Performance of spectral amplitude coded multiple pulse position modulation with non-uniform energy slots and fading

D. Bykhovsky; S. Arnon

doi:10.1364/OE.26.029225

1. Introduction

Multiple-pulse position modulation (MPPM) is highly attractive for optical wireless communication (OWC) since it does not require channel state information (CSI) [1,2]. One of the natural options for MPPM schemes is based on the application of spectral-amplitude coding (SAC) [3–5]. In this case, MPPM is applied by on-off keying (OOK) modulation of spectral components.

The application of SAC requires the use of a wide-band spectral source. However, the spectral characteristic of such a light source is typically far from flat (e.g. Gaussian or Lorentzian [6]). Hence, when combined with a typical spectral encoder/decoder with uniformly spaced diffractive gratings and line detectors, the resultant spectral components are unequal in amplitude/energy. Instead, the amplitude/energy of each spectral component follows the spectral shape of the modulated light source, as illustrated in Figure 1. Alternately, equal amplitude/energy components can be obtained with a carefully designed SAC with unequal slot widths.

Fig. 1 An illustration of different slot energies of SAC-MPPM symbol.

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OWC through the atmosphere inevitably suffers from signal scintillations and consequent signal fading that result from turbulence. This effect may be non-negligible, even for short-distance communication, under certain environmental conditions [7–9]. Turbulence effects have a similar influence on all spectral components due to the relatively low statistical dependence of scintillations on wavelength [10]. Therefore, even for wide-band signals, scintillations may be assumed to be wavelength-independent. In a similar manner, another source of signal fading is transceiver pointing errors that deteriorate the link budget performance [11–15].

2. Preliminaries

The SAC-MPPM scheme is based on simultaneously blocking multiple spectral bands, as illustrated in Figure 1. Each MPPM symbol includes k unblocked and n blocked slots out of a total of M = k + n available slots. The special case of k = 1 is termed PPM (or M-ary PPM).

The receiver comprises a dispersive component followed by an array of direct-detection (DD) detectors. The resulting electrical signal output is linearly related to the energy of the corresponding spectral component. A received symbol is determined by choosing the k highest values from a M -length vector. Such a receiver does not require any CSI, but only time-synchronization between consecutive symbols. Consecutive symbols are assumed to be independent, i.e. the symbol rate is slower than the channel coherence time.

A number of statistical models have been developed to describe the distribution of turbulence-induced signal fluctuations, e.g. the lognormal distribution for weak turbulence conditions and the gamma-gamma distribution for strong turbulence [16]. For the following theoretical analysis, an arbitrary chosen probability density function (PDF), f_I(I), is applied. Later, in Section 4, a particular numerical example using the lognormal distribution is provided. Finally, the use of pointing errors distributions is discussed in Section 5.

3. Symbol error rate

The received symbol, S, may be described by a combination of blocked and unblocked slots. Blocked slots values, [Y₁, …,Y_n], are channel independent. Assuming the shot-noise is negligible, the receiver noise can be modeled by additive white Gaussian noise (AWGN). Therefore, the received values may be assumed to be Gaussian identically and independently distributed, $Y_{i} ~ N (0, σ_{n}^{2})$ , where $σ_{n}^{2}$ is the noise variance.

The unblocked slot values, [X₁, …, X_k], are dependent on the non-uniform spectral slot energy, s_i, the channel gain value, I, and the above-mentioned receiver noise. These values may be modeled by independently Gaussian distributed values with an expectation s_iI and a variance $σ_{n}^{2}$ , i.e. $X_{i} ~ N (s_{i} I, σ_{n}^{2})$ .

The analysis of symbol error rate (SER) starts with the calculation of the probability of correct symbol detection, p_cr, as shown in [5]. Using the newly defined random variables,

W = \max (Y_{1}, \dots, Y_{n})

V = \min (X_{1}, \dots, X_{k})

Z = W - V,

this probability may be defined as

p_{c r} = P (W < V) = P (Z < 0) = F_{Z} (z = 0),

where F_Z(z) is the cumulative distribution function (CDF) of Z.

