PERIODIC TIME SPREAD PULSE SIGNALS WITH PERFECT AUTOCORRELATION

PERIODIC TIME SPREAD PULSE SIGNALS WITH PERFECT AUTOCORRELATION A. V. Titov 1 and G. J. Kazmierczak 2 1 Electro Technical University LETI in St. Petersburg, Russia 2 Cybersecurity Consultant, Princeton, New Jersey, United States of America E-mail: alvastitov@gmail.com ABSTRACT This paper considers periodic single pulse signals (PSPS) of pulse duration T and time repetition NT, where N (N>1) are positive integers. This paper examines a new previously unknown property of these signals that have perfect periodic autocorrelation on the interval NT. It is shown that PSPS can be represented as the linear composition of several orthogonal periodic pulse signals with zero cross correlation, and once represented as such, these signals can be transformed into periodic time spread pulse signals where the pulses are spread over the entire interval NT. These transformed signals have the same energy and autocorrelation properties as the corresponding original PSPS. The results of this study advance the theory of pulse signals and can be applied to any electronic periodic pulse signal systems including radar, radio navigation, communication and telemetric systems. Keywords: periodic single pulse signals, orthogonal signals, perfect periodic autocorrelation function, cross correlation function, radar, radio navigation and communication systems. INTRODUCTION Periodic single pulse signals (PSPS), sometimes called pulse train signals [1], with rectangular pulse duration of T and time repetition NT (Figure-1a), can be represented in the following manner: S NT (t) = N S (1) NT (t), (1) where N (N > 1) are positive integers, T is the duration of a signal pulse, and NT is the period of repetition. Signals S (1) NT (t) are periodic unit pulse signals, i.e. periodic rectangle single pulse signals of duration T with the amplitude of each pulse equal to 1 and with a period of repetition NT. For signals S NT (t) (1), the amplitude N of each pulse, i.e. the amplitude of the pulse signals, is related with the time repetition NT (Figure. 1a). This relation was made in order to simplify subsequent calculations. These types of periodic single pulse signals (1) (Figure-1a) are well known and widely used in electronic systems. Many electronic telemetric, communication and radar systems utilize these signals as the properties of PSPS are well understood in time and frequency domains. The main property of PSPS in the time domain is its perfect periodic autocorrelation function (ACF) R NT (τ) which has only one maximum (without any side lobes) on time interval NT (Figure. 1b). The perfect ACF of PSPS combined with the simplicity of generation and processing the signals makes PSPS so attractive for use in radar, telemetric and communication systems for synchronization, time delay measurements, etc. purposes. The primary disadvantage of PSPS is a large peak factor, since the entire signal energy is concentrated in the narrow time interval duration of T. Another disadvantage of PSPS is the high probability of signals detection and estimation of its parameters. As a result of signal detection, an adversary can destroy the proper functioning of a system by different types of organized jamming and parameter estimation may enable unauthorized access [2]. In this paper we examine a new previously unknown property of PSPS. This property is related with the representation of PSPS as a linear composition of orthogonal, i.e. not correlated, periodic pulse signals with zero cross correlation. It is shown that after this representation, it is possible to transform periodic single pulse signals S NT (t) into periodic time spread pulse signals S NT (t) where the energy of one PSPS pulse is spread over the entire time interval NT (Figure. 1c). Both periodic signals S NT (t) and S NT (t) (Figure. 1a and Figure. 1c) have the same energy and the same perfect ACF on interval NT (Figure-1b). Figure-1. Signals S NT (t) and S NT (t). 8854

