# Training Sequence Design of TDS-OFDM Signal in Joint Radar and Communication System

Authors: Jiajun Zuo; Ruijuan Yang; Shaohua Luo; Xiaobai Li
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

In the joint radar and communication system, using orthogonal frequency division multiplexing (OFDM) signals, cyclic prefix (CP) and pilots lead to the problem of high peak at the sidelobe (PSL) level in autocorrelation function (ACF), which deteriorates the radar detection performance seriously. To solve this problem, first, a new RadCom signal based on time-domain synchronization OFDM (TDS-OFDM) was proposed. TDS-OFDM adopts training sequence (TS) for guard interval, as well as synchronization and channel estimation, so that CP and pilots can be avoided. And then, ambiguity function (AF) of TDS-OFDM RadCom signal was analyzed. Finally, TS are optimized to suppress PSL of TDS-OFDM signal and maintain the autocorrelation properties of TS simultaneously. The results show that the autocorrelation performance of designed TDS-OFDM RadCom signal is much better than that of CP-OFDM RadCom signal. Considering the importance of radar target detection, TDS-OFDM is more appropriate than CP-OFDM for the RadCom system.


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## Body

## 1. Introduction

Joint radar and communication (RadCom) is proposed as a technology using a single signal to accomplish both radar and communication functions, which can not only effectively reduce the load of the platform, energy consumption, and electromagnetic interference, but also greatly improve the utilization of energy and spectrum, thus receiving much attention from military and civil fields [1–4].Orthogonal frequency division multiplexing (OFDM) is a multicarrier data transmission technology which has been widely used in communication system. Due to the flexibility in waveform design, OFDM has also been gradually utilized in radar [5–7], thus becoming a suitable RadCom signal waveform. To improve data transmission rate, the pulse signal consisted of multiple OFDM symbols that were used in RadCom system [8–12]. To mitigate intersymbol interference (ISI), cyclic prefix (CP) of appropriate length has been used among OFDM symbols as guard interval (GI).The application of CP suppresses the intercarrier interference (ICI) and improves the orthogonality of subcarriers. However, when CP-OFDM signal is utilized in the RadCom system, CP, the replication of the later section of OFDM data block, will inevitably cause the high level sidelobes of autocorrelation function (ACF). Moreover, the CP sidelobe level increases with the length ratio of CP and OFDM data block. In addition, to achieve communication synchronization and channel estimation, pilots must be used in CP-OFDM RadCom signal, but the pilot symbols will also lead to the problem of pilot sidelobes. Those sidelobes deteriorate the performance of radar target detection; thus the application of OFDM radar-communication integration will be severely restricted.To address the problem, a solution is put forward in [13, 14]; that is, zero padding is performed for the CP and pilot tones of the reference signal before the correlation with echo signal. However, despite the effective elimination of the CP and pilot sidelobes, such approach will greatly deteriorate the target detection performance result of the decline of pulse gain caused by the loss of signal energy. In this paper, a new form of RadCom signal based on TDS-OFDM is proposed, where CP is replaced by training sequences (TS) as guard interval while being employed in communication synchronization and channel estimation. Accordingly, CP and pilot sidelobes can be avoided as neither cyclic prefix nor pilot tones is required.Being matured in communication, TDS-OFDM is a key technology in the standard of digital television terrestrial broadcasting (DTTB), whose application is successful in China, Cuba, Cambodia, etc. [15–17], so what is mainly studied in this paper is the performance of TDS-OFDM RadCom signal in radar application. Firstly, the ambiguity function (AF) of TDS-OFDM RadCom signal is deduced. And then, a TS design criterion is proposed, based on which, genetic algorithm is used to search optimal TS; finally, the TDS-OFDM RadCom signal with good performance both in radar and communication is generated.The rest of the paper is organized as follows. The TDS-OFDM RadCom signal model is presented in Section2 and the derivation of AF of TDS-OFDM RadCom waveform is described in Section 3. Then the optimization procedure and simulation results analysis of TDS-OFDM waveform follow in Sections 4 and 5. Finally, conclusions are drawn in Section 6.


