FACTA UNIVERSITATIS 
Series: Electronics and Energetics Vol. 32, N

o
 4, December 2019, pp. 539-554 

https://doi.org/10.2298/FUEE1904539P 

© 2019 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND 

ON THE NODE ORDERING OF PROGRESSIVE POLYNOMIAL 

APPROXIMATION FOR THE SENSOR LINEARIZATION

 

Aneta Prijić, Aleksandar Ilić, Zoran Prijić, Emilija Živanović,  

Branislav Randjelović  

University of Niš, Faculty of Electronic Engineering, Niš, Serbia  

Abstract. Many sensors exhibit nonlinear dependence between their input and output 

variables and specific techniques are often applied for the linearization of their transfer 

characteristics. Some of them include additional analog circuits, while the others are 

based on different numerical procedures. One commonly used software solution is 

Progressive Polynomial Approximation. This method for sensor transfer function 

linearization shows strong dependence on the order of selected nodes in the linearization 

vector. There are several modifications of this method which enhance its effectiveness but 

require extensive computational time. This paper proposes the methodology that shows 

improvement over Progressive Polynomial Approximation without additional increase of 

complexity. It concerns the order of linearization nodes in linearization vector. The 

optimal order of nodes is determined on the basis of sensor transfer function concavity. 

The proposed methodology is compared to the previously reported methods on a set of 

analytical functions. It is then implemented in the temperature measurement system using 

a set of thermistors with negative temperature coefficients. It is shown that its 

implementation in the low-cost microcontrollers integrated into the nodes of reconfigurable 

sensor networks is justified. 

Key words: Sensor Linearization, Progressive Polynomial Approximation, 

Reconfigurable Sensor Networks, NTC Thermistor 

1. INTRODUCTION  

Transfer functions of sensors used in measurement systems usually do not have linear 

dependence between input and output variables. In addition, transfer functions often 

change with time. For these reasons, measurement systems based on the sensors exhibit 

various errors such as offset, gain, hysteresis, cross-sensitivity, drift, and non-linearity 

[1], [2]. In order to achieve reliable measurement, these errors should be compensated. 

One approach is to use additional analog circuits to condition sensors output signal [3], 

[4], [5]. However, analog compensation is not always appropriate for sensors integrated 

                                                           
Received February 5, 2019; received in revised form July 1, 2019 

Corresponding author: Aneta Prijić 
Faculty of Electronic Engineering, Aleksandra Medvedeva 14, 18000 Niš, Serbia 

(E-mail: aneta.prijic@elfak.ni.ac.rs) 

  



540 A. PRIJIĆ, A. ILIĆ, Z. PRIJIĆ, E. ŽIVANOVIĆ, B. RANDJELOVIĆ 

into reconfigurable sensor networks [1], [6], [7]. A more flexible solution is to convert 

sensor output into the digital domain, where various numerical linearization methods may 

be applied in the form of compensation algorithms. These methods rely on a set of 

correction functions applied on a so-called linearization vector composed of linearization 

nodes. The effectiveness of the linearization method is evaluated on the basis of the 

number of nodes required to reduce non-linearity below a specific value, computation 

time, and implementation complexity. 

The simplest linearization method is based on a look-up table (LUT) which is also the 

fastest one. However, to obtain a high accuracy of the estimated input value, a high 

number of linearization nodes should be implemented in the LUT, making it memory 

consuming. To reduce memory requirement, a sparse LUT can be combined with an 

interpolation method [8]. A simple method is piece-wise linear interpolation which 

connects each two adjacent LUT values with an appropriate linear function. This type of 

interpolation can be also used for linearization of the whole transfer function. For N 

linearization nodes, sensors inverse transfer function will be represented by N−1 first 

order polynomials [9]. The main disadvantage of this method is a high number of nodes 

required for the linearization of highly nonlinear functions. This implies either a large 

memory requirement or a slow response time of the system [8]. More advanced methods 

are Lagrange, Newton [10] and spline [11] interpolations. However, Lagrange interpolation 

suffers from overfitting effect for polynomials of higher degree and it is generally not 

applicable to highly nonlinear sensors [10], [12]. The Newton method is more flexible 

and efficient when additional linearization nodes are introduced, but it is primarily 

applied for equidistant nodes [13]. On the other hand, spline interpolation is effective, but 

it comes at high implementation costs [14]. 

