factors: SF; Bw;
CR and SNR. On the contrary, values of Preamble Length
(PL), Header, CRC, and Payload, have been fixed to the following values: PL=1 byte; Header=Explicit; CRC=Enabled;
Payload=1 byte, respectively.
For a CCD at 4 factors, ±α = ±2k/4 = ±2. Therefore, the arbitrary unit interval is: [-2, -1, 0, 1, 2]. The execution of CCD at 4
factors requires to perform only 36 experiments instead of the
625 associated with a full factorial experiment, with a consequent notable reduction of the experimental burden. However,
the application of CCD to the characterization of LoRa wireless
technology, in presence of noise, implies some approximation
since the involved factors can assume only discrete values in
the definition interval. As an example, factor CR can assume

only 4 actual values: 1; 2; 3; 4, while CCD require at least five
values. According to the CCD analysis, the maximum value
of the actual interval (equal to 4) corresponds to the maximum encoded value of the arbitrary interval (equal to 2), and
the minimum value of the actual interval (equal to 1) is associated with the minimum encoded value of the arbitrary interval
(equal to -2); the CCD analysis returns five experimental configurations that are not acceptable and selectable for the factor
CR : [1, 1.75, 2.5, 3.25, 4]. To overcome the considered limitation, these values have been rounded to the closest available
integer, obtaining thus, for the factor CR, the five-level of actual interval: [1, 2, 3, 3, 4]. The same approach has been applied
to carry out suitable values for the Bw factors. In the following
Table 3, the allowed values, the returned values by the CCD
analysis and the rounded values of the Bw and CR factors,
used to carry out the design points, are shown.
As for the configuration parameter SF, which can assume
6 different values, unlike Bw and CR, the five-level interval
has been obtained by matching the values 11 of the SF with the
maximum encoded value of the arbitrary interval (equal to 2),
and the value 7 of the SF with the minimum encoded value of
the arbitrary interval (equal to -2). Therefore, the five levels of
the SF factor used in the CCD analysis are: 7; 8; 9; 10; 11. The
five levels adopted for the SNR factor are: -15; -10; -5; 0; 5; (dB).
In the following, CCD's results are analyzed and discussed
in detail.
In Table 4, the experimental results obtained for each axial
point (2*4=8 runs) of the CCD analysis, are shown. As stated
above, axial points involve experimental configuration characterized by one parameter at a time, equal to the extreme value
while all others are set to their mean value.
The runs indicated as 1 and 2 in Table 4 refer to results carried out by varying the SNR level from the minimum value of

Table 4 - Experimental results obtained for each axial point design of the CCD analysis
RUNS

SNR

LOST
PACKETS

% LOST
PACKETS

-15

250

10.000

3460

34.6

250

10.000

-5

250

10.000

2610

26.1

-5

250

10.000

0.1

-5

125

10.000

-5

500

10.000

2620

26.2

-5

250

10.000

1150

11.5

-5

250

10.000

680

6.8

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Instrumentation & Measurement Magazine 23-9

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