## 2. Signal Model

As shown in Figure1, designed TDS-OFDM RadCom system adopts pulse radar signal, where each pulse consists of N OFDM symbols. Each OFDM symbol is composed of a TS and an OFDM data block, with the duration of Tg and Tb, respectively. Then, the envelope of a TDS-OFDM RadCom pulse is presented as the sum of the following two parts:(1)st=A1s1t+A2s2twhere s1(t) is the TS-string composed of N TS, s2(t) is the data blocks of N OFDM symbols, A1 and A2 are their amplitudes, respectively, and there is A12+A22=1.(2)s1t=∑n=0N-1∑k=0K-1cn,krectt-kT1-nT2T1(3)s2t=∑n=0N-1∑m=0M-1dn,mej2πmΔft-nT2-Tgrectt-nT2-TgTbwhere cn,k is the nth code element of kth TS, which is assigned as ±c0 and satisfies the energy normalized condition of ∫s1(t)2dt=1. K is the number of code elements in a TS, T1=Tg/K is the duration of a code element satisfying T1=1/B, and B is the bandwidth of TDS-OFDM signal. T2=Tg+Tb is the duration of a complete OFDM symbol, and rect(∙) is the rectangular function. In (3), M is the number of subcarriers, Δf is the frequency interval between adjacent subcarriers, and Δf=1/Tb to keep the orthogonality of subcarriers. dn,m is the data transmitted on the mth subcarrier of the nth OFDM symbol, which can be modulated by phase shift keying (PSK) modulation or quadrature amplitude modulation (QAM). dn,m at different locations is independent and satisfies the expectation E[dn,m]=0 and the energy normalized condition of E∫s2(t)2dt=1.Figure 1
Structure of TDS-OFDM RadCom signals.


## 3. Derivation of Ambiguity Function

Ambiguity function demonstrates the delay-Doppler characteristics of the radar signal, making it an important tool for studying radar signal and waveform design. In this section, AF of TDS-OFDM RadCom signal is deduced in detail.


### 3.1. Ambiguity Function of TDS-OFDM RadCom Signal

For general point targets, the following narrowband AF can be used for analysis:(4)χτ,ξ=∫-∞+∞sts∗t-τej2πξtdtwhere τ is time delay and ξ is Doppler shift. Substituting (1) into (4), AF of TDS-OFDM RadCom signal χs(τ,ξ) can be derived(5)χsτ,ξ=A12χ1,1τ,ξ+A22χ2,2τ,ξ+A1A2χ1,2τ,ξ+A1A2χ2,1τ,ξwhere χi,j(τ,ξ)=∫-∞+∞si(t)sj∗(t-τ)ej2πξtdt, i,j=1,2. χ1,1(τ,ξ) and χ2,2(τ,ξ) are the self-ambiguity functions of TS-string and OFDM data blocks, respectively; χ1,2(τ,ξ) and χ2,1(τ,ξ) are cross-ambiguity functions of TS-string and OFDM data blocks. Since E[dn,m]=0, the expectation of cross-ambiguity function E[χ1,2(τ,ξ)]=E[χ2,1(τ,ξ)]=0. Thus, if we take the expected value of both sides of (5), there is the following.(6)Eχsτ,ξ=A12χ1,1τ,ξ+A22Eχ2,2τ,ξIn fact, increasing the time-bandwidth product of OFDM pulse can effectively reduce the variance ofχs(τ,ξ) [18]. Thus, the following approximation is reasonable for the signal with high time-bandwidth product.(7)χsτ,ξ≈Eχsτ,ξIn addition, phase-coherent accumulation technology is generally used in modern radar, which can lower the variance of ACF further. Combining (6) and (7), χs(τ,ξ) approximately equals the weighted sum of χ1,1(τ,ξ) and E[χ2,2(τ,ξ)] about energy ratio. The following two subsections will derive these two AFs, respectively.