We decided to call the periodic pulse signals S NT (t) (Figure. 1c) periodic time spread pulse signals (PTSPS) as an analogy to spread spectrum signals for frequency hopping signals and leaner frequency modulated (LFM) signals [2]. By transforming PSPS into periodic time spread pulse signals (PTSPS), it is possible to mitigate the disadvantages of PSPS mentioned above, though generation and processing of PTSPS becomes more complicated compared to PSPS. Magnitude: p 1 =N 1, p 2 =N 1 (N 2-1), p 3 =N 1 N 2 (N 3-1). REPRESENTATION OF PSPS AS A COMPOSITION OF PERIODIC ORTHOGONAL PULSE SIGNALS WITH ZERO CROSS CORRELATION We will show that for any N (N > 1) (1), which are not prime numbers N p, i.e. N N p (N p = 2, 3, 5, 7, 11, etc.), PSPS can be represented as a composition of periodic orthogonal pulse signals with zero cross correlation. Any N (N > 1) (1), which is not a prime number N p (N N p ), can be represented as the product N = N 1 N 2 N n = N i, (2) where i = 1, 2, 3,, n, and N i are positive integers. Notice that value of i in (2) is not related with the value of N i at all, and it is only related with the order of product values (2). For instance, the number N = 30 can be represented as products: 1) N = 2 3 5, 2) N = 3 2 5 and 3) N = 5 2 3; and the N 1 values are different in all of these three cases. The N 1 values are N 1 = 2, N 1 = 3, and N 1 = 5 for case 1, case 2, and case 3 respectively. After presenting N as a product (2), periodic single pulse signals (1) can be represented as the composition of n independent, i.e. non-correlated, periodic pulse signals with zero cross correlation: S NT (t) = N S (1) NT (t) = P Ni (t), (3) i where i = 1, 2, 3,, n, and where P Ni (t) are periodic pulse signals of duration T with zero cross correlation between them. The characteristics of periodic pulse signals P Ni (t) are defined, as will be shown below, by the values of N i (2). Consider the following examples: Figure-2.Signals S NT (t) and S NT (t) for N=4. Example 2: N=10, N=N 1 N 2 N n =ПN i =2 5=10, where N 1 =2, N 2 =5 and n=2. For this example, periodic pulse signals P N1 (t) (i=1) and P N2 (t) (i=2) are plotted in Figure-3a and Figure- 3b respectively and the periodic single pulse signal S NT (t) = P N1 (t)+p N2 (t) is represented in Figure-3c. Example 1: N=4, N=N=N 1 N 2... N n =П N i =2 2=4, where N 1 =2, N 2 =2 and n=2. Periodic pulse signals P N1 (t) (i=1) and P N2 (t) (i=2) for example 1 are plotted in Figure-2a and Figure-2b respectively and the periodic single pulse signal S NT (t) = P N1 (t)+p N2 (t) is represented in Figure-2c. In Figures 2, 3, 4, and 5, we have the following values of T 1, T 2, T 3, p 1, p 2, and p 3 ; Time duration: T 1 =N 1 T, T 2 =(N 1 N 2 ) T, T 3 =(N 1 N 2 N 3 ) T, (4) 8855

Figure-3. Signals S NT (t) and S NT (t) for N=10. Example 3: N=10, N=N 1 N 2 N n =П N i =5 2=10, where N 1 =5, N 2 =2and n=2. For this example, periodic pulse signals P N1 (t) (i=1) and P N2 (t) (i=2) are plotted in Figure-4a and Figure- 4b respectively and periodic single pulse signal S NT (t) = P N1 (t) + P N2 (t) is represented in Figure-4c. Figure-5. Signals S NT (t) and S NT (t) for N=18. Example4: N=18;N=N 1 N 2 N n =ПN i =2 3 3=18, where N 1 =2, N 2 =3, N 3 =3 and n=3. For this example, periodic pulse signals P N1 (t) (i = 1), P N2 (t) (i = 2) and P N3 (t) (i = 3) are plotted in Figure-5a, Figure-5b and Figure-5c respectively and periodic single pulse signal S NT (t) = P N1 (t) + P N2 (t) + P N3 (t) is shown in Figure-5d. The number of periodic pulse signals P Ni (t) in the representations above is defined by the number of factors in (2), i.e. by value of n. Figure-2c, Figure-3c and Figure- 4c show that periodic single pulse signals S NT (t) (1), (3) for N = 4 (n = 2) and for N = 10 (n = 2) are the sum of two periodic pulse signals P N1 (t) and P N2 (t). Figure. 5d shows that periodic single pulse signals S NT (t) (1), (3) for N = 18 (n = 3) are the sum of three periodic pulse signals P N1 (t), P N2 (t) and P N3 (t). Observe (Figure-2, Figure-3, Figure-4 and Figure-5) that all periodic pulse signals P Ni (t) for the same value of N are orthogonal, i.e. non-correlated, signals with zero cross correlation. It is easy to show that cross correlation functions of all periodic pulse signals P Ni (t) for N = 4, N = 10, and N = 18 (Figure. 2, Figure. 3, Figure. 4 and Figure. 5) are zero for all values of τ on interval NT: NT R p(ij) (τ) = P Ni (t) P Nj (t-τ) dt = 0 (5) 0 Figure-4. Signals S NT (t) and S NT (t) for N=10. 8856

