# High Order Differential Frequency Hopping: Design and Analysis

Authors: Yong Li; Fuqiang Yao; Ba Xu
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

This paper considers spectrally efficient differential frequency hopping (DFH) system design. Relying on time-frequency diversity over large spectrum and high speed frequency hopping, DFH systems are robust against hostile jamming interference. However, the spectral efficiency of conventional DFH systems is very low due to only using the frequency of each channel. To improve the system capacity, in this paper, we propose an innovative high order differential frequency hopping (HODFH) scheme. Unlike in traditional DFH where the message is carried by the frequency relationship between the adjacent hops using one order differential coding, in HODFH, the message is carried by the frequency and phase relationship using two-order or higher order differential coding. As a result, system efficiency is increased significantly since the additional information transmission is achieved by the higher order differential coding at no extra cost on either bandwidth or power. Quantitative performance analysis on the proposed scheme demonstrates that transmission through the frequency and phase relationship using two-order or higher order differential coding essentially introduces another dimension to the signal space, and the corresponding coding gain can increase the system efficiency.


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## Body

## 1. Introduction

As one of the two basic modulation techniques used in spread spectrum communications, frequency hopping (FH) was originally designed to be inherently secure and reliable under adverse battle conditions for military applications. However, there are three major bottlenecks: low data rate, narrow bandwidth frequency hopping, and a low hopping rate when using frequency hopping in the high frequency (HF) band. Currently, the data rate of the existing HF frequency hopping radio is limited to 2400 bps or less, the reliable data rate is 1200 bps, the highest hopping rate is only tens of hops per sec, and the antijamming capability is weak. Under current conditions, it is difficult to achieve high-capacity communications with HF communications technology when there are harsh electromagnetic environments.In the 1990s, the Sanders Company, from the United States of America, pioneered an enhanced hopping spread spectrum radio (Correlated Hopping Enhanced Spread Spectrum), referred to as CHESS Radio [1, 2]. It was based on the differential frequency hopping system and was a good solution to improve the data rate and anti-tracking-jamming abilities, among other issues. Its hopping rate is 5000 hops per sec and the frequency hopping bandwidth is 2 MHz; these were breakthroughs compared to conventional HF communication systems. However, the conventional DFH system has low spectral efficiency over large bandwidth. Typically, conventional DFH systems require large bandwidth, which is proportional to the requirement on the frequency number. Along with the ever increasing demand on inherently secure high data rate wireless communications, new techniques that are more efficient and reliable have to be developed. In literature [3], a FH system called modulated differential frequency hopping was proposed, in this system, some information was used to produce a pseudo-random carrier frequency slot (as in the DFH technology), and others were changed into narrowband modulated waveforms (as in the conventional FH technology). The message-driven frequency hopping (MDFH) scheme proposed in literature [4] possesses higher spectral efficiency than conventional FH communication system. In literature [5], based on the idea of MDFH, a new scheme was proposed to improve the ability of anti-fixed-frequency interference. In literature [6, 7], the antijamming performance of the MDFH was analyzed in depth.In this paper, we propose an innovative high order differential frequency hopping (HODFH) scheme. The basic idea is that the frequencies and the phases of the signal are numbered and then using the two-order or higher order differential coding to code the message and the signal number. The message is parsed into several parts; each part will be transmitted by different order differential coding. In other words, the serial message stream is converted into parallel data stream transmitted by each order differential coding. At the receiver, the transmitting frequency is captured using a filter bank as in the FSK receiver rather than using the frequency synthesizer. As a result, the frequency can be blindly detected at each hop. This relaxes the burden of strict frequency synchronization at the receiver. And the phase is estimated as in the PSK receiver. Then the frequency and the phase are used to decode the message. The system spectral efficiency is significantly improved, because the high order differential coding and the phases do not require extra cost on bandwidth and power.In this paper, in-depth performance analysis is conducted based on both theoretical derivation and simulation examples. Our analysis demonstrates that phases essentially add another dimension to the signal space, and the high order differential coding can increase the system spectral efficiency.