### 3.2. Ambiguity Function of OFDM Data Blocks

First, we deduce the AF of OFDM data blocks. According to the definition of AF, (4) can be represented as(8)χτ,ξ=∫-∞+∞st+Tgs∗t+Tg-τej2πξt+Tgdtand, substituting (3) into (8), AF of s2(t) is as follows.(9)χ2,2τ,ξ=∑n=0N-1∑n′=0N-1∑m=0M-1∑m′=0M-1dn,mdn′,m′∗ej2πΔfn′m′T2-nmT2+m′τ∫-∞+∞ej2πmΔf-m′Δf+ξtrectt-nT2Tbrectt-n′T2-τTbdtej2πξTgLet F=mΔf-m′Δf+ξ, and the integral term in (9) is as follows.(10)I=∫-∞+∞ej2πFtrectt-nT2Tbrectt-n′T2-τTbdtThe integral is discussed in two cases:① when(n-n′)T2<τ≤(n-n′)T2+Tb,(11)I=∫n′T2+τnT2+Tbej2πFtdt=Tb+nT2-n′T2-τej2πFTb+nT2+n′T2+τsincFTb+nT2-n′T2-τ② when(n-n′)T2-Tb<τ≤(n-n′)T2,(12)I=∫nT2n′T2+Tb+τej2πFtdt=Tb+n′T2-nT2+τej2πFTb+nT2+n′T2+τsincFTb+n′T2-nT2+τand, combining (11) and (12), we can get the analytical expression of the integral as(13)I=Tb-nT2-n′T2-τejπFTb+nT2+n′T2+τsincFTb-nT2-n′T2-τ,n-n′T2-Tb<τ≤n-n′T2+Tbwhere sinc(x)=sin⁡(πx)/πx. Substituting (12) into (9) yields the following.(14)χ2,2τ,ξ=∑n=0N-1∑n′=0N-1∑m=0M-1∑m′=0M-1Tb-nT2-n′T2-τdn,mdn′,m′∗ej2πΔfn′m′T2-nmT2+m′τejπFTb+nT2+n′T2+τsincmΔf-m′Δf+ξTb-nT2-n′T2-τej2πξTg,n-n′T2-τ≤TbEquation (14) indicates that expectation value of the OFDM data segment’s AF depends on the cross-correlation function Edn,mdn′,m′∗. When n≠n′ or m≠m′, dn,m and dn′,m′ represent modulation data in different symbols or subcarriers, which are independent according to the hypothesis above, hence the cross-correlation function equals zero. Edn,mdn′,m′∗ will only have a nonzero value in case n=n′ and m=m′. This results in the following expression.(15)Edn,mdn′,m′∗=dn,m2,n=n′andm=m′0,othersSpecifically, when QAM modulation is adopted, the right side of (15) should be Edn,m2. For simplicity, PSK modulation is assumed here. combining (14) and (15), the expectation of AF of s2(t) can be finally expressed as follows.(16)Eχ2,2τ,ξ=Tb-τsincξTb-τej2πξTg∑n=0N-1∑m=0M-1dn,m2ej2πmΔfτejπξTb+2nT2+τ,-Tb<τ≤TbEquation (16) indicates that E[χ2,2(τ,ξ)] is nonzero only within the delay range of a single symbol, where the rest are all zero. Figure 2(a) shows a function graph of the OFDM data segment’s AF, which is generally ideally thumbtack-typed.Figure 2
Ambiguity function of TDS-OFDM RadCom signal (M=128,N=8,K=32).
(a)
E [ χ 2,2 ( τ , ξ ) ] (b)
χ 1,1 ( τ , ξ ) (c)
E [ χ s ( τ , ξ ) ]


### 3.3. Ambiguity Function of TS-String

According to the definition, substituting (2) into (4), AF of TS-string is as follows.(17)χ1,1τ,ξ=∑n=0N-1∑n′=0N-1∑k=0K-1∑k′=0K-1cn,kcn′,k′∗∫-∞+∞ej2πξtrectt-kT1-nT2T1rectt-k′T1-n′T2-τT1dtSimilar to the deriving process of (10) to (13), we can derive the integral term in (17); the final expression of TS-string’s AF is presented as(18)χ1,1τ,ξ=∑n=0N-1∑n′=0N-1∑k=0K-1∑k′=0K-1cn,kcn′,k′∗T1-ΔkT1+ΔnT2-τejπξk+k′+1T1+n+n′T2+τsincξT1-ΔkT1+ΔnT2-τ,ΔkT1+ΔnT2-τ≤T1where Δk=k-k′ and Δn=n-n′. And the value range of τ is determined by the value of Δk and Δn. Let τ=0; hence Δk=0 and Δn=0. Therefore we can get the zero-delay cut of AF.(19)χ1,10,ξ=T1∑n=0N-1∑k=0K-1cn,k2sincξT1ejπξ2k+1T1+2nT2As the amplitude ofcn,k is constant, (19) indicates that Doppler tolerance of TS-string is independent of the TS. Therefore, the impact on Doppler tolerance is out of consideration when designing TS-string. Moreover, assigning τ=ΔkT1+ΔnT2 and letting ξ=0, we can get the range autocorrelation sequence of TS-string.(20)χ1,1Δk+MΔn=T1∑n=0N-1∑n′=0N-1∑k=0K-1∑k′=0K-1cn,kcn′,k′∗As shown in (20), when Δn=0, χ1,1 equals the sum of N ACF of TS in the range of τ<T2; when Δn>0, χ1,1 equals the sum of cross-correlation function of different TS in the range of τ>T2. Thus, as shown in Figure 2(b), grating lobes are found on the AF along the delay axis. To reduce PSL, training sequences should be designed to ensure both good autocorrelation and cross-correlation.Substituting (16) and (18) into (6), AF of TDS-OFDM RadCom signal is obtained as shown in Figure 2(c), whose mathematical expression is no more described here. As can be seen from Figure 2, AF sidelobes of TDS-OFDM RadCom signal are mainly caused by TS-string, and thus it is necessary to optimize it. And on the zero-delay cut of AF, the function value is about 0.9 at the position of normalized Doppler shift of ξ=0.25, which indicates that its Doppler tolerance is consistent with conventional OFDM RadCom signal [19].


