跳频系统是指系统通信频率受跳频图案控制随时间不断改变的通信系统,不管在军用还是民用中,跳频技术都得到了广泛应用。相比其他通信系统,跳频系统具有一些明显的优点:抗干扰能力强;截获概率低;抗衰落能力强。但是,跳频系统在应用过程中也逐渐发现一些问题:为了实现跳频同步过程,增加了接收端复杂度;频谱利用率低,传输效率低;不易与其他宽带通信系统相结合等。
为了满足日益增长的传输效率的需求和降低接收端同步过程的负担,Ling等[1]创新的提出了一种消息驱动跳频(message driven frequency hopping, MDFH)系统。与传统跳频系统用跳频图案控制频点不同,消息驱动利用部分消息来控制频点的选择,可以极大地提升跳频系统的频谱效率[2-4]。为了进一步提升频谱效率,Qi等[5-6]在消息驱动跳频的基础上提出了增强型消息驱动跳频系统,即选用多个子信道同时传递信息,可以实现更高的频谱效率。对这种增强型消息驱动跳频系统分析可知,通过跳频点传递信息本质上是增加了一个用来传递信息的维度[7-8]。但是,现有的消息驱动跳频系统通常采用频率合成器及带通滤波器组实现调制解调的过程,收发端结构十分复杂[9-10]。Scholand等[11]提出了一种跳频OFDM的数学表达式,为后续跳频OFDM的研究打下了坚实的数学基础。美国Locheed Sanders公司在1995年研制出CHESS系统,这种跳频通信系统采用了多种先进技术,包括差分跳频、异步跳频等,同时,这种系统与OFDM系统类似,采用了IFFT/FFT来实现调制解调的过程,这种结构的跳频系统收端相对简单,经过FFT之后即可获得全频带内的信息[12-14]。
基于此点,本文借鉴CHESS系统提出了一种基于IFFT/FFT结构的消息驱动跳频系统,利用IFFT/FFT算法实现调制解调的过程。其次,目前的消息驱动跳频并没有详细讨论映射方式对系统性能的影响,都是以QAM映射来讨论的。本文在基于IFFT/FFT结构上提出了一种正交分组二进制频移键控(orthogonal grouped binary frequency shift keying, OG-2FSK)的映射方式,并与QAM中的特例BPSK映射进行对比,分析了2种映射方式在不同信道下的误码性能。最后,目前缺少消息驱动跳频系统在衰落信道下的性能分析,本文对此也作出了一些补充。
1 基于IFFT/FFT结构的消息驱动跳频系统 1.1 系统模型基于IFFT/FFT结构的消息驱动跳频系统框图如图 1所示。
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基带信息经串并转换、分帧之后根据驱动比特和常规比特的值进行星座点映射,对映射结果进行IFFT变换和并串转换后得到基带时域波形,经上变频后发送到信道中。
接收时,接收端对收到的信号进行采样及下变频处理得到基带时域波形,串并转换后进行FFT变换,对得到的结果进行星座点解映射,解调出驱动比特及常规比特。再将相应的驱动比特及常规比特进行组帧,经并串转换后得到解调出的基带信息。
1.2 OG-2FSK映射本系统中,信息主要分为2部分进行传输,分别是驱动比特和常规比特。二进制基带信息经串并转换后进行分帧处理,每帧长度为(No+Nc)bit,其中No为常规比特个数,Nc为驱动比特个数。
第j帧信息为:
$ {m_j} = \{ {O_j}, {C_j}\} $ | (1) |
式中:Oj为第j帧的常规比特;Cj为第j帧的驱动比特。则:
$ {{O_j} = \{ {a_{{N_{\rm{c}}} + 1 + (j - 1) \cdot ({N_{\rm{c}}} + {N_{\rm{o}}})}}, \cdots , {a_{j \cdot ({N_{\rm{c}}} + {N_{\rm{o}}})}}\} } $ | (2) |
$ {{C_j} = \{ {a_{1 + (j - 1) \cdot ({N_{\rm{c}}} + {N_{\rm{o}}})}}, \cdots , {a_{{N_{\rm{c}}} + (j - 1) \cdot ({N_{\rm{c}}} + {N_{\rm{o}}})}}\} } $ | (3) |
式中an为第n个基带码元。驱动比特用来取代传统跳频系统中的跳频图案,决定频点的选择。常规比特与传统跳频系统中的基带码元一样。由于通信频点本身携带一定的信息,具有随机性,因此不需要传统跳频系统的跳频图案同步,简化了通信系统收端复杂度。
为了适应衰落信道,本文提出了OG-2FSK这种映射方式。采用OG-2FSK映射时,如图 2系统将整个带宽分为N个互相重叠50%的子信道,其中前N/2个子信道用来传“1”,后N/2个子信道用来传“0”。前后两部分子信道中,相对应的子信道为一对子信道,例如图 2中第1子信道和第(N+1)/2子信道即为第1对子信道。
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对于OG-2FSK映射:
