Bussgang theorem

Let $\backslash left\backslash \{X(t)\backslash right\backslash \}$ be a zero-mean stationary Gaussian random process and $\backslash left\; \backslash \{\; Y(t)\; \backslash right\backslash \}\; =\; g(X(t))$ where $g(\backslash cdot)$ is a nonlinear amplitude distortion.

If $R\_X(\backslash tau)$ is the autocorrelation function of $\backslash left\backslash \{\; X(t)\; \backslash right\backslash \}$, then the cross-correlation function of $\backslash left\backslash \{\; X(t)\; \backslash right\backslash \}$ and $\backslash left\backslash \{\; Y(t)\; \backslash right\backslash \}$ is

:$R\_\{XY\}(\backslash tau)\; =\; CR\_X(\backslash tau),$

where $C$ is a constant that depends only on $g(\backslash cdot)$.

It can be further shown that

: $C\; =\; \backslash frac\{1\}\{\backslash sigma^3\backslash sqrt\{2\backslash pi\}\}\backslash int\_\{-\backslash infty\}^\backslash infty\; ug(u)e^\{-\backslash frac\{u^2\}\{2\backslash sigma^2\}\}\; \backslash ,\; du.$

It is a property of the two-dimensional normal distribution that the joint density of $y\_1$ and $y\_2$ depends only on their covariance and is given explicitly by the expression

: $p(y\_1,y\_2)\; =\; \backslash frac\{1\}\{2\; \backslash pi\; \backslash sqrt\{1-\backslash rho^2\}\}\; e^\{-\backslash frac\{y\_1^2\; +\; y\_2^2\; -\; 2\; \backslash rho\; y\_1\; y\_2\}\{2(1-\backslash rho^2)\}\}$

where $y\_1$ and $y\_2$ are standard Gaussian random variables with correlation $\backslash phi\_\{y\_1y\_2\}=\backslash rho$.

Assume that $r\_2\; =\; Q(y\_2)$, the correlation between $y\_1$ and $r\_2$ is,

: $\backslash phi\_\{y\_1r\_2\}\; =\; \backslash frac\{1\}\{2\; \backslash pi\; \backslash sqrt\{1-\backslash rho^2\}\}\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; y\_1\; Q(y\_2)\; e^\{-\backslash frac\{y\_1^2\; +\; y\_2^2\; -\; 2\; \backslash rho\; y\_1\; y\_2\}\{2(1-\backslash rho^2)\}\}\; \backslash ,\; dy\_1\; dy\_2$.

Since

: $\backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; y\_1\; e^\{-\backslash frac\{1\}\{2(1-\backslash rho^2)\}\; y\_1^2\; +\; \backslash frac\{\backslash rho\; y\_2\}\{1-\backslash rho^2\}\; y\_1\; \}\; \backslash ,\; dy\_1\; =\; \backslash rho\; \backslash sqrt\{2\; \backslash pi\; (1-\backslash rho^2)\}\; y\_2\; e^\{\; \backslash frac\{\backslash rho^2\; y\_2^2\}\{2(1-\backslash rho^2)\}\; \}$,

the correlation $\backslash phi\_\{y\_1\; r\_2\}$ may be simplified as

: $\backslash phi\_\{y\_1\; r\_2\}\; =\; \backslash frac\{\backslash rho\}\{\backslash sqrt\{2\; \backslash pi\}\}\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; y\_2\; Q(y\_2)\; e^\{-\backslash frac\{y\_2^2\}\{2\}\}\; \backslash ,\; dy\_2$.

The integral above is seen to depend only on the distortion characteristic $Q()$ and is independent of $\backslash rho$.

Remembering that $\backslash rho=\backslash phi\_\{y\_1\; y\_2\}$, we observe that for a given distortion characteristic $Q()$, the ratio $\backslash frac\{\backslash phi\_\{y\_1\; r\_2\}\}\{\backslash phi\_\{y\_1\; y\_2\}\}$ is $K\_Q=\backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\}\}\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; y\_2\; Q(y\_2)\; e^\{-\backslash frac\{y\_2^2\}\{2\}\}\; \backslash ,\; dy\_2$.

Therefore, the correlation can be rewritten in the form

$\backslash phi\_\{y\_1\; r\_2\}\; =\; K\_Q\; \backslash phi\_\{y\_1\; y\_2\}$.

If $Q(x)\; =\; \backslash text\{sign\}(x)$, or called one-bit quantization, then $K\_Q=\; \backslash frac\{2\}\{\backslash sqrt\{2\backslash pi\}\}\; \backslash int\_\{0\}^\{\backslash infty\}\; y\_2\; e^\{-\backslash frac\{y\_2^2\}\{2\}\}\; \backslash ,\; dy\_2\; =\; \backslash sqrt\{\backslash frac\{2\}\{\backslash pi\}\}$.