## 2. The Concept of High Order Differential Frequency Hopping

### 2.1. Transmitter Design

Unlike the conventional DFH systems in which only the frequencies are coded by the one-order coding, the HODFH systems code the frequencies and their phases by high order coding. That is to say, in the conventional DFH system, only the carrier frequencies carry the message; however, in the HODFH system, not only frequencies but also the phases carry the information. In a HODFH system, we add several phases to the frequencies; therefore, the signal number is significantly increased almost without increase in bandwidth.LetNc be the total number of frequencies, with f1,f2,…,fNc being the set of all frequencies. Let Mk be the number of initial phases where the frequency is fk in each hop, with ϕ1,ϕ2,…,ϕMk being the set of all phases. Define s(fn,i,ϕn,j) as the combinated signal of these frequencies and phases in the nth hop; it is clear that there are Ns=∑n=1NcMk signals. The set S, with s1,s2,…,sNs being its elements, is the set of all the combinated signals.LetNo be the order of the HODFH system; we start by reshaping the one-dimensional set S into No-dimensional set SD1×D2×⋯×DNo, where the subscript denotes element number of each dimension, and D1×D2×⋯×DNo≤Ns; denote sd1,d2,…,dNoas the element of the set SD1×D2×⋯×DNo. We assume that D1×D2×⋯×DNo=Ns. Divide the data stream into blocks of length L≜∑i=1NoBi, where Bi(≤log2Di) is the number of bits that is transmitted through the ith order differential coding. Denote the nth block by Xn. The block Xn can be grouped into No vectors, denoted by [Xn,1,Xn,2,…,Xn,No], where Xn,i denotes the ith order bits, as shown in Figure 1. We will transmit Xn within one hop.Figure 1
Thenth block of the information data.The transmitter block diagram of the proposed HODFH scheme is illustrated in Figure2. Each input data block, Xn, is fed into a serial-to-parallel (S/P) converter, where the data bits are split into No parallel data streams. Denote the signals in the nth and (n-1)th hop by sd1n,d2n,…,dNonn and sd1n-1,d2n-1,…,dNon-1n-1. The index of the sd1n,d2n,…,dNonn is obtained by No-order differential coding the index of the sd1n-1,d2n-1,…,dNon-1n-1 and information data block Xn. The differential processing is called a function G and expressed by the following equations:(1)d1n=G1d1n-1,d2n-1,…,dNon-1,Xn,1dmn=Gmdm-1n,Xn,m,whenm=2,…,No.Figure 2
The transmitter block diagram of the proposed HODFH scheme.Using the index vectors to select the combinated signal, the signal transmitted over the channel issd1n,d2n,…,dNonn.Recall thats(fn,i,ϕn,j) is the combination of the frequency and phase. The transmitted signal in the nth hop can be expressed as(2)sd1n,d2n,…,dNonn=sfn,i,ϕn,j=Acos⁡2πfn,it+ϕn,j0≤t<Th,where A is the signal amplitude, fn,i is the center frequency of the nth hop, ϕn,j is the possible phases of the nth hop, and Th is the hop duration. The passband waveform of the transmitted signal may be expressed as(3)st=2ThRe∑n=-∞∞e-j2πfn,i+ϕn,jtgt-nTh,where gt is the signal pulse-shaping filter.


### 2.2. Receiver Design

The receiver block diagram of the proposed HODFH scheme is illustrated in Figure3. Recall that there are Nc frequencies f1,f2,…,fNc that are extended out Ns signals. First, we must detect which frequency is transmitted. For this purpose, a bank of Nc bandpass filters (BPF), each centered at fi(i=1,2,…,Nc), with the same channel bandwidth as the transmitter, is deployed at the receiver front end. Since only one frequency band is occupied at any given hop, we simply measure the outputs of bandpass filters at each possible signaling frequency. The actual carrier frequency at a certain hopping period can be detected by selecting the one that captures the strongest signal. As a result, blind detection of the carrier frequency is achieved at the receiver.Figure 3
The receiver block diagram of the proposed HODFH scheme.More specifically, the received signal is a convolution of the transmitted signalst with the channel impulse response, and it can be expressed as(4)rt=∫-∞+∞hτst-τdτ+nt,where hτ is the channel impulse response and nt denotes additive white Gaussian noise (AWGN). Accordingly, the outputs of bandpass filters are given by(5)zit=∫-∞+∞qiτrt-τdτ,fori=1,…,Nc,where qit is the ideal bandpass filter centered at frequency fi. If the channel is ideal, that is, ht=δt, then(6)zit=∫-∞+∞qiτst-τdτ+uit,fori=1,…,Nc,where uit=∫-∞+∞qiτnt-τ is the filtered version of WGN. If the signal-to-noise ratio is sufficiently high, and there is no strong jamming, as in most communication systems, there is one and only one significantly strong signal among the outputs of the filter bank. Suppose that the lth filter captures this distinctive signal during the nth hop; then the estimated hopping frequency f^n,i=fl. The same procedures can be carried out to determine the carrier frequency at each hop.Next, the estimated hopping frequencyf^n,i is used to control NCO. The output signal multiplied by the received signal and NCO is used to estimate the phase of the transmitted signal. Denote the estimated phase as ϕ^n,j.Then, the estimated hopping frequencyf^n,i and the estimated phase ϕ^n,j are used to estimate the transmitted signal. Denote the estimated signal as sd^1n,d^2n,…,d^Nonn. The estimated index d^1n,…,d^Non at the nth hop and the estimated index d^1n-1,…,d^Non-1 at the (n-1)th hop are fed into the function Gm-1(1≤m≤No), the inverse function of Gm, to recover the information data blocks. This processing can be expressed by the following equations:(7)X^n,1=G1-1d^1n-1,d^2n-1,…,d^Non-1,d^1n,X^n,m=Gm-1d^m-1n,d^mn,whenm=2,…,No.It then follows that the estimate of thenth block Xn can be obtained as X^n=[X^n,1,X^n,2,…,X^n,No].