$ \left\{ {\begin{array}{*{20}{l}} {{N_{\rm{c}}} = {\rm{log}}{\kern 1pt} {\kern 1pt} 2(N) - 1}\\ {{N_{\rm{o}}} = {N_{{\rm{nph}}}}} \end{array}} \right. $ | (4) |
Nnph为一跳传输的符号数,本系统中Nnph=1,N为子信道个数。第j帧信息映射为一个N×Nnph的矩阵Xj:
$ {\mathit{\boldsymbol{X}}_j} = \{ {x_{(j - 1) \cdot {N_{{\rm{nph}}}} + 1}}(n), \cdots , {x_{j \cdot {N_{{\rm{nph}}}}}}(n)\} $ | (5) |
式中:xk(n)是一个N×1的矩阵,k=(j-1)·Nnph+1, 2, …, j·Nnph, n=1, 2, …, N。则当第j帧第k个常规比特值为“1”时:
$ {\mathit{\boldsymbol{x}}_k}(n) = \left\{ {\begin{array}{*{20}{l}} {1, \quad n = {\rm{bin 2 dec}} ({C_j}) + \frac{N}{2} + 1}\\ {0, \quad n = 1, 2, \cdots , N, n \ne {\rm{bin 2 dec}} ({C_j}) + \frac{N}{2} + 1} \end{array}} \right. $ | (6) |
当第j帧第k个常规比特值为“0”时:
$ {x_k}(n) = \left\{ {\begin{array}{*{20}{l}} {1, n = {\rm{bin}} (2 {\rm{dec}} ({C_j}) + 1)}\\ {0, n = 1, 2, \cdots , N, n \ne {\rm{bin 2 dec}} ({C_j}) + 1} \end{array}} \right. $ | (7) |
式中bin2dec(·)表示二进制数转换为十进制的数。则:
$ X = \{ {X_1}, {X_2}, \cdots , {X_j}, \cdots \} $ | (8) |
即为最终的映射结果。第j帧基带时域波形可以表示为:
$ \begin{array}{*{20}{l}} {{s_j}(t) = \sum\limits_{m = 1}^{{N_{{\rm{nph}}}}} {\frac{1}{{\sqrt N }}} \sum\limits_{n = 1}^N {\rm{A}} \cdot {X_j}(n, m){\rm{exp}}\left( {{\rm{i}}\frac{{2\pi n\Delta ft}}{N}} \right) = }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{1}{{\sqrt N }}\sum\limits_{m = 1}^{{N_{{\rm{nph}}}}} {\rm{A}} \cdot {\rm{exp}}({\rm{i}}{\omega _c}(m)t)} \end{array} $ | (9) |
式中:Δf为各子信道的频差;A为载频幅值;ωc(m)为:
$ {\omega _c}(m) = \left\{ {\begin{array}{*{20}{l}} {2\pi \Delta f \cdot ( {\rm{bin 2 dec}} ({C_j}) + \frac{N}{2} + 1), {O_j}(m) = 1}\\ {2\pi \Delta f \cdot ( {\rm{bin 2 dec}} ({C_j}) + 1), {O_j}(m) = 0} \end{array}} \right. $ | (10) |
设发送端第j帧第n子信道发射的信号为sj, n(t),接收端收到对应的信号为rj, n(t),则收发端表达式为:
$ {s_{j, n}}(t) = \left\{ {\begin{array}{*{20}{l}} {\sum\limits_{m = 1}^{{N_{{\rm{nph}}}}} A \cdot {\rm{exp}}({\rm{i}}{\omega _c}t), \quad n = {n_{s0}}}\\ {0, \quad n = 1, 2, \cdots , N, n \ne {n_{s0}}} \end{array}} \right. $ | (11) |
$ {r_{j, n}}(t) = \left\{ {\begin{array}{*{20}{l}} {\sum\limits_{m = 1}^{{N_{{\rm{ nph }}}}} A \cdot {\rm{exp}}({\rm{i}}{\omega _c}t) + {n_i}(t), \quad n = {n_{s0}}}\\ {{n_n}(t), \quad n = 1, 2, \cdots , N, n \ne {n_{s0}}} \end{array}} \right. $ | (12) |
