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I'm reading from Communication Systems, 5th Edition by Simon Haykin. At some point we derived that enter image description here

Which is how the effect due to AWGN is minimized. Meanwhile, to minimize ISI we need $P(f)$ to be raised cosine spectrum which is attainable under the choices of $H(f)=\sqrt {P(f)}$ and $G(f)=\sqrt {P(f)}$.

My problem is that under these chis choice of $H(f)$ it no longer seems like the optimal version that minimized the probability of error due to noise. So either one of these is true:

1 - The new choice of $H(f)$ somehow as well minimizes the probability of error due to noise.

2 - It doesn't. We choose $H(f)$ based on whether we're interested in minimizing noise or ISI.

In case, it's 1 then I have no idea why that would be the case.

The book's notation is that $P(f)$ is the pulse in frequency after passing through the channel and the receive filter. $G(f)$ is the transmit filter (pulse shape) and $H(f)$ is the receive filter.

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Under condition of white noise, the optimum SNR is reached by using a matched filter in the receiver that is matched to the transmit signal pulse (assuming we have properly equalized for channel distortions so that only white noise remains). So given the use of a Raised Cosine filter to eliminate ISI and constrain the transmitter bandwidth (which is the only reason we do the pulse shaping filter), we can achieve bandwidth constraint, zero-ISI AND matched filtering by using a ROOT-raised cosine filter in the transmitter, and matching that with a root-raised cosine filter in the receiver. This comes at a penalty in achieved rejection at the transmitter output as depicted in the graphic below (RRC would be at transmitter and RC would be after second RRC in the receiver), but we can design the filter accordingly and then achieve these objectives.

Raised and Root-Raised Filter

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  • $\begingroup$ If we match $H(f)$ to that then the final pulse becomes $$P(f)=k(G(f))^2e^{-j2\pi fT}$$ rather than $$P(f)=k(G(f))^2$$ is the delay there okay? I also couldn't understand what you meant by 3 db penalty. $\endgroup$
    – Essam
    Apr 20, 2022 at 8:18
  • $\begingroup$ There is no issue with a linear phase delay: see dsp.stackexchange.com/questions/31726/…. My 3 dB was a mistake, let me fix that to see if it's clearer to you. $\endgroup$ Apr 20, 2022 at 11:46

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