{{Cite journal|last=Vleck|first=J. H. Van|date=1966|title=The Spectrum of Clipped Noise|url=|journal=Radio Research Laboratory Report of Harvard University|volume=54 |issue=51|page=2 |doi=10.1109/PROC.1966.4567 |bibcode=1966IEEEP..54....2V }}{{Cite journal|last1=Vleck|first1=J. H. Van|last2=Middleton|first2=D.|date=January 1966|title=The spectrum of clipped noise|url=https://ieeexplore.ieee.org/document/1446497|journal=Proceedings of the IEEE|volume=54|issue=1|pages=2–19|doi=10.1109/PROC.1966.4567|bibcode=1966IEEEP..54....2V |issn=1558-2256}}{{Cite journal|last=Price|first=R.|date=June 1958|title=A useful theorem for nonlinear devices having Gaussian inputs|url=https://ieeexplore.ieee.org/document/1057444/;jsessionid=p7xQfWaG1zLvg43lhpnzWz6pUrVRPwQvTk_5Z-KclUPBlln2I6MR!144025597|journal=IRE Transactions on Information Theory|volume=4|issue=2|pages=69–72|doi=10.1109/TIT.1958.1057444|bibcode=1958ITIT....4...69P |issn=2168-2712}}

If the two random variables are both distorted, i.e., $r\_1\; =\; Q(y\_1),\; r\_2\; =\; Q(y\_2)$, the correlation of $r\_1$ and $r\_2$ is

$\backslash phi\_\{r\_1\; r\_2\}=\backslash int\_\{-\backslash infty\}^\{\backslash infty\}\backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; Q(y\_1)\; Q(y\_2)\; p(y\_1,\; y\_2)\; \backslash ,\; dy\_1\; dy\_2$.

$\backslash phi\_\{r\_1\; r\_2\}=\backslash frac\{1\}\{2\backslash pi\; \backslash sqrt\{1-\backslash rho^2\}\}\; \backslash left[\; \backslash int\_\{0\}^\{\backslash infty\}\; \backslash int\_\{0\}^\{\backslash infty\}\; e^\{-\backslash alpha\}\; \backslash ,\; dy\_1\; dy\_2\; +\; \backslash int\_\{-\backslash infty\}^\{0\}\; \backslash int\_\{-\backslash infty\}^\{0\}\; e^\{-\backslash alpha\}\; \backslash ,\; dy\_1\; dy\_2\; -\; \backslash int\_\{0\}^\{\backslash infty\}\; \backslash int\_\{-\backslash infty\}^\{0\}\; e^\{-\backslash alpha\}\; \backslash ,\; dy\_1\; dy\_2\; -\; \backslash int\_\{-\backslash infty\}^\{0\}\; \backslash int\_\{0\}^\{\backslash infty\}\; e^\{-\backslash alpha\}\; \backslash ,\; dy\_1\; dy\_2\; \backslash right]$

Noticing that

$\backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; p(y\_1,y\_2)\; \backslash ,\; dy\_1\; dy\_2$

= \frac{1}{2\pi \sqrt{1-\rho^2}}

\left[ \int_{0}^{\infty} \int_{0}^{\infty} e^{-\alpha} \, dy_1 dy_2

+ \int_{-\infty}^{0} \int_{-\infty}^{0} e^{-\alpha} \, dy_1 dy_2

+ \int_{0}^{\infty} \int_{-\infty}^{0} e^{-\alpha} \, dy_1 dy_2

+ \int_{-\infty}^{0} \int_{0}^{\infty} e^{-\alpha} \, dy_1 dy_2 \right]=1,

and $\backslash int\_\{0\}^\{\backslash infty\}\; \backslash int\_\{0\}^\{\backslash infty\}\; e^\{-\backslash alpha\}\; \backslash ,\; dy\_1\; dy\_2\; =\; \backslash int\_\{-\backslash infty\}^\{0\}\; \backslash int\_\{-\backslash infty\}^\{0\}\; e^\{-\backslash alpha\}\; \backslash ,\; dy\_1\; dy\_2$, $\backslash int\_\{0\}^\{\backslash infty\}\; \backslash int\_\{-\backslash infty\}^\{0\}\; e^\{-\backslash alpha\}\; \backslash ,\; dy\_1\; dy\_2$

