dsocWebbGaussianChannel

Create deep space optical communications Webb Gaussian or Gaussian channel

Since R2024b

Description

The dsocWebbGaussianChannel System object™ creates a deep space optical communications (DSOC) channel, based on the value set for the ApproximationMethod property.

• DSOC Webb Gaussian channel, as defined in NASA article Section C [1].

• DSOC Gaussian channel, as defined in NASA article Section D [1].

Use dsocWebbGaussianChannel to model a full avalanche photo detector (APD) output, which includes thermal noise and surface leakage current, using Webb Gaussian or Gaussian approximation, and to obtain the estimate photon count in each slot.

For more information, see Algorithms.

To create a DSOC Webb Gaussian or Gaussian channel:

1. Create the dsocWebbGaussianChannel object and set its properties.

2. Call the object with arguments, as if it were a function.

To learn more about how System objects work, see What Are System Objects?

Creation

Syntax

chan = dsocWebbGaussianChannel

chan = dsocWebbGaussianChannel(Name=Value)

Description

chan = dsocWebbGaussianChannel creates a default DSOC Webb Gaussian channel System object.

chan = dsocWebbGaussianChannel(Name=Value) sets properties using one or more name-value arguments. For example, dsocWebbGaussianChannel(ApproximationMethod="Gaussian") sets the approximation method to "Gaussian" and creates a DSOC Gaussian channel.

Properties

Unless otherwise indicated, properties are nontunable, which means you cannot change their values after calling the object. Objects lock when you call them, and the release function unlocks them.

If a property is tunable, you can change its value at any time.

For more information on changing property values, see System Design in MATLAB Using System Objects.

ApproximationMethod — Approximation method

"Webb-Gaussian" (default) | "Gaussian"

Approximation method to determine the distribution used to generate the slot counts, specified as one of these values.

• "Webb-Gaussian" — Use this option to create a Webb Gaussian channel, as defined in NASA article Section C [1].

• "Gaussian" — Use this option to create a Gaussian channel, as defined in NASA article Section D [1].

Data Types: char | string

NumSignalPhotons — Average signal photons per pulsed slot

100 (default) | positive real scalar

Average signal photons per pulsed slot, specified as a real positive scalar.

NumSignalPhotons also denotes the average number of signal photons per symbol, as the pulse energy of a symbol is captured in a single slot.

Data Types: double

NumNoisePhotons — Average noise photons per slot

0.001 (default) | nonnegative real scalar

Average noise photons per slot, specified as a nonnegative real scalar.

Data Types: double

Gain — APD gain

100 (default) | positive integer

APD gain, specified as a positive integer.

Data Types: double | uint32

IonizationRatio — Ionization ratio

0.007 (default) | scalar in range (0, 1)

Ionization ratio (keff), specified as a scalar in the range (0, 1).

This equation shows the relationship between keff and the ExcessNoiseFactor read-only property.

ExcessNoiseFactor = k_eff×G + ((2 – 1∕G)×(1 – k_eff)), where G is Gain.

Data Types: double

SurfaceLeakageCurrent — Surface leakage current

2e-9 (default) | nonnegative real scalar

Surface leakage current in amperes, specified as a nonnegative real scalar.

Data Types: double

SlotTime — Width of PPM slot required for laser

3.125e-8 (default) | positive real scalar

Width of the PPM slot required for the laser (T_s) in seconds, specified as a positive real scalar.

• This equation shows the relationship between SlotTime and LoadResistance read-only property.

  Load resistance = 5.75e12×T_s

• This equation shows the relationship between SlotTime and Bandwidth read-only property.

  Bandwidth = ½×T_s

Data Types: double

NoiseTemperature — Noise temperature

300 (default) | positive real scalar

Noise temperature in kelvin, specified as a positive real scalar.

