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I'm implementing DBPSK (differential binary phase-shift keying) over audio (44.1kHz sample-rate) using software. Eventually I'll use multiple audio frequencies at once, so I'm using FFTs and inverse FFTs.

As expected, as the window size (symbol length) gets smaller, ISI becomes more and more of a problem and the BER (bit error rate) increases.

It seems like using RRC (root-raised cosine) filters is an established way to reduce ISI.

If as part of my data transmission I send out a special known signal that can be detected, then presumably I can determine the impulse response of the channel and apply an inverse transformation to re-construct (ignoring noise) the original signal without multi-path distortion. Why is RRC useful if this transformation can be done?

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    $\begingroup$ I think historically, the computations for system identification for the use of inverting the effects of a channel were too costly as compared with shaping the data in advance to avoid ISI. $\endgroup$
    – user2718
    Commented Jan 30, 2013 at 16:07
  • $\begingroup$ As your question indicates, just being smart about the pulse shape that you choose is a much simpler process than trying to perform online system identification and deconvolution. If you don't expect to have some unknown channel frequency response when your system is operating, there's no need to design in that capability. You can instead characterize ISI due to pulse shaping at design time and accommodate it appropriately in your receiver. $\endgroup$
    – Jason R
    Commented Jan 30, 2013 at 18:14

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There are three reasons to avoid ISI through pulse shaping rather than correcting it via channel correction methods.

  1. As Bruce pointed out, it is a simpler solution and requires fewer computations.
  2. Channel correction usually amplifies the received noise.
  3. Received noise will corrupt the channel model, making the ISI removal imperfect to some degree.
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I think it can be misleading to think RRC are used to eliminate ISI. The only purpose of Raised Cosine (RC) and Root-Raised Cosine (RRC) filtering in the transmitter specifically is to constrain the transmission bandwidth when spectral efficiency and out of band emissions are concerns (as they are in most wireless communication solutions!). They achieve this purpose without introducing ISI in the process.

If we must reduce bandwidth, improper filtering can introduce ISI. A Raised Cosine filter is one solution that constrains the bandwidth as needed with zero ISI. By splitting the filter into two matches RC filters, we achieve the desired filtering of the spectrum at the transmitter and a matched filter in the receiver (which then optimizes SNR under white noise conditions).

If we have an equalizer in the receiver, we could theoretically use other filtering options to achieve the bandwidth reduction, knowing that we still want to have a matched filter in the receiver to optimize SNR. However the effective delay spread of the filtering will rss with the delay spread of the channel and therefore increase the required length and therefore complexity of the equalizer. Another way to say this is we are losing degrees of freedom in the equalizer by (needlessly) introducing ISI when an RRC solution can easily provide what we need. So, given we need a filter in the transmitter (and that need has nothing to do with ISI, we just don’t want to add ISI in the process if we can help it), and given we want a matched filter in the receiver, the RRC filter achieves both of these goals.

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  • $\begingroup$ I like this response. Given that the channel will have some finite bandwidth and accompanying IR that will often be significant compared to symbol period, any fixed filtering at sender or receiver would typically increase the effective IR length that an equalizer will try to invert? There is little reason to expect that those nice zeros will survive a radio channel, but the effective RC smearing will. So would it be better to pick a Tx filter that only minimize effective smearing, ignoring ISI, and let the Rx equalizer do its thing? $\endgroup$
    – Knut Inge
    Commented Sep 21, 2022 at 17:59
  • $\begingroup$ @KnutInge Often the impulse response of the filtering will be significant or even exceed that of the channel. Consider a single carrier terrestrial system with 2 MHz symbol rate at a 2 GHz carrier. Typical delay spreads at this carrier frequency can range from 100 ns to 2 us or longer (the shortest for open space and the longest for urban environments). The symbol rate is 0.5 us and the filter could span 20 symbols or more depending on our filtering objectives. $\endgroup$
    – Dan Boschen
    Commented Sep 21, 2022 at 19:26

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