Maximizing delay-constrained throughput in multi-channel DS-CDMA ALOHA networks through power diversity and successive decoding

Abstract

We study multi-channel ALOHA networks (e.g., satellite-based networks) for online transaction processing, striving to maximize attainable throughput while meeting a deadline with near certainty. This captures the service provider’s fixed costs and per-transaction revenue, the user’s delay consciousness and ALOHA’s probabilistic nature. Specifically, we consider CDMA channels and successive-decoding receivers. Interestingly, judicious use of power diversity is shown to be extremely effective: with a single transmission, capacity is doubled relative to that with power equalization. With the deadline permitting as few as one or two retransmission attempts upon failure, the probability of not meeting it can be virtually diminished (10⁻⁵ and 10⁻⁸, respectively) while approaching the throughput attainable without delay constraints. This also holds for limited mean transmission power. Thus, the effect of power diversity in conjunction with CDMA depends strongly on the type of receiver and on the exact performance measure, and the proposed approach is worth considering for next-generation systems.

Appendix

1.1 A. Random spreading sequences

The performance of various demodulation strategies depends on the signal-to-noise ratio, A _k /σ, and on the similarity among the spreading sequences, quantified by their cross-correlations:

$$ \rho_{ij}(\tau)= \int s_i(t)s_j(t-\tau)dt . $$

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In the random signature model

$$ s_{km}(t) = \frac{1}{\sqrt{N}} \sum_{i=1}^{N} d_{kmi}p_{T_c}(t-iT_c+T_c), $$

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where N is the spreading gain. $\{d_{kmi}\}_{{1\leq i \leq N \atop 1 \leq k \leq K} \atop 1 \leq m \leq M}$ are i.i.d. random variables and can be 1 or −1 with equal probabilities [18].

For further analysis of the interference among users, it is useful to know the statistical properties of the cross-correlation between the spreading sequences of each two users i, j in the mth bit: ρ _ijm . In [18] and [14], it is shown that

$$ E\left[\rho_{ijm}^2\right]=\frac{1}{3N} i \neq j. $$

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1.2 B. Packet collision model

1.2.1 Packet decoding

We assume that packets are encoded using an error-correction code. Thus, the probability that the kth user’s packet is not decoded correctly, P _c,k , can be closely approximated as a unit step function of the SIR (see for example in [4]):

$$ P_{c,k}=\left\{ \begin{array}{lc} 1, & \Upgamma_k \geq \Upgamma_{thr} \\ 0, & \Upgamma_k < \Upgamma_{thr} \end{array} \right. , $$

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where Γ _thr is a parameter that depends on the specific error-correcting code used and Γ _k is the SIR of the kth user.

Let $\{b_i\}_{i=1,\ldots,M}$ denote the M bits of a packet. The output of the decoder, denoted $\{\hat{b}_i\}_{i=1,\ldots,M}$ is an estimate of these bits. It should be emphasized that the estimation of $\{\hat{b}_i\}_{i=1,\ldots,M}$ is not a simple per-bit hard-decision but a decoding that takes into account many other neighboring bits. If decoding is successful (which can be easily determined by a simple CRC) then we assume that all of the bit estimates in the packets are accurate. (The amplitude estimation may nonetheless be imperfect). This assumption is an approximation, suitable for long coded packets (longer than 1 kb).

The spreading sequences are assumed to be independent from bit to bit, but the relative delay shifts, τ _k , do not change significantly during a given packet. This bit-to-bit error dependence was studied in [13], where the results show that the simple model that ignores inter-bit relations is quite accurate. It is also shown that resorting to a simpler model of synchronous spreading sequences (τ = 0) results in a very pessimistic assessment of capacity. We therefore ignore the bit-to-bit error dependence within each packet, but do not assume bit-level synchronization among different transmissions.

1.3 C. Successive decoding SIR analysis

Let r ^(k) denote the signal used to detect data from user k, which is the received signal, r(t), after k − 1 subtractions of previously decoded user signals:

$$ r^{(k)}(t)=r(t) - \sum_{i=1}^{k-1}\hat{x}_i(t), $$

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where $\hat{x}_i(t)$ is an estimate of the received signal from user i, defined as x _i (t) = A _i b _i s _i (t − τ _i ).

