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I have changed the way that the BPSK modulation is added to the carrier signal to smooth inversions using a Raised Root Cosine shape. This has the effect of reducing the bandwidth of the signal and greatly reducing the sidebands. Spectral plots are included for before and after.
Note that, instead of smoothing the bit transitions on phase boundries, the inversion was moved to the center of the bit cell. This results in the same signal but shifted one half bit time. Since this is a self clocking signal, that should not make any difference in the decoded data, but the test decoder fails to correctly decode the signal. There seems to be some instability in it which needs to be addressed. It looks like an overflow of some type. This can be seen in the waveform plot in the screenshot attached.