# 0.5 Discussion and future work

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## Discussion and future work

In any digital communication scheme there exist design parameters that are independent of the scheme itself: digital sampling rate and system baud rate. These parameters directly determine the values of $N$ and $B$ in the DMT communication scheme. It is therefore plausible to have many different values of $N$ and $B$ .

From the results, the FFAST algorithm outperformed the mixed-radix FFT for all signals with lengths greater than ${2}^{14}$ , with the additional condition that $2B<{N}^{\frac{1}{3}}$ . Thus, if a communication scheme takes at least ${2}^{14}$ samples in the time it takes to send $B=⌊\frac{1}{2}{2}^{\left(14,×,\frac{1}{3}\right)}⌋=12$ simultaneous bits, a sparse FFT will require fewer computations than the tested existing frequency domain schemes, reducing receiver bottlenecking, and will therefore be practical to use with some system designs.

Ultimately, the best way to address the viability of the sparse FFT (and therefore expand on the goals of this paper) is to physically implement a communications system compatible with the algorithm itself. While this paper has attempted to address concerns about the possibility of implementation there are still further matters to consider before a physical interpretation of this algorithm can arise.

The first and foremost matter to consider is that the version of the FFAST algorithm that we implemented only works when the signal is exactly sparse. Practically, the communications scheme would have to work with a noisy channel. A noisy version of the FFAST algorithm does exist [link] , however, and should be tested to verify our results in a noisy case.

Second, it would be useful to devise a more efficient communication scheme that takes into consideration the fact that the sparse FFT converges even though it does not“know” where the signal is not sparse. In our experiment, we allotted the first $B$ “slots” of the frequency domain of our signal to the sinusoids, a way to guarantee that the frequency sparsity of our signal would not exceed $2B$ . This does not take into consideration that for any given $N$ there are

$\left(\genfrac{}{}{0pt}{}{\frac{N}{2}}{B}\right)>>{2}^{B}$

different ways to have a sparse signal of density $2B$ . Finding a coherent way of organizing these different possibilities and using them will give transmitted signals a much higher density and also allow for a higher baud rate of the system (in the example above, $B$ would be increased from 12 bits to 127 simultaneous bits!).

Ultimately, once these considerations are taken into account, a coherent sparse communication system seems much more plausible.

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