为简化分析过程,以每跳1符号为例进行推导,即Nnph=1。其中A为信号幅度,ns0为选定的子信道。ni(t),i=1, …, N是复窄带高斯白噪声,其均值为0,方差为σn2,且可表示为:
$ {n_s}(t) = {n_{sc}}(t){\rm{cos}}({\omega _i}t) - {n_{ss}}(t){\rm{sin}}({\omega _c}t), s = 1, 2, \cdots , N $ | (13) |
式中nsc(t)、nss(t)与ns(t)具有相同的功率,其抽样值均服从高斯分布。接收端信号经采样及FFT变换后可表示为:
$ {R_j}(n) = \left\{ {\begin{array}{*{20}{l}} {A + {N_n}, n = {n_{s0}}}\\ {{N_n}, n = 1, 2, \cdots , N, n \ne {n_{s0}}} \end{array}} \right. $ | (14) |
式中Nn为窄带高斯白噪声的抽样值。
2.2 OG-2FSK映射AWGN信道下误码率推导设在N个子信道中选取第ns0子信道传输信息,第ns1子信道与之构成一对子信道,其中ns0与ns1满足:
$ \left\{ {\begin{array}{*{20}{l}} {{n_{s1}} = {n_{s0}} - N/2, {n_{s0}} > N/2}\\ {{n_{s1}} = {n_{s0}} + N/2, {n_{s0}} \le N/2} \end{array}} \right. $ | (15) |
则第ns0子信道与第ns1子信道可以完整传递1 bit信息,即选择ns0子信道代表常规比特值为“0”,选择ns1子信道代表常规比特值为“1”。以选定ns0子信道为例进行推导。FFT变换后的结果为:
$ {R_j}(n) = \left\{ {\begin{array}{*{20}{l}} {{\rm{A}} + {N_n}, \quad n = {n_{s0}}}\\ {{N_n}, n = 1, 2, \cdots , N, \quad n \ne {n_{s0}}} \end{array}} \right. $ | (16) |
能量检测的判决方式是找出abs(Rj(n)), n=1, 2, …, N中最大值所对应的i作为解映射的依据。根据随机过程的相关知识,abs(Rj(n)), n=1, 2, …, N, n≠ns0服从瑞利分布,abs(Rj(ns0))服从莱斯分布,它们的概率密度函数为:
$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} p({R_j}(n)) = \\ \left\{ \begin{array}{l} \frac{{{R_j}(n)}}{{\sigma _n^2}}{\rm{exp}}( - \frac{{{R_j}(n)}}{{2\sigma _n^2}}), \quad n = 1, \cdots , N, n \ne {n_{s0}}\\ \frac{{{R_j}(n)}}{{\sigma _n^2}}{{\rm{I}}_0}\left( {\frac{{{\rm{A}}{R_j}(n)}}{{\sigma _n^2}}} \right){\rm{exp}}\left( { - \frac{1}{{2\sigma _n^2}}({R_j}{{(n)}^2} + {{\rm{A}}^2})} \right), n = {n_{s0}} \end{array} \right. \end{array} $ | (17) |
式中:I0(·)为第一类零阶修正贝塞尔函数;Ri为信号和噪声和的包络;A为输出信号码元振幅;σn2为输出噪声功率。
根据前文叙述的解映射过程可知,对于OG-2FSK映射,在判决时若判定的选用子信道为第ns0子信道或第ns1子信道,即可正确解调出这一个符号的全部驱动比特,则驱动比特误符号率为:
$ {P_{{\rm{cse\_2FSK}}}} = 1 - {P_{{\rm{ sc0}}}} - {P_{{\rm{ sc1 }}}} $ | (18) |
式中:Psc0为判定选用子信道为第ns0子信道的概率; Psc1为判定选用ns1子信道为第子信道的概率。
$ \left\{ {\begin{array}{*{20}{l}} {{P_{{\rm{sc0}}}} = \int_0^\infty P ({R_j}({n_{s0}}) > {R_j}(1), \cdots , {R_j}({n_{s0}}) > {R_j}(N)|{R_j}({n_{s0}}))}\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} p({R_j}({n_{s0}})){\rm{d}}{R_j}({n_{s0}})}\\ {{P_{{\rm{sc1}}}} = \int_0^\infty P ({R_j}({n_{s1}}) > {R_j}(1), \cdots , {R_j}({n_{s1}}) > {R_j}(N)|{R_j}({n_{s1}}))}\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} p({R_j}({n_{s1}})){\rm{d}}{R_j}({n_{s1}})} \end{array}} \right. $ | (19) |
