*3.3.1 Temporal dispersion*

One of the most useful information regarding the underwater channel that may be provided by the Monte Carlo simulation, which is unavailable in most simple models like the Beer law, is the channel temporal dispersion. As seen before in this chapter, seawater is a dispersive medium in which light will suffer the effect of multiple scattering events. Scattering will cause the photons to arrive at the receiver at different time intervals causing temporal dispersion and a penalty in the channel bandwidth. In recent years, several researchers employed the double gamma functions to model the impulse response in underwater channels [15, 28]. Despite the accuracy of the double gamma function models, we find that the method described

**Figure 10.** *CDF comparison for selected phase functions.*

below consists in a simpler method by means of tracking the individual time of propagation of each photon.

By tracking the propagation distance of each photon (*dprop*), the time of propagation (TOP), may be estimated by:

$$t\_{op} = \frac{d\_{prop}}{c\_o},\tag{69}$$

Making a histogram of the received photons and distributing the received inten-

When using laser diodes in UOWC systems, it is desirable to modulate the light source at very high frequencies. The CIR generated by the Monte Carlo simulation can be converted to frequency domain by the means of a Fourier Transform algo-

Given that the impulse response is approximated by a discrete histogram of the time of arrival, it can be transformed to a frequency response, as proposed in [18, 19] by means of a discrete Fourier Transform (DFT), with the usual

> *N*�1 *k*¼0

To numerically implement this, the fast Fourier transform (FFT) algorithm can

where *dt* is the time step, or bin width, used when making the histogram of the

*t*delay½ � *k e*

�*j* 2π

*F f* ð Þ¼ abs *fft t*delayð Þ*<sup>t</sup> dt ,* (73)

*<sup>N</sup>nk,* (72)

sity in time slots defined as the bin widths, one can get a result for the channel impulse response (CIR) of the underwater channel. The results will be presented in

rithm, from which we can obtain the channel transfer function.

*Monte Carlo Radiative Transfer Modeling of Underwater Channel*

*DOI: http://dx.doi.org/10.5772/intechopen.85961*

F½ �¼ *n* DFT½ �¼ *CIR* ∑

be used to convert the *tdelay* data into the frequency domain:

Section 4.

**Figure 11.**

definition:

time delay data.

**117**

*3.3.2 Underwater channel transfer function*

*Log-linear plot of selected phase functions.*

where *co* is the speed of light in the water given by:

$$
\omega\_o = \frac{c\_{vacuum}}{n\_{water}},
\tag{70}
$$

with the index of refraction of sea water at 20° C and λ ¼ 500 nm being *nwater* ¼ 1*:*34295 [3].

For a better visualization of the impulse response, the time delay may be normalized by the straight-line distance, or direct distance, from source to the receiver using:

$$\mathbf{t}\_{delay} = \mathbf{t}\_{op} - \mathbf{t}\_{direct}.\tag{71}$$

*Monte Carlo Radiative Transfer Modeling of Underwater Channel DOI: http://dx.doi.org/10.5772/intechopen.85961*

**Figure 11.** *Log-linear plot of selected phase functions.*

Making a histogram of the received photons and distributing the received intensity in time slots defined as the bin widths, one can get a result for the channel impulse response (CIR) of the underwater channel. The results will be presented in Section 4.

### *3.3.2 Underwater channel transfer function*

When using laser diodes in UOWC systems, it is desirable to modulate the light source at very high frequencies. The CIR generated by the Monte Carlo simulation can be converted to frequency domain by the means of a Fourier Transform algorithm, from which we can obtain the channel transfer function.