We also define z _k to be the matched filter output after k − 1 previous subtractions, $z_k = \langle r^{(k)}(t), s_k(t) \rangle.$

Following a similar analysis to [7], the variance of z _k in the Hybrid SIC receiver model is

$$ \hbox{var} [z_k^{hyb}] = \frac{1}{3N} \sum_{i=k+1}^{K} A_i^2 + \frac{1}{3N} \sum_{i < k \atop \Upgamma_i < \Upgamma_{thr}} \hbox{var} [z_i^{hyb}] +\sigma^2. $$

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1.3.1 Modeling the amplitude estimation inaccuracy

We use a simple model to account for inaccurate amplitude estimation of successfully decoded signals, by assuming a fixed relative error,

$$ \frac{\left|\hat{A}_k-A_k\right|}{A_k} \approx \alpha . $$

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Adding this to (31) yields

$$ \begin{aligned} \hbox{var} [z_k^{hyb}] & = \frac{1}{3N} \sum_{i=k+1}^{K} A_i^2 + \frac{1}{3N} \sum_{i < k \atop \Upgamma_i < \Upgamma_{thr}} \hbox{var} [z_i^{hyb}] \\ &\quad + \frac{1}{3N}\sum_{i < k \atop \Upgamma_i \geq \Upgamma_{thr}} \alpha^2 A_i^2 +\sigma^2. \end{aligned} $$

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By iteratively calculating E ²[z _k ] and var[z _k ] for all the users, $\Upgamma_k=\frac{E^2[z_k]}{var[z_k]}$ can be compared to the decoding threshold, Γ _thr to determine the fate of each of the concurrent transmissions.

1.4 D. A numerical method for approximating $P_{c_{W_i}}$

We assume that in all the successive cancellation schemes, transmissions are decoded in decreasing-power order. Same-power transmissions are decoded in random order.

Our method hinges on the observation that whether a “probe” transmission collides or not depends only on its SIR, and not on the numbers of concurrent transmissions K _i . The SIR of the “probe” depends on its power, W _i , and on the aggregate interference, var[z]. Consequently, we do not need to go over the many possibilities of K _i vectors as was suggested by the summation in (15); rather, we merely need to calculate the probability distribution of the SIR seen by a transmission at power level W _i .⁵ $P_{c_{W_i}}$ is calculated given the power W _i , so W _i is a parameter. var[z] is a random variable, because it depends on the interfering transmissions, their power levels and the cross-correlations between their spreading sequences and the probe’s spreading sequence.

Let I _i denote the interference (contribution to var[z]) from higher- and lower-power transmissions than W _i . The interference from transmissions with the same power level W _i is excluded intentionally (The interference from these transmissions will be taken care of by the definition of the function $P_{c_{W_i}}(n,v)).$

We can simplify (15):

$$ P_{c_{W_i}} = \sum_{n=0}^{\infty} \Pr(K_i=n) \int_{v=0}^{\infty} P_{c_{W_i}}(n,v) dP_{I_i}(v), $$

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where $P_{I_i}(v)$ is the cumulative probability function of $I_i,\,\, P_{I_i}(v) = \Pr(I_i\leq v).$ The function $P_{c_{W_i}}(n,v)$ is the probability of collision for a W _i -power transmission, given that n transmissions with W _i occurred and the interference from other transmissions (lower + higher) equals v. This function can be pre-calculated and saved in a table for reference when calculating the integral in (34).