Data Types: double

TimingOffset — Normalized timing offset

0 (default) | scalar in the range [0, 1)

Normalized timing offset, specified as a scalar in the range [0, 1). TimingOffset indicates the fractional timing offset in each slot.

Data Types: double

RandomStream — Source of random number stream

"Global stream" (default) | "mt19937ar with seed"

Source of the random number stream, specified as "Global stream" or "mt19937ar with seed".

• "Global stream" — The object uses the current global random number stream for Webb Gaussian or Gaussian distributed random number generation. In this case, the reset object function resets only the filters.

• "mt19937ar with seed" — The object uses the mt19937ar algorithm for Webb Gaussian or Gaussian distributed random number generation. In this case, the reset object function resets the filters and reinitializes the random number stream to the value of the Seed property.

Data Types: char | string

Seed — Initial seed of mt19937ar random number stream generator algorithm

73 (default) | nonnegative integer

Initial seed of the mt19937ar random number stream generator algorithm, specified as a nonnegative integer. When you call the reset object function, it reinitializes the mt19937ar random number stream to the Seed value.

Dependencies

To enable this property, set the RandomStream property to "mt19937ar with seed".

Data Types: double | uint32

LoadResistance — Load resistance

Read-only: real scalar

This property is read-only.

Load resistance in ohms, returned as a real scalar.

Data Types: double

Bandwidth — Noise equivalent one-sided bandwidth

Read-only: real scalar

This property is read-only.

Noise equivalent one-sided bandwidth in hertz, returned as a real scalar.

Data Types: double

ExcessNoiseFactor — Excess noise factor

Read-only: real scalar

This property is read-only.

Excess noise factor, returned as a real scalar.

Data Types: double

Usage

Syntax

slotCount = chan(modIn)

Description

slotCount = chan(modIn) adds Webb Gaussian or Gaussian distributed noise to modIn, transmitted through the channel chan, and returns the estimated photon count in slotCount.

Input Arguments

modIn — Modulated input data

N-element binary column vector

Modulated input data, specified as an N-element binary column vector. N is the number of samples. The vector contains the modulated data, with 1 representing a laser pulse and 0 representing the absence of a laser pulse.

Data Types: double | int8 | logical

Output Arguments

slotCount — Estimate of photo-electron count per slot

N-element column vector

Estimate of the photo-electron count per slot at APD output, returned as an N-element column vector. The size of this vector is equal to the size of modIn.

Data Types: double

Object Functions

To use an object function, specify the System object as the first input argument. For example, to release system resources of a System object named obj, use this syntax:

release(obj)

Common to All System Objects

step	Run System object algorithm
release	Release resources and allow changes to System object property values and input characteristics
clone	Create duplicate System object
isLocked	Determine if System object is in use
reset	Reset internal states of System object

Examples

Estimate Photon Count per Slot at Receiver by Configuring DSOC Webb Gaussian Channel

To estimate photon count per slot at the receiver, configure a DSOC Webb Gaussian channel and pass the input signal through the channel.

Generate data.

rng("default")
symLen = 1000;
m = 6;
M = 2^m;                         % M-ary PPM
data = randi([0 M-1],symLen,1);

Create a Webb Gaussian channel object and set the properties.

chanObj = dsocWebbGaussianChannel(NumSignalPhotons=16, ...
             NumNoisePhotons=0.1, ...
             RandomStream="mt19937ar with seed",Seed=64);

Modulate the data using the pulse position modulation (PPM) technique.

modOut = zeros(symLen*M,1);
mapIndex = (0:symLen-1)'*M + data + 1; % Mapping Index
modOut(mapIndex) = 1;

Pass the modulated data through a Webb Gaussian channel.

receivedData = chanObj(modOut);

Plot the distribution for pulsed slots.

histogram(receivedData(modOut == 1))
xlabel("Slot")
ylabel("Photon Count")
title("Distribution for Pulsed Slots")

Add Timing Offset to Signal and Estimate Photon Count per Slot at Receiver by Configuring DSOC Gaussian Channel

Estimate photon counts per slot by configuring a DSOC Gaussian channel and adding a timing offset to the signal.

