What is the effect of a passband slope on a BPSK constellation? (This is a companion question to the related "DSP Puzzle" for QPSK)

Specifically given a linear slope (in magnitude, not dB), and no phase distortion (linear phase or even zero-phase) how would the following constellation appear after being passed through such a channel?

BPSK Constellation

This is a "DSP Puzzle". Please preface your answer with "spoiler" notation by typing the following two characters first ">!" (see other DSP Puzzle tagged questions for examples of what this means and does). Please do not confirm or get to your result with a simulation but post your best conjecture. An analysis is allowed and welcomed but not required. I am looking for the simplest intuitive explanation as to what the resulting shape, line or curve would be for the resulting samples in the constellation with a quantified range related to the slope in the channel (α).

It is common to get a passband slope in wide-band single-carrier solutions (D/A converter Sinc with digital IF, analog filters, amplifiers and transmission lines). Recognizing this possible distortion source from the constellation alone is a useful signal forensics tool.

Here are additional simplifying assumptions:

The BPSK waveform is perfectly pulse shaped to have an occupied bandwidth B with a rectangular magnitude shape in the frequency domain, and the data is independent and randomly distributed uniformly over all symbols in the constellation. The waveform as created prior to the channel distortion has zero inter-symbol interference (ISI).

I am not looking for detailed distributions, EVM, etc, just what the final shape of the constellation would be after many, many symbols.

You need not know anything else about the receiver. In this case all pulse shaping is done for zero ISI at the transmitter to achieve the constellation as shown with no other noise added or subsequent processing.

If this helps to clarify, consider the perfect root-raised cosine waveform at the transmitter in cascade with the perfect root-raised cosine matched filter that would normally be in the receiver which together would provide the desired zero-ISI raised cosine response and for a band-limited signal. This would have the constellation as shown when that waveform is sampled at the optimum symbol sampling locations in time. In this case the pulse shaping filter is an infinite duration Sinc function with scaled complex impulses at the input representing the desired BPSK symbols for a random stream of symbols. The result would be a rectangular magnitude B Hz wide for a symbol rate of B Symbols/sec.

Adding some hints since this hasn’t been answered yet:

  • There will be no ISI in this case! Why?

  • Use the fundamental properties of the Fourier Transform, in particular decomposing the passband into even and odd functions.

  • $\begingroup$ Very interesting puzzles! (I can only comment, I’m tempted but I don’t know where to start) $\endgroup$
    – Gilles
    Oct 13, 2022 at 18:47
  • $\begingroup$ @Gilles start with the basic fundamental properties of Fourier Transforms. Assuming that interests you, your review and recollection of that will make the final result more fun and interesting (and illuminate its practical use toward intuition with DSP). This also assumes you have familiarity with basic wireless communication waveforms such as BPSK and QPSK and the Nyquist Criterion for Zero-ISI - and know what all those terms and abbreviations mean. $\endgroup$ Oct 13, 2022 at 18:55
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    $\begingroup$ Thanks @DanBoschen , I’ll give it a try. $\endgroup$
    – Gilles
    Oct 13, 2022 at 22:08


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