Given that the impulse response is approximated by a discrete histogram of the time of arrival, it can be transformed to a frequency response, as proposed in [18, 19] by means of a discrete Fourier Transform (DFT), with the usual definition:

$$\mathbf{F}[n] = \text{DFT}[\text{CIR}] = \sum\_{k=0}^{N-1} t\_{\text{delay}}[k] e^{-j\frac{2\pi nk}{N}},\tag{72}$$

To numerically implement this, the fast Fourier transform (FFT) algorithm can be used to convert the *tdelay* data into the frequency domain:

$$F(f) = \mathbf{abs}\left(\mathcal{f}\mathcal{f}\left(t\_{\text{delay}}(t)dt\right)\right),\tag{73}$$

where *dt* is the time step, or bin width, used when making the histogram of the time delay data.

below consists in a simpler method by means of tracking the individual time of

By tracking the propagation distance of each photon (*dprop*), the time of

*top* <sup>¼</sup> *dprop co*

*co* <sup>¼</sup> *cvacuum nwater*

For a better visualization of the impulse response, the time delay may be normalized by the straight-line distance, or direct distance, from source to the

*,* (69)

*,* (70)

C and λ ¼ 500 nm being

*tdelay* ¼ *top* � *tdirect:* (71)

propagation of each photon.

*CDF comparison for selected phase functions.*

**Figure 10.**

*nwater* ¼ 1*:*34295 [3].

receiver using:

**116**

propagation (TOP), may be estimated by:

where *co* is the speed of light in the water given by:

*Wireless Mesh Networks - Security, Architectures and Protocols*

with the index of refraction of sea water at 20°

As expected, due to Nyquist sampling theorem, the maximum frequency that can be approximated in the frequency domain will be related to the time step by:

$$f\_{\max} = \frac{1}{2t\_{step}}.\tag{74}$$

**4.1 Clear and coastal waters**

*DOI: http://dx.doi.org/10.5772/intechopen.85961*

In **Figure 12**, we compare the received power for clear ocean waters with varying beam divergence parameters of 0, 10, 20, and 30 mrad for an emitted beam with a FWHM width of 1 mm. For the phase function, we used the FF, since, as seen in Section 3.2.4, it provides a good fit for Petzold's phase function. However, as mentioned in the "Introduction," when discussing simulation, the impact of different phase functions, especially those which are a good approximation for the Petzold's average phase function, is small for the clear and coastal waters due to the

Our simulation matches very well the Beer-Lambert law for the case of an emitted Gaussian beam with no divergence, since one of the main assumptions of the law is that the emitted beam is perfectly collimated, and as the beam divergence

We note that for a beam with no divergence, after 10 m, the normalized received optical power would be 2*:*<sup>2</sup> � <sup>10</sup>�<sup>1</sup> for a receiver aperture of 10 cm. Since

*w z*ð Þ¼ <sup>2</sup>*<sup>z</sup>* � tan *<sup>ϕ</sup>*

2

*,* (75)

low contribution of scattered photons on the final received power.

increases the power loss becomes more accentuated.

*Monte Carlo Radiative Transfer Modeling of Underwater Channel*

the beam spread after a distance *z* can be calculated by:

*Received power for clear ocean waters with varying beam divergence parameter.*

*4.1.1 Simulation results*

**Figure 12.**

**119**

## **4. Simulation**

In this section, we will provide some simulation results for several emitter parameters, underwater conditions, and different phase functions.

In the first part, we will simulate the received optical power for clear and coastal waters for different beam divergence values and compare it with the popular Beer-Lambert law. It can be seen in [14] that in these water conditions, the delay spread is so small that they can be considered as a nondispersive medium even for high data rate communications, and for this reason the channel impulse response (CIR) will be studied only for conditions where the medium is dispersive.

In the second part, we will study the effect of different phase functions in Harbor waters with a perfectly collimated beam. We believe that this is an interesting approach since in these waters', due to the higher scattering coefficient *b*, scattering will be a much larger factor and the amount of scattering that occurs at a given angle is an important factor for the received optical power and the delay spread. We will also provide numerical results for the CIR and an estimative for the 3-dB underwater channel bandwidth. The simulation parameters for Sections 4.1 and 4.2 are listed in **Tables 3** and **4**, respectively.


### **Table 3.**

*Simulation parameters for Section 4.1.*


### **Table 4.**

*Simulation parameters for Section 4.2.*
