Bussgang theorem

In mathematics, the Bussgang theorem is a theorem of stochastic analysis. The theorem states that the cross-correlation between a Gaussian signal before and after it has passed through a nonlinear operation are equal to the signals auto-correlation up to a constant. It was first published by Julian J. Bussgang in 1952 while he was at the Massachusetts Institute of Technology.

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

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


 * $$ R_{XY}(\tau) = CR_X(\tau), $$

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

It can be further shown that


 * $$ C = \frac{1}{\sigma^3\sqrt{2\pi}}\int_{-\infty}^\infty ug(u)e^{-\frac{u^2}{2\sigma^2}} \, du. $$

Derivation for One-bit Quantization
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


 * $$ p(y_1,y_2) = \frac{1}{2 \pi \sqrt{1-\rho^2}} e^{-\frac{y_1^2 + y_2^2 - 2 \rho y_1 y_2}{2(1-\rho^2)}} $$

where $$ y_1 $$ and $$ y_2 $$ are standard Gaussian random variables with correlation $$ \phi_{y_1y_2}=\rho $$.

Assume that $$ r_2 = Q(y_2) $$, the correlation between $$ y_1 $$ and $$ r_2 $$ is,


 * $$ \phi_{y_1r_2} = \frac{1}{2 \pi \sqrt{1-\rho^2}} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} y_1 Q(y_2) e^{-\frac{y_1^2 + y_2^2 - 2 \rho y_1 y_2}{2(1-\rho^2)}} \, dy_1 dy_2 $$.

Since


 * $$ \int_{-\infty}^{\infty} y_1 e^{-\frac{1}{2(1-\rho^2)} y_1^2 + \frac{\rho y_2}{1-\rho^2} y_1 } \, dy_1 = \rho \sqrt{2 \pi (1-\rho^2)} y_2 e^{ \frac{\rho^2 y_2^2}{2(1-\rho^2)} } $$,

the correlation $$\phi_{y_1 r_2}$$ may be simplified as


 * $$ \phi_{y_1 r_2} = \frac{\rho}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} y_2 Q(y_2) e^{-\frac{y_2^2}{2}} \, dy_2 $$.

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

Remembering that $$\rho=\phi_{y_1 y_2}$$, we observe that for a given distortion characteristic $$Q$$, the ratio $$\frac{\phi_{y_1 r_2}}{\phi_{y_1 y_2}}$$ is $$K_Q=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} y_2 Q(y_2) e^{-\frac{y_2^2}{2}} \, dy_2$$.

Therefore, the correlation can be rewritten in the form"$\phi_{y_1 r_2} = K_Q \phi_{y_1 y_2}$."The above equation is the mathematical expression of the stated "Bussgang‘s theorem".

If $$Q(x) = \text{sign}(x)$$, or called one-bit quantization, then $$K_Q= \frac{2}{\sqrt{2\pi}} \int_{0}^{\infty} y_2 e^{-\frac{y_2^2}{2}} \, dy_2 = \sqrt{\frac{2}{\pi}}$$.

Arcsine law
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 "$\phi_{r_1 r_2}=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} Q(y_1) Q(y_2) p(y_1, y_2) \, dy_1 dy_2$."When $$Q(x) = \text{sign}(x)$$, the expression becomes,"$\phi_{r_1 r_2}=\frac{1}{2\pi \sqrt{1-\rho^2}} \left[ \int_{0}^{\infty} \int_{0}^{\infty} e^{-\alpha} \, dy_1 dy_2 + \int_{-\infty}^{0} \int_{-\infty}^{0} e^{-\alpha} \, dy_1 dy_2 - \int_{0}^{\infty} \int_{-\infty}^{0} e^{-\alpha} \, dy_1 dy_2 - \int_{-\infty}^{0} \int_{0}^{\infty} e^{-\alpha} \, dy_1 dy_2 \right]$|undefined"where $$\alpha = \frac{y_1^2 + y_2^2 - 2\rho y_1 y_2}{2 (1-\rho^2)}$$.

Noticing that

,

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

we can simplify the expression of $$\phi_{r_1r_2}$$ as"$\phi_{r_1 r_2}=\frac{4}{2\pi \sqrt{1-\rho^2}} \int_{0}^{\infty} \int_{0}^{\infty} e^{-\alpha} \, dy_1 dy_2-1 $|undefined"Also, it is convenient to introduce the polar coordinate. It is thus found that

.

Integration gives $$\phi_{r_1 r_2}=\frac{2\sqrt{1-\rho^2}}{\pi} \int_{0}^{\pi/2} \frac{d\theta}{1-\rho \sin 2\theta} - 1= - \frac{2}{\pi} \arctan \left( \frac{\rho-\tan\theta} {\sqrt{1-\rho^2}} \right) \Bigg|_{0}^{\pi/2} -1 =\frac{2}{\pi} \arcsin(\rho) $$， This is called "Arcsine law", which was first found by J. H. Van Vleck in 1943 and republished in 1966. The "Arcsine law" can also be proved in a simpler way by applying Price's Theorem.

The function $$ f(x)=\frac{2}{\pi} \arcsin x $$ can be approximated as $$ f(x) \approx \frac{2}{\pi} x $$ when $$ x $$ is small.