where i = 1, 2, 3,, n; j = 1, 2, 3,, n; and i j. Similar properties of periodic single pulse signals S NT (t) (1), (2) and (3) exist for all N which are non-prime numbers N p, i.e. N N p, where N p = 2, 3, 5, 7, 11, etc. In the general case, the two main rules for representing periodic single pulse signals S NT (t) (1) of pulse duration T and with period of repetition NT (N > 1, N N p ) as the sum of n periodic orthogonal pulse signals P Ni (t) (3) are: Rule 1: for i = 1, P Ni (t) = P N1 (t) = S N1T (t), where S N1T (t) are periodic single pulse signals of duration T with period of repetition of N 1 T and a amplitude of N 1, i.e. p 1 = N 1 and T 1 = N 1 T (Figure-2a, Figure-3a, Figure-4a and Figure-5a). Rule 2: for i > 1, i = 2, 3, 4,, n, P Ni (t) are periodic pulse signals with period of repetition T i, where T i =(N 1 N 2 N 3 N i ) T. Each period of repetition of duration T i consists of N i pulses divided by time intervals T (i-1), where T (i-1) =(N 1 N 2 N 3 N i-1 ) T. The first pulse p i of the N i pulses has a positive value p i =(N 1 N 2 N 3 N (i-1) ) (N i -1), and the remaining N i -1 pulses have negative values equal to -(p 1 + p 2 + + p (i-1) )=- (N 1 N 2 N 3 N (i-1) ) (Figure. 2b, Figure-3b, Figure-4b, and Figures 5b and 5c). Thus, periodic single pulse signals S NT (t) (1) duration of T with period of repetition NT ((N > 1, N N p ) can be represented as linear composition, i.e. the sum, of n orthogonal periodic pulse signals P Ni (t) (3), where N=П N i and i=1, 2, 3,, n (Figure-2c, Figure-3c, Figure-4c, and Figure-5d). And all of these orthogonal periodic pulse signals P Ni (t) (3) have zero cross correlation. Occasionally it is more convenient to represent periodic pulse signals as numeric sequences such as: S NT (t) S N, where S N ={s i }=(s 1,s 2,, s N ) and P Ni (t) P Ni [3]. Applying this representation to Figure-2, where n=2 and N = N 1 N 2 =2 2=4, we obtain: N=4 i = 1 P N1 = (p 1, 0, p 1, 0, p 1, 0,p 1, 0) i = 2 P N2 = (p 2, 0, -p 1, 0, p 2, 0,-p 1, 0) - - - - - - - - - - - - - - - - - - - - - - - S N = (N,0, 0, 0, N, 0, 0, 0), where S N = P N1 + P N2, p 1 = N 1, p 2 = N 1 (N 2-1) and N = N 1 N 2, i.e. p 1 = 2, p 2 = 2 and S N = (4,0,0,0,4,0,0,0) (Figure. 2c). Applying this representation to Figure. 3, where n=2 and N = N 1 N 2 = 2 5 = 10, we obtain: N=10 i = 1 P N1 = (p 1, 0, p 1, 0, p 1, 0,p 1, 0,p 1,0) i = 2 P N2 = (p 2, 0, -p 1, 0, -p 1, 0,-p 1, 0, -p 1, 0) - - - - - - - - - - - - - - - - - - - - - - - S N = (N,0, 0, 0, 0, 0, 0, 0, 0, 0), (6) (7) where S N = P N1 + P N2, p 1 = N 1, p 2 = N 1 (N 2-1) and N = N 1 N 2, i.e. p 1 = 2, p 2 = 8 and S N = (10,0,0,0,0, 0,0,0,0,0) (Figure. 3c). Applying this representation to Figure. 4, where n=2 and N = N 1 N 2 = 5 2 = 10, we obtain: N=10 i = 1 P N1 = (p 1, 0, 0, 0, 0, p 1, 0, 0, 0, 0) i = 2 P N2 = (p 2, 0, 0, 0, 0, -p 1,0, 0, 0, 0) - - - - - - - - - - - - - - - - - - - - - - - S N = (N,0, 0, 0, 0,0, 0, 0, 0, 0), where, S N = P N1 +P N2, p 1 = N 1, p 2 = N 1 (N 2-1) and N = N 1 N 2, i.e. p 1 = 5, p 2 = 5 and S N = (10,0,0,0,0, 0,0,0,0,0) (Figure-4c). Applying this representation to Figure. 5, where n=3 and N = N 1 N 2 N 3 =2 3 3=18, we obtain: N=18 i = 1 P N1 = (p 1,0,p 1, 0,p 1, 0, p 1, 0,p 1, 0,p 1, 0, p 1, 0,p 1, 0,p 1, 0) i = 2 P N2 = (p 2, 0,-p 1, 0,-p 1, 0, p 2, 0,-p 1, 0,-p 1, 0, p 2, 0,-p 1, 0,-p 1, 0) i = 3 P N3 = (p 3,0,0,0,0,0, -(p 1 +p 2 ), 0, 0, 0, 0, 0, -(p 1 +p 2 ), 0, 0, 0, 0, 0) - - - - - - - - - - - - - - - - - - - - - - - - - - - S N = (N,0, 0, 0, 0,0, 0, 0,0, 0, 0, 0, 0, 0, 0, 0, 0, 0), where S N = P N1 + P N2 + P N1, p 1 = N 1, p 2 = N 1 (N 2-1), p 3 = N 1 N 2 (N 3-1) and (p 1 + p 2 ) = N 1 N 2, i.e. p 1 = 2, p 2 = 4, p 3 = 12, and S N = (18,0,0,0,0,0, 0,0,0,0,0,0, 0,0,0,0,0,0) (Figure-5c). From (6), (7), (8) and (9) we observe that the total energy of all sequences P Ni is the same as the energy of sequences S N, i.e. (P Ni ) 2 = (S N ) 2. PERIODIC TIME SPREAD PULSE SIGNALS WITH PERFECT AUTOCORRELATION PSPS S NT (t) (1) have perfect autocorrelation function R NT (τ) with the one main lobe equal to N 2 T (where N 2 T is the energy of PSPS on time interval NT) and no side lobes on the entire time interval NT (Figure. 