= \int_{-\infty}^{0} \int_{0}^{\infty} e^{-\alpha} \, dy_1 dy_2,

we can simplify the expression of $\backslash phi\_\{r\_1r\_2\}$ as

$\backslash phi\_\{r\_1\; r\_2\}=\backslash frac\{4\}\{2\backslash pi\; \backslash sqrt\{1-\backslash rho^2\}\}\; \backslash int\_\{0\}^\{\backslash infty\}\; \backslash int\_\{0\}^\{\backslash infty\}\; e^\{-\backslash alpha\}\; \backslash ,\; dy\_1\; dy\_2-1$

$\backslash phi\_\{r\_1\; r\_2\}\; =\backslash frac\{4\}\{2\backslash pi\; \backslash sqrt\{1-\backslash rho^2\}\}\; \backslash int\_\{0\}^\{\backslash pi/2\}\; \backslash int\_\{0\}^\{\backslash infty\}\; e^\{-\backslash frac\{R^2\; -\; 2R^2\; \backslash rho\; \backslash cos\; \backslash theta\; \backslash sin\; \backslash theta\; \backslash \; \}\{2(1-\backslash rho^2)\}\}\; R\; \backslash ,\; dR\; d\backslash theta-1=\backslash frac\{4\}\{2\backslash pi\; \backslash sqrt\{1-\backslash rho^2\}\}\; \backslash int\_\{0\}^\{\backslash pi/2\}\; \backslash int\_\{0\}^\{\backslash infty\}\; e^\{-\backslash frac\{R^2\; (1-\backslash rho\; \backslash sin\; 2\backslash theta\; )\}\{2(1-\backslash rho^2)\}\}\; R\; \backslash ,\; dR\; d\backslash theta\; -1$

.

Integration gives

$\backslash phi\_\{r\_1\; r\_2\}=\backslash frac\{2\backslash sqrt\{1-\backslash rho^2\}\}\{\backslash pi\}\; \backslash int\_\{0\}^\{\backslash pi/2\}\; \backslash frac\{d\backslash theta\}\{1-\backslash rho\; \backslash sin\; 2\backslash theta\}\; -\; 1=\; -\; \backslash frac\{2\}\{\backslash pi\}\; \backslash arctan\; \backslash left(\; \backslash frac\{\backslash rho-\backslash tan\backslash theta\}\; \{\backslash sqrt\{1-\backslash rho^2\}\}\; \backslash right)\; \backslash Bigg|\_\{0\}^\{\backslash pi/2\}\; -1\; =\backslash frac\{2\}\{\backslash pi\}\; \backslash arcsin(\backslash rho)$

，

Given two jointly normal random variables $y\_1$ and $y\_2$ with joint probability function

$\{\backslash displaystyle\; p(y\_\{1\},y\_\{2\})=\{\backslash frac\; \{1\}\{2\backslash pi\; \{\backslash sqrt\; \{1-\backslash rho\; ^\{2\}\}\}\}\}e^\{-\{\backslash frac\; \{y\_\{1\}^\{2\}+y\_\{2\}^\{2\}-2\backslash rho\; y\_\{1\}y\_\{2\}\}\{2(1-\backslash rho\; ^\{2\})\}\}\}\}$,

$I(\backslash rho)=E(g(y\_1,y\_2))=\backslash int\_\{-\backslash infty\}^\{+\backslash infty\}\; \backslash int\_\{-\backslash infty\}^\{+\backslash infty\}\; g(y\_1,\; y\_2)\; p(y\_1,\; y\_2)\; \backslash ,\; dy\_1\; dy\_2$

$\backslash frac\{\backslash partial^n\; I(\backslash rho)\}\{\backslash partial\; \backslash rho^n\}=\backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}$

\frac{\partial ^{2n} g(y_1, y_2)}{\partial y_1^n \partial y_2^n} p(y_1, y_2) \, dy_1 dy_2

=E \left(\frac{\partial ^{2n} g(y_1, y_2)}{\partial y_1^n \partial y_2^n} \right).

$\backslash Phi(\backslash omega\_1,\; \backslash omega\_2)=\backslash int\_\{-\backslash infty\}^\{\backslash infty\}\backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; p(y\_1,\; y\_2)$

e^{j (\omega_1 y_1 + \omega_2 y_2 )} \, dy_1 dy_2

= \exp \left\{-\frac{\omega_1^2 + \omega_2^2 + 2\rho \omega_1 \omega_2}{2} \right\}.

$p(y\_1,\; y\_2)\; =\; \backslash frac\{1\}\{4\; \backslash pi^2\}\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; \backslash Phi(\backslash omega\_1,\; \backslash omega\_2)$

e^{-j (\omega_1 y_1 + \omega_2 y_2)} \, d\omega_1 d\omega_2

=\frac{1}{4 \pi^2} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}

\exp \left\{-\frac{\omega_1^2 + \omega_2^2 + 2\rho \omega_1 \omega_2}{2} \right\}

e^{-j (\omega_1 y_1 + \omega_2 y_2)} \, d\omega_1 d\omega_2.