式中N个子信道间可视为相互独立,互不影响。
$ \left\{ {\begin{array}{*{20}{l}} {{P_{{\rm{sc0}}}} = \int_0^\infty {\prod\limits_{i = 1}^{N - 1} P } ({R_j}({n_{s0}}) > {R_j}(i)|{R_j}({n_{s0}}))p({R_j}({n_{s0}})){\rm{d}}{R_j}({n_{s0}})}\\ {{P_{{\rm{sc1}}}} = \int_0^\infty {\prod\limits_{i = 1}^{N - 1} P } ({R_j}({n_{s1}}) > {R_j}(i)|{R_j}({n_{s1}}))p({R_j}({n_{s1}})){\rm{d}}{R_j}({n_{s1}})} \end{array}} \right. $ | (20) |
根据概率论相关知识,有:
$ \begin{array}{*{20}{c}} {P({R_j}({n_{s0}}) > {R_j}(i)|{R_j}({n_{s0}})) = }\\ {\int_0^{{R_j}({n_{s0}})} {\frac{{{R_j}(i)}}{{\sigma _n^2}}} {\rm{exp}}\left( { - \frac{{{R_j}{{(i)}^2}}}{{2\sigma _n^2}}} \right){\rm{d}}{R_j}(i), i = 1, 2, \cdots , N, i \ne {n_{s0}}} \end{array} $ | (21) |
$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} P({R_j}({n_{s1}}) > {R_j}(i)|{R_j}({n_{s1}})) = \\ \left\{ \begin{array}{l} \int_0^{{R_j}({n_{s1}})} {\frac{{{R_j}(i)}}{{\sigma _n^2}}} {\rm{exp}}\left( { - \frac{{{R_j}{{(i)}^2}}}{{2\sigma _n^2}}} \right){\rm{d}}{R_j}(i), \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} i = 1, 2, \cdots , N, i \ne {n_{s0}}, {n_{s1}}\\ \int_0^{{R_j}({n_{s1}})} {\frac{{{R_j}(i)}}{{\sigma _n^2}}} {{\rm{I}}_0}\left( {\frac{{{\rm{A}}{R_j}(i)}}{{\sigma _n^2}}} \right){\rm{exp}}\left( { - \frac{{{{({R_j}(i) - {\rm{A}})}^2}}}{{2\sigma _n^2}}} \right){\rm{d}}{R_i}, \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} i = n \end{array} \right. \end{array} $ | (22) |
化简后,
$ \begin{array}{*{20}{l}} {{P_{{\rm{sc0}}}} = 1 - \sum\limits_{i = 1}^{N - 1} {{{( - 1)}^i}} \left( {\begin{array}{*{20}{c}} {N - 1}\\ i \end{array}} \right) \cdot \frac{1}{{i + 1}} \cdot }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{exp}}\left( { - \frac{i}{{i + 1}} \cdot r} \right)} \end{array} $ | (23) |
$ \begin{array}{*{20}{l}} {{P_{{\rm{sc1}}}} = 2 \cdot \int_0^\infty t {{(1 - {\rm{exp}}( - {t^2}))}^{N - 2}}(1 - }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{ }}{{\rm{Q}}_1}(\sqrt {2r} , \sqrt 2 t)){\rm{exp}}( - {t^2}){\rm{d}}t} \end{array} $ | (24) |
