Digital Algebraic Method for Processing Complex Signals for Radio Monitoring Systems

*Sergey G. Rassomakhin*

### **Abstract**

The methods of processing digital samples of complex structure signals with unknown parameters are considered. With the use of algebraic methods, the following tasks are sequentially solved: clock synchronization, determining the range of carrier frequencies, the multiplicity of phase modulation, and obtaining a stream of information bits. The methods for improving the quality of processing digital samples of signals based on solving special overdetermined systems of linear equations are proposed. The estimation of efficiency of the offered method is carried out by an imitation statistical modeling. The advantages of the proposed methods of signal processing for the telecommunications and radio monitoring systems are shown.

**Keywords:** orthogonal frequency-division multiplexing, digital sampling, linear algebraic equations, correlation convolutions

### **1. Introduction**

The construction of effective information transmission systems is inextricably linked with the problem of intensifying the usage of the time and frequency-energy resource of communication channels. One of the ways to solve this problem is using the complex signals with combined types of modulation in combination with the methods of spectrum narrowing and noise-resistant coding [1–5]. In this connection, the structure of the signals used to transmit information is becoming complicated, and, consequently, the algorithms of their processing are becoming complicated as well.

The most promising type of signal code constructions in wireless networks is orthogonal frequency-division multiplexing (OFDM) [2–5]. The basic idea of building such signals is arranging a set of mutually orthogonal frequency subchannels so that, on the one hand, one subchannel does not interfere with the other and, on the other hand, the spectra of the subchannels overlap. Due to the orthogonality of the linear subchannels, each of them can be considered independently of the others. Errors caused by the interference in one of the subchannels do not lead to errors in the other. As a result, only a small part of the transmitted information is distorted. Error-correcting coding being used, the errors can be corrected. The structure of signals with multiple simultaneously operating subcarrier frequencies has well established itself in conditions of heterogeneity of the propagation medium. In recent years, the capabilities of systems with OFDM

signals have evolved significantly. Such signals began to be used in a wide variety of telecommunication systems operating in different radio frequency bands.

The complex structure of such signals and the a priori uncertainty of the channel properties cause significant difficulties in solving the problems of radio control and radio monitoring [6–11]. Similar tasks also arise when processing images and signals in biotechnical systems [12–16]. A distinctive feature of such tasks is the absence of data on the structure and informative parameters of the measured signals. This information should be obtained from the results of the study, with high accuracy and as soon as possible. Therefore, the tasks of developing mathematical methods for analyzing complex signals based on digital measurement sequences are highly relevant. This is evidenced by the intensive development of special software systems [17].

### **2. Mathematical model of OFDM signals**

For correct choice of the methods for digital analysis of the primary parameters of OFDM signals, a brief description of their basic properties is necessary. Arbitrary OFDM signal *Sj*ð Þ*t* on *j*-th modulation interval *Tp* is formed by algebraic summation of the several harmonic oscillations of the same amplitude. Each of the oscillations has *m* options of modulation phase shift. The value *m* determines the multiplicity of the used phase (PM) modulation and corresponds to the base of the numerical source code. Commonly, *<sup>m</sup>* <sup>¼</sup> <sup>2</sup>*<sup>k</sup>* where *<sup>k</sup>* is the number of binary symbols (bits) represented by the elementary signal on one modulation interval. When using relative phase coding and a unit value of the amplitude of the oscillation subcarriers, the mathematical model of the signal can be represented as the following sequence:

$$S\_{\hat{j}}(t) = \sum\_{i=0}^{n\_f - 1} \sin \left\{ 2\pi \left( f\_0 + \frac{i}{T} \right) \cdot \left( t - T\_p \left\lfloor \frac{t}{T\_p} \right\rfloor \right) + \rho\_{j,i} \right\},\tag{1}$$

*P* ¼ *f* <sup>0</sup>modð Þ Δ*f :* (3)

For existing OFDM standards, parameter (3) can take two values that determine the sign of the cyclic prefix: when *P* ¼ 0 the prefix is positive and when *P* ¼ Δ*f =*2 inverse. **Figure 1** gives a qualitative idea of the form of the signal envelope constructed in accordance with model (1) on two adjacent modulation intervals with *nf* ¼ 16 and *Tp* ¼ 1*,* 47 � *T*. At the end of each of the intervals *Tp*, the inverse cyclic continuation of the signal with duration Δ*T* is located, which repeats, up to a sign, the shape of the initial segment of the signal on the modulation interval. For the example in question, *P* ¼ Δ*f =*2; therefore the prefix part is the inverse of the

*Digital Algebraic Method for Processing Complex Signals for Radio Monitoring Systems*

**3. The main stages of the structural analysis of OFDM signals**

A comprehensive analysis of the properties of complex signals is advisable to implement on the basis of phased processing. At each stage, only a part of the signal parameters is determined. Given the fact that OFDM signals contain a prefix, it is advisable to use the correlation method for determining structural time parameters *Tp* and *T* at the first stage. This technique is based on the principle of "sliding" time window. This makes it possible to determine the following parameters of an OFDM signal: the value of the orthogonality interval, the duration of the modulation interval, and the value of the frequency spacing between the channels.

At the second stage of the analysis, the tasks of determining the number and the values of service and information channel frequencies, as well as, the signal phase

The two-stage processing results in the possibility to extract an information flow from signals of an a priori unknown structure without using traditional fast Fourier

**3.1 Correlation method for determining the time parameters of OFDM signals**

assessment of *Tp*—the most likely period of the emergence of "bursts" of

We propose a correlation method for determining structural time parameters *Tp* and *T*. The basis is the "sliding time window" principle. The most probable value of the time interval between the most correlated segments (with the "+" or "�" sign) of the segments from the digital sample of signal measurements is determined. The

initial part of the signal.

**Figure 1.**

*Example of the OFDM signal.*

*DOI: http://dx.doi.org/10.5772/intechopen.85590*

demodulation.

**29**

transform (FFT) algorithms.

where *t* is the current time, *f* <sup>0</sup> is the lowest subcarrier frequency in the signal spectrum, *T* ¼ 1*=*Δ*f* is inverse of the minimum subcarrier spacing Δ*f*, *nf* is the number of frequencies used, and *φj,i* is the value of the manipulation angle of *i*-th fluctuation on *j*-th modulation interval. This angle can take one of the *m* values depending on the manipulation code used. The informative features in the signal described by model (1) are relative phase jumps in carrier frequencies. These jumps are measured for each of the frequency subcarriers separately: *φj,i* � *φ<sup>j</sup>*�1*,i*, *i* ¼ 0*,* …*, nf* � 1. The time parameters of the modulation interval used in model (1) are tied by the relation

$$T\_p = T + \Delta T = \frac{1}{\Delta f} + \Delta T,\tag{2}$$

where Δ*T* is the duration of the prefix part of the signal. The prefix part (hereinafter the prefix) is a repeating (to the exact sign) initial part of the signal added at the end of the modulation interval *Tp*. Prefix structure is used to facilitate synchronization in the presence of channel irregularities. For OFDM synchronization violations, the signal can be correctly received and processed for any segment with the duration of *T* within the full modulation interval *TP*. Commonly, the choice of the prefix duration on the modulation interval corresponds to the ratio Δ*T* ¼ ð Þ 0*;* 1 ÷ 0*;* 5 *T*. The sign of the prefix depends on the value of the following parameter:

*Digital Algebraic Method for Processing Complex Signals for Radio Monitoring Systems DOI: http://dx.doi.org/10.5772/intechopen.85590*

**Figure 1.** *Example of the OFDM signal.*

signals have evolved significantly. Such signals began to be used in a wide variety of

The complex structure of such signals and the a priori uncertainty of the channel properties cause significant difficulties in solving the problems of radio control and radio monitoring [6–11]. Similar tasks also arise when processing images and signals in biotechnical systems [12–16]. A distinctive feature of such tasks is the absence of data on the structure and informative parameters of the measured signals. This information should be obtained from the results of the study, with high accuracy and as soon as possible. Therefore, the tasks of developing mathematical methods for analyzing complex signals based on digital measurement sequences are highly relevant. This is evidenced by the intensive development of special software

For correct choice of the methods for digital analysis of the primary parameters of OFDM signals, a brief description of their basic properties is necessary. Arbitrary OFDM signal *Sj*ð Þ*t* on *j*-th modulation interval *Tp* is formed by algebraic summation of the several harmonic oscillations of the same amplitude. Each of the oscillations has *m* options of modulation phase shift. The value *m* determines the multiplicity of the used phase (PM) modulation and corresponds to the base of the numerical source code. Commonly, *<sup>m</sup>* <sup>¼</sup> <sup>2</sup>*<sup>k</sup>* where *<sup>k</sup>* is the number of binary symbols (bits) represented by the elementary signal on one modulation interval. When using relative phase coding and a unit value of the amplitude of the oscillation subcarriers, the mathematical model of the signal can be represented as the following sequence:

> *i T*

where *t* is the current time, *f* <sup>0</sup> is the lowest subcarrier frequency in the signal spectrum, *T* ¼ 1*=*Δ*f* is inverse of the minimum subcarrier spacing Δ*f*, *nf* is the number of frequencies used, and *φj,i* is the value of the manipulation angle of *i*-th fluctuation on *j*-th modulation interval. This angle can take one of the *m* values depending on the manipulation code used. The informative features in the signal described by model (1) are relative phase jumps in carrier frequencies. These jumps

are measured for each of the frequency subcarriers separately: *φj,i* � *φ<sup>j</sup>*�1*,i*,

*Tp* <sup>¼</sup> *<sup>T</sup>* <sup>þ</sup> <sup>Δ</sup>*<sup>T</sup>* <sup>¼</sup> <sup>1</sup>

where Δ*T* is the duration of the prefix part of the signal. The prefix part (hereinafter the prefix) is a repeating (to the exact sign) initial part of the signal added at the end of the modulation interval *Tp*. Prefix structure is used to facilitate synchronization in the presence of channel irregularities. For OFDM synchronization violations, the signal can be correctly received and processed for any segment with the duration of *T* within the full modulation interval *TP*. Commonly, the choice of the prefix duration on the modulation interval corresponds to the ratio Δ*T* ¼ ð Þ 0*;* 1 ÷ 0*;* 5 *T*. The sign of the prefix depends on the value of the following

*i* ¼ 0*,* …*, nf* � 1. The time parameters of the modulation interval used in model (1)

Δ*f*

� *t* � *Tp*

*t Tp*

þ *φj,i*

þ Δ*T,* (2)

*,* (1)

telecommunication systems operating in different radio frequency bands.

systems [17].

**2. Mathematical model of OFDM signals**

*Modulation in Electronics and Telecommunications*

*Sj*ðÞ¼ *t* ∑

are tied by the relation

parameter:

**28**

*nf* �1

*i*¼0

sin 2*π f* <sup>0</sup> þ

$$P = f\_0 \bmod(\Delta f). \tag{3}$$

For existing OFDM standards, parameter (3) can take two values that determine the sign of the cyclic prefix: when *P* ¼ 0 the prefix is positive and when *P* ¼ Δ*f =*2 inverse. **Figure 1** gives a qualitative idea of the form of the signal envelope constructed in accordance with model (1) on two adjacent modulation intervals with *nf* ¼ 16 and *Tp* ¼ 1*,* 47 � *T*. At the end of each of the intervals *Tp*, the inverse cyclic continuation of the signal with duration Δ*T* is located, which repeats, up to a sign, the shape of the initial segment of the signal on the modulation interval. For the example in question, *P* ¼ Δ*f =*2; therefore the prefix part is the inverse of the initial part of the signal.

### **3. The main stages of the structural analysis of OFDM signals**

A comprehensive analysis of the properties of complex signals is advisable to implement on the basis of phased processing. At each stage, only a part of the signal parameters is determined. Given the fact that OFDM signals contain a prefix, it is advisable to use the correlation method for determining structural time parameters *Tp* and *T* at the first stage. This technique is based on the principle of "sliding" time window. This makes it possible to determine the following parameters of an OFDM signal: the value of the orthogonality interval, the duration of the modulation interval, and the value of the frequency spacing between the channels.