1b). Because PSPS S NT (t) (1) can be represented as the sum of orthogonal pulse signals P Ni (t) (3) with zero cross correlation, autocorrelation function of S NT (t) can be represented as the autocorrelation function of the sum of orthogonal pulse signals P Ni (t). But, as it is well known, the autocorrelation function of the sum of orthogonal pulse signals with zero cross correlation is the sum of the autocorrelation functions of these signals [3], i.e. the sum of the autocorrelation functions of orthogonal pulse signals P Ni (t) NT (8) (9) 8857

R NT (τ) = R pi (τ) = P Ni (t) P Nj (t-τ) dt = 0, (10) ii 0 where i=1,2,3,,n; j=1,2,3,,n; and i=j; and where the difference in the integrals between (5) and (10) is that i j (cross correlation) in (5), but i=j (autocorrelation) in (10). From Figure-2, Figure-3, Figure-4, and Figure-5, we observe that the autocorrelation functions of each of the orthogonal pulse signals P Ni (t) are not perfect. We obtain perfect periodic autocorrelation functions for the sum of the signals P Ni (t) only after the summation procedure (10). The sum of periodic autocorrelation functions Rpi(τ) (10) is the same as the perfect autocorrelation function R NT (τ) with the one main lobe equal to N 2 T and zero side lobes on the entire time interval NT (Figure-1b). As an example, in Figure-6 depicts the autocorrelation functions of signals Rpi (τ) (Figure-6a, Figure-6b) and the sum of these ACF RNT (τ) = Rpi (τ) (Figure-6c) for case 2) N = 10, N = N 1 N 2 = 5 2, and n = 2 (Figure-4). A similar property of autocorrelation functions will exist for any value of N (N>1, N Np). 1) N=4 N=N=N 1 N 2 N n =П N i =2 2=4, where N 1 =2, N 2 =2, n=2 and sequence P N1 (6) has a cyclic shift = 1 (equivalent to the time shift of P N1 (t) on τ = T, Figure. 2). i = 1 P N1 ( =1)= (0,p 1, 0, p 1, 0, p 1, 0,p 1 ) i = 2 P N2 = (p 2, 0,-p 1, 0, p 2, 0,-p 1,0) - - - - - - - - - - - - - - - - - - - - - - - - - - - - S N = (p 2, p 1,-p 1, p 1, p 2, p 1,-p 1, p 1 ), (11) where S N =P N1 ( =1)+P N2, p 1 =N 1, p 2 =N 1 (N 2-1), and N=N 1 N 2 ; i.e. p 1 =2, p 2 =2 and S N = (2,2,-2,2,2,2,- 2,2). The periodic signal S NT (t) is represented in Figure. 2d. 2) N=10 N=N 1 N 2... N n = П N i = 2 5 = 10, where N 1 = 2, N 2 =5, n=2 and sequence P N1 (7) has a cyclic shift =1 (equivalent to time shift of P N1 (t) on τ = T, Figure. 3). i = 1 P N1 ( =1)= (0,p 1, 0, p 1, 0, p 1, 0, p 1, 0, p 1 ) i = 2 P N2 = (p 2, 0,-p 1, 0,-p 1, 0,-p 1,0,-p 1, 0) (12) S N =(p 2,p 1,-p 1,p 1,-p 1,p 1,-p 1,p 1,-p 1, p 1 ), where S N = P N1 ( = 1) + P N2, p 1 = N 1, p 2 = N 1 (N 2-1) and N = N 1 N 2 ; i.e. p 1 = 2, p 2 = 8 and S N = (8,2,-2,2,- 2,2,-2,2,-2,2). Periodic signal S NT (t) is shown in Figure. 3d. 3) N=10 N=N 1 N 2... N n = П N i = 5 2 = 10, where N 1 = 5, N 2 = 2, n = 2 and sequence P N2 (8) has cyclic shift = 2 (equivalent to the time shift of P N2 (t) on τ= 2T, Figure. 4). i = 1 P N1 = (p 1,0, 0, 0, 0,p 1, 0, 0, 0, 0) i = 2 P N2 ( =2) = (0, 0,p 2,0, 0, 0, 0,-p 1, 0, 0) S N = (p 1,0, p 2, 0, 0,p 1,0,-p 1, 0,0), (13) Figure-6.Autocorrelation functions R p1 (τ), R p2 (τ), and R NT (τ) for N=10. Recall another property of periodic orthogonal signals with zero cross correlation, whereby the periodic autocorrelation functions of orthogonal signals with zero cross correlation are unaffected by changes of the mutual (relative) time delay (cyclic shift) between orthogonal signals. This is a very important property of periodic signals. When combined with the previous properties of periodic single pulses signals, this property leads to the formation of new types of signals S NT (t) (PTSPS) from S NT (t) (PSPS) that have the same energy and the same perfect periodic autocorrelation functions as PSPS (Figure-2d, Figure-3d, Figure-4d, and Figure-5e). Applying this property to our examples above (6), (7), (8) and (9), we obtain: wheres N = P N1 + P N2 ( = 2),p 1 = N 1,p 2 = N 1 (N 2-1)andN = N 1 N 2 ;i.e.p 1 = 5,p 2 = 5and S N = (5,0,5,0,0,5,0,-5,0,0). Periodic signal S NT (t) is shown in Figure. 4d. 4) N=18 N = N 1 N 2 N n = П N i = 2 3 3 = 18, where N 1 = 2, N 2 = 3, N 3 = 3, n = 3 and sequence P N1 (9) has a cyclic shift = 1 (equivalent to a time shift of P N1 (t) on τ = T, Figure. 5), 8858

