【循环自相关和循环谱系列2】实信号、复信号模型下的BPSK信号循环谱推导

本文详细介绍了BPSK信号在实信号和复信号模型下的谱相关密度函数推导,包括主峰个数和切面特征。实信号模型下,信号在f=0和f=±f0处各有2个主峰,而复信号模型下只有一个主峰在f=f0。切面分析可用于估计载波频率、符号速率和初始相位。成形滤波器会影响谱相关密度函数,矩形成形会有较多小峰,而升余弦成形则减少旁瓣,使得谱更集中。

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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 ( n T ) q ( t − n T − t 0 ) cos ⁡ ( 2 π f 0 t + θ )  +  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(tnTt0)cos(2πf0t+θ) + n(t)(22)

其中, t 0 {t_0} t0为起始时间, T T T为符号速率, a ( n ) a(n) a(n)为基带符号序列, f 0 {f_0} f0为载波频率, θ \theta θ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 / 2 0 , ∣ t ∣ > T / 2 q(t)=\left\{\begin{array}{ll}1, & |t| \leq T / 2 \\ 0, & |t|>T / 2\end{array}\right. q(t)={1,0,tT/2t>T/2 (23)

Q ( f ) = T Sa ⁡ ( π f T ) Q(f) = T\operatorname{Sa} (\pi fT) Q(f)=TSa(πfT) (24)

s ( t ) = ∑ n = − ∞ ∞ a ( n T ) q ( t − n T − t 0 ) s(t) = \sum\limits_{n = - \infty }^\infty {a(nT)q(t - nT - {t_0})} s(t)=n=a(nT)q(tnTt0)

= q ( t − t 0 ) ⊗ ∑ n a ( t ) δ ( t − n T ) = q(t - {t_0}) \otimes \sum\limits_n {a(t)\delta (t - nT)} =q(tt0)na(t)δ(tnT)

= q ( t − t 0 ) ⊗ a ^ ( t ) = q(t - {t_0}) \otimes \hat a(t) =q(tt0)a^(t) (25)

p ( t ) = cos ⁡ ( 2 π f 0 t + θ ) p(t) = \cos (2\pi {f_0}t + \theta ) p(t)=cos(2πf0t+θ) (26)

由(21)知,基带脉冲序列 a ( n T ) a(nT) a(nT)的谱相关密度函数为

S ~ a α ( f ) = 1 T ∑ n = − ∞ ∞ ∑ m = − ∞ ∞ S a α  +  m / T ( f − m 2 T − n T ) \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(f2TmTn) (27)

由(19)可知, a ( t ) a(t) a(t)以周期 T T T理想抽样后的谱相关密度函数为

S a ^ α ( f ) = 1 T 2 ∑ n = − ∞ ∞ ∑ m = − ∞ ∞ S a ^ α  +  m / T ( f − m 2 T − n T ) 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(f2TmTn) (28)

根据傅里叶变换的时移性质, q ( t − t 0 ) q(t - {t_0}) q(tt0)的傅里叶变换为 Q ( f ) e − j 2 π f t 0 Q(f){e^{ - j2\pi f{t_0}}} Q(f)ej2πft0,则(5)由可得 s ( t ) s(t) s(t)的谱相关密度函数为

S s α ( f ) = 1 T Q ( f + α / 2 ) Q ∗ ( f − α / 2 ) e − j 2 π α t 0 S ~ a α ( 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(f+α/2)Q(fα/2)ej2παt0S~aα(f) (29)

考虑 p ( t ) p(t) p(t)的二次变换

v τ ( 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(tτ/2)

= 1 4 ( e j 2 π f 0 τ + e − j 2 π f 0 τ + e j ( 4 π f 0 t + 2 θ ) + e − j ( 4 π f 0 t + 2 θ ) ) = \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(ej2πf0τ+ej2πf0τ+ej(4πf0t+2θ)+ej(4πf0t+2θ)) (30)

其Fourier级数系数为

⟨ v τ ( t ) e − j 2 π α t ⟩ t = 1 4 e j 2 π f 0 τ ⟨ e − j 2 π α t ⟩ t + 1 4 e − j 2 π f 0 τ ⟨ e − j 2 π α t ⟩ 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)ej2παtt=41ej2πf0τej2παtt+41ej2πf0τej2παtt

