网站建设托管合同,iis7 添加php网站,网站项目建设的必要性,财务软件哪个好说明#xff1a;接上一节循环自相关函数和谱相关密度#xff08;一#xff09;——公式推导
7 BPSK信号谱相关密度函数
7.1 实信号模型
BPSK实信号表达式可以写为 r(t)y(t)n(t)r(t) y(t) n(t)r(t)y(t)n(t) s(t)p(t)n(t) s(t)p(t) n(t)s(t)p(t)n(t) ∑n−∞∞a(nT)q(t−…说明接上一节循环自相关函数和谱相关密度一——公式推导
7 BPSK信号谱相关密度函数
7.1 实信号模型
BPSK实信号表达式可以写为
r(t)y(t)n(t)r(t) y(t) n(t)r(t)y(t)n(t)
s(t)p(t)n(t) s(t)p(t) n(t)s(t)p(t)n(t)
∑n−∞∞a(nT)q(t−nT−t0)cos(2πf0tθ) n(t) \sum\limits_{n - \infty }^\infty {a(nT)q(t - nT - {t_0})} \cos (2\pi {f_0}t \theta ){\text{ }}n(t)n−∞∑∞a(nT)q(t−nT−t0)cos(2πf0tθ) n(t)(22)
其中t0{t_0}t0为起始时间TTT为符号速率a(n)a(n)a(n)为基带符号序列f0{f_0}f0为载波频率θ\thetaθ为初始相位n(t)n(t)n(t)为高斯白噪声q(t)q(t)q(t)为矩形脉冲其表达式和傅里叶变换为
q(t){1,∣t∣≤T/20,∣t∣T/2q(t)\left\{\begin{array}{ll}1, |t| \leq T / 2 \\ 0, |t|T / 2\end{array}\right.q(t){1,0,∣t∣≤T/2∣t∣T/2 (23)
Q(f)TSa(πfT)Q(f) T\operatorname{Sa} (\pi fT)Q(f)TSa(πfT) (24)
且
s(t)∑n−∞∞a(nT)q(t−nT−t0)s(t) \sum\limits_{n - \infty }^\infty {a(nT)q(t - nT - {t_0})}s(t)n−∞∑∞a(nT)q(t−nT−t0)
q(t−t0)⊗∑na(t)δ(t−nT) q(t - {t_0}) \otimes \sum\limits_n {a(t)\delta (t - nT)}q(t−t0)⊗n∑a(t)δ(t−nT)
q(t−t0)⊗a^(t) q(t - {t_0}) \otimes \hat a(t)q(t−t0)⊗a^(t) (25)
p(t)cos(2πf0tθ)p(t) \cos (2\pi {f_0}t \theta )p(t)cos(2πf0tθ) (26)
由(21)知基带脉冲序列a(nT)a(nT)a(nT)的谱相关密度函数为
S~aα(f)1T∑n−∞∞∑m−∞∞Saα m/T(f−m2T−nT)\tilde S_a^\alpha (f) \frac{1}{T}\sum\limits_{n - \infty }^\infty {\sum\limits_{m - \infty }^\infty {S_a^{\alpha {\text{ }}m/T}(f - \frac{m}{{2T}} - \frac{n}{T})} }S~aα(f)T1n−∞∑∞m−∞∑∞Saα m/T(f−2Tm−Tn) (27)
由(19)可知a(t)a(t)a(t)以周期TTT理想抽样后的谱相关密度函数为
Sa^α(f)1T2∑n−∞∞∑m−∞∞Sa^α m/T(f−m2T−nT)S_{\hat a}^\alpha (f) \frac{1}{{{T^2}}}\sum\limits_{n - \infty }^\infty {\sum\limits_{m - \infty }^\infty {S_{\hat a}^{\alpha {\text{ }}m/T}(f - \frac{m}{{2T}} - \frac{n}{T})} }Sa^α(f)T21n−∞∑∞m−∞∑∞Sa^α m/T(f−2Tm−Tn) (28)
根据傅里叶变换的时移性质q(t−t0)q(t - {t_0})q(t−t0)的傅里叶变换为Q(f)e−j2πft0Q(f){e^{ - j2\pi f{t_0}}}Q(f)e−j2πft0则(5)由可得s(t)s(t)s(t)的谱相关密度函数为