## 2.1. Transmitter Design

Unlike the conventional DFH systems in which only the frequencies are coded by the one-order coding, the HODFH systems code the frequencies and their phases by high order coding. That is to say, in the conventional DFH system, only the carrier frequencies carry the message; however, in the HODFH system, not only frequencies but also the phases carry the information. In a HODFH system, we add several phases to the frequencies; therefore, the signal number is significantly increased almost without increase in bandwidth.LetNc be the total number of frequencies, with f1,f2,…,fNc being the set of all frequencies. Let Mk be the number of initial phases where the frequency is fk in each hop, with ϕ1,ϕ2,…,ϕMk being the set of all phases. Define s(fn,i,ϕn,j) as the combinated signal of these frequencies and phases in the nth hop; it is clear that there are Ns=∑n=1NcMk signals. The set S, with s1,s2,…,sNs being its elements, is the set of all the combinated signals.LetNo be the order of the HODFH system; we start by reshaping the one-dimensional set S into No-dimensional set SD1×D2×⋯×DNo, where the subscript denotes element number of each dimension, and D1×D2×⋯×DNo≤Ns; denote sd1,d2,…,dNoas the element of the set SD1×D2×⋯×DNo. We assume that D1×D2×⋯×DNo=Ns. Divide the data stream into blocks of length L≜∑i=1NoBi, where Bi(≤log2Di) is the number of bits that is transmitted through the ith order differential coding. Denote the nth block by Xn. The block Xn can be grouped into No vectors, denoted by [Xn,1,Xn,2,…,Xn,No], where Xn,i denotes the ith order bits, as shown in Figure 1. We will transmit Xn within one hop.Figure 1
Thenth block of the information data.The transmitter block diagram of the proposed HODFH scheme is illustrated in Figure2. Each input data block, Xn, is fed into a serial-to-parallel (S/P) converter, where the data bits are split into No parallel data streams. Denote the signals in the nth and (n-1)th hop by sd1n,d2n,…,dNonn and sd1n-1,d2n-1,…,dNon-1n-1. The index of the sd1n,d2n,…,dNonn is obtained by No-order differential coding the index of the sd1n-1,d2n-1,…,dNon-1n-1 and information data block Xn. The differential processing is called a function G and expressed by the following equations:(1)d1n=G1d1n-1,d2n-1,…,dNon-1,Xn,1dmn=Gmdm-1n,Xn,m,whenm=2,…,No.Figure 2
The transmitter block diagram of the proposed HODFH scheme.Using the index vectors to select the combinated signal, the signal transmitted over the channel issd1n,d2n,…,dNonn.Recall thats(fn,i,ϕn,j) is the combination of the frequency and phase. The transmitted signal in the nth hop can be expressed as(2)sd1n,d2n,…,dNonn=sfn,i,ϕn,j=Acos⁡2πfn,it+ϕn,j0≤t<Th,where A is the signal amplitude, fn,i is the center frequency of the nth hop, ϕn,j is the possible phases of the nth hop, and Th is the hop duration. The passband waveform of the transmitted signal may be expressed as(3)st=2ThRe∑n=-∞∞e-j2πfn,i+ϕn,jtgt-nTh,where gt is the signal pulse-shaping filter.


## 2.2. Receiver Design

The receiver block diagram of the proposed HODFH scheme is illustrated in Figure3. Recall that there are Nc frequencies f1,f2,…,fNc that are extended out Ns signals. First, we must detect which frequency is transmitted. For this purpose, a bank of Nc bandpass filters (BPF), each centered at fi(i=1,2,…,Nc), with the same channel bandwidth as the transmitter, is deployed at the receiver front end. Since only one frequency band is occupied at any given hop, we simply measure the outputs of bandpass filters at each possible signaling frequency. The actual carrier frequency at a certain hopping period can be detected by selecting the one that captures the strongest signal. As a result, blind detection of the carrier frequency is achieved at the receiver.Figure 3
The receiver block diagram of the proposed HODFH scheme.More specifically, the received signal is a convolution of the transmitted signalst with the channel impulse response, and it can be expressed as(4)rt=∫-∞+∞hτst-τdτ+nt,where hτ is the channel impulse response and nt denotes additive white Gaussian noise (AWGN). Accordingly, the outputs of bandpass filters are given by(5)zit=∫-∞+∞qiτrt-τdτ,fori=1,…,Nc,where qit is the ideal bandpass filter centered at frequency fi. If the channel is ideal, that is, ht=δt, then(6)zit=∫-∞+∞qiτst-τdτ+uit,fori=1,…,Nc,where uit=∫-∞+∞qiτnt-τ is the filtered version of WGN. If the signal-to-noise ratio is sufficiently high, and there is no strong jamming, as in most communication systems, there is one and only one significantly strong signal among the outputs of the filter bank. Suppose that the lth filter captures this distinctive signal during the nth hop; then the estimated hopping frequency f^n,i=fl. The same procedures can be carried out to determine the carrier frequency at each hop.Next, the estimated hopping frequencyf^n,i is used to control NCO. The output signal multiplied by the received signal and NCO is used to estimate the phase of the transmitted signal. Denote the estimated phase as ϕ^n,j.Then, the estimated hopping frequencyf^n,i and the estimated phase ϕ^n,j are used to estimate the transmitted signal. Denote the estimated signal as sd^1n,d^2n,…,d^Nonn. The estimated index d^1n,…,d^Non at the nth hop and the estimated index d^1n-1,…,d^Non-1 at the (n-1)th hop are fed into the function Gm-1(1≤m≤No), the inverse function of Gm, to recover the information data blocks. This processing can be expressed by the following equations:(7)X^n,1=G1-1d^1n-1,d^2n-1,…,d^Non-1,d^1n,X^n,m=Gm-1d^m-1n,d^mn,whenm=2,…,No.It then follows that the estimate of thenth block Xn can be obtained as X^n=[X^n,1,X^n,2,…,X^n,No].