At the second stage of the analysis, the tasks of determining the number and the values of service and information channel frequencies, as well as, the signal phase demodulation.

The two-stage processing results in the possibility to extract an information flow from signals of an a priori unknown structure without using traditional fast Fourier transform (FFT) algorithms.

#### **3.1 Correlation method for determining the time parameters of OFDM signals**

We propose a correlation method for determining structural time parameters *Tp* and *T*. The basis is the "sliding time window" principle. The most probable value of the time interval between the most correlated segments (with the "+" or "�" sign) of the segments from the digital sample of signal measurements is determined. The assessment of *Tp*—the most likely period of the emergence of "bursts" of

**Figure 2.** *Calculation scheme.*

correlation in the process of moving the viewing window on the samples of the array of measurements *<sup>Q</sup>* <sup>¼</sup> *<sup>q</sup>*0*; <sup>q</sup>*1*;* … � �—is determined as well. The scheme of the calculation procedure is presented in **Figure 2**. For the correlation analysis, the two vectors, each containing *K* elements of array *Q* in two non-overlapping time observation windows of the signal, are formed:

$$\begin{aligned} Y\mathbf{0} &= \{q\_j, q\_{j+1}, \dots q\_{j+K-1}\}, \\ Y\mathbf{1} &= \{q\_{j+i+K}, q\_{j+i+K+1}, \dots q\_{j+i+2K-1}\}, \end{aligned} \tag{4}$$

*сos Y*ð Þ¼ 0*н; Y*1*<sup>н</sup>* ∑

*v*1*<sup>j</sup>* ¼ *match min Vj*

*v*2*<sup>j</sup>* ¼ *match max Vj*

elements of the vectors *V*1 and *V*2 are formed:

*Num* ¼

8 ><

>:

orthogonality interval of the signal *T*. It gives an opportunity to find two

correlation coefficient between these parts is �1.

*DOI: http://dx.doi.org/10.5772/intechopen.85590*

the rules

**Figure 3.**

**31**

*Cross-correlation coefficients.*

*K*�1 *i*¼0

*Digital Algebraic Method for Processing Complex Signals for Radio Monitoring Systems*

Since the prefix is a repetitive (up to sign) part of the OFDM signal, ideally the

**Figure 3** shows the distribution of the values of the elements of the vector *Vj* calculated according to a specific implementation OFDM signal (16 carrier frequencies, a modulation rate—75 bauds) at *M* ¼ 200. The presence of pronounced extreme values which are close in magnitude to unity is obvious. According to the results of calculations when *j* ¼ 0…*L*, the two new vectors *V*1 ¼ f g *v*10*; v*11*;* …*; v*1*<sup>L</sup>* and *V*2 ¼ f g *v*20*; v*21*;* …*; v*2*<sup>L</sup>* are formed. Their elements are calculated according to

> � �*;Vj* � �

� �*;Vj* � �

*X* equal to *x*, where the index 0 in the function (8) indicates a selection of the element with a minimum sequence number, if there are several such elements in the vector. The elements of the vector *V*1 represent the number of sampling intervals that fit between the initial elements *Y*0 and *Y*1 with minimal (negative) correlation on *j*-th step of moving the observation window. Accordingly, the elements of the vector *V*2 are calculated for the maximum (positive) correlation of the vectors *Y*0 and *Y*1. Simultaneous determination of the maximum and minimum is necessary to reveal the value of function (3). It is obvious that the elements of the vectors *V*1 and *V*2 defined by expression (8) can take values only in the range *K*…*K* þ *M*. To study the statistical distribution of the values of the elements, the histograms for the

Here the function *match x*½ � *;X* calculates the indices of the elements of the vector

*H*1 ¼ f g *h*10*; h*11*;* …*; h*1*<sup>K</sup>*þ*<sup>M</sup> ,*

Obtaining distributions (9) gives an opportunity to estimate the most likely value *Num*—the number of sampling intervals between the initial measurements of the segments of the digital sample *Q* with maximal (positive or negative) correlation:

> *match max H* ½ � ð Þ1 *; H*1 <sup>0</sup> h i *max H*ð Þ1 ≥ *max H*ð Þ2

> *match max H* ½ � ð Þ2 *; H*2 <sup>0</sup> h i *max H*ð Þ1 < *max H*ð Þ2

The value *Num* determines the number of sampling intervals that fit on the

*<sup>H</sup>*<sup>2</sup> <sup>¼</sup> f g *<sup>h</sup>*20*; <sup>h</sup>*21*;* …*; <sup>h</sup>*2*<sup>K</sup>*þ*<sup>M</sup> :* (9)

*;*

*:*

(10)

<sup>0</sup> þ *K;*

*Y*0*н<sup>i</sup>* � *Y*1*н<sup>i</sup>* ¼ *r Y*ð Þ 0*н; Y*1*<sup>н</sup> :* (7)

<sup>0</sup> <sup>þ</sup> *K; <sup>j</sup>* <sup>¼</sup> <sup>0</sup>…*L:* (8)

removed from each other by *i, i* ¼ 0*,* …*, M*. The position of the second time window corresponding to the vector *Y*1 is determined by the successive change in the offset index *i* ¼ 0*,* …*, M* that ensures its "slip" along the signal sample *Q* at each of the values *j* ¼ 0…*L*.

For a wide range of analyzed signals, for example, for the 0.3–3.4 KHz frequency band with minimum quality ADC, the most universal limits of the values of these parameters, resulting in a quick and accurate assessment, are *K* ¼ 10 ÷ 30, *M* ¼ 200 ÷ 300, and *L* ¼ 1000.

At each value of index *j* (moving the window slip area), the *M* þ 1 dimensional vector is being formed:

$$\mathbf{V}\_{j} = \left\{ \boldsymbol{v}\_{0}^{j}, \boldsymbol{v}\_{1}^{j}, \dots, \boldsymbol{v}\_{M}^{j} \right\},\tag{5}$$

the elements of which are the coefficients of mutual correlation of vectors *Y*0 and *Y*1. The calculations (according to **Figure 2**) are performed after centering and normalizing the vectors by the formulas

$$\begin{aligned} Y\mathbf{0}\_N &= \left[ Y\mathbf{0}\_j - \frac{\mathbf{1}}{K} \sum\_{i=0}^{K-1} Y\mathbf{0}\_i \right] \cdot \left[ \frac{1}{K} \sum\_{i=0}^{K-1} Y\mathbf{0}\_i \right]^{-1}; \\ Y\mathbf{1}\_N &= \left[ Y\mathbf{1}\_j - \frac{\mathbf{1}}{K} \sum\_{i=0}^{K-1} Y\mathbf{1}\_i \right] \cdot \left[ \frac{1}{K} \sum\_{i=0}^{K-1} Y\mathbf{1}\_i \right]^{-1}. \end{aligned} \tag{6}$$

The resulting vectors *Y*0*<sup>N</sup>* and *Y*1*<sup>N</sup>* in normalized space have the same length, equal to ffiffiffi *К* <sup>p</sup> , and the cosine of the angle between these vectors is equal to the cross-correlation coefficient:

*Digital Algebraic Method for Processing Complex Signals for Radio Monitoring Systems DOI: http://dx.doi.org/10.5772/intechopen.85590*

$$\cos(Y\mathbf{0}\_n, Y\mathbf{1}\_n) = \sum\_{i=0}^{K-1} Y\mathbf{0}\_{ni} \cdot Y\mathbf{1}\_{ni} = r(Y\mathbf{0}\_n, Y\mathbf{1}\_n). \tag{7}$$

Since the prefix is a repetitive (up to sign) part of the OFDM signal, ideally the correlation coefficient between these parts is �1.

**Figure 3** shows the distribution of the values of the elements of the vector *Vj* calculated according to a specific implementation OFDM signal (16 carrier frequencies, a modulation rate—75 bauds) at *M* ¼ 200. The presence of pronounced extreme values which are close in magnitude to unity is obvious. According to the results of calculations when *j* ¼ 0…*L*, the two new vectors *V*1 ¼ f g *v*10*; v*11*;* …*; v*1*<sup>L</sup>* and *V*2 ¼ f g *v*20*; v*21*;* …*; v*2*<sup>L</sup>* are formed. Their elements are calculated according to the rules

$$\begin{aligned} v1\_j &= \operatorname{match} \left[ \min \left( V\_j \right), V\_j \right]\_0 + \mathbf{K};\\ v2\_j &= \operatorname{match} \left[ \max \left( V\_j \right), V\_j \right]\_0 + \mathbf{K}; \end{aligned} \quad j = \mathbf{0}...\mathbf{L}. \tag{8}$$

Here the function *match x*½ � *;X* calculates the indices of the elements of the vector *X* equal to *x*, where the index 0 in the function (8) indicates a selection of the element with a minimum sequence number, if there are several such elements in the vector. The elements of the vector *V*1 represent the number of sampling intervals that fit between the initial elements *Y*0 and *Y*1 with minimal (negative) correlation on *j*-th step of moving the observation window. Accordingly, the elements of the vector *V*2 are calculated for the maximum (positive) correlation of the vectors *Y*0 and *Y*1. Simultaneous determination of the maximum and minimum is necessary to reveal the value of function (3). It is obvious that the elements of the vectors *V*1 and *V*2 defined by expression (8) can take values only in the range *K*…*K* þ *M*. To study the statistical distribution of the values of the elements, the histograms for the elements of the vectors *V*1 and *V*2 are formed:

$$\begin{aligned} H1 &= \{h1\_0, h1\_1, \dots, h1\_{K+M}\}, \\ H2 &= \{h2\_0, h2\_1, \dots, h2\_{K+M}\}. \end{aligned} \tag{9}$$

Obtaining distributions (9) gives an opportunity to estimate the most likely value *Num*—the number of sampling intervals between the initial measurements of the segments of the digital sample *Q* with maximal (positive or negative) correlation:

$$Num = \begin{cases} \operatorname{match}[\max(H1), H1]\_0; \\ \quad \langle \max(H1) \ge \max(H2) \rangle \\ \quad \operatorname{match}[\max(H2), H2]\_0. \end{cases} \tag{10}$$

The value *Num* determines the number of sampling intervals that fit on the orthogonality interval of the signal *T*. It gives an opportunity to find two

**Figure 3.** *Cross-correlation coefficients.*

correlation in the process of moving the viewing window on the samples of the array of measurements *<sup>Q</sup>* <sup>¼</sup> *<sup>q</sup>*0*; <sup>q</sup>*1*;* … � �—is determined as well. The scheme of the calculation procedure is presented in **Figure 2**. For the correlation analysis, the two vectors, each containing *K* elements of array *Q* in two non-overlapping time obser-

> *; qj*þ*<sup>1</sup>;* …*qj*þ*K*�*<sup>1</sup>* n o

*Y1* <sup>¼</sup> *qj*þ*i*þ*<sup>K</sup>; qj*þ*i*þ*K*þ<sup>1</sup>*;* …*qj*þ*i*þ2*K*�<sup>1</sup> n o

removed from each other by *i, i* ¼ 0*,* …*, M*. The position of the second time window corresponding to the vector *Y*1 is determined by the successive change in the offset index *i* ¼ 0*,* …*, M* that ensures its "slip" along the signal sample *Q* at each

parameters, resulting in a quick and accurate assessment, are *K* ¼ 10 ÷ 30,

*Vj* <sup>¼</sup> *<sup>v</sup> <sup>j</sup>*

*<sup>Y</sup>*0*<sup>N</sup>* <sup>¼</sup> *<sup>Y</sup>*0*<sup>j</sup>* � <sup>1</sup>

*<sup>Y</sup>*1*<sup>N</sup>* <sup>¼</sup> *<sup>Y</sup>*1*<sup>j</sup>* � <sup>1</sup>

For a wide range of analyzed signals, for example, for the 0.3–3.4 KHz frequency band with minimum quality ADC, the most universal limits of the values of these

At each value of index *j* (moving the window slip area), the *M* þ 1 dimensional

n o

� 1 *<sup>K</sup>* ∑ *K*�1 *i*¼0 *Y*0*<sup>i</sup>* � ��<sup>1</sup>

� 1 *<sup>K</sup>* ∑ *K*�1 *i*¼0 *Y*1*<sup>i</sup>* � ��<sup>1</sup>

<sup>0</sup>*; <sup>v</sup> <sup>j</sup>* <sup>1</sup> *;* …*; <sup>v</sup> <sup>j</sup> M*

the elements of which are the coefficients of mutual correlation of vectors *Y*0 and *Y*1. The calculations (according to **Figure 2**) are performed after centering and

> *<sup>K</sup>* <sup>∑</sup> *K*�1 *i*¼0 *Y*0*<sup>i</sup>*

*<sup>K</sup>* <sup>∑</sup> *K*�1 *i*¼0 *Y*1*<sup>i</sup>*

The resulting vectors *Y*0*<sup>N</sup>* and *Y*1*<sup>N</sup>* in normalized space have the same length,

<sup>p</sup> , and the cosine of the angle between these vectors is equal to the

� �

� �

*,*

*,*

*,* (5)

*;*

*:*

(4)

(6)

vation windows of the signal, are formed:

*Modulation in Electronics and Telecommunications*

of the values *j* ¼ 0…*L*.