Price's Theorem
Given two jointly normal random variables $$y_1$$ and $$y_2$$ with joint probability function "${\displaystyle p(y_{1},y_{2})={\frac {1}{2\pi {\sqrt {1-\rho ^{2}}}}}e^{-{\frac {y_{1}^{2}+y_{2}^{2}-2\rho y_{1}y_{2}}{2(1-\rho ^{2})}}}}$,|undefined"we form the mean"$I(\rho)=E(g(y_1,y_2))=\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} g(y_1, y_2) p(y_1, y_2) \, dy_1 dy_2$"of some function $$g(y_1,y_2)$$ of $$(y_1, y_2)$$. If $$g(y_1, y_2) p(y_1, y_2) \rightarrow 0$$ as $$(y_1, y_2) \rightarrow 0$$, then $$\frac{\partial^n I(\rho)}{\partial \rho^n}=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{\partial ^{2n} g(y_1, y_2)}{\partial y_1^n \partial y_2^n} p(y_1, y_2) \, dy_1 dy_2 =E \left(\frac{\partial ^{2n} g(y_1, y_2)}{\partial y_1^n \partial y_2^n} \right)$$. Proof. The joint characteristic function of the random variables $$y_1$$ and $$y_2$$ is by definition the integral $$\Phi(\omega_1, \omega_2)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} p(y_1, y_2) e^{j (\omega_1 y_1 + \omega_2 y_2 )} \, dy_1 dy_2 = \exp \left\{-\frac{\omega_1^2 + \omega_2^2 + 2\rho \omega_1 \omega_2}{2} \right\}$$. From the two-dimensional inversion formula of Fourier transform, it follows that $$p(y_1, y_2) = \frac{1}{4 \pi^2} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \Phi(\omega_1, \omega_2) e^{-j (\omega_1 y_1 + \omega_2 y_2)} \, d\omega_1 d\omega_2 =\frac{1}{4 \pi^2} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \exp \left\{-\frac{\omega_1^2 + \omega_2^2 + 2\rho \omega_1 \omega_2}{2} \right\} e^{-j (\omega_1 y_1 + \omega_2 y_2)} \, d\omega_1 d\omega_2$$. Therefore, plugging the expression of $$p(y_1, y_2)$$ into $$I(\rho)$$, and differentiating with respect to $$\rho$$, we obtain $$\begin{align} \frac{\partial^n I(\rho)}{\partial \rho^n} & = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} g(y_1, y_2) p(y_1, y_2) \, dy_1 dy_2 \\ & = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} g(y_1, y_2) \left(\frac{1}{4 \pi^2} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{\partial^ {n}\Phi(\omega_1, \omega_2)}{\partial \rho^n} e^{-j(\omega_1 y_1 + \omega_2 y_2)} \, d\omega_1 d\omega_2 \right) \, dy_1 dy_2 \\ & = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} g(y_1, y_2) \left(\frac{(-1)^n}{4 \pi^2} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \omega_1^n \omega_2^n \Phi(\omega_1, \omega_2) e^{-j(\omega_1 y_1 + \omega_2 y_2)} \, d\omega_1 d\omega_2 \right) \, dy_1 dy_2 \\ & = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} g(y_1, y_2) \left(\frac{1}{4 \pi^2} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \Phi(\omega_1, \omega_2) \frac{\partial^{2n} e^{-j(\omega_1 y_1 + \omega_2 y_2)}}{\partial y_1^n \partial y_2^n} \, d\omega_1 d\omega_2 \right) \, dy_1 dy_2 \\ & = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} g(y_1, y_2) \frac{\partial^{2n} p(y_1, y_2)}{\partial y_1^n \partial y_2^n} \, dy_1 dy_2 \\ \end{align}$$ After repeated integration by parts and using the condition at $$\infty$$, we obtain the Price's theorem. $$\begin{align} \frac{\partial^n I(\rho)}{\partial \rho^n} & = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} g(y_1, y_2) \frac{\partial^{2n} p(y_1, y_2)}{\partial y_1^n \partial y_2^n} \, dy_1 dy_2 \\ & = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{\partial^{2} g(y_1, y_2)}{\partial y_1 \partial y_2} \frac{\partial^{2n-2} p(y_1, y_2)}{\partial y_1^{n-1} \partial y_2^{n-1}} \, dy_1 dy_2 \\ &=\cdots \\ &=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{\partial ^{2n} g(y_1, y_2)}{\partial y_1^n \partial y_2^n} p(y_1, y_2) \, dy_1 dy_2 \end{align}$$

Proof of Arcsine law by Price's Theorem
If $$g(y_1, y_2) = \text{sign}(y_1) \text{sign} (y_2)$$, then $$\frac{\partial^2 g(y_1, y_2)}{\partial y_1 \partial y_2} = 4 \delta(y_1) \delta(y_2)$$ where $$\delta$$ is the Dirac delta function.

Substituting into Price's Theorem, we obtain, $$\frac{\partial E(\text{sign} (y_1) \text{sign}(y_2))}{\partial \rho} = \frac{\partial I(\rho)}{\partial \rho}= \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} 4 \delta(y_1) \delta(y_2) p(y_1, y_2) \, dy_1 dy_2=\frac{2}{\pi \sqrt{1-\rho^2}}$$. When $$\rho=0$$, $$I(\rho)=0$$. Thus"$E \left(\text{sign}(y_1) \text{sign}(y_2) \right) = I(\rho)=\frac{2}{\pi} \int_{0}^{\rho} \frac{1}{\sqrt{1-\rho^2}} \, d\rho=\frac{2}{\pi} \arcsin(\rho)$,|undefined"which is Van Vleck's well-known result of "Arcsine law".

Application
This theorem implies that a simplified correlator can be designed. Instead of having to multiply two signals, the cross-correlation problem reduces to the gating of one signal with another.