## 3. Bit Error Probability for HODFH

In HODFH system, the input bit stream is carried by hopping frequency and its phase. So, firstly, we analyze hopping frequency symbol error probability and phase symbol error probability, respectively, and then deduce the bit error rate (BER) of the HODFH.


### 3.1. Symbol Error Probability

Firstly, we analyze the hopping frequency symbol error probability. Based on the receiver design in HODFH, analysis of the hopping frequency symbol error probability is analogous to that of noncoherent FSK demodulation. For noncoherent detection ofMF-ary FSK signals, the probability of symbol error is given by [8, eqn. (5-4-46), page 310]:(8)ps,FSKεsN0=∑n=1MF-1-1n+1MF-1n1n+1exp⁡-nεsn+1N0,where εs/N0 is the SNR per symbol. For a HODFH system with Nc frequencies, MF=Nc. Let ps,HODFHF denote the probability of the hopping frequency symbol detection error (corresponding to the symbol error in FSK) in HODFH; then we have(9)ps,HODFHFεsFN0=∑n=1Nc-1-1n+1Nc-1n1n+1exp⁡-nεsFn+1N0,where εsF/N0 is the SNR of hopping frequency per symbol.Next, we focus on the phase symbol error probability. The phase symbol error probability can be calculated in a similar manner as that of theMP-ary PSK. The probability of phase symbol error probability is given by [8, eqn. (5-2-56) & (5-2-55), page 268]:(10)ps,PSKεsN0=1-∫-π/MPπ/MPpΘrΘrdΘr,(11)pΘrΘr=12πexp⁡-εsN0sin2Θr·∫0∞Vexp⁡-V-2εs/N0cos⁡Θr22dV.For a HODFH system, signal phase detection is implemented at the condition that the hopping frequency is detected correctly, for which the probability is1-ps,HODFHF; the probability of phase symbol error can be calculated as that of the MP-ary PSK. Let ps,HODFHP denote the probability of the phase symbol detection error (corresponding to the symbol error in PSK). Recall that there are Mk possible phases in each hopping frequency; we have(12)ps,HODFHPεsPN0=1-∫-π/Mkπ/MkpΘrΘrdΘr,where εsP/N0 is the SNR of phase per symbol and it will be substituted in (11) when we calculate ps,HODFHP.Only when the hopping frequency and the phase are detected correctly, the transmitted signal can be estimated correctly. Since, in a HODFH system, the transmitted signal carries the frequency and phase information, the SNR of the hopping frequency per symbol is the same as the SNR of the phase; that is,εsF/N0=εsP/N0. We denote the SNR per symbol by εs/N0. The overall symbol error probability of the HODFH scheme is calculated as the combination of ps,HODFHF and ps,HODFHP, defined as ps,HODFH, which can be found in (13)ps,HODFHεsN0=1-1-ps,HODFHFεsN01-ps,HODFHPεsN0=ps,HODFHFεsN0+ps,HODFHPεsN0-ps,HODFHFεsN0·ps,HODFHPεsN0.


### 3.2. Bit Error Probability

It is reasonable to assume that the probability of the indexd^in1≤i≤No estimated incorrectly is inversely proportionate to the number of elements at the ith dimensionality, when the symbol error occurs. Let pIndex,i be the error probability of the ith dimensional index; we have(14)pIndex,iεsN0=Di-1Dips,HODFHεsN0.For the first order bitsXn,1, it is only decided by d^1n-1,…,d^Non-1 and d^1n. The correct probability of d^1n-1,…,d^Non-1 is equal to the symbol correct probability 1-ps,HODFH. Recall that there are B1 bits at the first order. The BER of the first order bits, denoted as pb1,HODFH, is obtained:(15)pb1,HODFHεsN0=2B1-12B1-11-1-pIndex,1εsN01-ps,HODFHεsN0=2B1-12B1-12D1-1D1ps,HODFHεsN0-D1-1D1ps,HODFHεsN02.For themth order bits Xn,m(2≤m≤No), it is decided by d^m-1n and d^mn. Only the d^m-1n and d^mn are estimated correctly; the information bits can be recovered correctly. Assume that the error probability of d^m-1n and d^mn is independent. This is insured by the processing of reshaping the 1-dimensional set S into No-dimensional set SD1×D2×⋯×DNo. Let pIndexi,j be the correct probability of the ith and the jth dimensional index simultaneity; we have(16)pIndexi,jεsN0=1-pIndex,iεsN0·1-pIndex,jεsN0=1-Di-1Dips,HODFHεsN0·1-Dj-1Djps,HODFHεsN0=1-Dj-1Dj·ps,HODFHεsN0-Di-1Dips,HODFHεsN0+Di-1Dj-1DiDjps,HODFHεsN02.Recall that there areBm bits at the mth order. The BER of the mth order bits, denoted as pbm,HODFH, is obtained:(17)pbm,HODFH=2Bm-12Bm-11-pIndexm-1,m=2Bm-12Bm-1Dm-1-1Dm-1ps,HODFHεsN0+Dm-1Dmps,HODFHεsN0-Dm-1-1Dm-1Dm-1Dmps,HODFHεsN02.Therefore, the total BER of a HODFH system, denoted aspb,HODFH, can be obtained as(18)pb,HODFHεsN0=B1∑n=1NoBn·pb1,HODFHεsN0+∑m=2NoBm∑n=1NoBn·pbm,HODFHεsN0.Since∑n=1NoBn=L, (18) can be transformed as(19)pb,HODFHεsN0=1LB1pb1,HODFHεsN0+∑m=2NoBmpbm,HODFHεsN0.We assume that one symbol not correctly detected will incur theNo dimensional index being not correctly estimated. That is to say, pIndex,i and pIndexi,j will satisfy the following equations:(20)pIndex,iεsN0=ps,HODFHεsN0pIndexi,jεsN0=1-ps,HODFHεsN01-ps,HODFHεsN0=1-2ps,HODFHεsN0+ps,HODFHεsN02.Substitute (20) into (15) and (17); the uniform of the upper bound of BER for the first and the mth order bits, denoted as pbm,HODFHU, can be obtained as(21)pbm,HODFHU=2Bm-12Bm-12ps,HODFHεsN0-ps,HODFHεsN02,1≤m≤No.When the symbol error probabilityps,HODFH is small enough, the bit error probability pbm,HODFHU can be approximated as(22)pbm,HODFHUεsN0=2Bm2Bm-1ps,HODFHεsN0,1≤m≤No.Substitute (22) into (19); the upper bound of total BER for the HODFH system is obtained(23)pb,HODFHUεsN0=1L∑m=1NoBmpbm,HODFHUεsN0=1L∑m=1NoBm2Bm2Bm-1ps,HODFHεsN0.The lower bound of total BER, denoted aspb,HODFHL, can be obtained by assuming that the symbol error only incurs bits error in that hopping. The error probability of the ith dimensional index is expressed as (14), and the ith dimensional index error only incurs the ith order bits. Therefore, the lower bound of total BER is(24)pb,HODFHLεsN0=1L∑m=1NoBm2Bm-1Dm-12Bm-1Dmps,HODFHεsN0.To summarize our discussions above, we have the following.Proposition. In HODFH, the total BER, pb,HODFHεs/N0, is bounded by(25)1L∑m=1NoBm2Bm-1Dm-12Bm-1Dmps,HODFHεsN0≤pb,HODFHεsN0≤1L∑m=1NoBm2Bm2Bm-1ps,HODFHεsN0.