Generate data.

rng("default")
symLen = 2500;
m = 3;
M = 2^m;                         % M-ary PPM
data = randi([0 M-1],symLen,1);

Create a Gaussian channel object and set the properties.

chanObj = dsocWebbGaussianChannel(ApproximationMethod="Gaussian", ...
             SurfaceLeakageCurrent=1e-9,TimingOffset=0.3);

Modulate the data using the PPM technique.

modOut = zeros(symLen*M,1);
mapIndex = (0:symLen-1)'*M + data + 1; % Mapping Index
modOut(mapIndex) = 1;

Pass the modulated data through the Gaussian channel.

receivedData = chanObj(modOut);

Plot the distribution for pulsed slots.

histogram(receivedData(modOut == 1))
xlabel("Slot")
ylabel("Photon Count")
title("Distribution for Pulsed Slots")

Algorithms

Optical Wireless Communication

Optical wireless communications (OWC) is an emerging technology that provides superior bandwidth capabilities, high-speed data transmission, and highly secure links. The system modulates the light signals, which then traverse through free space to reach the receiver, where the signals are subsequently demodulated.

Photodectectors

When the light signals reach the receiver, it processes the captured light from the optical signal to recover the transmitted data. The filtered signal is then sent to the photodetector, which is a semiconductor photodiode that converts the optical signal into an electrical signal. Photodetectors help to capture and amplify weak signals that have been attenuated and distorted by atmospheric channel conditions. The prominent photodetectors are APDs and positive-intrinsic-negative (PIN) diodes. APDs are incorporated in many high-performance OWC applications because they enable a high signal-to-noise ratio (SNR) and high-speed operation. APDs are preferred over PIN diodes due to their high internal gain characteristics and greater SNR capability.

Model APD Output Using Various Distributions

In an OWC system using an APD detector, the number of absorbed photons n is a Poisson random variable with probability distribution p(n), which is given by: $$p(n)=\frac{{(n^{\text{'}})}^n}{n!}{e}^{-n^{\text{'}}}$$, where n^' is the mean absorbed photons. In response to these n absorbed photons, the APD generates an avalanche of q electrons with a conditional probability distribution p(q|n) derived by McIntyre-Conradi model. The McIntyre-Conradi probability density function of q, given an average n^' absorbed photons, is obtained by averaging the conditional probability distribution p(q|n) over the Poisson-distributed absorbed photons count n:

$$\begin{array}{l}{P}_{MC}(q)={\displaystyle \sum _{n=0}^{q}p(q/n)\frac{{(n^{\text{'}})}^n}{n!}{e}^{-n^{\text{'}}},q\ge 0}(Eq1)\\ \end{array}$$

The full APD output also contains additive Gaussian noise contributed by thermal noise and surface leakage current. Thus, the total electron count x = q + q^' at the output of the APD has a probability density function that is a convolution of the McIntyre–Conradi density, Eq1, for q with a Gaussian probability density function for q^':

$$\begin{array}{l}p(x)={\displaystyle \sum _{q=0}^{\infty }\frac{1}{{\sigma }^{\text{'}}}\varphi \left(\frac{x-q-m^{\text{'}}}{{\sigma }^{\text{'}}}\right)P_{MC}(q)}(Eq2)\\ \end{array}$$

Here:

• $$\varphi (y)=\frac{1}{\sqrt{2\Pi }}{e}^{-y^2/2}(Eq3)$$ denotes the standardized (zero-mean, unit-variance) Gaussian probability density.

• Mean m^' and variance σ^'2 of the convolving Gaussian density depend on the surface leakage current I_s, thermal noise temperature T_n, and other physical parameters. Here,

  • m^' = (I_sT_s)/e- (Eq4)

    where:

    • T_s represents the slot time.