$ {P_{{\rm{cse\_2FSK}}}} = 1 - {P_{{\rm{sc0}}}} - {P_{{\rm{sc1}}}} $ | (25) |
式中:
每个符号中错误比特的个数的期望为:
$ {E_{{\rm{cbe\_2FSK}}}} = \sum\limits_{n = 1}^{{\rm{log}}(2(N) - 1)} n \cdot \frac{{C_{{\rm{log}}(2(N) - 1)}^n}}{{\frac{N}{2} - 1}} $ | (26) |
则驱动比特误码率为:
$ {P_{{\rm{cbe\_2FSK}}}} = \frac{{{E_{{\rm{cbe\_2FSK}}}} \cdot {P_{{\rm{cbe\_2FSK}}}}}}{{{\rm{log}}{\kern 1pt} {\kern 1pt} 2(N) - 1}} $ | (27) |
常规比特误码率为:
$ {P_{{\rm{obe\_2FSK}}}} = \frac{N}{2} \cdot {P_{{\rm{sc1}}}} $ | (28) |
总误码率为:
$ {P_{{\rm{e\_2FSK}}}} = \frac{{{N_{\rm{c}}}}}{{{N_{\rm{c}}} + {N_{\rm{o}}}}} \cdot {P_{{\rm{cbe\_2FSK}}}} + \frac{{{N_{\rm{o}}}}}{{{N_{\rm{c}}} + {N_{\rm{o}}}}} \cdot {P_{{\rm{obe\_2FSK}}}} $ | (29) |
信号经过多普勒频移后,再做FFT相当于其采样值相对AWGN信道有2ε的偏差,导致能量由AWGN信道下聚集在一个子信道内变为分布在全部N个子信道上,AWGN信道下只有噪声的未被选用的子信道变为信号与噪声之和,每个子信道如图 3所示。假设N=21,选中第11子信道,其中实线为AWGN信道下FFT的结果,虚线为时间选择性衰落信道下,归一化多普勒频移为ε时FFT的结果。
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根据图 3的分析可知每一路的幅值为:
$ a(n) = {\rm{abs}} \left( {{\rm{sin}}{\kern 1pt} {\kern 1pt} c\left( {n - \left[ {\frac{N}{2}} \right] - 2\varepsilon } \right)} \right), n = 1, 2, \cdots , N $ | (30) |
式中:N为子信道个数;ε为归一化多普勒频偏。
2.3.2 OG-2FSK映射时间选择性衰落信道下误码率推导设在N个子信道中选取第ns0子信道传输信息,第ns1子信道与之构成一对子信道,其中ns0与ns1满足式(15)。
则第ns0子信道与第ns1子信道可以完整传递1 bit信息。FFT变换后的结果为:
$ {R_j}(n) = a(n) + {N_n}, n = 1, 2, \cdots , N $ | (31) |
服从莱斯分布,它们的概率密度函数为:
$ \begin{array}{l} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} p({R_j}(n)) = \\ \begin{array}{*{20}{l}} {\frac{{{R_j}(n)}}{{\sigma _n^2}}{I_0}\left( {\frac{{{\rm{A}}{R_j}(n)}}{{\sigma _n^2}}} \right){\rm{exp}}\left( { - \frac{1}{{2\sigma _n^2}}({R_j}{{(n)}^2} + {{\rm{A}}^2})} \right), }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} n = 1, 2, \cdots , N} \end{array} \end{array} $ | (32) |
具体推导过程与2.2节相同,这里给出结果:
$ \begin{array}{l} {P_{sc0}} = 2 \cdot \int_0^\infty {\prod\limits_{i - 1, i \ne \frac{N}{2}}^N {\left( {1 - {{\rm{Q}}_1}\left( {\sqrt {2r} a(i), \sqrt 2 t} \right)} \right)} } \cdot t \cdot \\ {\kern 1pt} {{\rm{I}}_0}\left( {2\sqrt r \cdot t \cdot a\left( {\frac{N}{2}} \right)} \right){\rm{exp}}\left( { - \left( {{t^2} + r \cdot a\left( {\frac{N}{2}} \right)} \right)} \right){\rm{d}}t \end{array} $ | (33) |