+ 1 4 e − j 2 θ ⟨ e − j 2 π ( α + 2 f 0 ) t ⟩ t + 1 4 e j 2 θ ⟨ e − j 2 π ( α − 2 f 0 ) 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} +41ej2θej2π(α+2f0)tt+41ej2θej2π(α2f0)tt(31)

p ( t ) p(t) p(t)的循环自相关函数和谱相关密度函数为

R p α ( τ ) = { 1 4 e ± j 2 θ α = ± 2 f 0 1 2 cos ⁡ ( 2 π f 0 τ ) α = 0 0  otherwise  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) & \alpha=0 \\ 0 & \text { otherwise }\end{array}\right. Rpα(τ)= 41e±j2θ21cos(2πf0τ)0α=±2f0α=0 otherwise (32)

S p α ( f ) = { 1 4 e ± j 2 θ δ ( f ) α = ± 2 f 0 1 4 [ δ ( f + f 0 ) + δ ( f − f 0 ) ] α = 0 0  otherwise  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(f+f_{0}\right)+\delta\left(f-f_{0}\right)\right] & \alpha=0 \\ 0 & \text { otherwise }\end{array}\right. Spα(f)= 41e±j2θδ(f)41[δ(f+f0)+δ(ff0)]0α=±2f0α=0 otherwise (33)

由(12)、(13)得 y ( t ) y(t) y(t)的循环自相关函数为

R y α ( τ ) = ∑ β R p β ( τ ) R s α − β ( τ ) R_y^\alpha (\tau ) = \sum\limits_\beta {R_p^\beta (\tau )R_s^{\alpha - \beta }(\tau )} Ryα(τ)=βRpβ(τ)Rsαβ(τ)

= 1 4 e j 2 θ R s α − 2 f 0 ( τ ) + 1 4 e − j 2 θ R s α + 2 f 0 ( τ ) + 1 2 cos ⁡ ( 2 π f 0 τ ) R s α ( 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 ) =41ej2θRsα2f0(τ)+41ej2θRsα+2f0(τ)+21cos(2πf0τ)Rsα(τ) (34)

S y α ( f ) = ∑ β S p β ( f ) ⊗ S s α − β ( f ) S_y^\alpha (f) = \sum\limits_\beta {S_p^\beta (f) \otimes S_s^{\alpha - \beta }(f)} Syα(f)=βSpβ(f)Ssαβ(f)

= 1 4 [ S s α ( f + f 0 ) + S s α ( f − f 0 ) + e j 2 θ S s α − 2 f 0 ( f ) + e − j 2 θ S s α + 2 f 0 ( 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(f+f0)+Ssα(ff0)+ej2θSsα2f0(f)+ej2θSsα+2f0(f)] (35)

将(29)代入(35),得 y ( t ) y(t) y(t)的谱相关密度函数为

S y α ( f ) = 1 4 T { [ Q ( f + f 0 + α / 2 ) Q ∗ ( f + f 0 − α / 2 ) S ~ a α ( f + f 0 ) 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(f+f0+α/2)Q(f+f0α/2)S~aα(f+f0)

Q ( f − f 0 + α / 2 ) Q ∗ ( f − f 0 − α / 2 ) S ~ a α ( f − f 0 ) ] e − j 2 π α t 0 Q(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(ff0+α/2)Q(ff0α/2)S~aα(ff0)]ej2παt0

Q ( f + f 0 + α / 2 ) Q ∗ ( f − f 0 − α / 2 ) S ~ a α + 2 f 0 ( f ) e − j [ 2 π ( α + 2 f 0 ) t 0 + 2 θ ] 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+f0+α/2)Q(ff0α/2)S~aα+2f0(f)ej[2π(α+2f0)t0+2θ]

Q ( f − f 0 + α / 2 ) Q ∗ ( f + f 0 − α / 2 ) S ~ a α − 2 f 0 ( f ) e − j [ 2 π ( α − 2 f 0 ) t 0 − 2 θ ] } 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(ff0+α/2)Q(f+f0α/2)S~aα2f0(f)ej[2π(α2f0)t02θ]} (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 ~ a α ( k T ) = lim ⁡ N → ∞ 1 2 N + 1 ∑ n = − N N a ( n T + k T ) a ( n T ) e − j 2 π α ( n + k / 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)=Nlim2N+11n=NNa(nT+kT)a(nT)ej2πα(n+k/2)T (37)

当且仅当 k = 0 k = 0 k=0 α = m / T \alpha = m/T α=m/T时, R ~ a α ( k T ) = 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 ~ a α ( f ) = { R ~ a ( 0 ) = 1 , α = m / T 0 , α ≠ m / T \tilde{S}_{a}^{\alpha}(f)=\left\{\begin{aligned} \tilde{R}_{a}(0)=1, & \alpha=m / T \\ 0, & \alpha \neq m / T \end{aligned}\right. S~aα(f)={R~a(0)=1,0,α=m/Tα=m/T(38)