More effective and more commonly used methods are Progressive Polynomial 

Approximation (PPA) [15], [16] and linearization methods based on the Artificial Neural 

Networks (ANN) [17], [18]. The effectiveness of the PPA method, besides the linearization 

nodes number, depends on the node ordering in the linearization vector. Results obtained 

using the same compensation algorithm, but with the different order of nodes (permutation) 

[19], may vary between almost perfect in some cases, to even increased non-linearity in the 

other. In the case of the ANN method, effectiveness depends on the neural network topology 

and the time needed for its training. 

This paper proposes the methodology that improves the accuracy of the PPA while 

keeping its simplicity. Theoretical background, a summary of PPA, and an overview of 

its modifications are presented in Section 2. Section 3 contains an analysis of the PPA 

method effectiveness considering different permutations of the linearization vector for 

four different functions. Two of these functions are convex and two concave. This is done 

in order to elaborate on the idea that the optimal order of nodes in the linearization vector 

is dependent on the transfer function shape. In such a sense, an extensive computation 

time needed to accomplish the desired linearity by analysis of all permutations may be 

avoided. Experimental support of presented numerical results is given in Section 4, using 

negative temperature coefficient (NTC) thermistors as sensors. 



 On the Node Ordering of Progressive Polynomial Approximation for the Sensor Linearization 541 

2. LINEARIZATION METHODS 

2.1. Underlying theory  

The transfer function of sensors is usually expressed as       , where x is sensor 
input and y is sensor output [20]. The linearization method calculates the desired output 

value     , using the workflow depicted in Fig. 1. Sensor output is first digitized, using 
an analog-to-digital converter (ADC), and then the linearization algorithm is applied. 

Obtained output value      should vary linearly with the sensor input, i.e.         
 , where k is the gain and n (usually 0) is the offset of the desired transfer function 
[1]. All operations are performed by a microcontroller, which is a part of the 

reconfigurable sensor node. 

 

Fig. 1 Workflow of a sensor linearization process 

There are two types of linearization methods, as illustrated in Fig. 2. The first 

involves an estimation of the sensor transfer function           and subsequent 
numerical determination of sensor inverse transfer function which is used to obtain the 

estimated input value          
     . The linearized output value is            .  

Methods of the second type modify sensors output y using a correction function  , so 
the linearized output is calculated as          . Estimated input value, in this case, is 
calculated as            . 

 

Fig. 2 Linearization of a sensor output for two distinct methods. 

The linearization node is a pair of two values: input x and corresponding output y. 

Values are determined experimentally by applying a known stimulus at the sensor input 

and measuring the value of its output. Input values are usually chosen equidistantly, 

starting at the minimal input that a sensor can detect, and ending at the full-scale. Nodes 

are then ordered to form the linearization vector. Linearization coefficients, implemented 

into the correction function     , are determined using the linearization vector, and then 
stored in the memory of a microcontroller. On each measurement, these coefficients are 

used to calculate the linear output value of the sensor. 



542 A. PRIJIĆ, A. ILIĆ, Z. PRIJIĆ, E. ŽIVANOVIĆ, B. RANDJELOVIĆ 

The effectiveness of linearization is evaluated using relative non-linearity [21]: 

              
    

    
     , (1) 

where      is full-scale input and      is the maximum deviation of the real input from 
the ideal transfer characteristic. Due to the nature of the linearization algorithm, non-

linearity will be equal to zero for any of the linearization nodes if a quantization error 

introduced by AD conversion is neglected. Therefore, additional measurements need to be 

performed to form a set of Ne evaluation nodes and calculate the maximum deviation as: 

          (|   
  

 
|)   for i=1,…Ne, (2) 

 

where    is the applied input value and    is the corresponding output value. 

2.2. Progressive Polynomial Approximation (PPA) 

Calculation of the correction function in PPA is a successive process [15]. For each 

linearization node, new correction function, denoted as      , is defined. Therefore, i-th 
function corrects the non-linearity around the i-th node, while keeping the corrections 

introduced by the previously defined functions. The final correction function is the one 

defined for the last node. 