Since W and V are independent, the resulting probability is given by [5]

F_{Z} (0) = \int_{- \infty}^{\infty} F_{W} (v) f_{V} (v) d v,

where F_W (w) is the CDF of W and f_V (v) is the PDF of V.

Since W is signal independent,

\begin{array}{l} F_{W} (w) = P (W \leq w) = \prod_{i = 1}^{n} P (Y_{i} \leq w) \\ = \prod_{i = 1}^{n} F_{Y_{i}} (y) = {[Q (- \frac{w}{σ_{n}})]}^{n} . \end{array}

Q (x) stands for the Q-function, given by

Q (x) = \frac{1}{\sqrt{2 π}} \int_{x}^{\infty} \exp (- \frac{t^{2}}{2}) d t .

Since V depends on the channel gain, I, its CDF, F_V(v), is given by

F_{V} (v) = \int_{0}^{\infty} F_{V} (v | I) f_{I} (I) d I,

where F_V (v|I) is a conditional CDF given by

\begin{array}{l} F_{V} (v | I) = P (W \leq w | I) \\ = 1 - \prod_{i = 1}^{k} Q (\frac{v - s_{i} I}{σ_{n}}) . \end{array}

The resulting PDF follows directly from the CDF in Eq. (6) and may be evaluated by

F_{V} (v) = \int_{0}^{\infty} f_{V} (v | I) f_{I} (I) d I,

where

\begin{array}{l} f_{V} (v | I) = \frac{\partial}{\partial v} F_{V} (v | I) \\ = - \frac{\partial}{\partial v} \prod_{i = 1}^{k} Q (\frac{v - s_{i} I}{σ_{n}}) . \end{array}

By applying the product rule of the form

\frac{d}{d x} [\prod_{i = 1}^{k} f_{i} (x)] = \sum_{i = 1}^{k} [(\prod_{j \neq i} f_{j} (x)) \frac{d}{d x} f_{i} (x)]

and the relation

\frac{d}{d x} Q (\frac{x - a}{b}) = - \frac{1}{b \sqrt{2 π}} \exp [- \frac{{(x - a)}^{2}}{2 b^{2}}],

the resulting conditional PDF in Eq. (9) is given by

f_{V} (v | I) = \frac{1}{\sqrt{2 π} σ_{n}} \sum_{i = 1}^{k} [\prod_{\begin{array}{l} j = 1 \\ i \neq j \end{array}}^{k} Q (\frac{v - s_{j} I}{σ_{n}})] \exp [- \frac{(v - s_{i} I)}{2 σ_{n}^{2}}] .

By substitution of all the expressions, the resulting probability of correct reception is given by

p_{c r} = \int_{- \infty}^{\infty} \int_{0}^{\infty} F_{W} (v) f_{V} (v | I) f_{I} (I) d I d v .

Finally, the SER may be evaluated from

p_{e} = - 1 - p_{c r} .

The resulting SER value is different for each symbol due to its dependence on non-uniform spectral slot energies. Therefore, the average SER requires an averaging of p_e probabilities of all possible symbols, $S_{j}$ , $j = 1, \dots, (\begin{matrix} M \\ n \end{matrix})$ .

3.1. Uniform-energy spectral slots

Uniform-energy spectral slots may be achieved, e.g. with a special carefully designed non-uniform SAC device. In this case, s_i = 1 ∀i, the expression F_V (v|I) in Eq. (7) may be rewritten as

F_{V} (v | I) = P (V \leq v | I) = 1 - {[Q (\frac{v - I}{σ_{n}})]}^{k},

the revised Eq. (12) is given by

f_{V} (v | I) = \frac{k}{\sqrt{2 π} σ_{n}} \exp [- \frac{{(v - I)}^{2}}{2 σ_{n}^{2}}] {[Q (\frac{v - I}{σ_{n}})]}^{k - 1},

and may be substituted in the resulting p_cr expression (Eq. (13)). Furthermore, the SER analysis may be substantially simplified, since no additional averaging of the SER over different symbols is required.

