Consider a random variable with a known distribution. I want to know if we quantized the variable the how the probability density function(PDF) would change? Specifically if the original PDF is Rayleigh distributed what would be the distribution would look like after the quantization?
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$\begingroup$ hi! nice question, but you don't actually mention anything related to channel coding in your question. Did you perhaps mean to tag channel-model instead of channel-coding, but couldn't (the tag doesn't already exist)? $\endgroup$– Marcus MüllerCommented Jun 9, 2017 at 10:17
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$\begingroup$ Nice question but a -1 because of the peevish response to Robert's answer which shows that you don't really understand the meaning of the terms that you are asking about. $\endgroup$– Dilip SarwateCommented Jun 9, 2017 at 11:38
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$\begingroup$ @MarcusMüller thank you for your interest. It is not about usual channel coding or channel model (broadly yes channel model). As you know the BLE gives you RSSI value in dBm, The user do not have a access to the original received signal but the RSSI. The RSSI value is reasonable at short distance but not at long distance say 10 m. So I was thinking of this probelm an asked the question. I can see that we may not find a solution as it like looking for 'missing information".. so just a try $\endgroup$– CreatorCommented Jun 9, 2017 at 18:46
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$\begingroup$ Then remove the unrelated tag. Don't use unrelated tags. Rssi is not a power in dBm. Otherwise it would be called "power". It's what it says: a reception strength indicator. That is influenced by many more things than received signal power. $\endgroup$– Marcus MüllerCommented Jun 9, 2017 at 19:02
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$\begingroup$ @MarcusMüller Yes, not power directly but it is expected to be proportional and in BLE ,we have nothing but RSSI to estimate power, so I used like that, sorry if it was misleading. $\endgroup$– CreatorCommented Jun 9, 2017 at 20:04
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1 Answer
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any discrete random variable has a p.d.f. that is a summation of dirac delta functions.
$$ p_\mathrm{y}(\alpha) = \sum\limits_i P_i \ \delta(\alpha - y_i) $$
where $\sum\limits_i P_i = 1 $.
if $y[n]$ is the quantization of $x[n]$:
$$ y[n] = \Delta \bigg\lfloor \frac{x[n]}{\Delta} \bigg\rfloor $$
where $ \lfloor \cdot \rfloor$ is the floor() function and $\Delta$ is the quantization step size.
then $y_i = i \Delta $ for some set of integers $i$. the p.d.f. weighting constants become
$$ P_i = \int\limits_{i \Delta}^{(i+1)\Delta} p_\mathrm{x}(\alpha) \, d \alpha $$
answered Jun 9, 2017 at 5:18
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$\begingroup$ @Creator you asked for value-quantized signals, and that usually also comes with a time-discretization (in all practical applications, at least; we call time- & value-discrete digital, usually). Thus, Robert took the time to show what the the prob. density function of a value-discrete, time-discrete $y$ looks like (instead of the value-discrete, time-continuous case, which you could infer from his third formula). $\endgroup$ Commented Jun 9, 2017 at 10:20
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$\begingroup$ +1 though in some instances, choosing the quantized value to be the center of the interval rather than the lower endpoint might be better. That is, $k\Delta$ is the quantized value corresponding to all $X$ in the range $\displaystyle \left(\left(k-\frac 12\right)\Delta, \left(k+\frac 12\right)\Delta \right)$ rather than the range $\left[k\Delta, (k+1)\Delta\right)$. This works better when $X$ has both positive and negative values where a quantized value of $0\Delta$ is not so lopsided a representation. (not an issue for the Rayleigh random variable that the OP wants to know about) $\endgroup$ Commented Jun 9, 2017 at 11:52
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1$\begingroup$ of course, there are many different p.d.f.'s of continuous random variable $p_\textrm{x}(\alpha)$ that will map to a single p.d.f. of discrete random variable, $p_\textrm{y}(\alpha)$. quantization destroys information. so you lose information with the p.d.f.'s. $\endgroup$ Commented Jun 9, 2017 at 18:36
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1$\begingroup$ Strictly speaking, for discrete random variables, delta functions may not be necessary because they can be used for categorical random variables like "heads" or "tails" or "goats" and "sheep" which doesn't have a meaningful interpretation on the real line. The between points aren't defined. Delta functions make sense when you have mixed continuous and discrete random variables or the discrete rv has a relationship to a continuos rv but to say a pmf is always a PDF of delta functions is not rigorous. $\endgroup$– user28715Commented Jun 9, 2017 at 19:34
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1$\begingroup$ "Cows" and "goats"? No. Even Oppenheimer and Schaefer don't use delta functions to represent data until the chapter on sampling. $\endgroup$– user28715Commented Jun 9, 2017 at 20:20