In order to calculate correction function, linear output value ti at each node should be 

defined as: 

            for i = 1, . . . , N. (3) 
 

These values are then used to calculate linearization coefficients for correction functions. 

The first correction function       is defined as: 

            (4) 

where    is the linearization coefficient calculated for the first linearization node: 

             . (5) 

Thus,       adds the value    to the sensor output, eliminating the offset (if present). The 
correction function at i-th node is defined as: 

                 ∏           
   
      for i = 2, . . . , N (6) 

 

and the linearization coefficient is calculated as: 
 

    
           

∏    (  )    
   
   

      for i = 2, . . . , N. (7) 

 

These correction functions eliminate the gain error and successively minimize the 

transfer function non-linearity. Since the final correction function includes all the 

linearization nodes, it will output the desired value for any of these nodes while between 

them linearized output will deviate from the ideal one to some extent. 



 On the Node Ordering of Progressive Polynomial Approximation for the Sensor Linearization 543 

2.3. Modifications of the PPA method 

In PPA, the first two nodes in the linearization vector are chosen from the ends of a 
sensor range, thus eliminating the offset and gain errors [15]. Then, each new node is 
added halfway between the previous two. When the linearization vector ordered in that 
way is used, achieved results are not optimal, but large non-linearity is avoided. Note that 
nodes are not necessarily equidistant. The main advantage of PPA lies in its simplicity, so 
it does not require intensive time-consuming operations. This makes it particularly 
suitable for the implementation in reconfigurable sensor networks. 

Improved Progressive Polynomial Approximation (IMPPA) is the method based on 
permutations of nodes in the initial linearization vector. Each permutation is tested in order 
to find the best one [19]. To reduce the number of arithmetic operations, IMPPA does not 
test all possible permutations of the initial vector. Rather, it fixes the first node from the 
beginning of the sensor range as the first, the node from the end of the range as the second, 
and then permutes the remaining ones. The effectiveness of each permutation is determined 
by the non-linearity obtained at nodes which are inserted between the nodes of the 
linearization vector. A major drawback of this method is increased implementation costs in 
terms of the complexity and computation time. It should be noted that this method finds the 
optimal permutation of the given linearization vector for equidistant nodes. The method can 
be further improved if linearization vector with non-equidistant nodes is used. 

A probability density function is used to improve PPA, as presented in [22], [23]. This 
approach proposes an accumulation of linearization nodes in the part of a transfer 
function that will be used most commonly during a sensor lifetime. This can significantly 
improve measurement system accuracy in some cases, but the problem of the further 
ordering of the selected nodes still remains unsolved. 

A different linearization method inspired by PPA, called Modified Progressive 
Polynomial Approximation (MPPA), is addressed in [24]. Methodology for selection of 
the nodes which does not consider their order in the linearization vector is introduced.  
The larger set of equidistant nodes is formed first. The linearization vector is not 
predefined, but it is populated at each step by the node from the original set at which 
current linearized function deviates most from the linear one. Consequently, selected 
nodes in the linearization vector are not equidistant. 

3. PROPOSED METHODOLOGY 

Proposed modification of the PPA method concerns the order of nodes in the 

linearization vector to obtain the desired transfer function linearity without increasing the 

algorithm complexity. Several analytic expressions commonly used to model sensors 

transfer functions are analyzed. In order to make a comparison of the results, transfer 

functions are normalized before the linearization methodology was applied. Both, input 

(argument) and output (function value) are normalized to range [0, 1]. If x
m
 is sensor input 

and y
m
 is sensor output, normalized input and output values are calculated using the 

following equations: 

   
       

 

    
      

 , (8) 

   
       

 

    
      

 , (9) 



544 A. PRIJIĆ, A. ILIĆ, Z. PRIJIĆ, E. ŽIVANOVIĆ, B. RANDJELOVIĆ 

where     
  and     

  are minimum and full range input values, while     
 and     

 are 
the corresponding output values. 