i = 1 P N1 ( =1)= (0,p 1, 0,p 1,0,p 1, 0, p 1, 0,p 1, 0,p 1, 0, p 1, 0,p 1, 0,p 1 ) i = 2 P N2 = (p 2, 0,-p 1, 0,-p 1, 0, p 2,0,-p 1, 0,-p 1, 0, p 2, 0,-p 1, 0,-p 1, 0) i = 3 P N3 = (p 3, 0, 0, 0, 0, 0, -(p 1 +p 2 ), 0, 0, 0, 0, 0, -(p+p 2 ),0, 0, 0, 0,0) S N = (p 2 +p 3,p 1,-p 1,p 1,-p 1,p 1, -p 1, p 1,-p 1,p 1,-p 1,p 1, -p 1, p 1,-p 1,p 1,-p 1,p 1 ), (14) where S N = P N1 ( = 1) + P N2 + P N3, p 1 = N 1, p 2 = N 1 (N 2-1), p 3 = N 1 N 2 (N 3-1), and (p 1 + p 2 ) = N 1 N 2 ; i.e. p 1 = 2, p 2 = 4,p 3 = 12 and S N = (16,2,-2,2,-2,2, -2,2,-2,2,-2,2, -2,2,-2,2,-2,2). 5) N=18 N=N 1 N 2 N n = П N i = 2 3 3 = 18, where N 1 = 2, N 2 = 3, N 3 = 3, n = 3 and sequence P N1 (9) has cyclic shift = 1 (equivalent to a time shift of P N1 (t) τ = T, Figure. 5) and negative polarity, i = 1 -P N1 ( =1)= (0,-p 1, 0,-p 1,0,-p 1, 0, -p 1,0,-p 1, 0,-p 1, 0, -p 1, 0,-p 1, 0,-p 1 ) i = 2 P N2 = (p 2, 0,-p 1, 0,-p 1, 0, p 2, 0,-p 1, 0,-p 1, 0, p 2, 0,-p 1, 0,-p 1, 0) i = 3 P N3 = (p 3, 0, 0, 0,0,0, -(p 1 +p 2 ),0, 0, 0, 0, 0, -(p+p 2 ),0, 0, 0, 0, 0) S N = (p 2 +p 3,-p 1,-p 1,-p 1,-p 1,-p 1, -p 1,-p 1,-p 1,-p 1,-p 1,-p 1, -p 1,-p 1,-p 1,-p 1,-p 1,-p 1 ), (15) Where S N = -P N1 ( = 1) + P N2 + P N3, p 1 = N 1,p 2 = N 1 (N 2-1), p 3 = N 1 N 2 (N 3-1), and (p 1 +p 2 ) = N 1 N 2 ; i.e. p 1 = 2, p 2 = 4, p 3 = 12 and S N = (16,-2,-2,-2,-2,-2,,-2,-2,-2,-2,-2) = ((N-2),-2,-2,,-2). Periodic signal S NT (t) is represented in Figure-5e. Notice that signal S NT (t)(15) was found in [4] using a completely different approach.from(11), (12), (13), (14), and (15), it is easy to calculate that for the same N the energy of S N is the same as the energy of S N (6), (7), (8) and (9). It is also easy to show that for the same N, sequences S N and S N, despite their differences, have the same perfect ACF on interval N. Thus, it has been shown that the primary features of periodic time spread signals S NT (t) (PTSPS) (11), (12), (13), (14) and (15) are: a) The energy of signals S NT (t) (PTSPS) on time interval NT are the same as energy of the corresponding single pulse periodic signals S NT (t) (PSPS), i.e. N 2 T (Figure. 1a, Figure. 1c). b) Signals S NT (t) (PTSPS) and the corresponding signals S NT (t) (PSPS) both have the same perfect periodic ACF on time interval NT, i.e. on the period of repetition interval NT (Figure. 1b). The examples (11), (12), (13), (14) and (15) above were used only to show the procedure of transforming PSPS into PTSPS. Observe that there are different ways, i.e. different combinations of multipliers, to represent N as the multiplication of two, three and more integers (2). For some N, for instance for N = 4 (11), there is only one way to represent N, i.e. the one combination N = 2 2. For other N, for instance for N = 10 (12), (13), there are only two ways to represent N, i.e. two different combinations N = 2 5 and N = 5 2. And for other N there are more than two ways to represent N, i.e. more than two different combinations. For instance, for N = 18 there are seven ways, i.e. seven different combinations, N = 2 3 3, N = 3 2 3, N = 3 3 2, N = 3 6, N = 6 3, N = 2 9 and N = 9 2. Each different combination of multipliers for the same N corresponds to a different combination of periodic orthogonal functions P i (t) (3) and subsequently different PTSPS (11), (12), (13), (14), and (15). For N = 18, the samples with only one combination from all nine combinations above were presented, namely N = 2 3 3 (9), (14) and (15). The number of different combinations of factors for N, which are not prime numbers, depends of number of factors n (2) and also