1 引言
R ^ x α ( f ) ≡ 0 \hat R_x^\alpha (f) \equiv 0 R^xα(f)≡0，对 ∀ α ≠ 0 , R ^ x ( τ ) ≠ 0 \forall \alpha \ne 0,{\hat R_x}(\tau ) \ne 0 ∀α=0,R^x(τ)=0，则 x ( t ) x(t) x(t)为纯平稳信号。
R ^ x α ( f ) ≠ 0 \hat R_x^\alpha (f) \ne 0 R^xα(f)=0，当 α = m T 0 \alpha = \frac{m}{ { {T_0}}} α=T0m，则 x ( t ) x(t) x(t)具备纯循环平稳性，周期为 T 0 {T_0} T0
R ^ x α ( f ) ≠ 0 \hat R_x^\alpha (f) \ne 0 R^xα(f)=0，其中 α \alpha α不全为 1 T 0 \frac{1}{ { {T_0}}} T01的整数倍，则 x ( t ) x(t) x(t)为循环平稳的。
R ^ y α ( τ ) = ∑ n , m = − ∞ ∞ tr { [ R ^ x α − ( n − m ) / T 0 ( τ ) e − j π ( n + m ) τ / T 0 ] ⊗ r n m α ( − τ ) } \hat R_y^\alpha (\tau ) = \sum\limits_{n,m = - \infty }^\infty {\operatorname{tr} \left\{ {\left[ {\hat R_x^{\alpha - (n - m)/{T_0}}(\tau ){ {\text{e}}^{ - j\pi (n + m)\tau /{T_0}}}} \right] \otimes r_{nm}^\alpha ( - \tau )} \right\}} R^yα(τ)=n,m=−∞∑∞tr{ [R^xα−(n−m)/T0(τ)e−jπ(n+m)τ/T0]⊗rnmα(−τ)}
r n m α ( τ ) ≜ ∫ − ∞ ∞ g ′ n ( t + 1 2 τ ) g m ∗ ( t − 1 2 τ ) e − i 2 π α t d t r_{nm}^\alpha (\tau ) \triangleq \int_{ - \infty }^\infty { { {g'}_n}} \left( {t + \frac{1}{2}\tau } \right)g_m^*\left( {t - \frac{1}{2}\tau } \right){ {\text{e}}^{ - {\text{i}}2\pi \alpha t}}{\text{d}}t rnmα(τ)≜∫−∞∞g′n(t+21τ)gm∗(t−21τ)e−i2παtdt
r n m α ( τ ) ↔ S ^ r α ( f ) = ∫ − ∞ ∞ ( ∫ − ∞ ∞ g ′ n ( t + 1 2 τ ) g m ∗ ( t − 1 2 τ ) e − j 2 π α t d t ) e − j 2 π f τ d τ r_{nm}^\alpha (\tau ) \leftrightarrow \hat S_r^\alpha (f) = \int_{ - \infty }^\infty {\left( {\int_{ - \infty }^\infty { { {g'}_n}\left( {t + \frac{1}{2}\tau } \right)g_m^*\left( {t - \frac{1}{2}\tau } \right){ {\text{e}}^{ - j2\pi \alpha t}}{\text{d}}t} } \right){e^{ - j2\pi f\tau }}d\tau } rnmα(τ)↔S^rα(f)=∫−∞∞(∫−∞∞g′n(t+21τ)gm∗(t−21τ)e−j2παtdt)e−j2πfτdτ
= ∫ − ∞ ∞ ∫ − ∞ ∞ g ′ n ( t + 1 2 τ ) e − j 2 π ( f + α / 2 ) ( t + τ / 2 ) d( t + 1 2 τ ) g m ∗ ( t − 1 2 τ ) e j 2 π ( f − α / 2 ) ( t − τ / 2 ) d ( t − 1 2 τ ) = \int_{ - \infty }^\infty {\int_{ - \infty }^\infty { { {g'}_n}\left( {t + \frac{1}{2}\tau } \right){e^{ - j2\pi (f + \alpha /2)(t + \tau /2)}}{\text{d(}}t + \frac{1}{2}\tau )g_m^*\left( {t - \frac{1}{2}\tau } \right)} {e^{j2\pi (f - \alpha /2)(t - \tau /2)}}d(t - \frac{1}{2}\tau )} =∫−∞∞∫−∞∞g′n(t+21τ)e−j2π(f+α/2)(t+τ/2)d(t+21τ)gm∗(t−21τ)ej2π(f−α/2)(t−τ/2)d(t−21τ)
= ∫ − ∞ ∞ g ′ n ( t + 1 2 τ ) e − j 2 π ( f + α / 2 ) ( t + τ / 2 ) d( t + τ / 2 ) ∫ − ∞ ∞ ( g m ( t − 1 2 τ ) e − j 2 π ( f − α / 2 ) ( t − τ / 2 ) ) ∗ d ( t − 1 2 τ ) = \int_{ - \infty }^\infty { { {g'}_n}\left( {t + \frac{1}{2}\tau } \right){e^{ - j2\pi (f + \alpha /2)(t + \tau /2)}}{\text{d(}}t + \tau /2)\int_{ - \infty }^\infty { { {\left( { {g_m}\left( {t - \frac{1}{2}\tau } \right){e^{ - j2\pi (f - \alpha /2)(t - \tau /2)}}} \right)}^*}d(t - \frac{1}{2}\tau )} } =∫−∞∞g′n(t+21τ)e−j2π(f+α/2)(t+τ/2)d(t+τ/2)∫−∞∞(gm(t−21τ)e−j2π(f−α/2)(t−τ/2))∗d(t−21τ)