### 3.3. Further Discussions about Bit Error Probability

Recall that there areNc available frequencies f1,f2,…,fNc for the HODFH system, and they distribute in broad band. It only occupies narrow band, centered at fi(i=1,2,…,Nc), available frequencies at any given moment. The SNR usually is unequal at each narrow band. In this occasion, the symbol error probability of the frequency and phase cannot be simply calculated by (9) and (12). The symbol correct decision probability pc,fi of the frequency fi can be deduced by [8, eqn. (5-4-41), page 309]. Because the random variables Rii=1,2,…,Nc are statistically independent and identically distributed and we assume that the signals centered at the Nc available frequencies f1,f2,…,fNc have equal energy and different noise, the probability pc,fi can be expressed as(26)pc,fi=∫0∞∏m=1m≠iNcPRm<σiσmRi∣Ri=xpRixdx,where σm2=1/2N0,m is the variance of the AWGN at frequency fi and(27)PRm<σiσmRi∣Ri=x=∫0σix/σmpRmrmdrm=1-e-σi2x2/2σm2=1-e-N0,ix2/2N0,m.Substitute (27) into (26); we obtain the general probability(28)pc,fi=∫0∞∏m=2m≠iNc1-e-N0,ix2/2N0,mpRixdx.We have known that, in a HODFH system, signal phase detection is implemented at the condition that the hopping frequency centred atfi is detected correctly, for which the probability is pc,fi; the probability of phase symbol error can be calculated as (10) and (11), but the SNR εsP/N0 of phase per symbol will be substituted by ε/N0,i, which is the SNR at frequency fi. Let ps,HODFHP,i denote the probability of the phase symbol detection error at frequency fi. In this occasion, the overall symbol error probability of the HODFH scheme is calculated by (29)ps,HODFH′=1-∑i=1Ncpi·pc,fi·1-ps,HODFHP,i=1-1Nc∑i=1Ncpc,fi·1-ps,HODFHP,i,where the pii=1,2,…,Nc is the probability that the signals centered at frequency fi and the p1,p2,…,pNc all equal 1/Nc. Substitute the probability ps,HODFH by ps,HODFH′ in (16) and (17); we can calculate the total BER of a HODFH system by (18) when it works in the occasion that the Nc available frequencies f1,f2,…,fNc distribute in broad band and the SNR is not equal at each narrow band centered at fi.