**Figure 2.** *Calculation scheme.*

vector is being formed:

equal to ffiffiffi

**30**

*К*

cross-correlation coefficient:

*M* ¼ 200 ÷ 300, and *L* ¼ 1000.

normalizing the vectors by the formulas

*Y0* ¼ *qj*

interrelated OFDM signal parameters: the orthogonality interval and the minimum carrier frequency spacing:

$$T = \frac{Num}{f\_{\vartheta}} \text{ and } \Delta f = T^{-1}. \tag{11}$$

series. Using the presented structure and values of the elements of the vector *V*3, it is possible to determine the number of sampling intervals that fit between adjacent pairs of mutually correlated segments of signal measurements, i.e., a period of

Using the obtained value *Num*1, we can determine the average value of the

For the final determination of the time-frequency structure of the signal, we must find the number of carrier frequencies *nf* and the vector of their nominal

values *Tp, T,* Δ*f* when the position of the element of the array *Q* corresponding to the beginning of the first full modulation interval is determined correctly. The beginning of a reliably identified clock interval could be most correctly associated with the beginning of the second series of maximal responses of correlators in the vector *V*3, since, due to the randomness of the beginning of the observation, the first series may be incomplete. It should be taken into account that the beginning of a series of maximal responses of correlation of the segments from *K* samples must appear before the next modulation interval actually begins. Therefore, to fall within the interval with the duration *Tp*(taking into account that demodulation can be performed on any segment *T* within *Tp*), it is necessary to add the number *K=*2 to the starting sample, at least. Then the beginning of the modulation interval can be assumed to coincide with the next element number in

*<sup>T</sup>* <sup>¼</sup> *round v*3*<sup>n</sup>*2*mod <sup>f</sup> <sup>д</sup>*

Here *round x*ð Þ is the rounding function to the nearest integer. The lowest possible frequency *f* <sup>0</sup> in the group of carrier frequencies is determined by the value of function (3) and the fulfillment of the corresponding condition in

*W*

þ *K* 2 � � (15)

(16)

� � � �

Δ*f, при max H*ð Þ1 ≥ *max H*ð Þ2 *;*

Δ*f, при max H*ð Þ1 < *max H*ð Þ2 *:*

*Fef* � *f* <sup>0</sup> <sup>Δ</sup>*<sup>f</sup>* <sup>þ</sup> <sup>1</sup>

� � (17)

The maximal number of subcarrier frequencies (or half the number of quadra-

� � � *<sup>v</sup>*3*n*ð Þ *nc*�<sup>1</sup> � *<sup>v</sup>*3*n*<sup>2</sup>

*,W* ¼ *Tp*

n o. This can be done on the basis of previously obtained

� �

<sup>2</sup>ð Þ *nc* � <sup>2</sup> (13)

�<sup>1</sup> (14)

*v*3*k*ð Þ *nc*�<sup>1</sup> � *v*3*k*<sup>2</sup>

*Digital Algebraic Method for Processing Complex Signals for Radio Monitoring Systems*

modulation interval *Tp* and, therefore, the average modulation rate *W*:

*Tp* <sup>¼</sup> *Num*<sup>1</sup> *f д*

"bursts" of correlation:

values *<sup>F</sup>* <sup>¼</sup> *<sup>f</sup>* <sup>0</sup>*;* …*; <sup>f</sup> nf* �<sup>1</sup>

the sequence

function (10):

**33**

*Num*1 ¼

*DOI: http://dx.doi.org/10.5772/intechopen.85590*

*n*0

*f* <sup>0</sup> ¼

1 2

ture components) that can fit in the channel band *Fef* is

*nf max* ¼ *round*

8 < :

Besides, choosing the corresponding method of detecting the *Num* value according to the condition specified in function (10) automatically determines the value of function (3) and, consequently, the ratio between the frequency parameters *f* <sup>0</sup> and Δ*f*. If the value *Num* is determined by the first line of expression (10), then *<sup>f</sup>* <sup>0</sup> <sup>¼</sup> *<sup>k</sup>* <sup>þ</sup> <sup>1</sup> 2 � � � <sup>Δ</sup>*f*. Otherwise the minimal carrier frequency is multiplied to the spacing of the carrier frequencies: *f* <sup>0</sup> ¼ *k* � Δ*f* where *k* is any positive integer.

As a result of processing histograms (9) according to function (10) based on the values of function (3), only one of the vectors *V*1 or *V*2 is left for the further analysis, hereinafter denoted *V* <sup>∗</sup> . This is possible because the prefix repetition sign is defined. Based on the elements of the vector *V* <sup>∗</sup> , another vector *V*3 is formed for the analysis:

$$V\mathfrak{J} = \operatorname{match}[\operatorname{Num}, V^\*].\tag{12}$$

The elements of this vector are equal to the numbers of the elements of the vector *V* <sup>∗</sup> in which the numbers *Num* are located. The feature of the vector *V* <sup>∗</sup> , provided that the analyzed signal belongs to the OFDM class, is that it contains a sequence of periodic series of numbers which are close or equal to *Num*. Therefore, the values of the elements of *V*3 in order of increasing their indices will be the segments (series) of an ordinary positive integer sequence with some gaps in the sequence. Small gaps can be observed inside the series too. A possible approximation of the sequence of the elements of *V*3 is illustrated by the following expression:

$$V\mathcal{B} = \left\{ \underbrace{11, 12, 14, 15, 16}\_{\text{series\ $1}}, \underbrace{105, 107, 108, 110}\_{\text{series\$ 2}}, \dots, \underbrace{620, 621, 622, 623, 624}\_{\text{series\ $8\$ n\_c}} \right\}$$

The length of series of consecutive numbers (position numbers) may differ due to measurement errors, features of the signal envelope, and rounding during calculations. However, when the vector length is sufficient, averaging results converges to the true estimation in accordance with the law of large numbers. To exclude "fragmentation" of the series, small gaps between adjacent numbers of the series must be ignored. It has been empirically found that in most cases, the number *Ni* should be considered to belong to the current series of numbers if *Ni* � *Ni*�<sup>1</sup> ≤*ε* where *ε* ¼ 3 ÷ 5. In general, the structure of the vector *V*3 can be depicted as shown in **Figure 4**.

In **Figure 4**, *v*3*nj* , *v*3*kj* are the initial and final elements of the *j*-th series of the consecutive numbers in the vector *V*3, and *nc* is the total number of the identified

$$\left\{ \underbrace{\begin{subarray}{c} \overline{\mathsf{v}\mathfrak{z}\_{0},\ldots,\mathsf{v}\mathfrak{z}\_{k\_{\mathrm{l}}}}, \\ \mathsf{s}\text{series}\mathcal{A}\Phi\text{l} \end{subarray}}\_{\text{series}\mathcal{A}\Phi\text{l}}, \underbrace{\left| \overbrace{\begin{subarray}{c} \overline{\mathsf{v}\mathfrak{z}\_{n\_{2}},\ldots,\mathsf{v}\mathfrak{z}\_{k\_{\mathrm{l}}}}, \\ \mathsf{s}\text{series}\mathcal{A}\Phi\text{l} \end{subarray}}\_{\text{series}\mathcal{A}\Phi\text{l}} \right|, \dots, \left| \overbrace{\begin{subarray}{c} \overline{\mathsf{v}\mathfrak{z}\_{n\_{c}-1}}, \\ \mathsf{s}\text{series}\mathcal{A}\mathfrak{z}\_{\mathrm{c}}(n\_{c}-1) \end{subarray}}^{\text{(c)}\mathcal{A}\mathfrak{z}\_{\mathrm{c}}(n\_{c}-1) \end{subarray}} \right|, \left| \overbrace{\begin{subarray}{c} \overline{\mathsf{v}\mathfrak{z}\_{n}(n\_{c}),\ldots,\mathsf{v}\mathfrak{z}\_{k\_{\mathrm{l}}}(n\_{c}) \\ \mathsf{s}\text{series}\mathcal{A}\mathfrak{z}\_{\mathrm{c}}(n\_{\mathrm{c}}) \end{subarray}}^{\text{(c)}\mathcal{A}\mathfrak{z}\_{\mathrm{c}}(n\_{\mathrm{c}}) \right|. \end{split}$$

**Figure 4.** *Example of the V*3 *series structure.*

*Digital Algebraic Method for Processing Complex Signals for Radio Monitoring Systems DOI: http://dx.doi.org/10.5772/intechopen.85590*

series. Using the presented structure and values of the elements of the vector *V*3, it is possible to determine the number of sampling intervals that fit between adjacent pairs of mutually correlated segments of signal measurements, i.e., a period of "bursts" of correlation:

$$Num1 = \frac{\left(v\mathfrak{Z}\_{k\_{(w-1)}} - v\mathfrak{Z}\_{k\_2}\right) - \left(v\mathfrak{Z}\_{n\_{(n\_c-1)}} - v\mathfrak{Z}\_{n\_2}\right)}{2(n\_c - 2)}\tag{13}$$

Using the obtained value *Num*1, we can determine the average value of the modulation interval *Tp* and, therefore, the average modulation rate *W*:

$$T\_p = \frac{Num\mathbf{1}}{f\_\vartheta}, W = T\_p^{-1} \tag{14}$$

For the final determination of the time-frequency structure of the signal, we must find the number of carrier frequencies *nf* and the vector of their nominal values *<sup>F</sup>* <sup>¼</sup> *<sup>f</sup>* <sup>0</sup>*;* …*; <sup>f</sup> nf* �<sup>1</sup> n o. This can be done on the basis of previously obtained values *Tp, T,* Δ*f* when the position of the element of the array *Q* corresponding to the beginning of the first full modulation interval is determined correctly. The beginning of a reliably identified clock interval could be most correctly associated with the beginning of the second series of maximal responses of correlators in the vector *V*3, since, due to the randomness of the beginning of the observation, the first series may be incomplete. It should be taken into account that the beginning of a series of maximal responses of correlation of the segments from *K* samples must appear before the next modulation interval actually begins. Therefore, to fall within the interval with the duration *Tp*(taking into account that demodulation can be performed on any segment *T* within *Tp*), it is necessary to add the number *K=*2 to the starting sample, at least. Then the beginning of the modulation interval can be assumed to coincide with the next element number in the sequence

$$m\_T^0 = round\left\{ \left[ v \mathfrak{Z}\_{n\_1} mod \left( \frac{f\_\vartheta}{W} \right) \right] + \frac{K}{2} \right\} \tag{15}$$

Here *round x*ð Þ is the rounding function to the nearest integer. The lowest possible frequency *f* <sup>0</sup> in the group of carrier frequencies is determined by the value of function (3) and the fulfillment of the corresponding condition in function (10):

$$f\_0 = \begin{cases} \frac{1}{2}\Delta f, & npu \quad \max(H1) \ge \max(H2); \\\\ \Delta f, & npu \quad \max(H1) < \max(H2). \end{cases} \tag{16}$$

