conversion of 1d signal to 2d signal for CNN operation
There are a couple things with that question. 

1) Why do you need 2D convolutions exactly? You could think of a 1D signal as an $Nx1$ 2d signal if you like ;) Then you can design your convolutional filters to be $7x1$, $5x1$, $3x1$ etc. The pooling you only apply in the x-direction. And there you go. It's a 2D convolutional neural network, which works on 1D signal. Just kidding, it just a CNN.

2) Simple fully-connected deep learning neural networks are better for handling 1D data, as far as I know. Of course it's best if your data is evenly sampled. CNN comes handy when the fully connected structure explodes the computation. In your case, that should not be a big issue.

If your 1d data vector is too large, just try subsampling, instead of a convolutional architecture.

3) Converting a 1d data to 2d is probably valid only if you know in advance that this 1d manifold carries non-uniform neighborhood information, which could be represented with a 2D matrix with nearby connections. The paper probably inserts a layer to the network which re-arranges the indices into a 32x32 matrix. The weights or the indices where to put each value maybe learnt by the network itself. You would need to write the backprop for this though.