## 3.1. Symbol Error Probability

Firstly, we analyze the hopping frequency symbol error probability. Based on the receiver design in HODFH, analysis of the hopping frequency symbol error probability is analogous to that of noncoherent FSK demodulation. For noncoherent detection ofMF-ary FSK signals, the probability of symbol error is given by [8, eqn. (5-4-46), page 310]:(8)ps,FSKεsN0=∑n=1MF-1-1n+1MF-1n1n+1exp⁡-nεsn+1N0,where εs/N0 is the SNR per symbol. For a HODFH system with Nc frequencies, MF=Nc. Let ps,HODFHF denote the probability of the hopping frequency symbol detection error (corresponding to the symbol error in FSK) in HODFH; then we have(9)ps,HODFHFεsFN0=∑n=1Nc-1-1n+1Nc-1n1n+1exp⁡-nεsFn+1N0,where εsF/N0 is the SNR of hopping frequency per symbol.Next, we focus on the phase symbol error probability. The phase symbol error probability can be calculated in a similar manner as that of theMP-ary PSK. The probability of phase symbol error probability is given by [8, eqn. (5-2-56) & (5-2-55), page 268]:(10)ps,PSKεsN0=1-∫-π/MPπ/MPpΘrΘrdΘr,(11)pΘrΘr=12πexp⁡-εsN0sin2Θr·∫0∞Vexp⁡-V-2εs/N0cos⁡Θr22dV.For a HODFH system, signal phase detection is implemented at the condition that the hopping frequency is detected correctly, for which the probability is1-ps,HODFHF; the probability of phase symbol error can be calculated as that of the MP-ary PSK. Let ps,HODFHP denote the probability of the phase symbol detection error (corresponding to the symbol error in PSK). Recall that there are Mk possible phases in each hopping frequency; we have(12)ps,HODFHPεsPN0=1-∫-π/Mkπ/MkpΘrΘrdΘr,where εsP/N0 is the SNR of phase per symbol and it will be substituted in (11) when we calculate ps,HODFHP.Only when the hopping frequency and the phase are detected correctly, the transmitted signal can be estimated correctly. Since, in a HODFH system, the transmitted signal carries the frequency and phase information, the SNR of the hopping frequency per symbol is the same as the SNR of the phase; that is,εsF/N0=εsP/N0. We denote the SNR per symbol by εs/N0. The overall symbol error probability of the HODFH scheme is calculated as the combination of ps,HODFHF and ps,HODFHP, defined as ps,HODFH, which can be found in (13)ps,HODFHεsN0=1-1-ps,HODFHFεsN01-ps,HODFHPεsN0=ps,HODFHFεsN0+ps,HODFHPεsN0-ps,HODFHFεsN0·ps,HODFHPεsN0.


## 3.2. Bit Error Probability

It is reasonable to assume that the probability of the indexd^in1≤i≤No estimated incorrectly is inversely proportionate to the number of elements at the ith dimensionality, when the symbol error occurs. Let pIndex,i be the error probability of the ith dimensional index; we have(14)pIndex,iεsN0=Di-1Dips,HODFHεsN0.For the first order bitsXn,1, it is only decided by d^1n-1,…,d^Non-1 and d^1n. The correct probability of d^1n-1,…,d^Non-1 is equal to the symbol correct probability 1-ps,HODFH. Recall that there are B1 bits at the first order. The BER of the first order bits, denoted as pb1,HODFH, is obtained:(15)pb1,HODFHεsN0=2B1-12B1-11-1-pIndex,1εsN01-ps,HODFHεsN0=2B1-12B1-12D1-1D1ps,HODFHεsN0-D1-1D1ps,HODFHεsN02.For themth order bits Xn,m(2≤m≤No), it is decided by d^m-1n and d^mn. Only the d^m-1n and d^mn are estimated correctly; the information bits can be recovered correctly. Assume that the error probability of d^m-1n and d^mn is independent. This is insured by the processing of reshaping the 1-dimensional set S into No-dimensional set SD1×D2×⋯×DNo. Let pIndexi,j be the correct probability of the ith and the jth dimensional index simultaneity; we have(16)pIndexi,jεsN0=1-pIndex,iεsN0·1-pIndex,jεsN0=1-Di-1Dips,HODFHεsN0·1-Dj-1Djps,HODFHεsN0=1-Dj-1Dj·ps,HODFHεsN0-Di-1Dips,HODFHεsN0+Di-1Dj-1DiDjps,HODFHεsN02.Recall that there areBm bits at the mth order. The BER of the mth order bits, denoted as pbm,HODFH, is obtained:(17)pbm,HODFH=2Bm-12Bm-11-pIndexm-1,m=2Bm-12Bm-1Dm-1-1Dm-1ps,HODFHεsN0+Dm-1Dmps,HODFHεsN0-Dm-1-1Dm-1Dm-1Dmps,HODFHεsN02.Therefore, the total BER of a HODFH system, denoted aspb,HODFH, can be obtained as(18)pb,HODFHεsN0=B1∑n=1NoBn·pb1,HODFHεsN0+∑m=2NoBm∑n=1NoBn·pbm,HODFHεsN0.Since∑n=1NoBn=L, (18) can be transformed as(19)pb,HODFHεsN0=1LB1pb1,HODFHεsN0+∑m=2NoBmpbm,HODFHεsN0.We assume that one symbol not correctly detected will incur theNo dimensional index being not correctly estimated. That is to say, pIndex,i and pIndexi,j will satisfy the following equations:(20)pIndex,iεsN0=ps,HODFHεsN0pIndexi,jεsN0=1-ps,HODFHεsN01-ps,HODFHεsN0=1-2ps,HODFHεsN0+ps,HODFHεsN02.Substitute (20) into (15) and (17); the uniform of the upper bound of BER for the first and the mth order bits, denoted as pbm,HODFHU, can be obtained as(21)pbm,HODFHU=2Bm-12Bm-12ps,HODFHεsN0-ps,HODFHεsN02,1≤m≤No.When the symbol error probabilityps,HODFH is small enough, the bit error probability pbm,HODFHU can be approximated as(22)pbm,HODFHUεsN0=2Bm2Bm-1ps,HODFHεsN0,1≤m≤No.Substitute (22) into (19); the upper bound of total BER for the HODFH system is obtained(23)pb,HODFHUεsN0=1L∑m=1NoBmpbm,HODFHUεsN0=1L∑m=1NoBm2Bm2Bm-1ps,HODFHεsN0.The lower bound of total BER, denoted aspb,HODFHL, can be obtained by assuming that the symbol error only incurs bits error in that hopping. The error probability of the ith dimensional index is expressed as (14), and the ith dimensional index error only incurs the ith order bits. Therefore, the lower bound of total BER is(24)pb,HODFHLεsN0=1L∑m=1NoBm2Bm-1Dm-12Bm-1Dmps,HODFHεsN0.To summarize our discussions above, we have the following.Proposition. In HODFH, the total BER, pb,HODFHεs/N0, is bounded by(25)1L∑m=1NoBm2Bm-1Dm-12Bm-1Dmps,HODFHεsN0≤pb,HODFHεsN0≤1L∑m=1NoBm2Bm2Bm-1ps,HODFHεsN0.


