Bivariate normal expectation of the sinus cardinal

Question

I would like to get an analytical expression for $$\mathbb{E}\left(\frac{\sin(aX)}{aX}\frac{\sin(bY)}{bY}\right)$$ or at least an analytical approximation thereof, when $a,b$ are positive reals, and

$$\begin{pmatrix}X\\Y\end{pmatrix} \sim \mathcal{N}(\mu,\Sigma) $$ where

$$\mu \equiv \begin{pmatrix}x^\circ \\ y^\circ\end{pmatrix} \quad \Sigma \equiv \begin{pmatrix} 1 & \rho \\ \rho & 1 \end{pmatrix}$$

Using either the probability density function or the characteristic function to compute the expectation leads me to an expression with a single integral, but I can't go further than this.

Answer 1

Here's some R code that does a numerical approximation (I assume it is clear how to change this for other values of the parameters):

require(MASS)
n<-10000
mux<-0
muy<-0
rho<-0.5
sig<-matrix(c(1,rho,rho,1), ncol=2)
dta<-mvrnorm(n = n, mu=c(mux, muy), Sigma=sig)

a<-1
b<- -1

appr<-mean(sin(a*dta[,1])/(a*dta[,1]) * sin(b*dta[,2])/(b*dta[,2]))
appr