## 3.1. Ambiguity Function of TDS-OFDM RadCom Signal

For general point targets, the following narrowband AF can be used for analysis:(4)χτ,ξ=∫-∞+∞sts∗t-τej2πξtdtwhere τ is time delay and ξ is Doppler shift. Substituting (1) into (4), AF of TDS-OFDM RadCom signal χs(τ,ξ) can be derived(5)χsτ,ξ=A12χ1,1τ,ξ+A22χ2,2τ,ξ+A1A2χ1,2τ,ξ+A1A2χ2,1τ,ξwhere χi,j(τ,ξ)=∫-∞+∞si(t)sj∗(t-τ)ej2πξtdt, i,j=1,2. χ1,1(τ,ξ) and χ2,2(τ,ξ) are the self-ambiguity functions of TS-string and OFDM data blocks, respectively; χ1,2(τ,ξ) and χ2,1(τ,ξ) are cross-ambiguity functions of TS-string and OFDM data blocks. Since E[dn,m]=0, the expectation of cross-ambiguity function E[χ1,2(τ,ξ)]=E[χ2,1(τ,ξ)]=0. Thus, if we take the expected value of both sides of (5), there is the following.(6)Eχsτ,ξ=A12χ1,1τ,ξ+A22Eχ2,2τ,ξIn fact, increasing the time-bandwidth product of OFDM pulse can effectively reduce the variance ofχs(τ,ξ) [18]. Thus, the following approximation is reasonable for the signal with high time-bandwidth product.(7)χsτ,ξ≈Eχsτ,ξIn addition, phase-coherent accumulation technology is generally used in modern radar, which can lower the variance of ACF further. Combining (6) and (7), χs(τ,ξ) approximately equals the weighted sum of χ1,1(τ,ξ) and E[χ2,2(τ,ξ)] about energy ratio. The following two subsections will derive these two AFs, respectively.


## 3.2. Ambiguity Function of OFDM Data Blocks

First, we deduce the AF of OFDM data blocks. According to the definition of AF, (4) can be represented as(8)χτ,ξ=∫-∞+∞st+Tgs∗t+Tg-τej2πξt+Tgdtand, substituting (3) into (8), AF of s2(t) is as follows.(9)χ2,2τ,ξ=∑n=0N-1∑n′=0N-1∑m=0M-1∑m′=0M-1dn,mdn′,m′∗ej2πΔfn′m′T2-nmT2+m′τ∫-∞+∞ej2πmΔf-m′Δf+ξtrectt-nT2Tbrectt-n′T2-τTbdtej2πξTgLet F=mΔf-m′Δf+ξ, and the integral term in (9) is as follows.(10)I=∫-∞+∞ej2πFtrectt-nT2Tbrectt-n′T2-τTbdtThe integral is discussed in two cases:① when(n-n′)T2<τ≤(n-n′)T2+Tb,(11)I=∫n′T2+τnT2+Tbej2πFtdt=Tb+nT2-n′T2-τej2πFTb+nT2+n′T2+τsincFTb+nT2-n′T2-τ② when(n-n′)T2-Tb<τ≤(n-n′)T2,(12)I=∫nT2n′T2+Tb+τej2πFtdt=Tb+n′T2-nT2+τej2πFTb+nT2+n′T2+τsincFTb+n′T2-nT2+τand, combining (11) and (12), we can get the analytical expression of the integral as(13)I=Tb-nT2-n′T2-τejπFTb+nT2+n′T2+τsincFTb-nT2-n′T2-τ,n-n′T2-Tb<τ≤n-n′T2+Tbwhere sinc(x)=sin⁡(πx)/πx. Substituting (12) into (9) yields the following.(14)χ2,2τ,ξ=∑n=0N-1∑n′=0N-1∑m=0M-1∑m′=0M-1Tb-nT2-n′T2-τdn,mdn′,m′∗ej2πΔfn′m′T2-nmT2+m′τejπFTb+nT2+n′T2+τsincmΔf-m′Δf+ξTb-nT2-n′T2-τej2πξTg,n-n′T2-τ≤TbEquation (14) indicates that expectation value of the OFDM data segment’s AF depends on the cross-correlation function Edn,mdn′,m′∗. When n≠n′ or m≠m′, dn,m and dn′,m′ represent modulation data in different symbols or subcarriers, which are independent according to the hypothesis above, hence the cross-correlation function equals zero. Edn,mdn′,m′∗ will only have a nonzero value in case n=n′ and m=m′. This results in the following expression.(15)Edn,mdn′,m′∗=dn,m2,n=n′andm=m′0,othersSpecifically, when QAM modulation is adopted, the right side of (15) should be Edn,m2. For simplicity, PSK modulation is assumed here. combining (14) and (15), the expectation of AF of s2(t) can be finally expressed as follows.(16)Eχ2,2τ,ξ=Tb-τsincξTb-τej2πξTg∑n=0N-1∑m=0M-1dn,m2ej2πmΔfτejπξTb+2nT2+τ,-Tb<τ≤TbEquation (16) indicates that E[χ2,2(τ,ξ)] is nonzero only within the delay range of a single symbol, where the rest are all zero. Figure 2(a) shows a function graph of the OFDM data segment’s AF, which is generally ideally thumbtack-typed.Figure 2
Ambiguity function of TDS-OFDM RadCom signal (M=128,N=8,K=32).
(a)
E [ χ 2,2 ( τ , ξ ) ] (b)
χ 1,1 ( τ , ξ ) (c)
E [ χ s ( τ , ξ ) ]