The initial linearization vector TI is formed as a set of N equidistant nodes starting 
from the beginning of the sensors range. It is expressed as: 

    [                         ], (10) 

or, using shorthand notation:    [       ]. Permutations of the linearization vector 
are denoted as   , i = 1, 2, . . . , N − 1!. 

3.1. Convex functions 

The PPA linearization methodology is applied to an exponential function: 

            ⁄  (11) 

where p is parameter used to adjust its non-linearity (0<p<1). The optimal permutation 
of the linearization vector depends on this parameter value [19]. 

For the basic permutation analysis, linearization vector with N=5 nodes is selected. 
Two permutations of the initial vector are specially considered: 

     [         ]  (12) 
     [         ]. (13) 

Following the IMPPA method, in both permutations, the first node is taken from the 
beginning and the second from the end of the input range. The remaining nodes in 

permutation    are ordered starting from the beginning, while in permutation    starting 
from the end of the range. The linearization method is applied and output non-linearity is 
calculated considering different input function non-linearities. Obtained results are shown 

in Fig. 3. It can be seen that permutation    is more effective since it can reduce the 
output non-linearity below 1% for the input non-linearity up to 24%, while    can 
achieve similar results for the input non-linearity up to 19%. The analysis is extended for 
other permutations and distribution shown in Fig. 4 is obtained. It is evident that output 
non-linearity is strongly dependent on the order of nodes in the linearization vector. For 
example, when the input function non-linearity is 24%, the output non-linearities 
achieved using all permutations vary between satisfactory 0.83%, and quite high 3.12%. 

 

Fig. 3 Output non-linearity vs. input non-linearity for convex exponential function (11). 



 On the Node Ordering of Progressive Polynomial Approximation for the Sensor Linearization 545 

 

Fig. 4 Output non-linearity distribution vs. input function non-linearity and linearization 

vector permutations for convex exponential function (11). 
 

Dependence of the output non-linearity on permutations of the linearization vector is 

also analyzed using a quadratic function: 

             (14) 

where p and q are parameters which determine the input non-linearity (0<p,q<1). 

Obtained results are shown in Fig. 5. A similar dependence of the output non-linearity on 

the order of nodes in the linearization vector is observed, as it was the case with function 

(11). It is important to note that permutation [         ] gives the best results for both 
analyzed functions for all input non-linearity values. 

 

Fig. 5 Output non-linearity distribution vs. input function non-linearity and linearization 

vector permutations for convex quadratic function (14). 



546 A. PRIJIĆ, A. ILIĆ, Z. PRIJIĆ, E. ŽIVANOVIĆ, B. RANDJELOVIĆ 

Permutations that will give the best results for both functions and all input non-

linearity values can be identified for other numbers of nodes in the linearization vector as 

well. Those permutations are listed below: 

             [       ] 
             [         ] 
             [           ] 
             [             ] 
             [               ] 

The proposed methodology is compared with the PPA and IMPPA methods. 

Maximum input non-linearity that can be compensated (reduced below 1%) is shown in 

Fig. 6 for function (11), and in Fig. 7 for function (14). These figures show that higher 

input non-linearity can be compensated by the proposed methodology than with PPA. 

The methodology is somewhat less efficient than IMPPA for the higher number of nodes 

since it is related to the transfer function shape and does not necessarily distinguish the 

best one from the set of all permutations. On the other hand, its implementation is much 

simpler. This is evident comparing the number of arithmetical operations required to 

calculate the linearization coefficients of the considered methods, as presented in Tab. 1. 

It is evident that improvement is achieved at a negligible cost regarding the number of 

required arithmetical operations. The linearization vectors with the maximum of eight 

nodes are considered, due to the fact that the IMPPA algorithm requires a large number 

of arithmetical operations that should be executed within the microcontroller. This makes 

it unsuitable for applications based on the low-cost microcontrollers. 

 

Fig. 6 Compensated non-linearity of the convex exponential function (11) vs. the number 

of nodes in the linearization vector. 



 On the Node Ordering of Progressive Polynomial Approximation for the Sensor Linearization 547 

 

Fig. 7 Compensated non-linearity of the convex quadratic function (14) vs. the number of 

nodes in the linearization vector. 