    • e- represents the electron charge and is equal to 1.6e^-19 Coulombs.

• $${\sigma }^{\text{'}2}=2B{T}_s\left(\frac{I_s{T}_s}{e-}+\frac{(k{T}_n)\times 2T_s}{R_L{(e-)}^2}\right)(Eq5)$$

  where:

  • B is the noise equivalent of one-sided bandwidth, and its nominal value as specified in [1] and as considered in the dsocWebbGaussianChannel object is ½ ✕ T_s hertz.

  • k is the Boltzmann constant.

  • R_L is the load resistance and its nominal value as specified in [1] and as considered in the dsocWebbGaussianChannel object is 5.75e¹² ✕ T_s ohms.

This section discusses the approximations to McIntrye-Conradi model when modeling an APD detector, as described in [1].

• Webb Approximation — Models the output of an APD in the absence of thermal noise and surface leakage current

• Webb Gaussian Approximation — Webb+Gaussian channel that models Gaussian thermal noise and surface leakage current added to Webb approximation channel outputs.

• Additive white Gaussian noise (AWGN) approximation

The Webb+Gaussian Approximation

x represents the full APD output, including thermal noise and surface leakage current.

$$x\approx m+m^{\text{'}}+\sqrt{{\sigma }^2+{\sigma }^{\text{'}2}}WG(0,1,{\delta }^2,\beta )$$

Here:

• m = Gn^'

• σ² = n^'G²F, where:

  • G is the APD gain

  • F is the excess noise factor and is given by:

    F = k_effG + (2 − 1/G)(1 − k_eff), and k_eff is the ionization ratio.

• WG(0,1,δ²,β) is a standardized zero-mean, unit-variance Webb+Gaussian random variable given by: $$WG(0,1,{\delta }^2,\beta )=\sqrt{\beta }W(0,1,{\delta }^2)+\sqrt{1-\beta }N(0,1)$$

  where:

  • N(0,1) is the standard Gaussian distribution, which is also denoted by ϕ(y) and is given by Eq3.

  • W(0,1,δ²) represents a standardized zero-mean, unit-variance Webb random variable w with δ² as the skewness of the density function around the origin.

    Here, δ² = n^'F/(F − 1)².

  • β is the blending parameter, which measures the proportion of the total variance contributed by the Webb component.

    β = σ²(σ² + σ^'2)

The probability density for the standardized Webb random variable w is given by:

$$\varphi (w;{\delta }^2)=\frac{1}{\sqrt{2\Pi }}{\left(1+\frac{w}{\delta }\right)}^{-3/2}{e}^{-w^2/[2(1+w/\delta )]},w>-\delta (Eq6)$$

AWGN Approximation

When the skewness parameter δ² is large, the Webb random variable q can be approximated by a Gaussian random variable N(m,σ²) with the same mean, m = Gn^', and variance, σ² = n^'G²F. The corresponding Gaussian probability density for q without any thermal noise is:

$$p_G(q)\approx \frac{1}{\sigma }\varphi \left(\frac{x-m}{\sigma }\right)$$

Furthermore, the full APD output is also Gaussian, and Eq2 simplifies to a single Gaussian probability density function which includes thermal noise:

$$p(x)\approx \frac{1}{\sqrt{{\sigma }^2+{\sigma }^{\text{'}2}}}\varphi \left(\frac{x-m-m^{\text{'}}}{\sqrt{{\sigma }^2+{\sigma }^{\text{'}2}}}\right)$$

References

[1] Dolinar, S., D. Divsalar, J. Hamkins, and F. Pollara. "Capacity of pulse-position modulation (PPM) on Gaussian and Webb Channels." August 15, 2020.

[2] Ascheid, G. “On the Generation of WMC-Distributed Random Numbers.” IEEE Transactions on Communications 38, no. 12 (December 1990): 2117–18.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Version History

Introduced in R2024b