depends from various combinations of these factors. For instance, for N consisting of three different factors N = A B C, there are 12 different combinations of multipliers: N = A B C, N = A C B, N = B A C, N = B C A, N = C A B, N = C B A, N = A (B C), N = (B C) A, N = B (A C), N = (A C) B, N = C (A B), and N = (A B) C. The total number of factors for N which are not prime numbers N p, i.e. N N p, can be found as [5]: N=b α c β d ɣ r y, (16) where b, c, d,, and r are prime numbers (factors), while α, β, ɣ,, and y are positive integers. The full set of different combinations of factors can be found using the general rules of combinatorics [6]. It is necessary to take into account the signs (plus or minus) of P i (t) (14), (15), and number of cyclic shifts of P i (t) (12), (13). Changes of signs (plus or minus) and changes of the number of cyclic shifts of P i (t) lead to formation of the different types of PTSPS for the same value of N. The Appendix provide examples for N = 12 with numerical calculations in order to show how many different combinations of PTSPS it is possible to obtain during the transformation of PSPS into PTSPS. In [4] it was shown that periodic sequences S (2) NT = ((N-2), -2, -2,, -2) (15) and corresponding signals S (2) NT (t) (Figure. 5e) with perfect periodic ACF on time interval NT exist for all values of N (N > 2), including prime numbers N p. 8859

This feature leads to construction of some additional sequences with perfect ACF for the same N. All of the previous examples (11), (12), (13), (14), (15) and the examples in the Appendix (cases 1-7) used P Ni (t) =P N1 (t)=s N1T (t) (i = 1) (Rule 1, above), where S N1T (t) are periodic single pulse signals of duration T, with value N 1 and period of repetition N 1 T, i.e. p 1 =N 1 and T 1 = N 1 T. If we used P (2) N1 = S (2) N1T [4] instead of P N1 = S N1T, then we obtain additional sequences with perfect periodic ACF. Some examples using P (2) N1 = S (2) N1T for N = 12 are provided in the Appendix (cases 8 and 9 corresponding to cases 4 and 7). We analyse a variety of sequences for N = 12 as examples from the Appendix: 1. S 1 N =(5, 1,-1, 1,-1, 1,-1, 1,-1, 1,-1, 1), 2. S 2 N =(4, 2, 1,-1, 1,-1,-2, 2,1,-1, 1,-1), 3. S 3 N =(2, 1, 1, 0, 1,-1,-2, 1,1, 0, 1,-1), 4. S 4 N =(2, 1, 0, 0,-1, 1, 0, 0,-1, 1, 0, 0), 5. S 5 N =(3, 1, 0,-1, 1, 0,-1, 1,0,-1, 1, 0), 6. S 6 N =(5,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1), 7. S 7 N =(1, 0, 0, 1, 0, 0, 1, 0, 0,-1, 0, 0), 8. S 8 N =(3, 1,-1,-1,-3, 1,-1,-1,-3, 1,-1,-1), 9. S 9 N = (2,2,-1,-1,-1,-1,-4, 2,-1,-1,-1,-1). (17) Notice that not all varieties of sequences for N = 12 are included in (17). In the samples in the Appendix (cases 1-7), all of the cyclic shifts of P Ni and changes of signs (pluses or minuses) (13), (15) were not used. At first glance, it is difficult to find the connections between the sequences (17) except their length, N=12. It is difficult because the sequences (17) are represented without coefficients a i (Appendix, cases 1-9) and, as result, the energy of the sequences are different. However, all of the sequences (17) and the periodic time spread pulse signals S NT (t) (PTSPS) corresponding to them have the same main properties: a) All of them were constructed using the same transformation procedure to convert PSPS into PTSPS for N=12. This transformation procedure was presented earlier in this paper. b) All of them have perfect periodic ACF on interval NT, i.e. they have ACF with the one main lobe and with zero side lobes on interval NT. Periodic signals with perfect periodic ACF are used in many telemetric, communication, radio navigation, and radar systems. Below we provide several examples of applications of the concepts presented in this paper in communication, radio navigation, and radar systems [1], [2]. As a first example, applications in communication systems utilize orthogonal signals to transmit information. All S N sequences can be used to make orthogonal matrices with size N x N (N > 2). For instance, for N = 6, M N=6 = M N=6 = 2 1 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 1, 1 1 1 2 1 1 1 1 1 1 2 1 [ 1 1 1 1 1 2] 2 1 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 1. 