We separate the interference I _i into two parts: $I_i=I_i^{H}+I_i^{L}.$ I ^L _i results from an aggregate of lower power level transmissions that are decoded after the “probe” transmission is decoded. Therefore, the interference that they induce does not depend upon the success/failure of their decoding. $I_i^{H},$ in contrast, is the residual interference seen by the “probe” transmission, left by imperfect subtractions of previously decoded transmissions. Therefore, it depends on the success/failure of previous decodings. Furthermore, the results of previous decoding depend on the SIR levels seen by higher power level transmissions, which depend on the interference $I_{i-1}^{L}.$ The analysis of $I_i^{H}$ must therefore take into account the dependence on $I_{i-1}^{L}.$

1.4.1 Calculation of the probability distribution of $I_i^{L}$

We begin with the calculation of $P_{I_i^{L}}(v) = \Pr(I_i^{L} \leq v).$ For i = m, the “probe” transmission is received with the lowest of power levels, so there is no interference from lower power levels by definition. Therefore, $I_m^{L}=0$ w.p. 1, and $P_{I_m^{L}}(v)$ is the unit step function:

$$ P_{I^{L}_m}(v) = \left\{ \begin{array}{ll} 1, & v \geq 0 \\ 0, & v < 0 \end{array} \right. $$

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$P_{I_i^{L}}(v)$ can be calculated iteratively using the following rule:

$$ \begin{aligned} P_{I_i^{L}}(v) & = \Pr(I_i^{L} \leq v) \\ & = \sum_{n=0}^{\infty} \Pr(K_{i+1}=n) \Pr(I^{L}_{i+1} + n W_{i+1} \leq v) \\ & = \sum_{n=0}^{\infty} \Pr(K_{i+1}=n) \Pr(I^{L}_{i+1} \leq v- n W_{i+1}) \\ & = \sum_{n=0}^{\infty} P_{I^{L}_{i+1}}( v- n W_{i+1} ) , \end{aligned} $$

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1.4.2 Calculation of the probability distribution of I ^H _i

Because of the dependency between I ^H _i and I ^L _i−1 , we calculate the conditional probability distribution function of I ^H _i given I ^L _i−1 . We define $P_{I_i^{H} | I_{i-1}^{L}}(u,v)$:

$$ P_{I_i^{H} | I_{i-1}^{L}}(u,v)= \Pr (I_i^{H} \leq u | I_{i-1}^{L}=v) . $$

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To facilitate the presentation, let I ⁺ _i be the residual interference left by the imperfect subtractions of the W _i transmissions. The residual interference left by imperfect subtractions of $W_1,\ldots,W_{i-1}$ is I ^H _i and is excluded intentionally. I ⁺ _i depends only on K _i , the number of transmissions with W _i , and on the total interference $I_i^{H}+I_i^{L}.$ The dependency on the interference in the linear receiver is due to the variance of the matched filter output, which depends on the interference. In the non-linear receiver, the success/failure of the decoding depends on the interference, and the decoding result affects the residual interference.

We will therefore use the notation $I_i^{+}(K_i,I_i)$ to denote the residual interference left by K _i imperfect subtractions (of transmissions with power level W _i ) given that the interference from other power levels (higher + lower) was I _i .

$I_i^{+}(K_i,I_i)$ and $P_{c_{W_i}}(K_i,I_i)$ (in (34)) can be pre-calculated and saved in a table using the analysis in Sect. C. As an example, we show how this precalculation should be done for the hybrid receiver defined in Sect. 2.

Example 1: Calculating $P_{c_{W_i}}(K_i,I_i)$ and $I_i^{+}(K_i,I_i)$ for the hybrid receiver case.

Set $P_{c_{W_i}}(K_i,I_i)=0$ and $I_i^{+}(K_i,I_i)=0$

Iterate on j from 1 to K i

$\Upgamma_j = \frac{W_i}{\frac{1}{3N} \left[ (K_i-j)W_i + I_i + I_i^{+}(K_i,I_i) \right] + \sigma^2};$

if (Γ j  > Γ thr ) /* Successful Decoding */

$I_i^{+}(K_i,I_i) = I_i^{+}(K_i,I_i) +$

α2 W i ; /* Power estimation error*/

else

$P_{c_{W_i}}=P_{c_{W_i}}+1/K_i;$

$I_i^{+}(K_i,I_i) = I_i^{+}(K_i,I_i)$

$+\frac{1}{3N} \left[ (K_i-j)W_i + I_i + I_i^{+}(K_i,I_i) \right]$

+ σ2;

end

end

1.4.3 Calculation of $P_{I^{H} | I^{L}}$

Now that we have $P_{c_{W_i}}(K_i,I_i)$ and $I_i^{+}(K_i,I_i)$ ready, we go on to calculate the probability distribution of $I_i^{H}$ given $I_{i-1}^{L}.$ By definition, $I^{H}_1=0$ w.p. 1. Therefore

$$ P_{I^{H}_1 | I^{L}_{0}}(u,v)=\left\{ \begin{array}{ll} 0, & u < 0 \\ 1, & u \geq 0 \end{array} \right. $$