对于高斯白噪声 n ( t ) n(t) n(t),当且仅当 α = 0 \alpha = 0 α=0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

S r α ( f ) = { S y α ( f ) + S n α ( f ) , α = 0 S y α ( f ) , α ≠ 0 S_{r}^{\alpha}(f)=\left\{\begin{array}{cc}S_{y}^{\alpha}(f)+S_{n}^{\alpha}(f), & \alpha=0 \\ S_{y}^{\alpha}(f), & \alpha \neq 0\end{array}\right. Srα(f)={Syα(f)+Snα(f),Syα(f),α=0α=0(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 ( n T ) q ( t − n T − t 0 ) e j ( 2 π f 0 t + θ ) = \sum\limits_{n = - \infty }^\infty {a(nT)q(t - nT - {t_0})} {e^{j(2\pi {f_0}t + \theta )}} =n=a(nT)q(tnTt0)ej(2πf0t+θ) (40)

同理, t 0 {t_0} t0为起始时间, T T T为符号速率, a ( n ) a(n) a(n)为基带符号序列, f 0 {f_0} f0为载波频率, θ \theta θ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 ) = e j ( 2 π f 0 t + θ ) p(t) = {e^{j(2\pi {f_0}t + \theta )}} p(t)=ej(2πf0t+θ) (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

⟨ v τ ( t ) e − j 2 π α t ⟩ t = ⟨ p ( t  +  τ / 2 ) p ∗ ( t − τ / 2 ) e − j 2 π α t ⟩ 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)ej2παtt=p(t + τ/2)p(tτ/2)ej2παtt

= e j 2 π f 0 τ ⟨ e − j 2 π α t ⟩ t = {e^{j2\pi {f_0}\tau }}{\left\langle {{e^{ - j2\pi \alpha t}}} \right\rangle _t} =ej2πf0τej2παtt (42)

p ( t ) p(t) p(t)的循环自相关函数和谱相关密度函数为

R p α ( τ ) = { e j 2 π f 0 τ α = 0 0 α ≠ 0 R_{p}^{\alpha}(\tau)=\left\{\begin{array}{cc}e^{j 2 \pi f_{0} \tau} & \alpha=0 \\ 0 & \alpha \neq 0\end{array}\right. Rpα(τ)={ej2πf0τ0α=0α=0(43)

S p α ( f ) = { δ ( f − f 0 ) α = 0 0 α ≠ 0 S_{p}^{\alpha}(f)=\left\{\begin{array}{cc}\delta\left(f-f_{0}\right) & \alpha=0 \\ 0 & \alpha \neq 0\end{array}\right. Spα(f)={δ(ff0)0α=0α=0(44)

由(12)、(13)得 y ( t ) y(t) y(t)的循环自相关函数为

R y α ( τ ) = ∑ β R p β ( τ ) R s α − β ( τ ) = e j 2 π f 0 τ R s α ( 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)=βRpβ(τ)Rsαβ(τ)=ej2πf0τRsα(τ) (45)

S y α ( f ) = ∑ β S p β ( f ) ⊗ S s α − β ( f ) S_y^\alpha (f) = \sum\limits_\beta {S_p^\beta (f) \otimes S_s^{\alpha - \beta }(f)} Syα(f)=βSpβ(f)Ssαβ(f)

= δ ( f − f 0 ) ⊗ S s α ( f ) = \delta (f - {f_0}) \otimes S_s^\alpha (f) =δ(ff0)Ssα(f)

= S s α ( f − f 0 ) = S_s^\alpha (f - {f_0}) =Ssα(ff0) (46)

将(29)代入(46),得 y ( t ) y(t) y(t)的谱相关密度函数为

S y α ( f ) = 1 T [ Q ( f − f 0 + α / 2 ) Q ∗ ( f − f 0 − α / 2 ) e − j 2 π α t 0 S ~ a α ( f − f 0 ) ] 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(ff0+α/2)Q(ff0α/2)ej2παt0S~aα(ff0)] (47)