1 1 1 2 1 1 1 1 1 1 2 1 [ 1 1 1 1 1 2] We observe that matrices M N of size N x N = 6 x 6 above are orthogonal matrices. These matrices can be used to create orthogonal signals to transmit information as used with orthogonal Walsh-Hadamard matrices in CDMA systems [2]. The second example application is related with coherent radars [1]. As mentioned in [4], coherent band pass signals with binary phase modulation corresponding to sequences S (2) N = ((N-2), -2, -2,, -2) in coherent radar application can provide high time delay (distance) resolution together with high Doppler frequency (speed) resolution. The same is true for signals S NT (t) which can provide high time delay (distance) resolution defined by T (T is the duration of a single pulse) together with high Doppler frequency (speed) resolution 1/NT (where NT is the duration of the entire periodic signal, which is the same as the period of repetition NT). CONCLUSIONS It has been shown that periodic single pulse signals S NT (t) (PSPS) of duration T with period of repetition NT (N > 1, N N p, where N p are prime numbers) can be represented as a linear composition of periodic orthogonal pulse signals. This study also formulates the main rules of the linear composition representation. It has been shown that after PSPS has been represented as a linear composition, these signals can be transformed into periodic time spread pulse signals S NT (t) (PTSPS) where the energy of the original single pulse, which was concentrated on a narrow interval T, is spread over the entire interval NT. During this transformation procedure the main property of PSPS, periodic perfect ACF on interval NT, does not change. This invariance means that the ACF of periodic time spread pulse signals (PTSPS) after transformation from PSPS is the same as the perfect periodic ACF for the original periodic single pulse signals (PSPS). The results of this study advance the theory of pulse signals and can be applied to any periodic pulse signals systems including radar, radio navigation, communication and telemetric systems. 8860

APPENDIX N=12 There are seven different combinations of multipliers for N=N 1 N 2 N n = ПN i =12. 1. N=N 1 N 2 N 3 =2 2 3=12 2. N=N 1 N 2 N 3 =2 3 2=12 3. N=N 1 N 2 N 3 =3 2 2=12 4. N=N 1 N 2 =4 3=12 5. N=N 1 N 2 =3 4=12 6. N=N 1 N 2 =2 6=12 7. N=N 1 N 2 =6 2=12. The main rules of the transformation procedure (6), (7), (8) and (9) are: T 1 =N 1 T, T 2 = (N 1 N 2 ) T, T 3 = (N 1 N 2 N 3 ) T, p 1 = N 1, p 2 = N 1 (N 2-1), p 3 = N 1 N 2 (N 3-1), S NT (t) = P N1 (t)+p N2 (t) (Figure-2, Figure. 3, Figure. 4) and S NT (t) = P N1 (t) + P N2 (t) + P N3 (t) (Figure-5). 1. N=N 1 N 2 N 3 =2 2 3=12, N 1 =2, N 2 =2, N 3 =3, i = 1 P N1 = (2,0,2,0,2,0, 2,0,2,0,2,0) i = 2 P N2 = (2,0,-2,0,2,0, -2,0,2,0,-2,0) i = 3 P N3 = (8,0,0, 0,-4,0, 0,0,-4,0,0,0) S N = (12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), where (9)S N =P N1 +P N2 +P N3, p 1 =N 1 =2,p 2 =N 1 (N 2-1)=2, p 3 =N 1 N 2 (N 3-1)=8, (p 1 +p 2 )=N 1 N 2 =4, S N = P N1 +P N2 +P N3, and S N =(10,2,-2,2,-2,2,-2,2,-2,2,-2,2) = a 1 (5,1,-1,1,-1,1, -1,1,-1,1, -1,1), where (14) S N =P N1 ( = 1)+P N2 +P N3 and coefficient a 1 =2. 2. N=N 1 N 2 N 3 =2 3 2=12, N 1 =2, N 2 =3, N 3 =2, i = 1 P N1 = (2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0) i = 2 P N2 = (4, 0,-2, 0,-2, 0, 4, 0,-2, 0,-2, 0) i = 3 P N3 = (6, 0, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0) - - - - - - - - - - - - - - - - - - - - - - - - - - - S N = (12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), where (9) S N =P N1 +P N2 +P N1, p 1 =N 1 =2, p 2 =N 1 (N 2-1)=4, p 3 =N 1 N 2 (N 3-1)=6, (p 1 +p 2 ) = N 1 N 2 =6, and S N =(8,4,2,-2,2,-2, -4, 4,2,-2, 2,-2) = a 2 (4,2,1,-1,1,-1, - 2,2,1,-1,1,-1), where (14) S N = P N1 +P N2 ( =1)+P N3 and coefficient a 2 =2. 