The maximal number of subcarrier frequencies (or half the number of quadrature components) that can fit in the channel band *Fef* is

$$n\_{f\max} = round\left(\frac{F\_{\text{ef}} - f\_0}{\Delta f} + 1\right) \tag{17}$$

interrelated OFDM signal parameters: the orthogonality interval and the minimum

Besides, choosing the corresponding method of detecting the *Num* value according to the condition specified in function (10) automatically determines the value of function (3) and, consequently, the ratio between the frequency parameters *f* <sup>0</sup> and Δ*f*. If the value *Num* is determined by the first line of expres-

multiplied to the spacing of the carrier frequencies: *f* <sup>0</sup> ¼ *k* � Δ*f* where *k* is any

values of function (3), only one of the vectors *V*1 or *V*2 is left for the further analysis, hereinafter denoted *V* <sup>∗</sup> . This is possible because the prefix repetition sign is defined. Based on the elements of the vector *V* <sup>∗</sup> , another vector *V*3 is formed for

The elements of this vector are equal to the numbers of the elements of the vector *V* <sup>∗</sup> in which the numbers *Num* are located. The feature of the vector *V* <sup>∗</sup> , provided that the analyzed signal belongs to the OFDM class, is that it contains a sequence of periodic series of numbers which are close or equal to *Num*. Therefore, the values of the elements of *V*3 in order of increasing their indices will be the segments (series) of an ordinary positive integer sequence with some gaps in the sequence. Small gaps can be observed inside the series too. A possible approximation of the sequence of the elements of *V*3 is illustrated by the following expression:

> *;* 105*,* 107*,* 108*,* 110 |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} series№2

to measurement errors, features of the signal envelope, and rounding during calculations. However, when the vector length is sufficient, averaging results converges to the true estimation in accordance with the law of large numbers. To exclude "fragmentation" of the series, small gaps between adjacent numbers of the series must be ignored. It has been empirically found that in most cases, the number *Ni* should be considered to belong to the current series of numbers if *Ni* � *Ni*�<sup>1</sup> ≤*ε* where *ε* ¼ 3 ÷ 5. In general, the structure of the vector *V*3 can be depicted as shown

The length of series of consecutive numbers (position numbers) may differ due

consecutive numbers in the vector *V*3, and *nc* is the total number of the identified

, *v*3*kj* are the initial and final elements of the *j*-th series of the

As a result of processing histograms (9) according to function (10) based on the

and <sup>Δ</sup>*<sup>f</sup>* <sup>¼</sup> *<sup>T</sup>*�<sup>1</sup>

� � � <sup>Δ</sup>*f*. Otherwise the minimal carrier frequency is

*<sup>V</sup>*<sup>3</sup> <sup>¼</sup> *match Num;<sup>V</sup>* <sup>∗</sup> ½ �*:* (12)

*;* …*;* 620*,* 621*,* 622*,* 623*,* 624 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} series№*nc*

9 = ;

*:* (11)

*<sup>T</sup>* <sup>¼</sup> *Num f д*

2

carrier frequency spacing:

*Modulation in Electronics and Telecommunications*

sion (10), then *<sup>f</sup>* <sup>0</sup> <sup>¼</sup> *<sup>k</sup>* <sup>þ</sup> <sup>1</sup>

*V*3 ¼ 11*,* 12*,* 14*,* 15*,* 16 |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} series№1

8 < :

positive integer.

the analysis:

in **Figure 4**.

**Figure 4.**

**32**

In **Figure 4**, *v*3*nj*

*Example of the V*3 *series structure.*

### **3.2 Determining the amount and nominal values of subcarrier frequencies of OFDM signals**

The correlation method, considered above, allows making a reliable assessment of the main structural parameters of OFDM—*<sup>T</sup>* and *TP*. The value of <sup>Δ</sup>*<sup>f</sup>* <sup>¼</sup> *<sup>T</sup>*�<sup>1</sup> uniquely defines the spacing of adjacent subcarrier frequencies. The minimal value of the subcarrier frequency and the maximal possible number of subcarriers placed within the signal bandwidth *nf max* are determined from Eqs. (16) and (17).

*<sup>Y</sup>* <sup>¼</sup> *<sup>y</sup>*0*;* …*; <sup>y</sup>*ð Þ *nf max*�<sup>1</sup>

mined by the ratio *A*<sup>1</sup> � *A*<sup>þ</sup>

tion *TP* is used:

*A*<sup>2</sup> ¼ *ai,j* � � �

*ai,j* ¼ Sin *ω<sup>j</sup>* � *ti*

*ai,j* ¼ Cos *ω<sup>j</sup>* � *ti*

SLAE has the form

ture components *X*<sup>∗</sup>

**35**

*B*<sup>2</sup> ¼ *b*0*;* …*; b*ð Þ *Num*�<sup>1</sup>

*A*þ

The representation of the matrix *A*<sup>þ</sup>

*A*<sup>1</sup> ¼ *D* � *C* ¼

which can be written in the form *X*<sup>1</sup> ¼ *A*<sup>þ</sup>

giving a zero residual: *X*<sup>1</sup> � *A*<sup>þ</sup> k k <sup>1</sup> � *B*<sup>1</sup> ¼ 0.

with the size of *N* � *r* and *r* � *N* is used:

n o*, yi* <sup>¼</sup> *<sup>x</sup>*<sup>1</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.85590*

*i* � �<sup>2</sup>

*Digital Algebraic Method for Processing Complex Signals for Radio Monitoring Systems*

Moore-Penrose can be used. It is known that there is a normal solution of an

underdetermined SLAE, and it is the only one. It is found by*X*<sup>1</sup> ¼ *A*<sup>þ</sup>

0

B@

� �*,* <sup>0</sup>≤*j*≤*nf max* � <sup>1</sup>*;*

*X*<sup>∗</sup> <sup>2</sup> ¼ *A*<sup>2</sup>

<sup>2</sup> as well as *X*<sup>∗</sup>

� �*, n*max <sup>≤</sup>*j*≤<sup>2</sup> � *nf max* � <sup>1</sup>*;*

<sup>þ</sup> *<sup>x</sup>*<sup>1</sup>

The case of insufficiently defined SLAE *<sup>N</sup>* <sup>&</sup>lt; <sup>2</sup> � *nf max* � � is interesting for analyzing small samples of the signal. To solve such a SLAE, the pseudoinverse matrix method

<sup>1</sup> <sup>¼</sup> *<sup>C</sup>*<sup>þ</sup> � *<sup>D</sup>*<sup>þ</sup> <sup>¼</sup> *<sup>C</sup>*<sup>∗</sup> � *<sup>C</sup>* � *<sup>C</sup>*<sup>∗</sup> ð Þ�<sup>1</sup> � *<sup>D</sup>* � *<sup>D</sup>*<sup>∗</sup> ð Þ�<sup>1</sup> � *<sup>D</sup>*<sup>∗</sup> *:*

With various skeletal decompositions of the matrix *A*, the same solution for *A*<sup>þ</sup>

The case of *<sup>N</sup>*><sup>2</sup> � *nf max* � � is the most advantageous for the maximal recording of signal information on the modulation interval. Due to using additional signal measurements from the sample *Q*, the system which contains more equations with the same number of unknowns is formed. To form the matrix *A*<sup>2</sup> and the vector *B*2, the maximal number of signal measurements determined by *Num*≈*TP=t<sup>д</sup>* on the dura-

�*, i* <sup>¼</sup> <sup>0</sup>*,* …*, Num* ð Þ � <sup>1</sup> *, j* <sup>¼</sup> <sup>0</sup>… <sup>2</sup> � *nf max* � <sup>1</sup> � �*;*

and, as a rule, has many solutions. To select the only one, we need to use some criteria. In practice, the maximum likelihood criterion is used more often. In the case of a normal distribution of the vector *B*2, it is equivalent of the least square's criterion:

> *<sup>T</sup>* � *<sup>A</sup>*<sup>2</sup> � ��<sup>1</sup>

An approximate solution of system (26) gives a more accurate result than a strict solution of system (25). The noise immunity of the solution is achieved by averaging the disturbing effect of interference when the number of signal measurements exceeds the required minimum. The obtained vector of amplitudes of the quadra-

� �*, bk* <sup>¼</sup> *qk, k* <sup>¼</sup> <sup>0</sup>*,* …*, Num* ð Þ � <sup>1</sup> *:* (24)

*A*2

<sup>1</sup> gives a possibility to calculate the power

1

0

B@

CA �

the Moore-Penrose pseudoinverse matrix of size 2 � *nf max* � 2 � *nf max* which is deter-

<sup>1</sup> � *A*<sup>1</sup> ¼ *A*1. In practice *A*<sup>þ</sup>

*d*1*,* <sup>1</sup> … *d*1*,r* ⋮⋱⋮ *dN,* <sup>1</sup> ⋯ *aN,r*

ð Þ *<sup>i</sup>*þ*nfmax* � �<sup>2</sup>

*, i* <sup>¼</sup> <sup>0</sup>*,* …*, nf max* � <sup>1</sup> � �

<sup>1</sup> can be found by the formula

1

CA*:*

<sup>1</sup> in the form of a product of two matrices

*c*1*,* <sup>1</sup> … *c*1*,N* ⋮⋱ ⋮ *cr,* <sup>1</sup> ⋯ *cr,N*

<sup>1</sup> � *B*<sup>1</sup> is derived. It is a pseudosolution

*A*<sup>2</sup> � *X*<sup>2</sup> ¼ *B*<sup>2</sup> (25)

*<sup>T</sup>* � *<sup>B</sup>*<sup>2</sup> (26)

(22)

(23)

<sup>1</sup> is

<sup>1</sup> � *B*<sup>1</sup> where *A*<sup>þ</sup>

The number of samples *N*, taken into account when analyzing a signal on one modulation interval, as well as the harmonic quadrature 2 � *nf max* � �, define the dimensions of the matrix of the linear algebraic equations system (SLAE) which can be compiled and solved to estimate the frequency range. Depending on the ratio of the vertical and horizontal dimensions of the matrix of coefficients, the system of equations can be overdetermined *<sup>N</sup>*><sup>2</sup> � *nf max* � �, determined *<sup>N</sup>* <sup>¼</sup> <sup>2</sup> � *nf max* � �, or underdetermined *<sup>N</sup>* <sup>&</sup>lt;<sup>2</sup> � *nf max* � �. The simplest one is the *<sup>N</sup>* <sup>¼</sup> <sup>2</sup> � *nf max* � � case because then the SLAE is a joint one almost every time. The number of equations that matches the number of used elements of the digital sample *Q* equals to the number of unknowns 2 � *nf max* � � determining the amplitudes of quadrature components in the spectrum of carrier frequencies OFDM. For the correct solution of SLAU 2 � *nf max* � �, uniformly spaced sample counts *<sup>Q</sup>* starting from the point of beginning of the observation of the first complete clock interval of signal *n*<sup>0</sup> *<sup>T</sup>* should be selected on *i*-m modulation interval. For this the following rule is used:

$$n\_T^i = n\_T^0 + round(i \cdot \Delta) \text{ where } \Delta = T\_P / t\_\vartheta. \tag{18}$$

Square matrix of coefficients for unknown SLAE with size <sup>2</sup> � *nf max* � � � <sup>2</sup> � *nf max* � � composed for quadrature components of subcarrier frequencies is formed according to the rule

$$A\_1 = \begin{vmatrix} \left| a\_{i,j} \right| \end{vmatrix}, \quad i, j = \mathbf{0}, \ \dots \ \left( \mathbf{2} \cdot \boldsymbol{n}\_{f\max} - \mathbf{1} \right);$$

$$a\_{i,j} = \text{Sin} \left[ \boldsymbol{o}\_{j} \cdot \boldsymbol{t}\_{i} \right], \qquad \mathbf{0} \le j \le \boldsymbol{n}\_{f\max} - \mathbf{1}; \tag{19}$$

$$a\_{i,j} = \text{Cost} \left[ \boldsymbol{o}\_{j} \cdot \boldsymbol{t}\_{i} \right], \qquad \boldsymbol{\eta}\_{j} \le j \le 2 \cdot \boldsymbol{\eta}\_{f\max} - \mathbf{1};$$

where *<sup>ω</sup><sup>j</sup>* <sup>¼</sup> *<sup>ω</sup>*ð Þ *<sup>j</sup>*þ*nf max* <sup>¼</sup> <sup>2</sup>*<sup>π</sup> <sup>f</sup>* <sup>0</sup> <sup>þ</sup> *<sup>j</sup>* � <sup>Δ</sup>*<sup>f</sup>* � �*, j* <sup>¼</sup> <sup>0</sup>*,* …*, nf max*, *ti* <sup>¼</sup> *<sup>n</sup><sup>i</sup> <sup>T</sup>* þ *i* � *tд*.