= G ′ n ( f + α / 2 ) G m ∗ ( f − α / 2 ) = { {G'}_n}(f + \alpha /2)G_m^*(f - \alpha /2) =G′n(f+α/2)Gm∗(f−α/2)(1)
S ^ y α ( f ) = ∑ n , m = − ∞ ∞ G n ( f + 1 2 α ) S ^ x α − ( n − m ) / T 0 ( f − n + m 2 T 0 ) G m ′ ( f − 1 2 α ) ∗ \hat S_y^\alpha (f){\text{ }} = \sum\limits_{n,m = - \infty }^\infty { {G_n}} \left( {f + \frac{1}{2}\alpha } \right)\hat S_x^{\alpha - (n - m)/{T_0}}\left( {f - \frac{ {n + m}}{ {2{T_0}}}} \right){G'_m}{\left( {f - \frac{1}{2}\alpha } \right)^*} S^yα(f) =n,m=−∞∑∞Gn(f+21α)S^xα−(n−m)/T0(f−2T0n+m)Gm′(f−21α)∗(2)
信号 x ( t ) x(t) x(t)通过线性时不变系统后的输出为
z ( t ) = h ( t ) ⊗ x ( t ) = ∫ − ∞ ∞ h ( u ) x ( t − u ) d u z(t) = h(t) \otimes x(t) = \int_{ - \infty }^\infty {h(u)x(t - u)du} z(t)=h(t)⊗x(t)=∫−∞∞h(u)x(t−u)du(3)
其功率谱为
S z ( f ) = ∣ H ( f ) ∣ 2 S x ( f ) {S_z}(f) = {\left| {H(f)} \right|^2}{S_x}(f) Sz(f)=∣H(f)∣2Sx(f)(4)
谱相关密度函数为
S z α ( f ) = H ( f + α / 2 ) H ∗ ( f − α / 2 ) S x α ( f ) S_z^\alpha (f) = H(f + \alpha /2){H^*}(f - \alpha /2)S_x^\alpha (f) Szα(f)=H(f+α/2)H∗(f−α/2)Sxα(f)(5)
2 周期抽样函数的循环自相关函数和谱相关密度函数
周期抽样函数可以表示为
$g(t) = \sum\limits_{n = - \infty }^\infty {\delta (t - nT)} $(6)
由 R x α ( τ ) ≜ lim x → ∞ 1 T ∫ − T / 2 T / 2 x ( t + τ / 2 ) x ( t − τ / 2 ) d t = ⟨ x ( t + τ / 2 ) x ( t − τ / 2 ) ⟩ t R_x^\alpha (\tau ) \triangleq \mathop {\lim }\limits_{x \to \infty } \frac{1}{T}\int_{ - T/2}^{T/2} {x(t + \tau /2)x(t - \tau /2)dt} = {\left\langle {x(t + \tau /2)x(t - \tau /2)} \right\rangle _t} Rxα(τ)≜x→∞limT1∫−T/2T/2x(t+τ/2)x(t−τ/2)dt=⟨x(t+τ/2)x(t−τ/2)⟩t，得
R g β ( τ ) = ⟨ ∑ n = − ∞ ∞ ∑ m = − ∞ ∞ δ ( t + τ / 2 − n T ) δ ( t − τ / 2 − m T ) e − j 2 π β ⋅ t ⟩ t R_g^\beta (\tau ) = {\left\langle {\sum\limits_{n = - \infty }^\infty {\sum\limits_{m = - \infty }^\infty {\delta (t + \tau /2 - nT)\delta (t - \tau /2 - mT){e^{ - j2\pi \beta \cdot t}}} } } \right\rangle _t} Rgβ(τ)=⟨n=−∞∑∞m=−∞∑∞δ(t+τ/2−nT)δ(t−τ/2−mT)e−j2πβ⋅t⟩t
= ⟨ ∑ n = − ∞ ∞ ∑ m = − ∞ ∞ δ ( ( τ / 2 + m T ) + τ / 2 − n T ) δ ( t − τ / 2 − m T ) e − j 2 π β ( t − τ / 2 + τ / 2 ) ⟩ t = {\left\langle {\sum\limits_{n = - \infty }^\infty {\sum\limits_{m = - \infty }^\infty {\delta ((\tau /2 + mT) + \tau /2 - nT)\delta (t - \tau /2 - mT){e^{ - j2\pi \beta (t - \tau /2 + \tau /2)}}} } } \right\rangle _t} =⟨n=−∞∑∞m=−∞∑∞δ((τ/2+mT)+τ/2−nT)δ(t−τ/2−mT)e−j2πβ(t−τ/2+τ/2)⟩t
= e − j π β τ ∑ n = − ∞ ∞ ∑ m = − ∞ ∞ δ ( τ + ( m − n ) T ) ⟨ ∑ m = − ∞ ∞ δ ( t − τ / 2 − m T ) e − j 2 π β ( t − τ / 2 ) ⟩ t = {e^{ - j\pi \beta \tau }}\sum\limits_{n = - \infty }^\infty {\sum\limits_{m = - \infty }^\infty {\delta (\tau + (m - n)T)} } {\left\langle {\sum\limits_{m = - \infty }^\infty {\delta (t - \tau /2 - mT){e^{ - j2\pi \beta (t - \tau /2)}}} } \right\rangle _t} =e−jπβτn=−∞∑∞m=−∞∑∞δ(τ+(m−n)T)⟨m=−∞∑∞δ(t−τ/2−mT)e−j2πβ(t−τ/2)⟩t