## 3.3. Ambiguity Function of TS-String

According to the definition, substituting (2) into (4), AF of TS-string is as follows.(17)χ1,1τ,ξ=∑n=0N-1∑n′=0N-1∑k=0K-1∑k′=0K-1cn,kcn′,k′∗∫-∞+∞ej2πξtrectt-kT1-nT2T1rectt-k′T1-n′T2-τT1dtSimilar to the deriving process of (10) to (13), we can derive the integral term in (17); the final expression of TS-string’s AF is presented as(18)χ1,1τ,ξ=∑n=0N-1∑n′=0N-1∑k=0K-1∑k′=0K-1cn,kcn′,k′∗T1-ΔkT1+ΔnT2-τejπξk+k′+1T1+n+n′T2+τsincξT1-ΔkT1+ΔnT2-τ,ΔkT1+ΔnT2-τ≤T1where Δk=k-k′ and Δn=n-n′. And the value range of τ is determined by the value of Δk and Δn. Let τ=0; hence Δk=0 and Δn=0. Therefore we can get the zero-delay cut of AF.(19)χ1,10,ξ=T1∑n=0N-1∑k=0K-1cn,k2sincξT1ejπξ2k+1T1+2nT2As the amplitude ofcn,k is constant, (19) indicates that Doppler tolerance of TS-string is independent of the TS. Therefore, the impact on Doppler tolerance is out of consideration when designing TS-string. Moreover, assigning τ=ΔkT1+ΔnT2 and letting ξ=0, we can get the range autocorrelation sequence of TS-string.(20)χ1,1Δk+MΔn=T1∑n=0N-1∑n′=0N-1∑k=0K-1∑k′=0K-1cn,kcn′,k′∗As shown in (20), when Δn=0, χ1,1 equals the sum of N ACF of TS in the range of τ<T2; when Δn>0, χ1,1 equals the sum of cross-correlation function of different TS in the range of τ>T2. Thus, as shown in Figure 2(b), grating lobes are found on the AF along the delay axis. To reduce PSL, training sequences should be designed to ensure both good autocorrelation and cross-correlation.Substituting (16) and (18) into (6), AF of TDS-OFDM RadCom signal is obtained as shown in Figure 2(c), whose mathematical expression is no more described here. As can be seen from Figure 2, AF sidelobes of TDS-OFDM RadCom signal are mainly caused by TS-string, and thus it is necessary to optimize it. And on the zero-delay cut of AF, the function value is about 0.9 at the position of normalized Doppler shift of ξ=0.25, which indicates that its Doppler tolerance is consistent with conventional OFDM RadCom signal [19].


## 4. Design of Training Sequences

PSL of radar signal is an important factor affecting radar target detection. In this section, PSL of average ACF of TDS-OFDM signals, denoted as PSL0, is used as one of the objective functions (21)PSL0=-20lg⁡maxk≠0⁡Eχskwhere E[χs(k)] is the expectation of autocorrelation sequence of TDS-OFDM RadCom signal. To reduce the sidelobes of OFDM data blocks within the delay range of Tb, Hamming window function was used in the autocorrelation operation.On the other hand, to realize robust communication synchronization and channel estimation, TS itself should be well autocorrelated. Letrn,n(k) represent the aperiodic autocorrelation sequence of nth TS.(22)rn,nk=∑m=kK-1cn,mcn,m-k∗And the other objective function is the maximal peak sidelobe ofNTS. (23)PSLTS=-20lg⁡maxk=1,⋯,K-1n=0,⋯,N-1⁡rn,nkThus, the final optimizing criterion is minimizing the weighted sum of average peak sidelobe of TDS-OFDM signal and maximal peak sidelobe of TS, which is presented as the following fitness function:(24)E=-ω1PSL0+ω2PSLTSwhere ω1 and ω2 are, respectively, the weights of two costs, and ω1+ω2=1. As (24) involves a nonlinear multivariable optimization problem, genetic algorithm (GA) is utilized for designing TS. The flow chart of GA is depicted in Figure 3, and the algorithm flow is as follows.Figure 3
Flow chart of GA.Step 1.
Initialize the populationS(0) with numbers randomly produced by 0 and 1. For example, if there are two TS in a TS-string, and each TS contains four code elements, the population size is four. Thus, S(0) can be represented as(25)S0=π10110010010111001100011101011100where each row of S(0) represents an individual.Step 2.
Calculate fitness value according to the fitness function represented by (24), and adjust the weights ω1 and ω2 to balance PSL0 and PSLTS.Step 3 (select operators).
According to the individual fitness, select superior individuals and eliminate inferior individuals by roulette selection.Step 4 (cross operators).
Recombine individuals with one-point crossover method.Step 5 (mutate operators).
Mutate individual gene values with 0.1 mutation probability.Step 6 (terminate).
When the iterations number reaches its maximum, the genetic algorithm completes and the optimization operation can be terminated. If the termination condition is not satisfied, go back to Step2 to continue the optimization process.Besides, the time complexity of GA is linearly increasing with iterative times and input size. When using the termination strategy above, the time complexity of GA can be approximately described asO(GmaxNK), where Gmax is the maximum iterative times. N and K represent the amount and the length of TS, respectively.