3. N=N 1 N 2 N 3 =3 2 2=12, N 1 =3, N 2 =2, N 3 =2, i = 1 P N1 =(3, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0) i = 2 P N2 =(3, 0, 0,-3, 0, 0, 3, 0, 0,-3, 0, 0) i = 3 P N3 =(6, 0, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0) - - - - - - - - - - - - - - - - - - - - - - - - - - S N =(12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), where (9) S N =P N1 +P N2 +P N1, p 1 =N 1 =3, p 2 =N 1 (N 2-1)=3, p 3 =N 1 N 2 (N 3-1) = 6, (p 1 +p 2 )=N 1 N 2 =6, S N =P N1 + P N2 + P N1, and S N =(6,3,3,0,3,-3, -6,3,3,0,3,- 3)=a 3 (2,1,1,0,1,-1, -2,1,1,0,1,-1), where (14) S N = P N1 ( = 1)+P N2 ( =2)+P N3 (14) and coefficient a 3 =3. 4. N=N 1 N 2 =4 3=12, N 1 =4, N 2 =3, i = 1 P N1 =(4, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0) i = 2 P N2 =(8, 0, 0, 0,-4, 0, 0, 0,-4, 0, 0, 0) - - - - - - - - - - - - - - - - - - - - - - - - - - S N =(12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), where (7), (8) S N =P N1 +P N2, p 1 =N 1 =4, p 2 =N 1 (N 2-1)=8, and S N =(8,4,0,0,-4,4, 0,0,-4,4,0,0) = a 4 (2,1,0,0,-1,1, 0,0,-1,1,0,0), where (12), (13) S N =P N1 ( =1)+P N2 and coefficient a 4 =4. 5. N=N 1 N 2 =3 4=12, N 1 =3, N 2 =4, i = 1 P N1 =(3, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0) i = 2 P N2 =(9, 0, 0,-3, 0, 0, -3, 0, 0,-3, 0, 0) - - - - - - - - - - - - - - - - - - - - - - - - - - S N =(12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), where (7) S N =P N1 +P N2, p 1 =N 1 =3, p 2 =N 1 (N 2-1)=9, and S N =(9,3,0,-3,3,0,-3,3,0,-3,3,0) = a 5 (3,1,0,-1,1,0, -1,1,0,-1,1,0), where (12) S N = P N1 ( =1)+P N2 and coefficient a 5 =3. 6. N=N 1 N 2 =2 6=12, N 1 =2, N 2 =6, i = 1 P N1 =( 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0) i = 2 P N2 =(10, 0,-2, 0, -2, 0,-2, 0,-2, 0,-2, 0) - - - - - - - - - - - - - - - - - - - - - - - - - - - - S N =(12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), where (7) S N =P N1 +P N2, p 1 =N 1 =2, p 2 =N 1 (N 2-1)=10, and S N = (10,-2,-2,2,-2,-2, -2,-2,-2,-2,-2,-2) = a 6 (5,-1,-1,-1,-1,-1, -1,-1,-1,-1,-1,-1), where (12) S N = -P N1 ( =1)+P N2 and coefficient a 6 =2. 7. N=N 1 N 2 =6 2=12, N 1 =6, N 2 =2, i = 1 P N1 =(6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0) i = 2 P N2 =( 6, 0, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0) - - - - - - - - - - - - - - - - - - - - - - - - - - S N =(12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), where (8) S N =P N1 +P N2, p 1 =N 1 =6, p 2 =N 1 (N 2-1)=6, and S N = (6,0,0,6,0,0, 6,0,0,-6,0,0) = a 7 (1,0,0,1,0,0, 1,0,0,-1,0,0), where (13) S N = P N1 + P N2 ( =3) and coefficient a 7 =6. 8. N=N 1 N 2 =4 3=12, N 1 =4, N 2 =3, i = 1 P (2) N1 =(2,-2,-2,-2,2,-2,-2,-2,2,-2,-2,-2) i = 2 P N2 =( 8,0,0,0,-4,0, 0,0,-4,0,0,0) - S (2) N =(10,-2,-2,-2,-2,-2, -2,-2,-2,-2,-2,-2), where (8) S (2) N =P (2) N1 +P N2, P (2) N1 (t)=(s (2) N1T, S (2) N1T, S (2) N1T), p 1 = N 1 = 4, p 2 = N 1 (N 2-1)=8, S (2) N1T = ((N 1-2),-2,-2,-2) = (2,-2,-2,-2), ands N = (6,2,-2,-2,-6,2, -2,-2,-6,2,-2,-2) = a 8 (3,1,-1,-1,-3,1, -1,-1,-3,1,-1,-1), where (13) S N =P (2) N1 ( =1)+P N2 and coefficient a 8 =2. 9. N=N 1 N 2 =6 2=12, N 1 =6, N 2 =2, i = 1 P (2) N1 =(4,-2,-2,-2,-2,-2, 4,-2,-2,-2,-2,-2) i = 2 P N2 =( 6,0,0,0,0,0, -6,0,0,0,0,0) S (2) N =(10,-2,-2,-2,-2,-2, -2,-2,-2,-2,-2,-2), 8861

where (8) S (2) N =P (2) N1 +P N2, P (2) N1(t)=(S (2) N1T, S (2) N1T), p 1 = N 1 = 6, p 2 = N 1 (N 2-1) = 6, S (2) N1T =((N 1-2),-2,-2,-2,-2,-2)=(4,-2,-2,-2,-2,-2), and S N = (4,4,-2,-2,-2,-2,-8,4,-2,-2,-2,-2) =a 9 (2,2,-1,-1,-1,-1, -4,2,-1,-1,-1,-1), where (13) S N =P (2) N1 ( =1)+P N2 and coefficient a 9 =2. REFERENCES [1] Nadav Levanon, Eli Mozeson. 2004. Radar Signals. John Wiley & Sons, Inc. USA. [2] Valery P. Ipatov. 2005. Spread Spectrum and CDMA: Principles and Applications. John Wiley & Sons Ltd., England. [3] Pingzhi Fan and Michael Darnell. 1996. Sequence Design for Communication Applications. Research Study Press Ltd., England. [4] A.V. Titov and G.J. Kazmierczak. 2017. Periodic Non-Binary Sequences with Perfect Autocorrelation. International Journal of Engineering Technology and Computer Research (IJETCR). 5(3): 135-145. [5] Hardy. G. H, Wright E. M. 1979. An Introduction to Theory of Numbers. (Fifth ed.) Oxford University Press, USA. [6] Erickson M. J. 1996. Introduction to Combinatorics. New York: Wiley. 8862