u = t − τ / 2 ‾ ‾ e − j π β τ ∑ n = − ∞ ∞ ∑ m = − ∞ ∞ δ ( τ + ( m − n ) T ) lim T ′ → ∞ 1 T ′ ∫ − ( T ′ + τ ) / 2 ( T ′ − τ ) / 2 ∑ m = − ∞ ∞ δ ( u − m T ) e − j 2 π β m T d u {\underline{\underline {u = t - \tau /2}} } {e^{ - j\pi \beta \tau }}\sum\limits_{n = - \infty }^\infty {\sum\limits_{m = - \infty }^\infty {\delta (\tau + (m - n)T)} } \mathop {\lim }\limits_{T' \to \infty } \frac{1}{ {T'}}\int_{ - (T' + \tau )/2}^{(T' - \tau )/2} {\sum\limits_{m = - \infty }^\infty {\delta (u - mT){e^{ - j2\pi \beta mT}}} du} u=t−τ/2e−jπβτn=−∞∑∞m=−∞∑∞δ(τ+(m−n)T)T′→∞limT′1∫−(T′+τ)/2(T′−τ)/2m=−∞∑∞δ(u−mT)e−j2πβmTdu
= e − j π β τ ∑ n = − ∞ ∞ δ ( τ − n T ) lim N → ∞ 1 2 N T + 2 ε ∫ − N T − ε N T + ε ∑ m = − N N δ ( u − m T ) e − j 2 π m ⋅ m d u = {e^{ - j\pi \beta \tau }}\sum\limits_{n = - \infty }^\infty {\delta (\tau - nT)} \mathop {\lim }\limits_{N \to \infty } \frac{1}{ {2NT + 2\varepsilon }}\int_{ - NT - \varepsilon }^{NT + \varepsilon } {\sum\limits_{m = - N}^N {\delta (u - mT){e^{ - j2\pi m \cdot m}}} du} =e−jπβτn=−∞∑∞δ(τ−nT)N→∞lim2NT+2ε1∫−NT−εNT+εm=−N∑Nδ(u−mT)e−j2πm⋅mdu
= e − j π β τ ∑ n = − ∞ ∞ δ ( τ − n T ) lim N → ∞ 1 2 N T + 2 ε ∫ − N T − ε N T + ε ∑ m = − N N δ ( u − m T ) e − j 2 π m ⋅ m d u = {e^{ - j\pi \beta \tau }}\sum\limits_{n = - \infty }^\infty {\delta (\tau - nT)} \mathop {\lim }\limits_{N \to \infty } \frac{1}{ {2NT + 2\varepsilon }}\int_{ - NT - \varepsilon }^{NT + \varepsilon } {\sum\limits_{m = - N}^N {\delta (u - mT){e^{ - j2\pi m \cdot m}}} du} =e−jπβτn=−∞∑∞δ(τ−nT)N→∞lim2NT+2ε1∫−NT−εNT+εm=−N∑Nδ(u−mT)e−j2πm⋅mdu
= e − j π β τ ∑ n = − ∞ ∞ δ ( τ − n T ) lim N → ∞ 1 2 N T + 2 ε ∑ m = − N N ∫ − N T − ε N T + ε δ ( u − m T ) d u = {e^{ - j\pi \beta \tau }}\sum\limits_{n = - \infty }^\infty {\delta (\tau - nT)} \mathop {\lim }\limits_{N \to \infty } \frac{1}{ {2NT + 2\varepsilon }}\sum\limits_{m = - N}^N {\int_{ - NT - \varepsilon }^{NT + \varepsilon } {\delta (u - mT)du} } =e−jπβτn=−∞∑∞δ(τ−nT)N→∞lim2NT+2ε1m=−N∑N∫−NT−εNT+εδ(u−mT)du
= e − j π β τ ∑ n = − ∞ ∞ δ ( τ − n T ) lim N → ∞ 2 N + 1 2 N T + 2 ε = {e^{ - j\pi \beta \tau }}\sum\limits_{n = - \infty }^\infty {\delta (\tau - nT)} \mathop {\lim }\limits_{N \to \infty } \frac{ {2N + 1}}{ {2NT + 2\varepsilon }} =e−jπβτn=−∞∑∞δ(τ−nT)N→∞lim2NT+2ε2N+1
= e − j π β τ ∑ n = − ∞ ∞ δ ( τ − n T ) = {e^{ - j\pi \beta \tau }}\sum\limits_{n = - \infty }^\infty {\delta (\tau - nT)} =e−jπβτn=−∞∑∞δ(τ−nT)(7)
则抽样函数的谱相关密度函数为
S g β ( f ) = ∫ − ∞ ∞ R g β ( τ ) e − j 2 π f τ d τ S_g^\beta (f) = \int_{ - \infty }^\infty {R_g^\beta (\tau ){e^{ - j2\pi f\tau }}d\tau } Sgβ(f)=∫−∞∞Rgβ(τ)e−j2πfτdτ
= ∫ − ∞ ∞ e − j π β τ ∑ n = − ∞ ∞ δ ( τ − n T ) e − j 2 π f τ d τ = \int_{ - \infty }^\infty { {e^{ - j\pi \beta \tau }}\sum\limits_{n = - \infty }^\infty {\delta (\tau - nT)} {e^{ - j2\pi f\tau }}d\tau } =∫−∞∞e−jπβτn=−∞∑∞δ(τ−nT)e−j2πfτdτ