Ssα(f)1TQ(fα/2)Q∗(f−α/2)e−j2παt0S~aα(f)S_s^\alpha (f) \frac{1}{T}Q(f \alpha /2){Q^*}(f - \alpha /2){e^{ - j2\pi \alpha {t_0}}}\tilde S_a^\alpha (f)Ssα(f)T1Q(fα/2)Q∗(f−α/2)e−j2παt0S~aα(f) (29)
考虑p(t)p(t)p(t)的二次变换
vτ(t)p(t τ/2)p∗(t−τ/2){v_\tau }(t) p(t{\text{ }}\tau /2){p^*}(t - \tau /2)vτ(t)p(t τ/2)p∗(t−τ/2)
14(ej2πf0τe−j2πf0τej(4πf0t2θ)e−j(4πf0t2θ)) \frac{1}{4}({e^{j2\pi {f_0}\tau }} {e^{ - j2\pi {f_0}\tau }} {e^{j(4\pi {f_0}t 2\theta )}} {e^{ - j(4\pi {f_0}t 2\theta )}})41(ej2πf0τe−j2πf0τej(4πf0t2θ)e−j(4πf0t2θ)) (30)
其Fourier级数系数为
⟨vτ(t)e−j2παt⟩t14ej2πf0τ⟨e−j2παt⟩t14e−j2πf0τ⟨e−j2παt⟩t{\left\langle {{v_\tau }(t){e^{ - j2\pi \alpha t}}} \right\rangle _t} \frac{1}{4}{e^{j2\pi {f_0}\tau }}{\left\langle {{e^{ - j2\pi \alpha t}}} \right\rangle _t} \frac{1}{4}{e^{ - j2\pi {f_0}\tau }}{\left\langle {{e^{ - j2\pi \alpha t}}} \right\rangle _t}⟨vτ(t)e−j2παt⟩t41ej2πf0τ⟨e−j2παt⟩t41e−j2πf0τ⟨e−j2παt⟩t
14e−j2θ⟨e−j2π(α2f0)t⟩t14ej2θ⟨e−j2π(α−2f0)t⟩t \frac{1}{4}{e^{ - j2\theta }}{\left\langle {{e^{ - j2\pi (\alpha 2{f_0})t}}} \right\rangle _t} \frac{1}{4}{e^{j2\theta }}{\left\langle {{e^{ - j2\pi (\alpha - 2{f_0})t}}} \right\rangle _t}41e−j2θ⟨e−j2π(α2f0)t⟩t41ej2θ⟨e−j2π(α−2f0)t⟩t(31)
则p(t)p(t)p(t)的循环自相关函数和谱相关密度函数为
Rpα(τ){14e±j2θα±2f012cos(2πf0τ)α00otherwise R_{p}^{\alpha}(\tau)\left\{\begin{array}{cc}\frac{1}{4} e^{\pm j 2 \theta} \alpha\pm 2 f_{0} \\ \frac{1}{2} \cos \left(2 \pi f_{0} \tau\right) \alpha0 \\ 0 \text { otherwise }\end{array}\right.Rpα(τ)⎩⎨⎧41e±j2θ21cos(2πf0τ)0α±2f0α0 otherwise (32)
Spα(f){14e±j2θδ(f)α±2f014[δ(ff0)δ(f−f0)]α00otherwise S_{p}^{\alpha}(f)\left\{\begin{array}{cc}\frac{1}{4} e^{\pm j 2 \theta} \delta(f) \alpha\pm 2 f_{0} \\ \frac{1}{4}\left[\delta\left(ff_{0}\right)\delta\left(f-f_{0}\right)\right] \alpha0 \\ 0 \text { otherwise }\end{array}\right.Spα(f)⎩⎨⎧41e±j2θδ(f)41[δ(ff0)δ(f−f0)]0α±2f0α0 otherwise (33)