The column matrix of free members is formed as a vector of signal measurements on the duration of one orthogonality interval:

$$B\_1 = \left\{ b\_0, \ldots, b\_{\left\{ 2 \cdot n\_{f\_{\max}} - 1 \right\}} \right\}; b\_i = q\_i, \quad i \cdot t\_\partial \in T\_P^{\;i} \tag{20}$$

Normal solution of normally defined SLAE

$$A\_1 \cdot X\_1 = B\_1 \quad \Rightarrow \qquad X\_1 = A\_1^{-1} \cdot B\_1 \tag{21}$$

gives an estimation of the amplitude vector of quadrature components *<sup>X</sup>*<sup>1</sup> <sup>¼</sup> *<sup>x</sup>*<sup>1</sup> <sup>0</sup>*;* …*; x*<sup>1</sup> ð Þ <sup>2</sup>�*nfmax*�<sup>1</sup> � � which corresponds to the permissible values of carrier frequencies.

On the basis of this solution, it is possible to determine the power distribution vector of the signal between the harmonic oscillations of the carrier frequencies:

*Digital Algebraic Method for Processing Complex Signals for Radio Monitoring Systems DOI: http://dx.doi.org/10.5772/intechopen.85590*

$$Y = \left\{ y\_0, \dots, y\_{\binom{\eta\_{fmax} - 1}{2}} \right\}, \quad y\_i = \left( x\_i^1 \right)^2 + \left( x\_{\binom{i + \eta\_{fmax}}{\binom{i + \eta\_{fmax}}}} \right)^2, \quad i = 0, \dots \\ \text{ } \left( \eta\_{fmax} - 1 \right) \tag{22}$$

The case of insufficiently defined SLAE *<sup>N</sup>* <sup>&</sup>lt; <sup>2</sup> � *nf max* � � is interesting for analyzing small samples of the signal. To solve such a SLAE, the pseudoinverse matrix method Moore-Penrose can be used. It is known that there is a normal solution of an underdetermined SLAE, and it is the only one. It is found by*X*<sup>1</sup> ¼ *A*<sup>þ</sup> <sup>1</sup> � *B*<sup>1</sup> where *A*<sup>þ</sup> <sup>1</sup> is the Moore-Penrose pseudoinverse matrix of size 2 � *nf max* � 2 � *nf max* which is determined by the ratio *A*<sup>1</sup> � *A*<sup>þ</sup> <sup>1</sup> � *A*<sup>1</sup> ¼ *A*1. In practice *A*<sup>þ</sup> <sup>1</sup> can be found by the formula

$$\mathcal{A}\_1^+ = \mathbf{C}^+ \cdot \mathbf{D}^+ = \mathbf{C}^\* \cdot (\mathbf{C} \cdot \mathbf{C}^\*)^{-1} \cdot (\mathbf{D} \cdot \mathbf{D}^\*)^{-1} \cdot \mathbf{D}^\*.$$

The representation of the matrix *A*<sup>þ</sup> <sup>1</sup> in the form of a product of two matrices with the size of *N* � *r* and *r* � *N* is used:

$$A\_1 = D \cdot C = \begin{pmatrix} d\_{1,1} & \dots & d\_{1,r} \\ \vdots & \ddots & \vdots \\ d\_{N,1} & \dots & a\_{N,r} \end{pmatrix} \cdot \begin{pmatrix} c\_{1,1} & \dots & c\_{1,N} \\ \vdots & \ddots & \vdots \\ c\_{r,1} & \dots & c\_{r,N} \end{pmatrix}.$$

With various skeletal decompositions of the matrix *A*, the same solution for *A*<sup>þ</sup> which can be written in the form *X*<sup>1</sup> ¼ *A*<sup>þ</sup> <sup>1</sup> � *B*<sup>1</sup> is derived. It is a pseudosolution giving a zero residual: *X*<sup>1</sup> � *A*<sup>þ</sup> k k <sup>1</sup> � *B*<sup>1</sup> ¼ 0.

The case of *<sup>N</sup>*><sup>2</sup> � *nf max* � � is the most advantageous for the maximal recording of signal information on the modulation interval. Due to using additional signal measurements from the sample *Q*, the system which contains more equations with the same number of unknowns is formed. To form the matrix *A*<sup>2</sup> and the vector *B*2, the maximal number of signal measurements determined by *Num*≈*TP=t<sup>д</sup>* on the duration *TP* is used:

$$\begin{aligned} A\_2 &= ||a\_{i,j}||, \quad i = 0, \ldots, \ (Num - 1), \quad j = 0 \ldots (2 \cdot n\_{f\max} - 1); \\\ a\_{i,j} &= \text{Sin} \left[ \boldsymbol{\alpha}\_{\boldsymbol{j}} \cdot \boldsymbol{t}\_i \right], \qquad 0 \le j \le n\_{f\max} - 1; \\\ a\_{i,j} &= \text{Cost} \left[ \boldsymbol{\alpha}\_{\boldsymbol{j}} \cdot \boldsymbol{t}\_i \right], \qquad n\_{\max} \le \boldsymbol{j} \le 2 \cdot n\_{f\max} - 1; \\\ B\_2 &= \left\{ b\_0, \ldots, \quad b\_{(\text{Num}-1)} \right\}, \quad b\_k = q\_k, \quad k = 0, \ldots, \ (Num - 1). \end{aligned} \tag{24}$$

SLAE has the form

$$\mathbf{A}\_2 \cdot \mathbf{X}\_2 = \mathbf{B}\_2 \tag{25}$$

and, as a rule, has many solutions. To select the only one, we need to use some criteria. In practice, the maximum likelihood criterion is used more often. In the case of a normal distribution of the vector *B*2, it is equivalent of the least square's criterion:

$$X\_2^\* = \left(A\_2^{\;T} \cdot A\_2\right)^{-1} A\_2^{\;T} \cdot B\_2 \tag{26}$$

An approximate solution of system (26) gives a more accurate result than a strict solution of system (25). The noise immunity of the solution is achieved by averaging the disturbing effect of interference when the number of signal measurements exceeds the required minimum. The obtained vector of amplitudes of the quadrature components *X*<sup>∗</sup> <sup>2</sup> as well as *X*<sup>∗</sup> <sup>1</sup> gives a possibility to calculate the power

**3.2 Determining the amount and nominal values of subcarrier frequencies**

of the main structural parameters of OFDM—*<sup>T</sup>* and *TP*. The value of <sup>Δ</sup>*<sup>f</sup>* <sup>¼</sup> *<sup>T</sup>*�<sup>1</sup> uniquely defines the spacing of adjacent subcarrier frequencies. The minimal value of the subcarrier frequency and the maximal possible number of subcarriers placed

within the signal bandwidth *nf max* are determined from Eqs. (16) and (17).

modulation interval, as well as the harmonic quadrature 2 � *nf max*

equations can be overdetermined *N*>2 � *nf max*

*Modulation in Electronics and Telecommunications*

*ni <sup>T</sup>* <sup>¼</sup> *<sup>n</sup>*<sup>0</sup>

frequencies is formed according to the rule

*A*<sup>1</sup> ¼ *ai,j* � � �

*ai,j* ¼ Sin *ω<sup>j</sup>* � *ti*

*ai,j* ¼ Cos *ω<sup>j</sup>* � *ti*

*<sup>B</sup>*<sup>1</sup> <sup>¼</sup> *<sup>b</sup>*0*;* …*; <sup>b</sup>*ð Þ <sup>2</sup>�*nf max*�<sup>1</sup> n o

ments on the duration of one orthogonality interval:

Normal solution of normally defined SLAE

underdetermined *N* <2 � *nf max*

number of unknowns 2 � *nf max*

SLAU 2 � *nf max*

2 � *nf max*

*<sup>X</sup>*<sup>1</sup> <sup>¼</sup> *<sup>x</sup>*<sup>1</sup>

**34**

frequencies.

<sup>0</sup>*;* …*; x*<sup>1</sup>

� �

ð Þ <sup>2</sup>�*nfmax*�<sup>1</sup>

� � � <sup>2</sup> � *nf max*

The correlation method, considered above, allows making a reliable assessment

The number of samples *N*, taken into account when analyzing a signal on one

dimensions of the matrix of the linear algebraic equations system (SLAE) which can be compiled and solved to estimate the frequency range. Depending on the ratio of the vertical and horizontal dimensions of the matrix of coefficients, the system of

because then the SLAE is a joint one almost every time. The number of equations that matches the number of used elements of the digital sample *Q* equals to the

nents in the spectrum of carrier frequencies OFDM. For the correct solution of

beginning of the observation of the first complete clock interval of signal *n*<sup>0</sup>

be selected on *i*-m modulation interval. For this the following rule is used:

where *<sup>ω</sup><sup>j</sup>* <sup>¼</sup> *<sup>ω</sup>*ð Þ *<sup>j</sup>*þ*nf max* <sup>¼</sup> <sup>2</sup>*<sup>π</sup> <sup>f</sup>* <sup>0</sup> <sup>þ</sup> *<sup>j</sup>* � <sup>Δ</sup>*<sup>f</sup>* � �*, j* <sup>¼</sup> <sup>0</sup>*,* …*, nf max*, *ti* <sup>¼</sup> *<sup>n</sup><sup>i</sup>*

The column matrix of free members is formed as a vector of signal measure-

*<sup>A</sup>*<sup>1</sup> � *<sup>X</sup>*<sup>1</sup> <sup>¼</sup> *<sup>B</sup>*<sup>1</sup> ) *<sup>X</sup>*<sup>1</sup> <sup>¼</sup> *<sup>A</sup>*�<sup>1</sup>

On the basis of this solution, it is possible to determine the power distribution vector of the signal between the harmonic oscillations of the carrier frequencies:

gives an estimation of the amplitude vector of quadrature components

Square matrix of coefficients for unknown SLAE with size

� �, uniformly spaced sample counts *Q* starting from the point of

� � composed for quadrature components of subcarrier

�*, i, j* <sup>¼</sup> <sup>0</sup>*,* …*,* <sup>2</sup> � *nf max* � <sup>1</sup> � �*;*

� �*, nf* <sup>≤</sup>*j*≤<sup>2</sup> � *nf max* � <sup>1</sup>*;*

*; bi* ¼ *qi*

*, i* � *t<sup>д</sup>* ∈*TP*

which corresponds to the permissible values of carrier

� �*,* <sup>0</sup><sup>≤</sup> *<sup>j</sup>*≤*nf max* � <sup>1</sup>*;*

� �. The simplest one is the *<sup>N</sup>* <sup>¼</sup> <sup>2</sup> � *nf max*

� �, determined *<sup>N</sup>* <sup>¼</sup> <sup>2</sup> � *nf max*

� � determining the amplitudes of quadrature compo-

*<sup>T</sup>* þ *round i*ð Þ � Δ where Δ ¼ *TP=tд:* (18)

� �, define the

� � case

� �, or

*<sup>T</sup>* should

(19)

*<sup>T</sup>* þ *i* � *tд*.