= ∑ n = − ∞ ∞ e − j π β n T e − j 2 π f n T ∫ − ∞ ∞ δ ( τ − n T ) d τ = \sum\limits_{n = - \infty }^\infty { {e^{ - j\pi \beta nT}}{e^{ - j2\pi fnT}}\int_{ - \infty }^\infty {\delta (\tau - nT)d\tau } } =n=−∞∑∞e−jπβnTe−j2πfnT∫−∞∞δ(τ−nT)dτ
= ∑ n = − ∞ ∞ e − j 2 π n T ( f + β 2 ) = \sum\limits_{n = - \infty }^\infty { {e^{ - j2\pi nT(f + \frac{\beta }{2})}}} =n=−∞∑∞e−j2πnT(f+2β)
= ∑ n = − ∞ ∞ e − j 2 π T ′ n ( f + β 2 ) = \sum\limits_{n = - \infty }^\infty { {e^{ - j\frac{ {2\pi }}{ {T'}}n(f + \frac{\beta }{2})}}} =n=−∞∑∞e−jT′2πn(f+2β)(8)
其中 T ′ = 1 / T T' = 1/T T′=1/T，由 ∑ n = − ∞ ∞ δ ( t − n T ) = 1 T ∑ n = − ∞ ∞ e j 2 π T n ⋅ t \sum\limits_{n = - \infty }^\infty {\delta (t - nT)} = \frac{1}{T}\sum\limits_{n = - \infty }^\infty { {e^{j\frac{ {2\pi }}{T}n \cdot t}}} n=−∞∑∞δ(t−nT)=T1n=−∞∑∞ejT2πn⋅t，得
S g β ( f ) = T ′ ∑ k = − ∞ ∞ δ ( f + β 2 − k T ′ ) = 1 T ∑ k = − ∞ ∞ δ ( f + β 2 − k T ) S_g^\beta (f) = T'\sum\limits_{k = - \infty }^\infty {\delta (f + \frac{\beta }{2} - kT')}= \frac{1}{T}\sum\limits_{k = - \infty }^\infty {\delta (f + \frac{\beta }{2} - \frac{k}{T})} Sgβ(f)=T′k=−∞∑∞δ(f+2β−kT′)=T1k=−∞∑∞δ(f+2β−Tk)(9)
3 信号相乘前后谱相关密度函数关系
假设 x ( t ) = r ( t ) s ( t ) x(t) = r(t)s(t) x(t)=r(t)s(t)，由
R x ( t , τ ) = E [ x ( t + τ 2 ) x ∗ ( t − τ 2 ) ] = R r ( t , τ ) R s ( t , τ ) {R_x}(t,\tau ) = E[x(t + \frac{\tau }{2}){x^*}(t - \frac{\tau }{2})] = {R_r}(t,\tau ){R_s}(t,\tau ) Rx(t,τ)=E[x(t+2τ)x∗(t−2τ)]=Rr(t,τ)Rs(t,τ)(10)
${R_x}(t,\tau ) = \sum\limits_\alpha {R_x^\alpha (\tau ){e^{j2\pi \alpha t}}} $(11)
得 x ( t ) x(t) x(t)的循环自相关函数为
R x α ( τ ) = lim T → ∞ 1 T ∫ − T / 2 T / 2 R x ( t , τ ) e − j 2 π α t d t R_x^\alpha (\tau ) = \mathop {\lim }\limits_{T \to \infty } \frac{1}{T}\int_{ - T/2}^{T/2} { {R_x}(t,\tau ){e^{ - j2\pi \alpha t}}dt} Rxα(τ)=T→∞limT1∫−T/2T/2Rx(t,τ)e−j2παtdt
= lim T → ∞ 1 T ∫ − T / 2 T / 2 R r ( t , τ ) R s ( t , τ ) e − j 2 π α t d t = \mathop {\lim }\limits_{T \to \infty } \frac{1}{T}\int_{ - T/2}^{T/2} { {R_r}(t,\tau ){R_s}(t,\tau ){e^{ - j2\pi \alpha t}}dt} =T→∞limT1∫−T/2T/2Rr(t,τ)Rs(t,τ)e−j2παtdt
= lim T → ∞ 1 T ∫ − T / 2 T / 2 ∑ β R r ( t , τ ) e − j 2 π ( α − β ) t R s ( t , τ ) e − j 2 π β t d t = \mathop {\lim }\limits_{T \to \infty } \frac{1}{T}\int_{ - T/2}^{T/2} {\sum\limits_\beta { {R_r}(t,\tau ){e^{ - j2\pi (\alpha - \beta )t}}{R_s}(t,\tau ){e^{ - j2\pi \beta t}}} dt} =T→∞limT1∫−T/2T/2β∑Rr(t,τ)e−j2π(α−β)tRs(t,τ)e−j2πβtdt
= ∑ β lim T → ∞ 1 T ∫ − T / 2 T / 2 R r ( t , τ ) e − j 2 π ( α − β ) t R s ( t , τ ) e − j 2 π β t d t = \sum\limits_\beta {\mathop {\lim }\limits_{T \to \infty } \frac{1}{T}\int_{ - T/2}^{T/2} { {R_r}(t,\tau ){e^{ - j2\pi (\alpha - \beta )t}}{R_s}(t,\tau ){e^{ - j2\pi \beta t}}dt} } =β∑T→∞limT1∫−T/2T/2Rr(t,τ)e−j2π(α−β)tRs(t,τ)e−j2πβtdt
= ∑ β R r α − β ( τ ) R s β ( τ ) = \sum\limits_\beta {R_r^{\alpha - \beta }(\tau )R_s^\beta (\tau )} =β∑Rrα−β(τ)Rsβ(τ)
= ∑ β R r β ( τ ) R s α − β ( τ ) = \sum\limits_\beta {R_r^\beta (\tau )R_s^{\alpha - \beta }(\tau )} =β∑Rrβ(τ)Rsα−β(τ) (12式）