## 3.3. Further Discussions about Bit Error Probability

Recall that there areNc available frequencies f1,f2,…,fNc for the HODFH system, and they distribute in broad band. It only occupies narrow band, centered at fi(i=1,2,…,Nc), available frequencies at any given moment. The SNR usually is unequal at each narrow band. In this occasion, the symbol error probability of the frequency and phase cannot be simply calculated by (9) and (12). The symbol correct decision probability pc,fi of the frequency fi can be deduced by [8, eqn. (5-4-41), page 309]. Because the random variables Rii=1,2,…,Nc are statistically independent and identically distributed and we assume that the signals centered at the Nc available frequencies f1,f2,…,fNc have equal energy and different noise, the probability pc,fi can be expressed as(26)pc,fi=∫0∞∏m=1m≠iNcPRm<σiσmRi∣Ri=xpRixdx,where σm2=1/2N0,m is the variance of the AWGN at frequency fi and(27)PRm<σiσmRi∣Ri=x=∫0σix/σmpRmrmdrm=1-e-σi2x2/2σm2=1-e-N0,ix2/2N0,m.Substitute (27) into (26); we obtain the general probability(28)pc,fi=∫0∞∏m=2m≠iNc1-e-N0,ix2/2N0,mpRixdx.We have known that, in a HODFH system, signal phase detection is implemented at the condition that the hopping frequency centred atfi is detected correctly, for which the probability is pc,fi; the probability of phase symbol error can be calculated as (10) and (11), but the SNR εsP/N0 of phase per symbol will be substituted by ε/N0,i, which is the SNR at frequency fi. Let ps,HODFHP,i denote the probability of the phase symbol detection error at frequency fi. In this occasion, the overall symbol error probability of the HODFH scheme is calculated by (29)ps,HODFH′=1-∑i=1Ncpi·pc,fi·1-ps,HODFHP,i=1-1Nc∑i=1Ncpc,fi·1-ps,HODFHP,i,where the pii=1,2,…,Nc is the probability that the signals centered at frequency fi and the p1,p2,…,pNc all equal 1/Nc. Substitute the probability ps,HODFH by ps,HODFH′ in (16) and (17); we can calculate the total BER of a HODFH system by (18) when it works in the occasion that the Nc available frequencies f1,f2,…,fNc distribute in broad band and the SNR is not equal at each narrow band centered at fi.