## 5. Simulation Result and Analysis

In the following simulations, GA parameters are as follows: the number of individuals is 100, maximum number of generations is 120, generation gap is 0.9, and weightω1=0.8.


### 5.1. Algorithm Simulation

We first optimize the TDS-OFDM RadCom signal with high time-bandwidth product whose parameters are shown in Table1. The iterative process is shown in Figure 4. As the number of iterations increases, fitness of the population decreases gradually. ACF of the designed TS is shown in Figure 5 which contains 4 curves, and each curve represents a TS of the TS-string. As shown in Figure 5, maximal PSL of all training sequences is 15.99 dB, enabling the signal to achieve robust communication synchronization and channel estimation performance.Table 1
Parameters of TDS-OFDM signal.
Parameters Value Number of OFDM symbols (N) 4 TS length (K) 63 Number of sub-carriers (M) 256 Bandwidth (B) 20MHz Duration of an OFDM data block (Tb) 12.8μs Duration of guard interval (Tg) 3.15μs Pulse width 63.8μs Time-bandwidth product 1276Figure 4
Fitness change of GA algorithm.Figure 5
ACF of 4 training sequences.By using the designed TS-string, average ACF of TDS-OFDM signal is shown in Figure6. As shown in the figure, PSL0 of the signal is 32.46 dB, which means the designed signal on average is able to detect weak targets.Figure 6
Average ACF of TDS-OFDM signal.


### 5.2. Analysis of Impact of Correlation Noise

In the ACF of TDS-OFDM signals, there are accompanying noise-like floors which are caused by the random OFDM data. Therefore, the instantaneous PSL of random TDS-OFDM signal will be lower than PSL0. The noise-like effect of the random components in the transmitted signal is described as correlation noise in [20]. In the presence of correlation noise, PSL is reduced as (26)PSL=PSL0S/N0PSL0+S/N0where (S/N)0 represents the power ratio of signal to correlation noise. For one OFDM symbol, (S/N)0 equals the subcarrier numberM[18]. Furthermore, after the use ofN symbols within each pulse and coherent integration ofP pulses, (S/N)0 becomes(27)SN0=MNPwhereMN is also the time-bandwidth product of OFDM data blocks. Equations (26) and (27) indicate that by increasing the number of time-bandwidth product or phase-coherent accumulation pulse, PSL will gradually approach PSL0.To verify this analysis, in this subsection, after phase-coherent accumulation ofP pulses, ACF of designed TDS-OFDM signals with low and high time-bandwidth product were simulated, respectively, andP equals 1, 4, 16, and 64.Figure7 shows ACF of TDS-OFDM RadCom signal with low time-bandwidth product, whose parameters areN=4,M=32,K=8,B=20MHz, and PSL0=28.52. As shown in the figure, correlation noise leads to a significant fluctuation of ACF, and PSL of a single pulse is only about 20dB. With the growth of accumulation pulse number, the fluctuation of ACF decreases gradually. When P is 64, PSL is about 28dB, which is very close to PSL0.Figure 7
ACF of TDS-OFDM signal (MN=128).Figure8 shows the ACF of TDS-OFDM RadCom signal with high time-bandwidth product, whose parameters are the same as Table 1 (PSL0=32.46dB). As the time-bandwidth product is high, the fluctuation of sidelobe decreases relatively. PSL reaches about 32dB after phase-coherent accumulation of just 16 pulses. In addition, by comparing Figures 6 and 8, it is obvious that, with the growth of pulse number of coherent accumulation, ACF of the TDS-OFDM signal approaches its expectation gradually.Figure 8
ACF of TDS-OFDM signal (MN=1024).This simulation indicates that, as the technology of phase-coherent accumulation is generally used in modern radar, even if the time-bandwidth product of the signal is not high enough, ACF can also approach its expectation by using more pulses in phase-coherent accumulation, which means the signal design method based on optimizing average PSL is effective.Furthermore, simulation of range ACF of CP-OFDM RadCom signal after phase-coherent accumulation of 16 pulses is also conducted in the same condition in Table1. As shown in Figure 9, the CP sidelobes are about 15dB and pilot sidelobes of different sizes are also found in ACF of CP-OFDM RadCom signal. By comparing Figures 8 and 9, we can find that the autocorrelation performance of optimized TDS-OFDM RadCom signal is much better than that of CP-OFDM RadCom signal.Figure 9
ACF of CP-OFDM signal (MN=1024,P=16).