由(12)可知，两个信号相乘后的循环自相关函数等于两个循环自相关函数在离散$\alpha $域的卷积。
其谱相关密度函数为
S x α ( f ) = ∫ − ∞ ∞ R x α ( τ ) e − j 2 π f τ d τ S_x^\alpha (f) = \int_{ - \infty }^\infty {R_x^\alpha (\tau ){e^{ - j2\pi f\tau }}d\tau } Sxα(f)=∫−∞∞Rxα(τ)e−j2πfτdτ
= ∫ − ∞ ∞ ∑ β R r β ( τ ) R s α − β ( τ ) e − j 2 π f τ d τ = \int_{ - \infty }^\infty {\sum\limits_\beta {R_r^\beta (\tau )R_s^{\alpha - \beta }(\tau )} {e^{ - j2\pi f\tau }}d\tau } =∫−∞∞β∑Rrβ(τ)Rsα−β(τ)e−j2πfτdτ
= ∑ β ∫ − ∞ ∞ ∫ − ∞ ∞ R r β ( τ ) e − j 2 π F τ R s α − β ( τ ) e − j 2 π ( f − F ) τ d τ d F = \sum\limits_\beta {\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {R_r^\beta (\tau ){e^{ - j2\pi F\tau }}R_s^{\alpha - \beta }(\tau ){e^{ - j2\pi (f - F)\tau }}d\tau dF} } } =β∑∫−∞∞∫−∞∞Rrβ(τ)e−j2πFτRsα−β(τ)e−j2π(f−F)τdτdF
= ∑ β ∫ − ∞ ∞ ∫ − ∞ ∞ R r β ( τ ) e − j 2 π F τ R s α − β ( τ ) e − j 2 π ( f − F ) τ d τ d F = \sum\limits_\beta {\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {R_r^\beta (\tau ){e^{ - j2\pi F\tau }}R_s^{\alpha - \beta }(\tau ){e^{ - j2\pi (f - F)\tau }}d\tau dF} } } =β∑∫−∞∞∫−∞∞Rrβ(τ)e−j2πFτRsα−β(τ)e−j2π(f−F)τdτdF
= ∫ − ∞ ∞ ∑ β S r β ( F ) S s α − β ( f − F ) d F = \int_{ - \infty }^\infty {\sum\limits_\beta {S_r^\beta (F)S_s^{\alpha - \beta }(f - F)} dF} =∫−∞∞β∑Srβ(F)Ssα−β(f−F)dF
= ∑ β S r β ( f ) ⊗ S s α − β ( f ) = \sum\limits_\beta {S_r^\beta (f) \otimes S_s^{\alpha - \beta }(f)} =β∑Srβ(f)⊗Ssα−β(f) (13式)
其中 ⊗ \otimes ⊗表示卷积，由此可知两个信号相乘的谱相关等于两个谱相关在连续 f f f域和离散 α \alpha α域的双重卷积。
4 周期信号（Almost periodic）的循环自相关与谱相关密度函数
设 p ( t ) p(t) p(t)为周期信号，其傅里叶级数可以表示为
p ( t ) = ∑ β P β e j 2 π β t p(t) = \sum\limits_\beta { {P_\beta }{e^{j2\pi \beta t}}} p(t)=β∑Pβej2πβt (14式)
其中 β = n T \beta = \frac{n}{T} β=Tn， T T T为周期，则
P β = ⟨ p ( t ) e − j 2 π β t ⟩ t {P_\beta } = {\left\langle {p(t){e^{ - j2\pi \beta t}}} \right\rangle _t} Pβ=⟨p(t)e−j2πβt⟩t (15式)
由循环自相关函数的定义得
R p α ( τ ) = ⟨ p ( t ) p ∗ ( t − τ ) e − j 2 π α ( t − τ / 2 ) ⟩ t R_p^\alpha (\tau ) = {\left\langle {p(t){p^*}(t - \tau ){e^{ - j2\pi \alpha (t - \tau /2)}}} \right\rangle _t} Rpα(τ)=⟨p(t)p∗(t−τ)e−j2πα(t−τ/2)⟩t
= ⟨ ∑ β p ( t ) e − j 2 π β t p ∗ ( t − τ ) e − j 2 π ( α − β ) ( t − τ ) ⋅ e j π ( 2 β − α ) τ ⟩ t = {\left\langle {\sum\limits_\beta {p(t){e^{ - j2\pi \beta t}}{p^*}(t - \tau ){e^{ - j2\pi (\alpha - \beta )(t - \tau )}} \cdot {e^{j\pi (2\beta - \alpha )\tau }}} } \right\rangle _t} =⟨β∑p(t)e−j2πβtp∗(t−τ)e−j2π(α−β)(t−τ)⋅ejπ(2β−α)τ⟩t
= ∑ β P β P α − β ∗ e j π ( 2 β − α ) τ = \sum\limits_\beta { {P_\beta }P_{\alpha - \beta }^*{e^{j\pi (2\beta - \alpha )\tau }}} =β∑PβPα−β∗ejπ(2β−α)τ (16式)
则其谱相关密度函数为
S p α ( f ) = ∫ − ∞ ∞ ∑ β P β P α − β ∗ e j π ( 2 β − α ) τ e − j 2 π f τ d τ S_p^\alpha (f) = \int_{ - \infty }^\infty {\sum\limits_\beta { {P_\beta }P_{\alpha - \beta }^*{e^{j\pi (2\beta - \alpha )\tau }}} {e^{ - j2\pi f\tau }}d\tau } Spα(f)=∫−∞∞β∑PβPα−β∗ejπ(2β−α)τe−j2πfτdτ