*<sup>i</sup>* (20)

<sup>1</sup> � *B*<sup>1</sup> (21)

**of OFDM signals**

distribution signal in carrier frequencies using expression (22), wherein *x*<sup>2</sup> *<sup>i</sup>* is used instead of *x*<sup>1</sup> *i* .

as well as a column matrix of *Num* signal measurements

*Digital Algebraic Method for Processing Complex Signals for Radio Monitoring Systems*

*, J* <sup>¼</sup> *<sup>n</sup>*<sup>0</sup>

Here *<sup>f</sup> <sup>д</sup>* <sup>¼</sup> ð Þ *<sup>t</sup><sup>д</sup>* �<sup>1</sup> is the sampling frequency of the digital signal sample (*t<sup>д</sup>* is the

subcarrier frequencies, *N<sup>и</sup>* is the number of the analyzed modulation interval, *W* is

the primary correlation analysis using the "sliding time window" method discussed

To find the solution of the overdetermined SLAE (28), (29) the method of least

digital signal sampling), corresponding to the reliably detected beginning of the

where *AT* is the transposed matrix (28) and *<sup>Z</sup>* <sup>¼</sup> *<sup>z</sup>*0*;* …*; znf* �<sup>1</sup>*; znf ;* …*; <sup>z</sup>*2*nf* �<sup>1</sup>

The decision vector *Z*contains the amplitudes of sines (first *nf* elements) and cosines (last *nf* elements) in frequencies determined by the vector *F*. It gives an opportunity to calculate the elements of the phase vector (expressed in degrees) of

• *Atan x*ð Þ *; y* —to calculate the angle in radians from the interval ½ � �*π; π* , between

The process of calculations by expressions (30) and (31) is the absolute phase demodulation of discrete measurements on an interval *T* located within the segment of modulation interval *TP*. It should be noted that with relative phase shift keying, the calculated absolute values of the phases do not matter for correct demodulation of the bit stream. Therefore, in this case, the error in determining the beginning of the orthogonality interval *T* within the modulation interval *TP* is not significant. The calculation of the sampling number of the next interval is based on the known modulation rate *W*, the starting number of the initial interval *n*<sup>0</sup>

the sampling frequency *F<sup>д</sup>* and performed by using the rounding formula from

The factor that reduces the effectiveness of group digital demodulation is the asynchrony of the sampling procedure when a sample of signal measurements is being obtained with the duration of repeated demodulation intervals. In general, the

n o in the subcarrier frequencies:

� � <sup>þ</sup> <sup>360</sup> � �*;* <sup>360</sup> � �*, i* <sup>¼</sup> <sup>0</sup>*,* …*, nf* � 1 (31)

the vector of the amplitudes of the harmonic quadrature components for all *nf*

*<sup>T</sup>* þ *round N<sup>и</sup>*

*<sup>T</sup>* is the initial measurement number (in the

*<sup>A</sup><sup>T</sup>* � *<sup>A</sup>* � *<sup>Z</sup>* <sup>¼</sup> *<sup>A</sup><sup>T</sup>* � *<sup>B</sup>* (30)

� � is the digital sample of measurements of the

n o is the vector of nominal values of the detected

*f д W*

*<sup>T</sup>* is one of the results of the stage of

n o is

*<sup>T</sup>*, and

� � (29)

*<sup>B</sup>* <sup>¼</sup> f g *<sup>b</sup>*0*;* …*; bNum*�<sup>1</sup> *, bi* <sup>¼</sup> *yJ*þ*<sup>i</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.85590*

first complete modulation interval. The value *n*<sup>0</sup>

squares is used to solve the following system:

sampling interval), *Y* ¼ *y*0*;* …*; yk*

the modulation rate in bauds, and *n*<sup>0</sup>

subcarriers of the signal frequencies.

harmonic oscillations Φ ¼ *ϕ*0*;* …*; ϕnf* �<sup>1</sup>

*<sup>π</sup> Atan zi; zi*þ*nf*

In this expression, the following functions are used:

• *mod a*ð Þ *; b* —to calculate the value of the number *a* by modulo *b*

the axis 0*x* and the vector drawn from point (0,0) to point ð Þ *x; y*

*<sup>ϕ</sup><sup>i</sup>* <sup>¼</sup> *mod* <sup>180</sup>

expression (29).

**37**

OFDM signal, *<sup>F</sup>* <sup>¼</sup> *<sup>f</sup>* <sup>0</sup>*;* …*; <sup>f</sup> nf* �<sup>1</sup>

in the previous section.

For any type of SLAE, determining the actual list of carrier frequencies in the OFDM spectrum is performed by comparing the elements of power distribution histograms with a threshold value. The obtained nominal values of frequencies determine the last structural time-frequency parameter of the analyzed signal—the vector of working subcarrier frequencies *F*.

Thus, the previously obtained signal parameters *Tp,T,* Δ*f*, *W*, and the obtained in this subsection vector *F* identify completely their structural properties and make signal demodulation possible.

## **4. Group algebraic demodulation**

#### **4.1 Determination of phase modulation multiplicity**

OFDM standards as an informative parameter on subcarriers assume the use of relative phase shift keying. The number of variants of phase angles can vary from 2 to 16 or more [2–5]. As a rule, absolute phase modulation methods are not applied because they are critical to the correctness of the subcarrier phase restoration, which is highly dependent on the accuracy of determining the beginning of the modulation interval. In case of digital demodulation, when a sample of signal measurements is formed for time points asynchronously with respect to a periodic sequence of modulation interval boundaries, the usage of absolute phase modulation methods is practically impossible, since it is theoretically impossible to provide absolutely accurate clock synchronization of modems. Therefore, as an axiomatic assumption for developing the stages of secondary analysis, the hypothesis of using the relative phase coding methods is adopted. Depending on the multiplicity of the applied modulation method *<sup>m</sup>* <sup>¼</sup> <sup>2</sup>*<sup>α</sup>*,*<sup>α</sup>* <sup>¼</sup> <sup>1</sup>*,* <sup>2</sup>*,* <sup>3</sup>*,* <sup>⋯</sup> is an integer, during the transition from interval to interval; on each of the carriers, a phase jump occurs, the value of which determines the corresponding combination of information symbols. Jumps occur even in the case of complete coincidence of combinations of symbols on adjacent intervals.

The general idea of a preliminary analysis of the PM multiplicity is to statistically identify the number of observed fixed values of the phase of harmonic oscillations in carrier frequencies. To perform the analysis, let us use the algebraic method for the case of overdetermined SLAEs considered in the previous section. Let *Num* be the maximum number of measurements performed on the modulation interval and *nf* the number of carriers detected during the primary analysis. Then the degree of overdetermination of SLAE is calculated from the expression

$$K\_{II} = \frac{Num}{2 \cdot n\_f} \text{>1} \tag{27}$$

In accordance with the findings, the increase of *K<sup>П</sup>* leads to the increase in the noise immunity of the system solution. SLAE, compiled for the entire spectrum of subcarriers, is determined by a rectangular (size *Num* � 2 � *nf* ) matrix of coefficients for the following unknowns:

$$A = ||a\_{i,j}||, \quad a\_{i,j} = \begin{cases} \text{Sin}\left(2\pi f\_j(i \cdot \Delta T)\right), \quad i = 0, \dots, Num - 1, \quad j = 0, \dots, n\_f - 1; \\\text{Cov}\left(2\pi f\_j(i \cdot \Delta T)\right), \quad i = 0, \dots, Num - 1, \quad j = n\_f, \dots, 2 \cdot n\_f - 1 \end{cases}, \quad j = n\_f, \dots, 2 \cdot n\_f - 1. \tag{28}$$

*Digital Algebraic Method for Processing Complex Signals for Radio Monitoring Systems DOI: http://dx.doi.org/10.5772/intechopen.85590*

as well as a column matrix of *Num* signal measurements

$$B = \{b\_0, \dots, b\_{Num-1}\}, \quad b\_i = y\_{j+i}, \quad J = n\_T^0 + round\left(N\_u \frac{f\_\cdot}{W}\right) \tag{29}$$

Here *<sup>f</sup> <sup>д</sup>* <sup>¼</sup> ð Þ *<sup>t</sup><sup>д</sup>* �<sup>1</sup> is the sampling frequency of the digital signal sample (*t<sup>д</sup>* is the sampling interval), *Y* ¼ *y*0*;* …*; yk* � � is the digital sample of measurements of the OFDM signal, *<sup>F</sup>* <sup>¼</sup> *<sup>f</sup>* <sup>0</sup>*;* …*; <sup>f</sup> nf* �<sup>1</sup> n o is the vector of nominal values of the detected subcarrier frequencies, *N<sup>и</sup>* is the number of the analyzed modulation interval, *W* is the modulation rate in bauds, and *n*<sup>0</sup> *<sup>T</sup>* is the initial measurement number (in the digital signal sampling), corresponding to the reliably detected beginning of the first complete modulation interval. The value *n*<sup>0</sup> *<sup>T</sup>* is one of the results of the stage of the primary correlation analysis using the "sliding time window" method discussed in the previous section.

To find the solution of the overdetermined SLAE (28), (29) the method of least squares is used to solve the following system:

$$\mathbf{A}^T \cdot \mathbf{A} \cdot \mathbf{Z} = \mathbf{A}^T \cdot \mathbf{B} \tag{30}$$

where *AT* is the transposed matrix (28) and *<sup>Z</sup>* <sup>¼</sup> *<sup>z</sup>*0*;* …*; znf* �<sup>1</sup>*; znf ;* …*; <sup>z</sup>*2*nf* �<sup>1</sup> n o is the vector of the amplitudes of the harmonic quadrature components for all *nf* subcarriers of the signal frequencies.

The decision vector *Z*contains the amplitudes of sines (first *nf* elements) and cosines (last *nf* elements) in frequencies determined by the vector *F*. It gives an opportunity to calculate the elements of the phase vector (expressed in degrees) of harmonic oscillations Φ ¼ *ϕ*0*;* …*; ϕnf* �<sup>1</sup> n o in the subcarrier frequencies:

$$\phi\_i = \text{mod}\left[\left(\frac{180}{\pi}\text{Atom}\left(z\_i, z\_{i+n\_f}\right) + \text{360}\right), \text{360}\right], \quad i = 0, \dots, n\_f - 1 \tag{31}$$

In this expression, the following functions are used:


The process of calculations by expressions (30) and (31) is the absolute phase demodulation of discrete measurements on an interval *T* located within the segment of modulation interval *TP*. It should be noted that with relative phase shift keying, the calculated absolute values of the phases do not matter for correct demodulation of the bit stream. Therefore, in this case, the error in determining the beginning of the orthogonality interval *T* within the modulation interval *TP* is not significant. The calculation of the sampling number of the next interval is based on the known modulation rate *W*, the starting number of the initial interval *n*<sup>0</sup> *<sup>T</sup>*, and the sampling frequency *F<sup>д</sup>* and performed by using the rounding formula from expression (29).