### 5.3. Performance Comparison

As pseudorandom noise (PN) sequences are usually used in conventional TDS-OFDM communication signal, in this subsection, we compare the optimized sequence with m sequence and Gold sequence about the performance ofPSLTS and PSL0. The data of three different sequence lengths is shown in Table 2, which is obtained in a random experiment. The table shows that PSLTS of m sequence is the highest while PSL0 is low, making it only suitable for communication. On the contrary, PSL0 of Gold sequence is higher because of the good cross-correlation, but PSLTS is the lowest. In contrast, the optimized sequence with different lengths is superior in both two properties, demonstrating the effectiveness of optimization algorithm. Furthermore, the performance of optimized sequence is relatively improved with the increase of sequence length. Because the increase of length leads to higher freedom of designing. In addition, the lengths of m sequence and Gold sequence cannot meet the needs of complex wireless transmission environment as their lengths are restricted by 2K-1, while our method is not limited by sequence length which has greater flexibility.Table 2
Performance comparison of different TS.
Length Sequence P S L T S (dB) PSL0 (dB) 31 m sequence 14.26 23.38 Gold sequence 10.74 27.72 optimized sequence 14.26 30.74  63 m sequence 15.99 17.36 Gold sequence 10.41 21.22 optimized sequence 15.99 32.46  127 m sequence 19.80 11.91 Gold sequence 15.63 19.66 optimized sequence 17.47 34.47The elapsed time of the GA program is shown in Table3. The specification of computer is as follows: CPU is quad core at 2.4 GHz, 8GB RAM, operating system is Windows 10, and platform is MATLAB 2016a. As the optimization method is independent of communication data, TS-string should be renewed only when the signal’s parameters change. Therefore, RadCom system does not demand much of the real-time characteristics, which means GA can almost meet the requirement of RadCom system in time complexity.Table 3
Elapsed time of GA (N=4).
Length of TS 20 40 60 80 100 120 Elapsed time (sec) 6.79 7.62 8.02 9.34 9.72 10.1


## 5.1. Algorithm Simulation

We first optimize the TDS-OFDM RadCom signal with high time-bandwidth product whose parameters are shown in Table1. The iterative process is shown in Figure 4. As the number of iterations increases, fitness of the population decreases gradually. ACF of the designed TS is shown in Figure 5 which contains 4 curves, and each curve represents a TS of the TS-string. As shown in Figure 5, maximal PSL of all training sequences is 15.99 dB, enabling the signal to achieve robust communication synchronization and channel estimation performance.Table 1
Parameters of TDS-OFDM signal.
Parameters Value Number of OFDM symbols (N) 4 TS length (K) 63 Number of sub-carriers (M) 256 Bandwidth (B) 20MHz Duration of an OFDM data block (Tb) 12.8μs Duration of guard interval (Tg) 3.15μs Pulse width 63.8μs Time-bandwidth product 1276Figure 4
Fitness change of GA algorithm.Figure 5
ACF of 4 training sequences.By using the designed TS-string, average ACF of TDS-OFDM signal is shown in Figure6. As shown in the figure, PSL0 of the signal is 32.46 dB, which means the designed signal on average is able to detect weak targets.Figure 6
Average ACF of TDS-OFDM signal.