= ∑ β P β P α − β ∗ ∫ − ∞ ∞ e − j 2 π ( f − β + α / 2 ) τ d τ = \sum\limits_\beta { {P_\beta }P_{\alpha - \beta }^*\int_{ - \infty }^\infty { {e^{ - j2\pi (f - \beta + \alpha /2)\tau }}d\tau } } =β∑PβPα−β∗∫−∞∞e−j2π(f−β+α/2)τdτ
= ∑ β P β P α − β ∗ δ ( f − β + α / 2 ) = \sum\limits_\beta { {P_\beta }P_{\alpha - \beta }^*\delta (f - \beta + \alpha /2)} =β∑PβPα−β∗δ(f−β+α/2) (17式)
5 连续平稳信号被理想抽样后的谱相关密度函数
连续平稳信号 x ( t ) x(t) x(t)被理想抽样后的信号可以表示为
y ( t ) = x ( t ) g ( t ) = ∑ n = − ∞ ∞ x ( n T ) δ ( t − n T ) y(t) = x(t)g(t) = \sum\limits_{n = - \infty }^\infty {x(nT)\delta (t - nT)} y(t)=x(t)g(t)=n=−∞∑∞x(nT)δ(t−nT) (18式)
其中， g ( t ) = ∑ n = − ∞ ∞ δ ( t − n T ) g(t) = \sum\limits_{n = - \infty }^\infty {\delta (t - nT)} g(t)=n=−∞∑∞δ(t−nT)，则由(13)得 y ( t ) y(t) y(t)的谱相关密度函数为
S y α ( f ) = ∫ − ∞ ∞ ∑ β S g β ( F ) S x α − β ( f − F ) d F S_y^\alpha (f) = \int_{ - \infty }^\infty {\sum\limits_\beta {S_g^\beta (F)S_x^{\alpha - \beta }(f - F)} dF} Syα(f)=∫−∞∞β∑Sgβ(F)Sxα−β(f−F)dF
= ∑ β ∫ − ∞ ∞ 1 T 2 ∑ n = − ∞ ∞ δ ( F + β 2 − n T ) S x α − β ( f − F ) d F = \sum\limits_\beta {\int_{ - \infty }^\infty {\frac{1}{ { {T^2}}}\sum\limits_{n = - \infty }^\infty {\delta (F + \frac{\beta }{2} - \frac{n}{T})} S_x^{\alpha - \beta }(f - F)dF} } =β∑∫−∞∞T21n=−∞∑∞δ(F+2β−Tn)Sxα−β(f−F)dF
= ∑ m = − ∞ ∞ ∫ − ∞ ∞ 1 T 2 ∑ n = − ∞ ∞ δ ( F + β 2 − n T ) S x α − β ( f − F ) d F = \sum\limits_{m = - \infty }^\infty {\int_{ - \infty }^\infty {\frac{1}{ { {T^2}}}\sum\limits_{n = - \infty }^\infty {\delta (F + \frac{\beta }{2} - \frac{n}{T})} S_x^{\alpha - \beta }(f - F)dF} } =m=−∞∑∞∫−∞∞T21n=−∞∑∞δ(F+2β−Tn)Sxα−β(f−F)dF
= 1 T 2 ∑ m = − ∞ ∞ ∑ n = − ∞ ∞ S x α − m T ( f + m 2 T − n T ) ∫ − ∞ ∞ δ ( F + m 2 T − n T ) d F = \frac{1}{ { {T^2}}}\sum\limits_{m = - \infty }^\infty {\sum\limits_{n = - \infty }^\infty {S_x^{\alpha - \frac{m}{T}}(f + \frac{m}{ {2T}} - \frac{n}{T})} \int_{ - \infty }^\infty {\delta (F + \frac{m}{ {2T}} - \frac{n}{T})dF} } =T21m=−∞∑∞n=−∞∑∞Sxα−Tm(f+2Tm−Tn)∫−∞∞δ(F+2Tm−Tn)dF
= 1 T 2 ∑ m = − ∞ ∞ ∑ n = − ∞ ∞ S x α − m T ( f + m 2 T − n T ) = \frac{1}{ { {T^2}}}\sum\limits_{m = - \infty }^\infty {\sum\limits_{n = - \infty }^\infty {S_x^{\alpha - \frac{m}{T}}(f + \frac{m}{ {2T}} - \frac{n}{T})} } =T21m=−∞∑∞n=−∞∑∞Sxα−Tm(f+2Tm−Tn) (19式)
6 连续平稳信号与其周期采样序列的谱相关密度函数关系
连续平稳信号x(t)非对称形式的循环自相关函数为
（1）
由(1)定义离散时间序列 x ( n T 0 ) x(nT_0) x(nT0)的循环自相关函数为
（2）
x(t)的谱相关密度函数为
（3）
由(3)定义 x ( n T 0 ) x(nT_0) x(nT0)的谱相关密度函数为
（4）
将恒等式
（5）
代入(2)，得
R ~ x α ( k T 0 ) ≜ lim N → ∞ 1 2 N + 1 ∑ n = − N N x ( n T 0 + k T 0 ) x ( n T 0 ) e − j 2 π α ( n + k / 2 ) T 0 \tilde R_x^\alpha (k{T_0}) \triangleq \mathop {\lim }\limits_{N \to \infty } \frac{1}{
{2N + 1}}\sum\limits_{n = - N}^N {x(n{T_0} + k{T_0})x(n{T_0})} {e^{ - j2\pi \alpha (n + k/2){T_0}}} R~xα(kT0)≜N→∞lim2N+11n=−N∑Nx(nT0+kT0)x(nT0)e−j2πα(n+k/2)T0