由(12)、(13)得y(t)y(t)y(t)的循环自相关函数为
Ryα(τ)∑βRpβ(τ)Rsα−β(τ)R_y^\alpha (\tau ) \sum\limits_\beta {R_p^\beta (\tau )R_s^{\alpha - \beta }(\tau )}Ryα(τ)β∑Rpβ(τ)Rsα−β(τ)
14ej2θRsα−2f0(τ)14e−j2θRsα2f0(τ)12cos(2πf0τ)Rsα(τ) \frac{1}{4}{e^{j2\theta }}R_s^{\alpha - 2{f_0}}(\tau ) \frac{1}{4}{e^{ - j2\theta }}R_s^{\alpha 2{f_0}}(\tau ) \frac{1}{2}\cos (2\pi {f_0}\tau )R_s^\alpha (\tau )41ej2θRsα−2f0(τ)41e−j2θRsα2f0(τ)21cos(2πf0τ)Rsα(τ) (34)
Syα(f)∑βSpβ(f)⊗Ssα−β(f)S_y^\alpha (f) \sum\limits_\beta {S_p^\beta (f) \otimes S_s^{\alpha - \beta }(f)}Syα(f)β∑Spβ(f)⊗Ssα−β(f)
14[Ssα(ff0)Ssα(f−f0)ej2θSsα−2f0(f)e−j2θSsα2f0(f)] \frac{1}{4}\left[ {S_s^\alpha (f {f_0}) S_s^\alpha (f - {f_0}) {e^{j2\theta }}S_s^{\alpha - 2{f_0}}(f) {e^{ - j2\theta }}S_s^{\alpha 2{f_0}}(f)} \right]41[Ssα(ff0)Ssα(f−f0)ej2θSsα−2f0(f)e−j2θSsα2f0(f)] (35)
将(29)代入(35)得y(t)y(t)y(t)的谱相关密度函数为
Syα(f)14T{[Q(ff0α/2)Q∗(ff0−α/2)S~aα(ff0)S_y^\alpha (f) \frac{1}{{4T}}\{ [Q(f {f_0} \alpha /2){Q^*}(f {f_0} - \alpha /2)\tilde S_a^\alpha (f {f_0})Syα(f)4T1{[Q(ff0α/2)Q∗(ff0−α/2)S~aα(ff0)
Q(f−f0α/2)Q∗(f−f0−α/2)S~aα(f−f0)]e−j2παt0Q(f - {f_0} \alpha /2){Q^*}(f - {f_0} - \alpha /2)\tilde S_a^\alpha (f - {f_0})]{e^{ - j2\pi \alpha {t_0}}}Q(f−f0α/2)Q∗(f−f0−α/2)S~aα(f−f0)]e−j2παt0
Q(ff0α/2)Q∗(f−f0−α/2)S~aα2f0(f)e−j[2π(α2f0)t02θ]Q(f {f_0} \alpha /2){Q^*}(f - {f_0} - \alpha /2)\tilde S_a^{\alpha 2{f_0}}(f){e^{ - j[2\pi (\alpha 2{f_0}){t_0} 2\theta ]}}Q(ff0α/2)Q∗(f−f0−α/2)S~aα2f0(f)e−j[2π(α2f0)t02θ]
Q(f−f0α/2)Q∗(ff0−α/2)S~aα−2f0(f)e−j[2π(α−2f0)t0−2θ]}Q(f - {f_0} \alpha /2){Q^*}(f {f_0} - \alpha /2)\tilde S_a^{\alpha - 2{f_0}}(f){e^{ - j[2\pi (\alpha - 2{f_0}){t_0} - 2\theta ]}}\}Q(f−f0α/2)Q∗(ff0−α/2)S~aα−2f0(f)e−j[2π(α−2f0)t0−2θ]} (36)
对于01先验等概的基带符号序列a(n)a(n)a(n)其循环自相关函数为
