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For a set of $N$ orthogonal sequences $\textbf{A}^i = \{a_1^i,a_2^i,...,a_M^i\}, i =1 ..N$, the detection of these sequences can be done by calculating the inner product (dot product) of receive sequence $\textbf{Y}$ with all $N$ sequences $\textbf{A}^i$ to find the one maximizing the result:

$$\hat{i}=\text{argmax}_{1\leq i \leq N} \sum_{k=1}^M a^i_k y_k^* \tag{1}$$

This (1) works perfectly in AWGN channels: $$\textbf{Y} = \textbf{X} + \textbf{W} \tag{2}$$ where $\textbf{X} \in\{\textbf{A}^i \}_{1\leq i\leq N}$ and $\textbf{W}$ is complex Gaussian noise.

However, in fading, specifically Rayleigh fading, channels, (2) becomes $$\textbf{Y} = \textbf{H}\textbf{X} + \textbf{W} \tag{3}$$ where $\textbf{H}$ is fading coefficient matrix. To be more specific, the system is OFDM based and (3) is channel model in frequency domain. In practical conditions, channel can be severely selective, hence coherence bandwidth can be small (not like coherence time which is usually relatively large because of normal to walking-pace velocity). Then I assume the worst case where there is one $x_k$ per coherence bandwidth, hence the associated channel coefficients are decorrelated.

Therefore, (1) becomes $$\hat{i}=\text{argmax}_{1\leq i \leq N} \sum_{k=1}^M a^i_k y_k^* =\text{argmax}_{1\leq i \leq N} \sum_{k=1}^M a^i_k h_k^*x_k^* +w_k^*\tag{4}$$ which simply fails because $\{h_k\}_{1\leq k\leq M}$ are decorrelated (assumption: one $x_k$ per coherence bandwidth).

How could we detect orthogonal sequences in channel model (3)? I am interested in Rayleigh fading. But I am wonder if there is universal method for fading channels.

I think this is basics of CDMA (coherence bandwidth may be replaced by coherence time) or sequence correlation in mobile networks such as 4G and 5G. I do appreciate any answer or any paper reference.

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  • $\begingroup$ You'll definitely want to research the terms slow fading and fast fading, and what a channel coherence time is. You're right, in the absence of a sufficiently "spread out" channel autocorrelation function, reception is impossible. $\endgroup$ – Marcus Müller Feb 29 '20 at 8:50
  • $\begingroup$ @MarcusMüller thank you. Actually $\textbf{X}$ is preamble sequence, which is also used for channel estimation, and I must deal with the worst case, i.e. one $x_k$ in one coherence period. Then (1) does not work for sure, but is there any other method? (to be continued ...) $\endgroup$ – Rokai Feb 29 '20 at 9:56
  • $\begingroup$ @MarcusMüller This worst case is justified because I am working with a OFDM based system, hence coherence period=coherence bandwith. I mean in practical conditions, channel can be severely selective, hence coherence bandwidth can be small, not like coherence time which is usually relatively large because of normal velocity. Then I assume the worst case must have been dealt with. Could you please give me some keywords? $\endgroup$ – Rokai Feb 29 '20 at 9:56
  • $\begingroup$ You're giving contradicting information. Could you please edit your question to include all the coherence information on the channel you have, and also system aspects? $\endgroup$ – Marcus Müller Feb 29 '20 at 10:20
  • $\begingroup$ @MarcusMüller I am sorry to not being clear. I have edited the question. Do you think it is clear enough? Thank you. $\endgroup$ – Rokai Feb 29 '20 at 11:14

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