$ \begin{array}{*{20}{l}} {{P_{{\rm{sc1}}}} = 2 \cdot \int_0^\infty {\prod\limits_{i = 1}^{N - 1} {(1 - {{\rm{Q}}_1}(} } \sqrt {2r} a(i), \sqrt 2 t)) \cdot t}\\ {{{\rm{I}}_0}(2\sqrt r \cdot t \cdot a(N)){\rm{exp}}( - ({t^2} + r \cdot a(N))){\rm{d}}t} \end{array} $ | (34) |
$ {P_{{\rm{cse\_2FSK}}}} = 1 - {P_{{\rm{sc0}}}} - {P_{{\rm{sc1}}}} $ | (35) |
由驱动比特误符号率Pcse_2FSK推导驱动比特误码率Pcbe_2FSK、常规比特误码率Pobe_2FSK及总误码率Pe_2FSK的过程与式(25)~(29)类似。
3 仿真结果 3.1 传输效率对比在传输相同比特数、码元速率相同、调制阶数相同的条件下,消息驱动跳频系统与普通跳频系统传输效率对比如图 4所示。高度越短对应的时间越短,传输效率越高。
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根据仿真结果可知:
1) 在相同调制阶数的条件下,不论是OG-2FSK映射还是BPSK映射的消息驱动跳频系统的传输效率都要高于普通跳频系统。而对于消息驱动跳频系统,BPSK映射的传输效率略高于OG-2FSK映射。
2) 消息驱动跳频系统的传输效率受子信道个数及每跳传输符号个数影响,子信道个数越高传输效率越高,每跳传输符号越多传输效率越低,而普通跳频系统的传输效率与子信道个数及每跳传输符号个数均无关。
3.2 AWGN信道下误码率性能仿真在AWGN信道下,OG-2FSK映射及BPSK映射的消息驱动误码性能基本一致,这里仅给出OG-2FSK映射的消息驱动跳频系统的误码率仿真结果。
仿真条件:码元速率1 Mbit/s,AWGN信道,子信道个数分别为8、16、32。每跳发送符号数为1。仿真结果如图 5~8所示。
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图 5、6中理论曲线分别由式(25)、式(27)计算得出,由仿真结果可知仿真曲线与理论曲线基本重合,证明推导的正确性。
图 7、8中理论曲线分别由式(28)、式(29)计算得出,由仿真结果可知仿真曲线与理论曲线基本重合,证明推导的正确性。并且随着子信道数的增大,误码性能随之下降。
3.3 多普勒频移下误码率性能仿真 3.3.1 OG-2FSK映射仿真条件:码元速率1 Mbps,AWGN信道,子信道个数分别为16、32。每跳发送符号数Nnph分别为1,映射方式选择OG-2FSK映射。归一化多普勒频偏为0.05、0.10、0.15。
图中理论曲线由式(35)及推导出的公式绘制得出,仿真结果验证了OG-2FSK映射的消息驱动跳频系统在多普勒频偏下的理论推导的正确性。随着归一化多普勒频偏及子信道数的增大,误码性能随之下降。
3.3.2 OG-2FSK映射与BPSK映射对比仿真条件:码元速率1 Mbit/s,AWGN信道,子信道个数分别为16、32。每跳发送符号数Nnph分别为1,映射方式选择OG-2FSK映射及BPSK映射。归一化多普勒频偏为0.05、0.10、0.15。
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根据仿真结果,在多普勒频移影响下OG-2FSK映射的误码性能要优于BPSK映射的误码性能。这是因为BPSK映射中,部分信息是调制在波形的相位上,部分信息调制到频点上。在时间选择性衰落信道中,调制在相位上的信息受多普勒频移影响较大,误码性能损失严重,调制在频点上的信息所受影响相对较小。因此,OG-2FSK映射的消息驱动跳频系统比BPSK映射要更适合在时间选择性衰落信道中应用。
4 结论1) 本文提出了一种基于IFFT/FFT架构的消息驱动跳频系统,解决了传统消息驱动跳频系统收端实现复杂度高的问题,且易于与现有OFDM体制通信系统结合。
2) 由于消息驱动跳频系统易于受衰落信道影响,本文基于该系统架构提出了一种新型映射方式OG-2FSK。
3) 对OG-2FSK误码性能进行了理论推导和分析,得到OG-2FSK在AWGN信道和时间选择性衰落信道下的误码率闭合表达式,根据结果可知系统误性能主要受信噪比、子信道个数、多普勒频移等因素的影响。
4) 最后对所设计的系统进行了仿真,误码率仿真结果与理论推导相符。同时仿真结果表明,在时间选择性衰落信道下,所提出的OG-2FSK映射误码性能要优于BPSK映射。
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