n T 0 → u ‾ ‾ ∑ m = − ∞ ∞ lim T → ∞ 1 T ∫ − T / 2 T / 2 x ( u + k T 0 ) x ( u ) e − j 2 π α ( u + k T 0 / 2 ) e − j 2 π m u / T 0 d u {\underline{\underline {n{T_0} \to u}} } \sum\limits_{m = - \infty }^\infty {\mathop {\lim }\limits_{T \to \infty } \frac{1}{T}\int_{ - T/2}^{T/2} {x(u + k{T_0})x(u)} } {e^{ - j2\pi \alpha (u + k{T_0}/2)}}{e^{ - j2\pi mu/{T_0}}}du nT0→um=−∞∑∞T→∞limT1∫−T/2T/2x(u+kT0)x(u)e−j2πα(u+kT0/2)e−j2πmu/T0du
= ∑ m = − ∞ ∞ lim T → ∞ 1 T ∫ − T / 2 T / 2 x ( u + k T 0 ) x ( u ) e − j 2 π α ( u + k T 0 / 2 ) e − j 2 π m / T 0 ( u + k T 0 / 2 ) e j π m k d u = \sum\limits_{m = - \infty }^\infty {\mathop {\lim }\limits_{T \to \infty } \frac{1}{T}\int_{ - T/2}^{T/2} {x(u + k{T_0})x(u)} } {e^{ - j2\pi \alpha (u + k{T_0}/2)}}{e^{ - j2\pi m/{T_0}(u + k{T_0}/2)}}{e^{j\pi mk}}du =m=−∞∑∞T→∞limT1∫−T/2T/2x(u+kT0)x(u)e−j2πα(u+kT0/2)e−j2πm/T0(u+kT0/2)ejπmkdu
= ∑ m = − ∞ ∞ lim T → ∞ 1 T ∫ − T / 2 T / 2 x ( u + k T 0 ) x ( u ) e − j 2 π ( α + m / T 0 ) ( u + k T 0 / 2 ) d u ⋅ e j π m k = \sum\limits_{m = - \infty }^\infty {\mathop {\lim }\limits_{T \to \infty } \frac{1}{T}\int_{ - T/2}^{T/2} {x(u + k{T_0})x(u)} } {e^{ - j2\pi (\alpha + m/{T_0})(u + k{T_0}/2)}}du \cdot {e^{j\pi mk}} =m=−∞∑∞T→∞limT1∫−T/2T/2x(u+kT0)x(u)e−j2π(α+m/T0)(u+kT0/2)du⋅ejπmk
= ∑ m = − ∞ ∞ R ^ x α + m / T 0 ( k T 0 ) e j π m k = \sum\limits_{m = - \infty }^\infty {\hat R_x^{\alpha {\text{ + }}m/{T_0}}(k{T_0})} {e^{j\pi mk}} =m=−∞∑∞R^xα + m/T0(kT0)ejπmk (6式)
将(6)代入(4)，得一种错误的结果如下
S ~ x α ( f ) = ∑ k = − ∞ ∞ [ ∑ m = − ∞ ∞ R ^ x α + m / T 0 ( k T 0 ) e j π m k ] e − j 2 π k T 0 f \tilde S_x^\alpha (f) = \sum\limits_{k = - \infty }^\infty {[\sum\limits_{m = - \infty }^\infty {\hat R_x^{\alpha {\text{ + }}m/{T_0}}(k{T_0})} {e^{j\pi mk}}]{e^{ - j2\pi k{T_0}f}}} S~xα(f)=k=−∞∑∞[m=−∞∑∞R^xα + m/T0(kT0)ejπmk]e−j2πkT0f
= ∑ k = − ∞ ∞ ∑ m = − ∞ ∞ R ^ x α + m / T 0 ( k T 0 ) e − j 2 π k T 0 ( f − m 2 T 0 ) = \sum\limits_{k = - \infty }^\infty {\sum\limits_{m = - \infty }^\infty {\hat R_x^{\alpha {\text{ + }}m/{T_0}}(k{T_0})} {e^{ - j2\pi k{T_0}(f - \frac{m}{ {2{T_0}}})}}} =k=−∞∑∞m=−∞∑∞R^xα + m/T0(kT0)e−j2πkT0(f−2T0m)
k T 0 → τ ‾ ‾ 1 T 0 ∫ − ∞ ∞ ∑ m = − ∞ ∞ R ^ x α + m / T 0 ( τ ) e − j 2 π ( f − m 2 T 0 ) τ d τ {\underline{\underline {k{T_0} \to \tau }} } \frac{1}{ { {T_0}}}\int_{ - \infty }^\infty {\sum\limits_{m = - \infty }^\infty {\hat R_x^{\alpha {\text{ + }}m/{T_0}}(\tau ){e^{ - j2\pi (f - \frac{m}{ {2{T_0}}})\tau }}} d\tau } kT0→τT01∫−∞∞m=−∞∑∞R^xα + m/T0(τ)e−j2π(f−2T0m)τdτ
= 1 T 0 ∑ m = − ∞ ∞ ∫ − ∞ ∞ R ^ x α + m / T 0 ( τ ) e − j 2 π ( f − m 2 T 0 ) τ d τ = \frac{1}{ { {T_0}}}\sum\limits_{m = - \infty }^\infty {\int_{ - \infty }^\infty {\hat R_x^{\alpha {\text{ + }}m/{T_0}}(\tau ){e^{ - j2\pi (f - \frac{m}{ {2{T_0}}})\tau }}d\tau } } =T01m=−∞∑∞∫−∞∞R^xα + m/T0(τ)e−j2π(f−2T0m)τdτ