Table 1 Number of arithmetical operations required to calculate the linearization coefficients 

for the different number of nodes. 

N PPA Proposed IMPPA 
5 121 146 726 
6 320 350 8400 
7 841 871 100920 
8 2205 2245 1587600 

3.2. Concave functions 

Permutations (12) and (13) are also used for the linearization of the concave function: 

          (15) 

with different values of the non-linearity parameter p and results are shown in Fig 8. In 

this case permutation    is more effective (opposite to the results shown in Fig. 3).  

 

Fig. 8 Output non-linearity vs. input non-linearity for concave exponential function (15). 



548 A. PRIJIĆ, A. ILIĆ, Z. PRIJIĆ, E. ŽIVANOVIĆ, B. RANDJELOVIĆ 

Analysis of the quadratic function: 

             (16) 

also gives permutation    as better for the linearization. 
Following the same procedure as for the convex functions, it is possible to identify 

the best permutations of the linearization vector for the concave functions: 

             [       ] 
             [         ] 
             [           ] 
             [             ] 
             [               ] 

The proposed methodology is again compared with PPA and IMPPA. In order to 

compare these methods, the maximum value of the input non-linearity that can be reduced 

below 1% using different lengths of the linearization vector is calculated. This value is 

plotted as a function of the number of linearization nodes in Fig. 9 for the function (15) and 

in Fig. 10 for the function (16). It can be seen that the improvement over the PPA is in this 

case even greater than it was in the case of convex functions. Also, the improvement 

offered by IMPPA over the proposed methodology remains small. 

 

Fig. 9 Compensated non-linearity of the concave exponential function (15) vs. the 

number of nodes in the linearization vector. 

 

Fig. 10 Compensated non-linearity of the concave quadratic function (16) vs. the number 

of nodes in the linearization vector. 



 On the Node Ordering of Progressive Polynomial Approximation for the Sensor Linearization 549 

The simplified flowchart of the calibration procedure by the proposed method is 

presented in Fig. 11. The sign of the second derivative of the transfer function is used to 

distinguish between the concave or convex property. It is easily determined by numerical 

differentiation at the set of normalized measured values xi and yi. The sensor transfer 

functions with the inflection point can be treated as a union of segments with the specific 

convex or concave property. Particular segments of the transfer function can be 

independently linearized by the proposed method, which encounters the increased number of 

linearization nodes. Therefore, it is cost effective to employ the PPA method for such 

functions, even though it gives less linearized output. However, in real measurements, a very 

few sensor transfer characteristics have the inflection point in their specific application range. 

 

Fig. 11 The simplified flowchart of the calibration procedure by the proposed method 



550 A. PRIJIĆ, A. ILIĆ, Z. PRIJIĆ, E. ŽIVANOVIĆ, B. RANDJELOVIĆ 

4. CASE STUDY 

A temperature measurement system using an NTC thermistor is chosen for testing of 

the proposed method. NTC thermistor has highly nonlinear dependence of the resistance 

on temperature: 

      
 (

 

 
 

 

  
)
 , (17) 

where R0 is resistance at the reference temperature T0 and β is material–specific constant. 

Block diagram and a photo of the experimental setup are shown in Fig. 12. The sensing 

part of the system is constructed using NTC thermistor in the voltage divider configuration 

with referent resistance Rref. It gives a convex shape of the transfer function [25]: 

      
 

      
, (18) 

where y is the normalized output voltage of the divider, x = 1/T, B = β and A is a 

constant dependent on the voltage divider configuration: 

   
   

  
 
  

    
. (19) 

The microcontroller is used to convert the output voltage of the sensing part into a digital 

domain and then to apply the linearization algorithm to the transfer function. In order to 

calculate non-linearity after the linearization, K–type thermocouple with ±0.75% accuracy 

connected through NI-USB-TC01 interface is used as a reference [26]. The temperature in 

the furnace is varied and measured using both, the NTC sensor and the referent 

thermocouple. Results are then transferred to a PC, where non-linearity is calculated by the 

software. 

 

a) 

 

b) 

Fig. 12 Experimental setup: a) Block diagram, b) Photograph. 