The factor that reduces the effectiveness of group digital demodulation is the asynchrony of the sampling procedure when a sample of signal measurements is being obtained with the duration of repeated demodulation intervals. In general, the

distribution signal in carrier frequencies using expression (22), wherein *x*<sup>2</sup>

For any type of SLAE, determining the actual list of carrier frequencies in the OFDM spectrum is performed by comparing the elements of power distribution histograms with a threshold value. The obtained nominal values of frequencies determine the last structural time-frequency parameter of the analyzed signal—the

Thus, the previously obtained signal parameters *Tp,T,* Δ*f*, *W*, and the obtained in this subsection vector *F* identify completely their structural properties and make

OFDM standards as an informative parameter on subcarriers assume the use of relative phase shift keying. The number of variants of phase angles can vary from 2 to 16 or more [2–5]. As a rule, absolute phase modulation methods are not applied because they are critical to the correctness of the subcarrier phase restoration, which is highly dependent on the accuracy of determining the beginning of the modulation interval. In case of digital demodulation, when a sample of signal measurements is formed for time points asynchronously with respect to a periodic sequence of modulation interval boundaries, the usage of absolute phase modulation methods is practically impossible, since it is theoretically impossible to provide absolutely accurate clock synchronization of modems. Therefore, as an axiomatic assumption for developing the stages of secondary analysis, the hypothesis of using the relative phase coding methods is adopted. Depending on the multiplicity of the applied modulation method *<sup>m</sup>* <sup>¼</sup> <sup>2</sup>*<sup>α</sup>*,*<sup>α</sup>* <sup>¼</sup> <sup>1</sup>*,* <sup>2</sup>*,* <sup>3</sup>*,* <sup>⋯</sup> is an integer, during the transition from interval to interval; on each of the carriers, a phase jump occurs, the value of which determines the corresponding combination of information symbols. Jumps occur even in the case

of complete coincidence of combinations of symbols on adjacent intervals.

overdetermination of SLAE is calculated from the expression

coefficients for the following unknowns:

8 ><

>:

*Sin* 2*πfj*

*Cos* 2*πfj*

ð Þ *i* � Δ*T* � �

ð Þ *i* � Δ*T* � �

�*; ai,j* ¼

*A* ¼ *ai,j* � � �

**36**

The general idea of a preliminary analysis of the PM multiplicity is to statistically identify the number of observed fixed values of the phase of harmonic oscillations in carrier frequencies. To perform the analysis, let us use the algebraic method for the case of overdetermined SLAEs considered in the previous section. Let *Num* be the maximum number of measurements performed on the modulation interval and *nf* the number of carriers detected during the primary analysis. Then the degree of

> *<sup>K</sup><sup>П</sup>* <sup>¼</sup> *Num* 2 � *nf*

In accordance with the findings, the increase of *K<sup>П</sup>* leads to the increase in the noise immunity of the system solution. SLAE, compiled for the entire spectrum of subcarriers, is determined by a rectangular (size *Num* � 2 � *nf* ) matrix of

>1 (27)

*, i* ¼ 0*,* …*, Num* � 1*, j* ¼ 0*,* …*, nf* � 1*;*

*, i* ¼ 0*,* …*, Num* � 1*, j* ¼ *nf ,* …*,* 2 � *nf* � 1

*,*

(28)

instead of *x*<sup>1</sup>

*i* .

signal demodulation possible.

vector of working subcarrier frequencies *F*.

*Modulation in Electronics and Telecommunications*

**4. Group algebraic demodulation**

**4.1 Determination of phase modulation multiplicity**

*<sup>i</sup>* is used

value *T=t<sup>д</sup>* ¼ *f <sup>д</sup>* � *T* is not an integer. This leads to a variation of the value of "indentation" *dTi* from the beginning of the *i*-th modulation interval. This phenomenon is illustrated in **Figure 5**. Differences in the values of *dTi* lead to the fact that the phase error periodically repeated on a combination of several adjacent modulation intervals is added to the correct solution of the system. The significant "blurring" of phase increments in subcarrier frequencies, as well as the departure of their average values from standardized nominal values, is an external manifestation of this asynchronous effect. Therefore, the phase errors of asynchronous sampling are an additional source of disturbing noise, which must be taken into account and, if possible, compensated.

A known correlation between magnitudes *W* and *t<sup>д</sup>* allows us to estimate the magnitude of these errors. On an arbitrary *i*-th interval, the error in determining the initial measurement is

$$dT\_i = t\_\vartheta \left[ i \frac{f\_\vartheta}{\mathcal{W}} - round \left( i \frac{f\_\vartheta}{\mathcal{W}} \right) \right] \tag{32}$$

The vector contains *N* series of *nf* elements. Each element in the *ψ<sup>j</sup>*

*Digital Algebraic Method for Processing Complex Signals for Radio Monitoring Systems*

*<sup>i</sup>* <sup>¼</sup> *mod <sup>ϕ</sup> <sup>j</sup>*

occurrence of various increments. When rounding calculations up to 1°

tion interval and is calculated (in degrees) as follows:

*DOI: http://dx.doi.org/10.5772/intechopen.85590*

where *ϕ <sup>j</sup>*

½ � *i* � 0*;* 5; *i* þ 0*;* 5 .

**Figure 6.**

**39**

*Tuple of reference functions.*

gram *Ψ* will contain 360 elements:

function based on the following expression:

*GF m*ð Þ¼ *; <sup>w</sup>; <sup>d</sup>* <sup>1</sup>

of modulation *m* ¼ 2*,* 4*,* 8 *и* 16 is presented in **Figure 6**.

*σ* ffiffiffiffiffi <sup>2</sup>*<sup>π</sup>* <sup>p</sup> <sup>∑</sup> *m*�1 *i*¼0

where *m* is the multiplicity of RPM, *w* the phase value in degrees, and *d* the offset parameter of the "reference comb" of the histogram maxima on the interval <sup>0</sup> � <sup>360</sup>° � �. This expression has no probabilistic sense, since the Gaussian distribution is used only as the convenient analytical definition of the multimodal structure of reference functions. An example of generation of references for the multiplicity

*ψ j*

phase increment measurement of the*j*-th subcarrier oscillation on the *i*-th modula-

modulation interval. Since the multiplicity of RPM in all frequencies must be the same, the vector *Ψ* can be considered as a single statistic characterizing the entire OFDM signal. Based on the vector of the statistics of phase increments *Ψ*, we form a vector histogram *Ψ*, the elements of which are proportional to the frequency of

where *nψ<sup>i</sup>* is the number of elements of the vector Φ that fall inside the interval

A visual analysis of the histogram (36) lets us reliably identify the type of modulation used. However, the automatic analysis requires the formal mathematical one. The cross-correlation which is the most reliable mathematical tool for calculating the degree of similarity of two functions should be the natural basis of such algorithm. A tuple of reference functions for calculating mutual correlation with a histogram vector *Ψ* can be obtained using Gaussian probability density

*<sup>i</sup>*þ<sup>1</sup> � *<sup>ϕ</sup> <sup>j</sup> i*

*<sup>i</sup>* is the absolute phase measurement of the *j*-th subcarrier on the *i*-th

� �*;* <sup>360</sup> h i (35)

*Ψ* ¼ *ψ*<sup>0</sup> f g *;* …*; ψ*<sup>359</sup> *, ψ<sup>i</sup>* ¼ *nψi,* (36)

*exp* � *<sup>w</sup>* � *<sup>d</sup>* <sup>360</sup>

*m* � �<sup>2</sup> 2*σ*<sup>2</sup> ! *<sup>i</sup>* series is the

, the histo-

*,* (37)

Obviously, for different subcarrier frequencies, the same time shift *dT* will cause phase determination errors which are different in magnitude. By knowing the range of carrier frequencies of the vector *F*, we can precisely calculate the corrections to the calculations for the current phase of the *L*-th carrier frequency on *i*-th modulation interval as follows:

$$d\phi\_{L,i} = 2\pi \cdot f\_L \cdot dT\_i = \mathbf{360} \cdot f\_L \cdot dT\_i \tag{33}$$

The multiplicity of the phase modulation used in the observed signal is uniquely determined by the minimal difference between the closest values of the vector *Φ* elements calculated on the basis of expression (31), taking into account corrections (32) and (33). Under the conditions of a priori uncertainty, the multiplicity of relative PM (RPM) applied in the subcarrier frequencies can be detected only by a statistical method based on an analysis of a series of SLAE solutions from the sample of consecutive modulation intervals which has a sufficiently large size *N*. For this the vector is created:

$$\Psi = \left\{ \boldsymbol{\psi}\_0^0, \dots, \boldsymbol{\psi}\_{N-1}^0; \ \boldsymbol{\psi}\_0^1, \dots, \boldsymbol{\psi}\_{N-1}^1; \dots; \boldsymbol{\psi}\_0^{n\_f - 1}, \dots, \boldsymbol{\psi}\_{N-1}^{n\_f - 1} \right\} \tag{34}$$

**Figure 5.** *Asynchronous sampling illustrations.*

*Digital Algebraic Method for Processing Complex Signals for Radio Monitoring Systems DOI: http://dx.doi.org/10.5772/intechopen.85590*

The vector contains *N* series of *nf* elements. Each element in the *ψ<sup>j</sup> <sup>i</sup>* series is the phase increment measurement of the*j*-th subcarrier oscillation on the *i*-th modulation interval and is calculated (in degrees) as follows:

$$\psi\_i^j = \text{mod}\left[\left(\phi\_{i+1}^j - \phi\_i^j\right), \mathbf{360}\right] \tag{35}$$

where *ϕ <sup>j</sup> <sup>i</sup>* is the absolute phase measurement of the *j*-th subcarrier on the *i*-th modulation interval. Since the multiplicity of RPM in all frequencies must be the same, the vector *Ψ* can be considered as a single statistic characterizing the entire OFDM signal. Based on the vector of the statistics of phase increments *Ψ*, we form a vector histogram *Ψ*, the elements of which are proportional to the frequency of occurrence of various increments. When rounding calculations up to 1° , the histogram *Ψ* will contain 360 elements:

$$\overline{\Psi} = \{\overline{\varphi}\_0, \dots, \overline{\varphi}\_{359}\}, \quad \overline{\varphi}\_i = n\mu\_i,\tag{36}$$

where *nψ<sup>i</sup>* is the number of elements of the vector Φ that fall inside the interval ½ � *i* � 0*;* 5; *i* þ 0*;* 5 .

A visual analysis of the histogram (36) lets us reliably identify the type of modulation used. However, the automatic analysis requires the formal mathematical one. The cross-correlation which is the most reliable mathematical tool for calculating the degree of similarity of two functions should be the natural basis of such algorithm. A tuple of reference functions for calculating mutual correlation with a histogram vector *Ψ* can be obtained using Gaussian probability density function based on the following expression:

$$GF(m, w, d) = \frac{1}{\sigma\sqrt{2\pi}}\sum\_{i=0}^{m-1} \exp\left(-\frac{\left(w - d\frac{360}{m}\right)^2}{2\sigma^2}\right),\tag{37}$$

where *m* is the multiplicity of RPM, *w* the phase value in degrees, and *d* the offset parameter of the "reference comb" of the histogram maxima on the interval <sup>0</sup> � <sup>360</sup>° � �. This expression has no probabilistic sense, since the Gaussian distribution is used only as the convenient analytical definition of the multimodal structure of reference functions. An example of generation of references for the multiplicity of modulation *m* ¼ 2*,* 4*,* 8 *и* 16 is presented in **Figure 6**.

**Figure 6.** *Tuple of reference functions.*

value *T=t<sup>д</sup>* ¼ *f <sup>д</sup>* � *T* is not an integer. This leads to a variation of the value of "indentation" *dTi* from the beginning of the *i*-th modulation interval. This phenomenon is illustrated in **Figure 5**. Differences in the values of *dTi* lead to the fact that the phase error periodically repeated on a combination of several adjacent modulation intervals is added to the correct solution of the system. The significant "blurring" of phase increments in subcarrier frequencies, as well as the departure of their average values from standardized nominal values, is an external manifestation of this asynchronous effect. Therefore, the phase errors of asynchronous sampling are an additional source of disturbing noise, which must be taken into account and,

A known correlation between magnitudes *W* and *t<sup>д</sup>* allows us to estimate the magnitude of these errors. On an arbitrary *i*-th interval, the error in determining the

*<sup>W</sup>* � *round i <sup>f</sup> <sup>д</sup>*

Obviously, for different subcarrier frequencies, the same time shift *dT* will cause phase determination errors which are different in magnitude. By knowing the range of carrier frequencies of the vector *F*, we can precisely calculate the corrections to the calculations for the current phase of the *L*-th carrier frequency on *i*-th modula-

The multiplicity of the phase modulation used in the observed signal is uniquely determined by the minimal difference between the closest values of the vector *Φ* elements calculated on the basis of expression (31), taking into account corrections (32) and (33). Under the conditions of a priori uncertainty, the multiplicity of relative PM (RPM) applied in the subcarrier frequencies can be detected only by a statistical method based on an analysis of a series of SLAE solutions from the sample of consecutive modulation intervals which has a sufficiently large size *N*. For this

� � � �

*W*

*dϕL,i* ¼ 2*π* � *f <sup>L</sup>* � *dTi* ¼ 360 � *f <sup>L</sup>* � *dTi* (33)

*<sup>N</sup>*�<sup>1</sup>; ……; *<sup>ψ</sup>nf* �<sup>1</sup>

<sup>0</sup> *;* …*; <sup>ψ</sup>nf* �<sup>1</sup> *N*�1

(32)

(34)

*f д*

*dTi* ¼ *t<sup>д</sup> i*

if possible, compensated.