R~aα(kT)limN→∞12N1∑n−NNa(nTkT)a(nT)e−j2πα(nk/2)T\tilde R_a^\alpha (kT) \mathop {\lim }\limits_{N \to \infty } \frac{1}{{2N 1}}\sum\limits_{n - N}^N {a(nT kT)a(nT)} {e^{ - j2\pi \alpha (n k/2)T}}R~aα(kT)N→∞lim2N11n−N∑Na(nTkT)a(nT)e−j2πα(nk/2)T (37)
当且仅当k0k 0k0且αm/T\alpha m/Tαm/T时R~aα(kT)R~a(0)\tilde R_a^\alpha (kT) {\tilde R_a}(0)R~aα(kT)R~a(0)则其谱相关密度函数为
S~aα(f){R~a(0)1,αm/T0,α≠m/T\tilde{S}_{a}^{\alpha}(f)\left\{\begin{aligned} \tilde{R}_{a}(0)1, \alpham / T \\ 0, \alpha \neq m / T \end{aligned}\right.S~aα(f){R~a(0)1,0,αm/Tαm/T(38)
对于高斯白噪声n(t)n(t)n(t)当且仅当α0\alpha 0α0时其谱相关密度函数不为零则BPSK实信号的谱相关密度函数为
Srα(f){Syα(f)Snα(f),α0Syα(f),α≠0S_{r}^{\alpha}(f)\left\{\begin{array}{cc}S_{y}^{\alpha}(f)S_{n}^{\alpha}(f), \alpha0 \\ S_{y}^{\alpha}(f), \alpha \neq 0\end{array}\right.Srα(f){Syα(f)Snα(f),Syα(f),α0α0(39)
由此可见噪声n(t)n(t)n(t)只影响循环频率为零时的截面。
7.2 复信号模型
BPSK复信号表达式可以写为
r(t)y(t)n(t)r(t) y(t) n(t)r(t)y(t)n(t) s(t)p(t)n(t){\text{ }}s(t)p(t) n(t) s(t)p(t)n(t)
∑n−∞∞a(nT)q(t−nT−t0)ej(2πf0tθ) \sum\limits_{n - \infty }^\infty {a(nT)q(t - nT - {t_0})} {e^{j(2\pi {f_0}t \theta )}}n−∞∑∞a(nT)q(t−nT−t0)ej(2πf0tθ) (40)
同理t0{t_0}t0为起始时间TTT为符号速率a(n)a(n)a(n)为基带符号序列f0{f_0}f0为载波频率θ\thetaθ为初始相位n(t)n(t)n(t)为高斯白噪声q(t)q(t)q(t)为矩形脉冲。令
p(t)ej(2πf0tθ)p(t) {e^{j(2\pi {f_0}t \theta )}}p(t)ej(2πf0tθ) (41)
同实数信号模型对比只有p(t)p(t)p(t)发生了改变其二次变换的其傅里叶级数系数为
⟨vτ(t)e−j2παt⟩t⟨p(t τ/2)p∗(t−τ/2)e−j2παt⟩t{\left\langle {{v_\tau }(t){e^{ - j2\pi \alpha t}}} \right\rangle _t} {\left\langle {p(t{\text{ }}\tau /2){p^*}(t - \tau /2){e^{ - j2\pi \alpha t}}} \right\rangle _t}⟨vτ(t)e−j2παt⟩t⟨p(t τ/2)p∗(t−τ/2)e−j2παt⟩t
ej2πf0τ⟨e−j2παt⟩t {e^{j2\pi {f_0}\tau }}{\left\langle {{e^{ - j2\pi \alpha t}}} \right\rangle _t}ej2πf0τ⟨e−j2παt⟩t (42)
则p(t)p(t)p(t)的循环自相关函数和谱相关密度函数为
Rpα(τ){ej2πf0τα00α≠0R_{p}^{\alpha}(\tau)\left\{\begin{array}{cc}e^{j 2 \pi f_{0} \tau} \alpha0 \\ 0 \alpha \neq 0\end{array}\right.Rpα(τ){ej2πf0τ0α0α0(43)