 On the Node Ordering of Progressive Polynomial Approximation for the Sensor Linearization 551 

Four different NTC thermistors are employed within the sensing part to achieve 
different non-linearities of the transfer function. Thermistors parameters are listed in 
Table 2, while Fig. 13 shows their transfer functions non-linearities as a function of 
temperature for Rref=R0. The proposed methodology of linearization is applied for the 
different number of linearization nodes. The results for the thermistor R1 calculated using 
five linearization nodes are shown in Fig. 14. Relative non-linearity of the linearized 
output as a function of the normalized input for PPA and the proposed methodology are 
presented. It can be seen that using the proposed methodology the non-linearity was 
reduced to a value of 0.49%, while PPA reduced it to a value of 1.39%. This is a 
significant improvement, achieved without increased complexity. 

Table 2 Parameters of the considered NTC thermistors  

Parameter R1 R2 R3 R4 

R0 (kΩ) at T0=298K 1 10 100 470 

 (K) 3730 4300 4600 5000 

 
Fig. 13 Sensor transfer function non-linearity vs. temperature. 

 

Fig. 14 Output non-linearity of the transfer function vs. the normalized input range for  

N = 5 linearization nodes and thermistor R1. 



552 A. PRIJIĆ, A. ILIĆ, Z. PRIJIĆ, E. ŽIVANOVIĆ, B. RANDJELOVIĆ 

Summary of the linearization results is shown in Tabs. 3 and 4. Maximal output non-

linearities of the transfer function obtained by three methods for the different numbers of 

linearization nodes are listed. It is evident that the proposed methodology performs better 

than PPA in almost all considered cases. Also, an interesting observation is that in the 

case of thermistor R3 and seven linearization nodes, PPA gives even better results than 

IMPPA. This is possible since PPA does not use equidistant nodes. Therefore, the 

combination of nodes used by PPA can, in some cases, give better results than the 

optimal permutation of the linearization vector with equidistant nodes, used by IMPPA. 

 

Table 3 Linearization results for NTC thermistors R1 and R2 

 

N 
R1 R2 

PPA Proposed IMPPA PPA Proposed IMPPA 

5 1.39% 0.49% 0.49% 12.5% 1.93% 1.93% 
6 0.34% 0.32% 0.31% 1.58% 1.53% 0.88% 
7 1.04% 0.91% 0.30% 1.15% 1.05% 0.20% 
8 0.31% 0.26% 0.08% 0.69% 0.37% 0.12% 

 

Table 4 Linearization results for NTC thermistors R3 and R4 

 

N 
R3 R4 

PPA Proposed IMPPA PPA Proposed IMPPA 

5 6.25% 1.13% 1.13% 5.23% 1.54% 1.54% 
6 1.14% 0.77% 0.56% 0.90% 0.48% 0.44% 
7 0.66% 0.84% 0.73% 0.59% 0.43% 0.37% 
8 0.42% 0.19% 0.15% 0.89% 0.40% 0.28% 

5. CONCLUSION 

Progressive Polynomial Approximation is analyzed as a method for linearization of 

sensors. Methods based on PPA could achieve low output non-linearity only by using a 

large number of permutations of the linearization vector. It is found that the number of 

permutations of the linearization vector, required to accomplish the desired output non-

linearity, may be significantly reduced by taking into account the shape of the sensors 

transfer function. By using numerical experiment, the optimal order of nodes in the 

linearization vector is determined for convex and concave exponential and quadratic 

functions. It is shown that non-linearity could be reduced to lower values than obtained 

using PPA, with the negligible increase of complexity in the linearization algorithm. 

Comparing to the IMPPA, the proposed methodology is somewhat less efficient for the 

higher number of nodes, but it requires much lower computational resources. Therefore, 

the proposed methodology is suitable for implementation in low-cost microcontrollers 

integrated into the nodes of reconfigurable sensor networks. 

Acknowledgement: This work was supported in part by the Serbian Ministry of Education, Science 

and Technological Development under Grant TR32026 and in part by Ei PCB Factory, Niš, Serbia.  



 On the Node Ordering of Progressive Polynomial Approximation for the Sensor Linearization 553 

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