4. Simulation results

In order to validate the theoretical results derived in the previous section, numerical simulations were performed. The SER was evaluated for the lognormal PDF [16]

f_{I} (I) = \frac{1}{I \sqrt{2 π} σ_{I}^{2}} \exp {- \frac{\ln^{2} (I / μ)}{2 σ_{I}^{2}}}, I > 0

where µ is the average intensity and

σ_{I}^{2}

is the scintillation index. The applied turbulence parameters are µ = 0 and σ_I = 0.1 and the modulation parameters are M = 12 and k = 3. The closed-form evaluation of the p_e expression (Eq. (14)) is non-trivial and complicated [17]. However, numerical integration is possible with sufficient accuracy and was carried out using Mathematica software. For the simulation of communication performance, the number of consecutive symbols was set to 5 × 10⁶.

An example of simulation results for uniform and non-uniform spectral energies is presented in Fig. 2. The SNR value in the figure is 〈I〉/σ_n in dB units (i.e. SNR per slot). The non-uniform results are for a single symbol with s₁ = 1.2, s₂ = 1, s₃ = 0.75 that were normalized such that $E [s_{i}^{2}] = 1$ in order to preserve the same average received energy as a uniform slot signal. Both results show high resemblance between the theory and the simulation. As expected, the resulting SER for the non-uniform signal is significantly worse than for the uniform case.

Fig. 2 Illustration of theoretical and simulation results for (U)niform and (N)on-uniform energy spectral slots.

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In order to demonstrate the discrepancy in SER for different symbols with non-uniform energy per slot, the spectral profile of a LED was used to yield the respective values of s_i [6, Sec. 17.1]. These values were normalized in order to preserve the same average received energy as a uniform slot signal. SER values were evaluated for all possible symbols, $(\begin{matrix} M \\ k \end{matrix}) = 200$ in number. The histogram of these SER values is presented in Figure 3. The results show a significant influence of the energy non-uniformity on the SER of different symbols. When a suboptimal set of MPPM symbols is of interest [18], the symbols with the highest energy that correspond to the highest s_i values should be chosen and may result in a significant improvement in the average SER. In the example above, the average SER for the 2⁷ = 128 lowest SER symbols is 1.09 × 10⁻⁶, which is significantly lower than the overall symbols’ average of 1.18 × 10⁻⁵.

Fig. 3 Histogram of p_e values that correspond to different non-uniform energy symbols. Inset shows s_i coefficients used in simulation.

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5. Pointing errors

Most long range optical wireless communication systems use a narrow beam-divergence angle laser transmitter. Hence any mispointing results in fading that deteriorates the link budget performance [11–15]. The commonly used pointing-error PDF is based on a Gaussian distribution with zero means for the elevation and the horizontal directions [11], and is given by [12,13]

f_{h} (h) = \frac{γ^{2}}{A_{0}^{γ^{2}}} h^{γ^{2} - 1}, 0 \leq h \leq A_{0}

where

γ = w_{z_{eq}} / 2 σ_{θ}

and the parameter

w_{z_{eq}}

may be evaluated by the relations

v = \frac{\sqrt{π} a}{\sqrt{2} w_{z}}, A_{0} = {[erf (v)]}^{2}, w_{z_{eq}}^{2} = w_{z}^{2} \frac{\sqrt{π} erf (v)}{2 v \exp (- v^{2})},

where a is the receiver radius and w_z is the beam waist (see [12] for more details).

The PDF of received energy distribution in the presence of pointing errors, f_h(h), may replace the fading distribution, f_I(I), in the expressions above to study the effect of pointing loss on SER. Moreover, a closed form PDF of combined turbulence and pointing loss can be used when both of these phenomena are present; the assumptions and methodology are described in [11–13].

6. Summary & conclusions

The main goal of this study is to evaluate a SER expression for SAC communication with non-uniform energy slots. A theoretical expression for SER is provided. The theoretical results were successfully validated by a numerical simulation.

In this study a general fading PDF, f_I(I), was examined. Therefore, any PDF describing different fading conditions may be applied, as required.

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Performance of spectral amplitude coded multiple pulse position modulation with non-uniform energy slots and fading

Abstract

1. Introduction

2. Preliminaries

3. Symbol error rate

3.1. Uniform-energy spectral slots

4. Simulation results

5. Pointing errors

6. Summary & conclusions

References

Cited By

Figures (3)

Equations (21)

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