## 5.2. Analysis of Impact of Correlation Noise

In the ACF of TDS-OFDM signals, there are accompanying noise-like floors which are caused by the random OFDM data. Therefore, the instantaneous PSL of random TDS-OFDM signal will be lower than PSL0. The noise-like effect of the random components in the transmitted signal is described as correlation noise in [20]. In the presence of correlation noise, PSL is reduced as (26)PSL=PSL0S/N0PSL0+S/N0where (S/N)0 represents the power ratio of signal to correlation noise. For one OFDM symbol, (S/N)0 equals the subcarrier numberM[18]. Furthermore, after the use ofN symbols within each pulse and coherent integration ofP pulses, (S/N)0 becomes(27)SN0=MNPwhereMN is also the time-bandwidth product of OFDM data blocks. Equations (26) and (27) indicate that by increasing the number of time-bandwidth product or phase-coherent accumulation pulse, PSL will gradually approach PSL0.To verify this analysis, in this subsection, after phase-coherent accumulation ofP pulses, ACF of designed TDS-OFDM signals with low and high time-bandwidth product were simulated, respectively, andP equals 1, 4, 16, and 64.Figure7 shows ACF of TDS-OFDM RadCom signal with low time-bandwidth product, whose parameters areN=4,M=32,K=8,B=20MHz, and PSL0=28.52. As shown in the figure, correlation noise leads to a significant fluctuation of ACF, and PSL of a single pulse is only about 20dB. With the growth of accumulation pulse number, the fluctuation of ACF decreases gradually. When P is 64, PSL is about 28dB, which is very close to PSL0.Figure 7
ACF of TDS-OFDM signal (MN=128).Figure8 shows the ACF of TDS-OFDM RadCom signal with high time-bandwidth product, whose parameters are the same as Table 1 (PSL0=32.46dB). As the time-bandwidth product is high, the fluctuation of sidelobe decreases relatively. PSL reaches about 32dB after phase-coherent accumulation of just 16 pulses. In addition, by comparing Figures 6 and 8, it is obvious that, with the growth of pulse number of coherent accumulation, ACF of the TDS-OFDM signal approaches its expectation gradually.Figure 8
ACF of TDS-OFDM signal (MN=1024).This simulation indicates that, as the technology of phase-coherent accumulation is generally used in modern radar, even if the time-bandwidth product of the signal is not high enough, ACF can also approach its expectation by using more pulses in phase-coherent accumulation, which means the signal design method based on optimizing average PSL is effective.Furthermore, simulation of range ACF of CP-OFDM RadCom signal after phase-coherent accumulation of 16 pulses is also conducted in the same condition in Table1. As shown in Figure 9, the CP sidelobes are about 15dB and pilot sidelobes of different sizes are also found in ACF of CP-OFDM RadCom signal. By comparing Figures 8 and 9, we can find that the autocorrelation performance of optimized TDS-OFDM RadCom signal is much better than that of CP-OFDM RadCom signal.Figure 9
ACF of CP-OFDM signal (MN=1024,P=16).


## 5.3. Performance Comparison

As pseudorandom noise (PN) sequences are usually used in conventional TDS-OFDM communication signal, in this subsection, we compare the optimized sequence with m sequence and Gold sequence about the performance ofPSLTS and PSL0. The data of three different sequence lengths is shown in Table 2, which is obtained in a random experiment. The table shows that PSLTS of m sequence is the highest while PSL0 is low, making it only suitable for communication. On the contrary, PSL0 of Gold sequence is higher because of the good cross-correlation, but PSLTS is the lowest. In contrast, the optimized sequence with different lengths is superior in both two properties, demonstrating the effectiveness of optimization algorithm. Furthermore, the performance of optimized sequence is relatively improved with the increase of sequence length. Because the increase of length leads to higher freedom of designing. In addition, the lengths of m sequence and Gold sequence cannot meet the needs of complex wireless transmission environment as their lengths are restricted by 2K-1, while our method is not limited by sequence length which has greater flexibility.Table 2
Performance comparison of different TS.
Length Sequence P S L T S (dB) PSL0 (dB) 31 m sequence 14.26 23.38 Gold sequence 10.74 27.72 optimized sequence 14.26 30.74  63 m sequence 15.99 17.36 Gold sequence 10.41 21.22 optimized sequence 15.99 32.46  127 m sequence 19.80 11.91 Gold sequence 15.63 19.66 optimized sequence 17.47 34.47The elapsed time of the GA program is shown in Table3. The specification of computer is as follows: CPU is quad core at 2.4 GHz, 8GB RAM, operating system is Windows 10, and platform is MATLAB 2016a. As the optimization method is independent of communication data, TS-string should be renewed only when the signal’s parameters change. Therefore, RadCom system does not demand much of the real-time characteristics, which means GA can almost meet the requirement of RadCom system in time complexity.Table 3
Elapsed time of GA (N=4).
Length of TS 20 40 60 80 100 120 Elapsed time (sec) 6.79 7.62 8.02 9.34 9.72 10.1


## 6. Conclusion

To address the problem of CP and pilot sidelobes in conventional CP-OFDM RadCom signal, a novel RadCom signal based on TDS-OFDM is proposed in this paper. AF of proposed signal was analyzed in detail. And then, TS are optimized to suppress PSL of TDS-OFDM signal and maintain the autocorrelation properties of TS simultaneously, thus achieving good performance in both radar and communication. Although, the TS and the OFDM data block cause mutual interference to each other, thus more calculation has to be costed to achieve reliable time-domain channel estimation and frequency-domain data detection in TDS-OFDM systems. Generally, considering the importance of autocorrelation of the signal to radar target detection, TDS-OFDM is more appropriate than CP-OFDM for the RadCom system.



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*Source: 3914314-2019-07-30.xml*