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To calculate $P_{I_i^{H}|I_{i-1}^{L}}$ iteratively, we would like to express it in terms of $P_{I_{i-1}^{H}|I^{L}_{i-2}}:$

$$ \begin{aligned} P_{I_i^{H} | I_{i-1}^{L}}& (u,v) = \Pr (I_i^{H} \leq u | I_{i-1}^{L}=v) \\ &= \sum_{n=0}^{\infty} \Pr(K_{i-1}=n) \cdot \\ & \int_{u^{\prime}|I^{+}_{i-1}(n,v+u^{\prime})+u^{\prime}\leq u} dP_{I_{i-1}^{H} | I^{L}_{i-2}}(u^{\prime},v+nW_{i-1}) \end{aligned} $$

The summation and integration in (39) are over all values of K _i−1 and u′, where u′ is the residual interference $I_{i-1}^{H}.$ Because $I_i^{H}=I_{i-1}^{H}+I^{+}_{i-1},$ we integrate over u′ values that yield a $I_i^{H}$ interference less or equal to u.

Let $u^{\bigstar}$ be the solution to the equation $I^{+}_{i-1}(K_i,I_{i-1}^{L}+u^{\bigstar})+u^{\bigstar}=I_i^{H}.$ $u^{\bigstar}$ can be pre-calculated and saved in a table, $u^{\bigstar}(K_i,I_{i-1}^{L},I_i^{H}).$

$I^{+}_{i-1}(K_i,I_{i-1}^{L}+u^{\prime})$ is a monotonically non-decreasing function of u′, so the set of values of u′ over which we integrate is simply $u^{\prime}\leq u^{\bigstar}.$ Using the table $u^{\bigstar}(K_i,I_{i-1}^{L},I_i^{H}),$ (39) can be simplified,

$$ \begin{aligned} P_{I_i^{H} | I_{i-1}^{L}}(u,v) & = \sum_{n=0}^{\infty}\Pr(K_{i-1}=n) \cdot \\ & \quad P_{I_{i-1}^{H}|I^{L}_{i-2}}\left(u^{\bigstar}(n,v,u),v+nW_{i-1}\right) . \end{aligned} $$

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Finally, $P_{c_{W_i}}$ can be calculated using the pre-calculated tables $P_{c_{W_i}}(K_i,I_{i-1}^{L}),\,\, P_{I_i^{H}|I_{i-1}^{L}}(u,v)$ and $P_{I_i^{L}}(v)$:

$$ \begin{aligned} P_{c_{W_i}} &= \sum_{n=0}^{\infty} \Pr(K_{i}=n) \int_{v=0}^{\infty} dP_{I_i^{L}}(v) \\ & \int_{u=0}^{\infty} P_{c_{W_i}}(n,u+v) dP_{I_i^{H}|I_{i-1}^{L}}(u,v+nW_i) . \end{aligned} $$

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1.4.4 Complexity analysis

Let m be the number of power levels and K _max be the maximum number of transmissions per power level.⁶ The probability functions described in the analysis above are actually approximated by histograms. Let N _I be the number of bins on the interference axis of the histograms that we keep for functions like $P_{I_i^{H} | I_{i-1}^{L}}(u,v).$ The table size required for this function is $N_I^2.$

The calculations that are done according to the analysis above are in the order of $m \cdot N_I^2 \cdot K_{\rm max}.$ If we had to do the brute-force summation described in (15), this would require calculations that are in the order of $K \cdot K_{\rm max}^{m} .$ For K _max = 100, m = 6 and N _I  = 1000, the brute force method is simply infeasible while the numerical approach leads to the results shown in this paper. Furthermore, our method is more scalable in the sense that the computation needed for larger K and m grows linearly rather than exponentially.