同(39),可得复BPSK信号的谱相关密度函数为

S r α ( f ) = { S y α ( f ) + S n α ( f ) , α = 0 S y α ( f ) , α ≠ 0 S_{r}^{\alpha}(f)=\left\{\begin{array}{cc}S_{y}^{\alpha}(f)+S_{n}^{\alpha}(f), & \alpha=0 \\ S_{y}^{\alpha}(f), & \alpha \neq 0\end{array}\right. Srα(f)={Syα(f)+Snα(f),Syα(f),α=0α=0(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 f = 0 f = 0 f=0 α = ± 2 f 0 \alpha = \pm 2{f_0} α=±2f0处各有一个主峰;在 α = 0 \alpha = 0 α=0 f = ± f 0 f = \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 f = f 0 f = {f_0} f=f0 α = 0 \alpha = 0 α=0处有一个谱峰。

7.3.2 切面特征

在式(36)中,令 f = 0 f = 0 f=0 α = ± 2 f 0 + m / T \alpha = \pm 2{f_0} + m/T α=±2f0+m/T

S y α ( f ) = { 1 4 T ∣ Q ( − f 0 + α / 2 ) ∣ 2 e − j [ 2 π n t 0 / T − 2 θ ] α = 2 f 0 + m / T 1 4 T ∣ Q ( f 0 + α / 2 ) ∣ 2 e − j [ 2 π n t 0 / T + 2 θ ] α = − 2 f 0 + m / T S_{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]} & \alpha=2 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} / T+2 \theta\right]} & \alpha=-2 f_{0}+m / T\end{array}\right. Syα(f)={4T1Q(f0+α/2)2ej[2πnt0/T2θ]4T1Q(f0+α/2)2ej[2πnt0/T+2θ]α=2f0+m/Tα=2f0+m/T(49)

特别地,当 m = 0 m = 0 m=0时,有

S y α ( f ) = { 1 4 T ∣ Q ( 0 ) ∣ 2 e j 2 θ α = 2 f 0 1 4 T ∣ Q ( 0 ) ∣ 2 e − j 2 θ α = − 2 f 0 S_{y}^{\alpha}(f)=\left\{\begin{array}{ll}\frac{1}{4 T}|Q(0)|^{2} e^{j 2 \theta} & \alpha=2 f_{0} \\ \frac{1}{4 T}|Q(0)|^{2} e^{-j 2 \theta} & \alpha=-2 f_{0}\end{array}\right. Syα(f)={4T1Q(0)2ej2θ4T1Q(0)2ej2θα=2f0α=2f0(50)

即在 f = 0 f = 0 f=0切面,其谱相关密度函数幅度最大值出现在循环频率为 α = ± 2 f 0 \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 = ± 2 f 0 \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 \theta θ

f = ± f 0 f = \pm {f_0} f=±f0 α = m / T \alpha = m/T α=m/T

S y α ( f ) = 1 4 T { [ Q ( 2 f 0 + α / 2 ) Q ∗ ( 2 f 0 − α / 2 ) + ∣ Q ( α / 2 ) ∣ 2 ] e − j 2 π α t 0 S_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(2f0+α/2)Q(2f0α/2)+Q(α/2)2]ej2παt0 (51)

即在 f = ± f 0 f = \pm {f_0} f=±f0切面,其谱相关密度函数幅度在循环频率为 α = m / T \alpha = m/T α=m/T即符号速率整数倍处出现峰值,在 α = 0 \alpha = 0 α=0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 t 0 {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 Δ f ≫ Δ α \Delta f \gg \Delta \alpha ΔfΔα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 α = 0 \alpha = 0 α=0,得

S y α ( f ) = 1 T ∣ Q ( f − f 0 ) | 2 S_y^\alpha (f) = \frac{1}{T}|Q(f - {f_0}){{\text{|}}^2} Syα(f)=T1Q(ff0)|2 (52)

即在 α = 0 \alpha = 0 α=0切面,其谱相关密度函数幅度只在 f = f 0 f = {f_0} f=f0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。

f = f 0 f = {f_0} f=f0,得

S y α ( f ) = 1 T ∣ Q ( α / 2 ) ∣ 2 e − j 2 π α t 0 S ~ a α ( 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)=T1Q(α/2)2ej2παt0S~aα(0) (53)

即在 f = f 0 f = {f_0} f=f0切面,其谱相关密度函数幅度在循环频率为 α = m / T \alpha = m/T α=m/T即符号速率整数倍处出现峰值,在 α = 0 \alpha = 0 α=0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 t 0 {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 α = m / T \alpha = m/T α=m/T α = m / T ± 2 f 0 \alpha = m/T \pm 2{f_0} α=m/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 α = 1 / T \alpha = 1/T α=1/T α = 1 / T ± 2 f 0 \alpha = 1/T \pm 2{f_0} α=1/T±2f0处有值。

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