= 1 T 0 ∑ m = − ∞ ∞ S ^ x α + m / T 0 ( f − m 2 T 0 ) = \frac{1}{ { {T_0}}}\sum\limits_{m = - \infty }^\infty {\hat S_x^{\alpha {\text{ + }}m/{T_0}}(f - \frac{m}{ {2{T_0}}})} =T01m=−∞∑∞S^xα + m/T0(f−2T0m) (20式)
正确结果应为 S x α ( f ) S_x^\alpha (f) Sxα(f)
S ~ x α ( f ) = ∑ k = − ∞ ∞ [ ∑ m = − ∞ ∞ R ^ x α + m / T 0 ( k T 0 ) e j π m k ] e − j 2 π k T 0 f \tilde S_x^\alpha (f) = \sum\limits_{k = - \infty }^\infty {[\sum\limits_{m = - \infty }^\infty {\hat R_x^{\alpha {\text{ + }}m/{T_0}}(k{T_0})} {e^{j\pi mk}}]{e^{ - j2\pi k{T_0}f}}} S~xα(f)=k=−∞∑∞[m=−∞∑∞R^xα + m/T0(kT0)ejπmk]e−j2πkT0f
= ∑ k = − ∞ ∞ ∑ m = − ∞ ∞ R ^ x α + m / T 0 ( k T 0 ) e − j 2 π k T 0 ( f − m 2 T 0 ) = \sum\limits_{k = - \infty }^\infty {\sum\limits_{m = - \infty }^\infty {\hat R_x^{\alpha {\text{ + }}m/{T_0}}(k{T_0})} {e^{ - j2\pi k{T_0}(f - \frac{m}{ {2{T_0}}})}}} =k=−∞∑∞m=−∞∑∞R^xα + m/T0(kT0)e−j2πkT0(f−2T0m)
= ∑ k = − ∞ ∞ ∑ n = − ∞ ∞ ∑ m = − ∞ ∞ R ^ x α + m / T 0 ( k T 0 ) e − j 2 π k T 0 ( f − m 2 T 0 ) ⋅ e j 2 π n k = \sum\limits_{k = - \infty }^\infty {\sum\limits_{n = - \infty }^\infty {\sum\limits_{m = - \infty }^\infty {\hat R_x^{\alpha {\text{ + }}m/{T_0}}(k{T_0})} {e^{ - j2\pi k{T_0}(f - \frac{m}{ {2{T_0}}})}} \cdot {e^{j2\pi nk}}} } =k=−∞∑∞n=−∞∑∞m=−∞∑∞R^xα + m/T0(kT0)e−j2πkT0(f−2T0m)⋅ej2πnk
= ∑ k = − ∞ ∞ ∑ n = − ∞ ∞ ∑ m = − ∞ ∞ R ^ x α + m / T 0 ( k T 0 ) e − j 2 π k T 0 ( f − m 2 T 0 − n T 0 ) = \sum\limits_{k = - \infty }^\infty {\sum\limits_{n = - \infty }^\infty {\sum\limits_{m = - \infty }^\infty {\hat R_x^{\alpha {\text{ + }}m/{T_0}}(k{T_0})} {e^{ - j2\pi k{T_0}(f - \frac{m}{ {2{T_0}}} - \frac{n}{ { {T_0}}})}}} } =k=−∞∑∞n=−∞∑∞m=−∞∑∞R^xα + m/T0(kT0)e−j2πkT0(f−2T0m−T0n)
k T 0 → τ ‾ ‾ 1 T 0 ∫ − ∞ ∞ ∑ m = − ∞ ∞ R ^ x α + m / T 0 ( τ ) e − j 2 π ( f − m 2 T 0 − n T 0 ) τ d τ {\underline{\underline {k{T_0} \to \tau }} } \frac{1}{ { {T_0}}}\int_{ - \infty }^\infty {\sum\limits_{m = - \infty }^\infty {\hat R_x^{\alpha {\text{ + }}m/{T_0}}(\tau ){e^{ - j2\pi (f - \frac{m}{ {2{T_0}}} - \frac{n}{ { {T_0}}})\tau }}} d\tau } kT0→τT01∫−∞∞m=−∞∑∞R^xα + m/T0(τ)e−j2π(f−2T0m−T0n)τdτ
= 1 T 0 ∑ n = − ∞ ∞ ∑ m = − ∞ ∞ ∫ − ∞ ∞ R ^ x α + m / T 0 ( τ ) e − j 2 π ( f − m 2 T 0 − n T 0 ) τ d τ = \frac{1}{ { {T_0}}}\sum\limits_{n = - \infty }^\infty {\sum\limits_{m = - \infty }^\infty {\int_{ - \infty }^\infty {\hat R_x^{\alpha {\text{ + }}m/{T_0}}(\tau ){e^{ - j2\pi (f - \frac{m}{ {2{T_0}}} - \frac{n}{ { {T_0}}})\tau }}d\tau } } } =T01n=−∞∑∞m=−∞∑∞∫−∞∞R^xα + m/T0(τ)e−j2π(f−2T0m−T0n)τdτ
= 1 T 0 ∑ n = − ∞ ∞ ∑ m = − ∞ ∞ S ^ x α + m / T 0 ( f − m 2 T 0 − n T 0 ) = \frac{1}{ { {T_0}}}\sum\limits_{n = - \infty }^\infty {\sum\limits_{m = - \infty }^\infty {\hat S_x^{\alpha {\text{ + }}m/{T_0}}(f - \frac{m}{ {2{T_0}}} - \frac{n}{ { {T_0}}})} } =T01n=−∞∑∞m=−∞∑∞S^xα + m/T0(f−2T0m−T0n) (21式)