Let $\Omega > 0$ and denote by $\mathcal{B}_\Omega$ the subspace of $L^2(\Bbb R)$ consisting of signals that are bandlimited to $(-\Omega, \Omega)$. Denote $\mathcal{L}_{\Omega} : L^2(\Bbb R) \rightarrow L^2(\Bbb R)$ the ideal lowpass filter that projects signals $x \in L^2(\Bbb R)$ into $\mathcal{B}_\Omega$: i.e., $\mathcal{L}_\Omega$ takes $x$ to the signal $y$ with

\begin{equation} \hat{y} =\begin{cases} \hat{x} & \text{for } |\omega| < \Omega\\ 0 & \text{otherwise} \end{cases} \end{equation}

A function $\gamma: \Bbb R \rightarrow \Bbb R$ is monotone increasing if $\gamma (t_2) > \gamma (t_1)$. The companding of a signal $x: \Bbb R \rightarrow \Bbb R$ by $\gamma$ is the signal $[\gamma \circ x](t) = \gamma(x(t))$.

  1. Denote $u(t) = sinc(2\Omega t)$. Find an explicit expression of $\mathcal{L}_\Omega (u)(t)$.

  2. If $x \in \mathcal{B}_\Omega$ and $\gamma$ is monotone increasing, then $\gamma \circ x$ is generally not bandlimited. In particular, $\mathcal{L}_\Omega (\gamma \circ x) \neq \gamma \circ x$. Suppose $x_1, x_2 \in \mathcal{B}_\Omega$ and $\mathcal{L}_\Omega (\gamma \circ x_1) = (\gamma \circ x_2)$. Show that $x_1 = x_2$ almost everywhere on $\Bbb R$.

My Attempt/Input

For the first part, since the sinc function is a low pass filter in the time domain, are they essentially just asking for its corresponding rectangular function?

As for the second part, I know I start with the integral of the product of $(x_1 - x_2)$ with $(\gamma \circ x_1 - \gamma \circ x_2)$ but not sure how to go about it.

  • $\begingroup$ Oops, that was a typo. $\endgroup$ – John Nov 4 '13 at 15:14
  • $\begingroup$ I assume for $t_2 > t_1$? $\endgroup$ – Peter K. Nov 4 '13 at 15:45
  • $\begingroup$ Yes, that is the case. $\endgroup$ – John Nov 4 '13 at 15:46

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