*Modulation in Electronics and Telecommunications*

initial measurement is

tion interval as follows:

the vector is created:

**Figure 5.**

**38**

*Asynchronous sampling illustrations.*

*<sup>Ψ</sup>* <sup>¼</sup> *<sup>ψ</sup>*<sup>0</sup>

<sup>0</sup>*;* …*; ψ*<sup>0</sup>

*<sup>N</sup>*�<sup>1</sup>; *<sup>ψ</sup>*<sup>1</sup>

<sup>0</sup>*;* …*; ψ*<sup>1</sup>

n o

The structure of the references specified by Eq. (37) ensures the symmetry of the structure of functions relative to the average value of the phase *π*. For the given multiplicity *m* based on Eq. (37), the vector *x* for different values of the shift of the reference comb is formed:

$$\mathbf{x} = \{\mathbf{x}\_0, \dots, \mathbf{x}\_{359}\}, \quad \mathbf{x}\_i = GF(m, i, j), \quad i = 0, \dots, 359; \quad j = 0, \dots, 180 \tag{38}$$

For each shift value *<sup>j</sup>*, the cross-correlation coefficient is calculated: *Xj* <sup>¼</sup> *<sup>x</sup>*�*<sup>Ψ</sup>* j j� *<sup>x</sup>* j j *<sup>Ψ</sup>* where "�" denotes a scalar product.

The maximal obtained value *Ym* ¼ *max Xj* � � is taken as an indicator of the similarity of the histogram *Ψ* with the reference of the given multiplicity. Then the next value *m* is taken and calculations are repeated. After a complete exhaustion of all possible values of multiplicity, the largest calculated value *Ym* is determined. The index of this element corresponds to the detected multiplicity of phase modulation *m*. The described computational process for calculating the correlation proximity is conveniently implemented using the mechanism of nested cycles. It is easy to program, and the numerous tests show that it guarantees the correct finding of the modulation multiplicity even under conditions of significant distortion of signal samples.

#### **4.2 Algebraic demodulation**

For the final demodulation, the SLAE of reduced dimensionality is compiled with a rectangular matrix of coefficients with unknowns:

$$\begin{aligned} A &= ||a\_{i,j}||, \quad i = 0, \ldots, Num - 1, \quad j \in \mathbb{N};\\ B &= \{b\_0, \ldots, b\_{\text{Num}-1}\}, \quad b\_i = \jmath\_{j+i}, \quad J = n\_T^0 + round((\mathcal{N} \cdot \mathcal{f}\_o)/\mathcal{W}). \end{aligned} \tag{39}$$

The solution of the system *<sup>A</sup><sup>T</sup>* � *<sup>A</sup>* � *<sup>Z</sup>* <sup>¼</sup> *<sup>A</sup><sup>T</sup>* � *<sup>B</sup>* for several modulation intervals gives a vector of absolute values of the phases which gives a sequence of corrected phase increments after transformations. To identify the combinations of binary symbols in accordance with the obtained modulation multiplicity and the corresponding keying code, it is necessary to calculate the decision boundaries. Since all combinations of binary symbols are assumed to be equally probable and phase errors are distributed normally and symmetrically around the standard positions, then, in accordance with the maximum likelihood rule, it is advisable to use equidistant decision-making boundaries:

$$G\_i = i\frac{360}{m}, \quad i = 0, \ldots m \tag{40}$$

The full stream of transmitted characters is formed with this rule. The result of the work of the considered method of automatic technical radio monitoring of the source of OFDM signals is represented as received by binary sequences, for

**Parameter Value**

Stage 1 Period of orthogonality 73 0.009 —

*Digital Algebraic Method for Processing Complex Signals for Radio Monitoring Systems*

Sign of the prefixes correlation—negative

Numbers of carriers—17

*DOI: http://dx.doi.org/10.5772/intechopen.85590*

channel =0

*k*∈1…16

*Analysis experiment results.*

Number of phase angles PSK—4

Number and nominal value of carrier of synchronization

Numbers and nominal values of information carriers

Subcarriers distance — — 110 Duration of the OFDM character 107.083 0.013 —

Starting number of the first full OFDM character 107 — — Time of analysis — 0.5 — Sample size of measurements 4001 — — Stage 2 The possible frequency grid *i*∈0…31 — — 55 þ *i* � 110

Nominals of carriers *j*∈ 0…16 825 þ *j* � 110

**Samples Seconds Hertz**

— — 825

— — 825 þ *k* � 110

The correctness of the considered method for analyzing and demodulating the OFDM signals under conditions of partial uncertainty has been tested using both

Based on the considered algebraic algorithms for automatic analysis, recognition, and demodulation of signals, a software package was developed. **Table 1** presents an example of the main results of the use of the software package for the analysis and demodulation of a complex signal of the standard MIL188-110A. The moment of the beginning of a signal observation was accidental. The test signal was sampled in the low-frequency band 0.3÷3.4 kHz after spectrum transfer by decimation of measurements. In this example, the signal contained 16 informational subcarrier frequencies and 1 synchronization subchannel at a modulation rate of 75 OFDM characters/s and relative PSK-4. The sampling rate was 8000 sample/s. From the data in the table, it follows that it took 0.5 seconds of observation to identify all the parameters of the signal and start demodulation. Note that signal processing can be done in real time or based on the results of previously stored

real and simulated digital measurements of signals.

example,

**Table 1.**

and so on.

digital samples.

**41**

Then the rule for forming a bit stream at the output of the investigated example of a digital demodulator when using, for example, four phase angles on *i*-th modulation interval for one carrier frequency can be written as a set of conditions for forming a pair of binary symbols:

$$\mathfrak{s}\_{2,i}, \mathfrak{s}\_{2\dot{i}+1} = \begin{cases} \mathfrak{l}, \mathfrak{0}; & \dot{\mathfrak{f}}^{\prime} \quad G\_0 \le \mathbb{P}\_i < G\_1; \\ \mathfrak{0}, \mathfrak{0}; & \dot{\mathfrak{f}}^{\prime} \quad G\_1 \le \mathbb{P}\_i < G\_2; \\ \mathfrak{0}, \mathfrak{1}; & \dot{\mathfrak{f}}^{\prime} \quad G\_2 \le \mathbb{P}\_i < G\_3; \\ \mathfrak{1}, \mathfrak{1}; & \dot{\mathfrak{f}}^{\prime} \quad G\_3 \le \mathbb{P}\_i < G\_4; \end{cases} \quad i = 0, \mathfrak{1}, \mathfrak{2}, \ldots \tag{41}$$


*Digital Algebraic Method for Processing Complex Signals for Radio Monitoring Systems DOI: http://dx.doi.org/10.5772/intechopen.85590*

#### **Table 1.** *Analysis experiment results.*

The structure of the references specified by Eq. (37) ensures the symmetry of the structure of functions relative to the average value of the phase *π*. For the given multiplicity *m* based on Eq. (37), the vector *x* for different values of the shift of the

*x* ¼ f g *x*0*;* …*; x*<sup>359</sup> *, xi* ¼ *GF m*ð Þ *; i; j , i* ¼ 0*,* …*,* 359*; j* ¼ 0*,* …*,* 180 (38)

For each shift value *<sup>j</sup>*, the cross-correlation coefficient is calculated: *Xj* <sup>¼</sup> *<sup>x</sup>*�*<sup>Ψ</sup>*

larity of the histogram *Ψ* with the reference of the given multiplicity. Then the next value *m* is taken and calculations are repeated. After a complete exhaustion of all possible values of multiplicity, the largest calculated value *Ym* is determined. The index of this element corresponds to the detected multiplicity of phase modulation *m*. The described computational process for calculating the correlation proximity is conveniently implemented using the mechanism of nested cycles. It is easy to program, and the numerous tests show that it guarantees the correct finding of the modulation multiplicity even under conditions of significant distortion of signal

For the final demodulation, the SLAE of reduced dimensionality is compiled

*, J* <sup>¼</sup> *<sup>n</sup>*<sup>0</sup>

The solution of the system *<sup>A</sup><sup>T</sup>* � *<sup>A</sup>* � *<sup>Z</sup>* <sup>¼</sup> *<sup>A</sup><sup>T</sup>* � *<sup>B</sup>* for several modulation intervals gives a vector of absolute values of the phases which gives a sequence of corrected phase increments after transformations. To identify the combinations of binary symbols in accordance with the obtained modulation multiplicity and the corresponding keying code, it is necessary to calculate the decision boundaries. Since all combinations of binary symbols are assumed to be equally probable and phase errors are distributed normally and symmetrically around the standard positions, then, in accordance with the maximum likelihood rule, it is advisable to use

*<sup>T</sup>* þ *round N* � *f <sup>д</sup>*

*, i* ¼ 0*,* …*m* (40)

*i* ¼ 0*,* 1*,* 2*,* …

(41)

� �*=<sup>W</sup>* � �*:* (39)

j j� *<sup>x</sup>* j j *<sup>Ψ</sup>*

� � is taken as an indicator of the simi-

reference comb is formed:

samples.

where "�" denotes a scalar product.

**4.2 Algebraic demodulation**

*A* ¼ *ai,j* � � �

The maximal obtained value *Ym* ¼ *max Xj*

*Modulation in Electronics and Telecommunications*

with a rectangular matrix of coefficients with unknowns:

*<sup>B</sup>* <sup>¼</sup> f g *<sup>b</sup>*0*;* …*; bNum*�<sup>1</sup> *, bi* <sup>¼</sup> *yJ*þ*<sup>i</sup>*

equidistant decision-making boundaries:

forming a pair of binary symbols:

**40**

*s*<sup>2</sup>�*i, s*<sup>2</sup>�*i*þ<sup>1</sup> ¼

8 >>>>><

>>>>>:

�*, i* ¼ 0*,* …*, Num* � 1*, j*∈ *N;*

*Gi* ¼ *i*

360 *m*

*,* 0*; if G*<sup>0</sup> ≤*Ψ<sup>i</sup>* < *G*1*; ,* 0*; if G*<sup>1</sup> ≤ *Ψ<sup>i</sup>* < *G*2*; ,* 1*; if G*<sup>2</sup> ≤ *Ψ<sup>i</sup>* < *G*3*; ,* 1*; if G*<sup>3</sup> ≤*Ψ<sup>i</sup>* < *G*4*;*

Then the rule for forming a bit stream at the output of the investigated example of a digital demodulator when using, for example, four phase angles on *i*-th modulation interval for one carrier frequency can be written as a set of conditions for

The full stream of transmitted characters is formed with this rule. The result of the work of the considered method of automatic technical radio monitoring of the source of OFDM signals is represented as received by binary sequences, for example,


and so on.

The correctness of the considered method for analyzing and demodulating the OFDM signals under conditions of partial uncertainty has been tested using both real and simulated digital measurements of signals.

Based on the considered algebraic algorithms for automatic analysis, recognition, and demodulation of signals, a software package was developed. **Table 1** presents an example of the main results of the use of the software package for the analysis and demodulation of a complex signal of the standard MIL188-110A. The moment of the beginning of a signal observation was accidental. The test signal was sampled in the low-frequency band 0.3÷3.4 kHz after spectrum transfer by decimation of measurements. In this example, the signal contained 16 informational subcarrier frequencies and 1 synchronization subchannel at a modulation rate of 75 OFDM characters/s and relative PSK-4. The sampling rate was 8000 sample/s.

From the data in the table, it follows that it took 0.5 seconds of observation to identify all the parameters of the signal and start demodulation. Note that signal processing can be done in real time or based on the results of previously stored digital samples.