Spα(f){δ(f−f0)α00α≠0S_{p}^{\alpha}(f)\left\{\begin{array}{cc}\delta\left(f-f_{0}\right) \alpha0 \\ 0 \alpha \neq 0\end{array}\right.Spα(f){δ(f−f0)0α0α0(44)
由(12)、(13)得y(t)y(t)y(t)的循环自相关函数为
Ryα(τ)∑βRpβ(τ)Rsα−β(τ)ej2πf0τRsα(τ)R_y^\alpha (\tau ) \sum\limits_\beta {R_p^\beta (\tau )R_s^{\alpha - \beta }(\tau )} {e^{j2\pi {f_0}\tau }}R_s^\alpha (\tau )Ryα(τ)β∑Rpβ(τ)Rsα−β(τ)ej2πf0τRsα(τ) (45)
Syα(f)∑βSpβ(f)⊗Ssα−β(f)S_y^\alpha (f) \sum\limits_\beta {S_p^\beta (f) \otimes S_s^{\alpha - \beta }(f)}Syα(f)β∑Spβ(f)⊗Ssα−β(f)
δ(f−f0)⊗Ssα(f) \delta (f - {f_0}) \otimes S_s^\alpha (f)δ(f−f0)⊗Ssα(f)
Ssα(f−f0) S_s^\alpha (f - {f_0})Ssα(f−f0) (46)
将(29)代入(46)得y(t)y(t)y(t)的谱相关密度函数为
Syα(f)1T[Q(f−f0α/2)Q∗(f−f0−α/2)e−j2παt0S~aα(f−f0)]S_y^\alpha (f) \frac{1}{T}[Q(f - {f_0} \alpha /2){Q^*}(f - {f_0} - \alpha /2){e^{ - j2\pi \alpha {t_0}}}\tilde S_a^\alpha (f - {f_0})]Syα(f)T1[Q(f−f0α/2)Q∗(f−f0−α/2)e−j2παt0S~aα(f−f0)] (47)
同(39)可得复BPSK信号的谱相关密度函数为
Srα(f){Syα(f)Snα(f),α0Syα(f),α≠0S_{r}^{\alpha}(f)\left\{\begin{array}{cc}S_{y}^{\alpha}(f)S_{n}^{\alpha}(f), \alpha0 \\ S_{y}^{\alpha}(f), \alpha \neq 0\end{array}\right.Srα(f){Syα(f)Snα(f),Syα(f),α0α0(48)
7.3 谱分析
7.3.1 主峰个数
对于实BPSK信号由(36)、(38)可知其谱相关密度函数在f0f 0f0且α±2f0\alpha \pm 2{f_0}α±2f0处各有一个主峰在α0\alpha 0α0且f±f0f \pm {f_0}f±f0处各有一个主峰即实BPSK信号共有4个主峰。
对于复BPSK信号由(47)、(48)可知其谱相关密度函数只有在ff0f {f_0}ff0且α0\alpha 0α0处有一个谱峰。
7.3.2 切面特征
在式(36)中令f0f 0f0α±2f0m/T\alpha \pm 2{f_0} m/Tα±2f0m/T得
Syα(f){14T∣Q(−f0α/2)∣2e−j[2πnt0/T−2θ]α2f0m/T14T∣Q(f0α/2)∣2e−j[2πnt0/T2θ]α−2f0m/TS_{y}^{\alpha}(f)\left\{\begin{array}{ll}\frac{1}{4 T}\left|Q\left(-f_{0}\alpha / 2\right)\right|^{2} e^{-j\left[2 \pi n t_{0} / T-2 \theta\right]} \alpha2 f_{0}m / T \\ \frac{1}{4 T}\left|Q\left(f_{0}\alpha / 2\right)\right|^{2} e^{-j\left[2 \pi n t_{0} / T2 \theta\right]} \alpha-2 f_{0}m / T\end{array}\right.Syα(f){4T1∣Q(−f0α/2)∣2e−j[2πnt0/T−2θ]4T1∣Q(f0α/2)∣2e−j[2πnt0/T2θ]α2f0m/Tα−2f0m/T(49)