$\backslash begin\{align\}$

\frac{\partial^n I(\rho)}{\partial \rho^n} & =

\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} g(y_1, y_2) p(y_1, y_2) \, dy_1 dy_2 \\

& =

\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} g(y_1, y_2)

\left(\frac{1}{4 \pi^2} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{\partial^ {n}\Phi(\omega_1, \omega_2)}{\partial \rho^n}

e^{-j(\omega_1 y_1 + \omega_2 y_2)} \, d\omega_1 d\omega_2 \right)

\, dy_1 dy_2 \\

& =

\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} g(y_1, y_2)

\left(\frac{(-1)^n}{4 \pi^2} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \omega_1^n \omega_2^n \Phi(\omega_1, \omega_2)

e^{-j(\omega_1 y_1 + \omega_2 y_2)} \, d\omega_1 d\omega_2 \right)

\, dy_1 dy_2 \\

& =

\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} g(y_1, y_2)

\left(\frac{1}{4 \pi^2} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \Phi(\omega_1, \omega_2)

\frac{\partial^{2n} e^{-j(\omega_1 y_1 + \omega_2 y_2)}}{\partial y_1^n \partial y_2^n} \, d\omega_1 d\omega_2 \right)

\, dy_1 dy_2 \\

& =

\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} g(y_1, y_2)

\frac{\partial^{2n} p(y_1, y_2)}{\partial y_1^n \partial y_2^n}

\, dy_1 dy_2 \\

\end{align}

$\backslash begin\{align\}$

\frac{\partial^n I(\rho)}{\partial \rho^n} & =

\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} g(y_1, y_2)

\frac{\partial^{2n} p(y_1, y_2)}{\partial y_1^n \partial y_2^n}

\, dy_1 dy_2 \\

& = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}

\frac{\partial^{2} g(y_1, y_2)}{\partial y_1 \partial y_2}

\frac{\partial^{2n-2} p(y_1, y_2)}{\partial y_1^{n-1} \partial y_2^{n-1}}

\, dy_1 dy_2

\\

&=\cdots \\

&=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty}

\frac{\partial ^{2n} g(y_1, y_2)}{\partial y_1^n \partial y_2^n} p(y_1, y_2) \, dy_1 dy_2

\end{align}

If $g(y\_1,\; y\_2)\; =\; \backslash text\{sign\}(y\_1)\; \backslash text\{sign\}\; (y\_2)$, then $\backslash frac\{\backslash partial^2\; g(y\_1,\; y\_2)\}\{\backslash partial\; y\_1\; \backslash partial\; y\_2\}\; =\; 4\; \backslash delta(y\_1)\; \backslash delta(y\_2)$ where $\backslash delta()$ is the Dirac delta function.

Substituting into Price's Theorem, we obtain,

$\backslash frac\{\backslash partial\; E(\backslash text\{sign\}\; (y\_1)\; \backslash text\{sign\}(y\_2))\}\{\backslash partial\; \backslash rho\}\; =\; \backslash frac\{\backslash partial\; I(\backslash rho)\}\{\backslash partial\; \backslash rho\}=\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}$

4 \delta(y_1) \delta(y_2) p(y_1, y_2) \, dy_1 dy_2=\frac{2}{\pi \sqrt{1-\rho^2}}.

$E\; \backslash left(\backslash text\{sign\}(y\_1)\; \backslash text\{sign\}(y\_2)\; \backslash right)\; =\; I(\backslash rho)=\backslash frac\{2\}\{\backslash pi\}\; \backslash int\_\{0\}^\{\backslash rho\}\; \backslash frac\{1\}\{\backslash sqrt\{1-\backslash rho^2\}\}\; \backslash ,\; d\backslash rho=\backslash frac\{2\}\{\backslash pi\}\; \backslash arcsin(\backslash rho)$,

This theorem implies that a simplified correlator can be designed.{{clarify|reason=compared to what?|date=December 2010}} Instead of having to multiply two signals, the cross-correlation problem reduces to the gating{{clarify|reason=undefined|date=December 2010}} of one signal with another.{{citation needed|date=December 2010}}

{{reflist}}

• E.W. Bai; V. Cerone; D. Regruto (2007) [https://web.archive.org/web/20110710123549/http://diegoregruto.com/P14.pdf "Separable inputs for the identification of block-oriented nonlinear systems"], Proceedings of the 2007 American Control Conference (New York City, July 11–13, 2007) 1548–1553

{{DEFAULTSORT:Bussgang Theorem}}

Category:Theorems about stochastic processes