## 4. Simulation Results

Without loss of generality, in this paper, the HODFH system in the AWGN channel with equal SNR at anywhere was simulated and its bit error rate (BER) performance was analyzed.Example 1.
BER performance of the HODFH system in AWGN channel: the performance of BER at the information rates of 9600 bps, 19200 bps, and 28800 bps in AWGN channel was compared. For the information rate of 9600 bps,No=2, Nc=128, Mk=2(k=1,…,Nc), and Bk=1(k=1,…,No). For the information rate of 19200 bps, No=2, Nc=64, Mk=4(k=1,…,Nc), and Bk=2(k=1,…,No). For the information rate of 28800 bps, No=2, Nc=64, Mk=4(k=1,…,Nc), B1=4, and B2=2.Figure4 is the BER performance of the HODFH system in AWGN channel. The requirement for the energy-per-bit noise rate (Eb/N0) becomes lower when the information rate increases. In practice, it is usually possible that only the energy-per-symbol noise rate (Es/N0) can be measured. According to formula Es/N0=Eb/N0+10log⁡k (k is the number of bits per symbol), convert Eb/N0 to Es/N0, and then the higher Es/N0 is required for the higher information rate, which is reasonable.Figure 4
The BER performance at information rates of 9600 bps, 19200 bps, and 28800 bps in AWGN channel.Example 2.
The BER performance of information rate 28800 bps at different number of phases: assumeNo=2 and Bk=3(k=1,…,No). The numbers of available frequencies Nc are 64, 32, and 16; the numbers of phases Mk are 4, 8, and 16, respectively.As seen in Figure5, when more phrases are used while keeping the information rate the same, there is poorer performance for the HODFH system. It will spend about 6 dB extra Eb/N0 to obtain the same BER performance of 10-3 when the number of phases increases from 8 to 16. Therefore, the number of phases should be kept to a minimum, if the number of transmission signals has reached the requirements.Figure 5
The BER performance of information rate 28800 bps at different number of phases.Example 3.
The BER performance of the HODFH system and the conventional DFH (CDFH) system: for the information rate of 9600 bps, the HODFH system has the following parameters:No=2, Nc=128, Mk=2(k=1,…,Nc), and Bk=1(k=1,…,No); for the information rate of 19200 bps, it has parameters: No=2, Nc=64, Mk=4(k=1,…,Nc), and Bk=2(k=1,…,No). The CDFH has the following parameters: for information rate of 9600 bps, Nc=128 and BPH=2; for information rate of 19200 bps, Nc=128 and BPH=4.Figure6 shows the BER performance comparison of the HODFH system and the conventional DFH system for information rates of 9600 bps and 19200 bps. At the same information rate, the HODFH system has a better BER performance than the conventional DFH system. The higher information rate is, the higher gain is obtained by the HODFH system when compared to the conventional DFH system.Figure 6
The BER performance of the HODFH system and the CDFH system.Example 4.
Compare the BER performances of the HODFH system at three information rates (9600 bps, 19200 bps, and 28800 bps) with the theoretical BER performance of QPSK, 8PSK, 16PSK, 32PSK, and 64PSK. Assume the HODFH system has the following parameters, respectively. For the information rate of 9600 bps,No=2, Nc=128, Mk=2(k=1,…,Nc), and Bk=1(k=1,…,No). For the information rate of 19200 bps, No=2, Nc=64, Mk=4(k=1,…,Nc), and Bk=2(k=1,…,No). For the information rate of 28800 bps, we have three combinations; the first is No=2, Nc=64, Mk=4(k=1,…,Nc), and Bk=3(k=1,…,No), the second is No=2, Nc=32, Mk=8(k=1,…,Nc), and Bk=3(k=1,…,No), and the third is No=2, Nc=16, Mk=16(k=1,…,Nc), and Bk=3(k=1,…,No).Figure7 presents the BER performance of the HODFH system and the theoretical BER performance of QPSK, 8PSK, 16PSK, 32PSK, and 64PSK. When the information rate is 9600 bps, the HODFH system carries two bits of information per hop (symbol), which is the same as the QPSK; the BER performance of the HODFH system is worse than the QPSK. When the information rate is 19200 bps, the HODFH system carries four bits of information per hop (symbol), which is the same as the 16PSK, but it has better BER performance than the 16PSK. When the information rate is 28800 bps, the HODFH system carries six bits of information per hop (symbol), which is the same as the 64PSK, but its performance is much better than the 64PSK no matter how many phases are used by the HODFH system. The BER performance of the HODFH system is better than the M-PSK when the information rate is high. The carrier frequencies of the HODFH systems are used to transmit information; the HODFH systems have higher information rate without extra cost on power and do not need extra cost on power, when the numbers of phases adopted by the HODFH systems and the M-PSK are equal. The HODFH systems have better BER performance at the same energy-per-bit noise rate compared to the PSK.Figure 7
The BER performance of the HODFH system and theoretical BER performance of the M-PSK.Example 5.
Compare the BER performance of information rate 19200 bps at the occasion that the SNR is equal and unequal at each narrow band. Assume that the SNRs atNc/2 narrow bands are different with the SNRs at the other Nc/2 narrow bands. Use SNRd to denote the difference among the Nc SNRs. The SNRd increases from 2 to 8 by increment 2. Use the average SNR in the broad band as the symbol signal-to-noise ratio. The HODFH system has the following parameters: No=2, Nc=64, Mk=4(k=1,…,Nc), and Bk=2(k=1,…,No).Example 6.
Compare the BER performance of information rate 28800 bps at different number of phases when the SNR differenceSNRd=4 and 6. The SNR is similar as Example 5. The HODFH system has the following parameters: No=2, Nc=64, Bk=3(k=1,…,No), and Mk=4, 8, and 16(k=1,…,Nc).As seen in Figure8, the BER performance is worse when the SNR is unequal at each narrow band, despite the average SNR of the unequal signal-to-noise ratio broad band being the same as the SNR of the equal signal-to-noise ratio. As the difference of the SNRs is increasing, the performance is degraded.Figure 8
The BER performance of information rate 19200 bps atSNRd=2, 4, 6, and 8.Figure9 shows that the BER performance is degraded when the number of phases used by the HODFH system increases. This recalls the conclusion in Example 2 that the number of phases should be kept to a minimum, if the number of transmission signals has reached the requirements.Figure 9
BER performance of information rate 28800 bps at different number of phases whenSNRd=4 and 6.


## 5. Summary

In the future, communication will have higher information rate requirements, and this paper proposed a solution for this with a novel differential frequency hopping, high order differential frequency hopping method. This was done by conumbering the frequencies and their phases, so that the system information rate was improved without having to increase the frequency resources. The BER performance of the HODFH system is better than the conventional DFH system and M-PSK in the AWGN channel. In this work, the number of phases can be adjusted to match the channel conditions to obtain the optimal performance.Although using phases to expand the signal set can improve the bandwidth efficiency, their introduction of the phases makes the HODFH system more sensitive to frequency offset and timing errors. That is to say, implementation of this system is more complex, but it is a price worth paying for the information rate improvement and increased performance.



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*Source: 586080-2015-06-25.xml*