特别地当m0m 0m0时有
Syα(f){14T∣Q(0)∣2ej2θα2f014T∣Q(0)∣2e−j2θα−2f0S_{y}^{\alpha}(f)\left\{\begin{array}{ll}\frac{1}{4 T}|Q(0)|^{2} e^{j 2 \theta} \alpha2 f_{0} \\ \frac{1}{4 T}|Q(0)|^{2} e^{-j 2 \theta} \alpha-2 f_{0}\end{array}\right.Syα(f){4T1∣Q(0)∣2ej2θ4T1∣Q(0)∣2e−j2θα2f0α−2f0(50)
即在f0f 0f0切面其谱相关密度函数幅度最大值出现在循环频率为α±2f0\alpha \pm 2{f_0}α±2f0处由此可估计实BPSK信号的载波频率在其左右偏移符号速率处出现次峰值可估计其符号速率且可根据α±2f0\alpha \pm 2{f_0}α±2f0处对应的谱相关密度函数的相位来估计初相θ\thetaθ。
令f±f0f \pm {f_0}f±f0αm/T\alpha m/Tαm/T得
Syα(f)14T{[Q(2f0α/2)Q∗(2f0−α/2)∣Q(α/2)∣2]e−j2παt0S_y^\alpha (f) \frac{1}{{4T}}\{ [Q(2{f_0} \alpha /2){Q^*}(2{f_0} - \alpha /2) |Q(\alpha /2){|^2}]{e^{ - j2\pi \alpha {t_0}}}Syα(f)4T1{[Q(2f0α/2)Q∗(2f0−α/2)∣Q(α/2)∣2]e−j2παt0 (51)
即在f±f0f \pm {f_0}f±f0切面其谱相关密度函数幅度在循环频率为αm/T\alpha m/Tαm/T即符号速率整数倍处出现峰值在α0\alpha 0α0处的峰值最大由此可估计实BPSK信号的符号速率此外还可根据符号速率处对应的谱相关密度函数的相位来估计时延t0{t_0}t0其中需要注意的是当频率分辨率远远小于循环频率分辨率即Δf≫Δα\Delta f \gg \Delta \alphaΔf≫Δα时符号速率处对应的峰值才比较明显。
对于复BPSK信号在式(47)中令α0\alpha 0α0得
Syα(f)1T∣Q(f−f0)|2S_y^\alpha (f) \frac{1}{T}|Q(f - {f_0}){{\text{|}}^2}Syα(f)T1∣Q(f−f0)|2 (52)
即在α0\alpha 0α0切面其谱相关密度函数幅度只在ff0f {f_0}ff0出现峰值由此可估计复BPSK信号的载波频率但此时噪声n(t)n(t)n(t)的谱相关密度函数不为零因此利用该切面进行载频估计受噪声影响较大。
令ff0f {f_0}ff0得
Syα(f)1T∣Q(α/2)∣2e−j2παt0S~aα(0)S_y^\alpha (f) \frac{1}{T}|Q(\alpha /2){|^2}{e^{ - j2\pi \alpha {t_0}}}\tilde S_a^\alpha (0)Syα(f)T1∣Q(α/2)∣2e−j2παt0S~aα(0) (53)
即在ff0f {f_0}ff0切面其谱相关密度函数幅度在循环频率为αm/T\alpha m/Tαm/T即符号速率整数倍处出现峰值在α0\alpha 0α0处的峰值最大由此可估计实BPSK信号的符号速率此外还可根据符号速率处对应的谱相关密度函数的相位来估计时延t0{t_0}t0。
7.4 成形滤波器对谱相关密度函数的影响
无论是BPSK还是QPSK调制信号对于矩形成形其频谱为Sa函数当∣f∣1/T\left| f \right| 1/T∣f∣1/T时存在衰减较慢的旁瓣因此在循环频率为αm/T\alpha m/Tαm/T或αm/T±2f0\alpha m/T \pm 2{f_0}αm/T±2f0处其谱相关密度函数仍然不为零即在主峰周围会有很多小峰。对于根升余弦成形当∣f∣1/T\left| f \right| 1/T∣f∣1/T时其频谱较快衰减为零因此其谱相关密度函数只在循环频率为α1/T\alpha 1/Tα1/T或α1/T±2f0\alpha 1/T \pm 2{f_0}α1/T±2f0处有值。
