19.3.1 Reflection from Earth’s Surface

Signal reflection occurs when an incident electrometric wave intersects a large surface with dimensions much greater than the carrier wavelength. In long‐distance communication links, signal reflections from the Earth’s surface will combine with a direct path signal resulting in received signal variations referred to as multipath fading. On a smaller scale, signals in a mobile communication system will reflect off of building and other structures often without an identifiable direct signal path. In these situations, the received signal will fluctuate rapidly with the dynamics of the mobile unit. Signal reflections are characterized in terms of the complex reflection coefficient

(19.14)

where ρ = |E_r/E_i| is the magnitude of the reflected‐to‐incident electric field ratio and Φ is the corresponding angle of E_r/E_i. The ranges of ρ and Φ are: 0 ≤ ρ ≤ 1 and |Φ| ≤ π and when Φ > 0 then the reflected electric field lags the incident electric field. Ramo and Whinnery [8] provide a detailed discussion of the reflection of electromagnetic wave forming the basis of the following results.

The reflection coefficient also depends on the grazing angle and polarization of the incident electric field and for horizontally and vertically polarized waves the results are expressed as [9]

(19.15)

and

(19.16)

where ε_c is the complex dielectric constant given by

(19.17)

where ε_r is the ordinary dielectric constant and ε_i is given by

(19.18)

where λ is the carrier wavelength and σ is surface conductivity.¹ When λ is in meters then σ has units of mhos per meter; values of ε_r and σ for various types of surfaces are given in Table 19.3.

The reflection coefficient for the vertically polarized wave is examined by substituting (19.17) into (19.16) and the resulting amplitude and phase functions are plotted in Figures 19.5 and 19.6 for the indicated conditions. The grazing angle corresponding to the minimum magnitude of the reflection coefficient in Figure 19.5 is referred to as the pseudo‐Brewster angle. With a perfect dielectric, the minimum magnitude of the reflection coefficient is |Γ_v| = 0, that is, no reflection occurs, and the corresponding grazing angle is referred to as the polarizing angle or, in geometrical optics, the Brewster angle. The Brewster angle occurs only with a vertically polarized wave. With horizontal polarization, the reflection coefficient decreases slowly with grazing angle and the angle Φ_h is very close to π radians over the entire range of grazing angles.

Circularly polarized waves are described in terms of horizontally and vertically polarized waves that are in quadrature with one another. Right‐hand circular polarization occurs when the horizontal polarized wave lags that of the vertically polarized wave, that is, when ϕ_h − ϕ_v > 0 and left‐hand circular polarization occurs when ϕ_v − ϕ_h > 0. For right‐hand or left‐hand circularly polarized incident waves, the reflection coefficient with the same‐sense as the incident wave is expressed as [2, 11]

(19.19)

and the reflection coefficient with the opposite‐sense as the incident wave is

(19.20)

Therefore, in general, both senses of the incident circular polarized wave appear at the receiver from the reflecting surface.

19.5.1 Tropospheric Refraction

Because the troposphere does not contain free‐electrons, signal propagation in the troposphere is influenced principally by water vapor content, air temperature, and pressure. Based on the standard atmosphere described by Kerr [1], the index of refraction in the troposphere is given by

(19.38)

where a and b are constants equal to 79°K/mb and 4800°K respectively; the respective values of a and b as determined by Smith and Weintraub [12] are 77.6°K/mb and 4810°K and by Campen and Cole [13] are 74.4°K/mb and 4973°K. The parameter T is the air temperature in °K, p is the air pressure in millibars (mb) and e is the partial pressure of water vapor in millibars.

Based on the analysis of Campen and Cole as outlined by Millman [14] the parameters describing the standard atmosphere are expressed as a function of altitude or height h in kilometers for h ≤ 10 km as

(19.39)

(19.40)

and

(19.41)

where p_o = 1013 mb is the standard temperature at sea level and e_o is the partial pressure of water vapor at the Earth’s surface in mb. These expressions are essentially independent of frequency. The normalized form, (n − 1)10⁶, of the refractive index is valid to within 0.5% for frequencies below about 30 MHz.

Upon curve fitting these results using a polynomial in h with units of km, Campen and Cole express the refractive index as a function of h through the troposphere for a completely wet environment with 100% relative humidity as

(19.42)

and for a completely dry environment with 0% relative humidity at refractive index is expressed as

(19.43)

Equations (19.42) and (19.43) apply for heights up to 10 km and decay exponentially from 10 km to the upper limit of the troposphere at about 30.5 km according to the relationships

(19.44)

and

(19.45)

These results are plotted in Figure 19.12 in terms of the parameter N = (n − 1)1e⁶ referred to refractivity.

Millman uses these results to evaluate the refractive bending along the propagation path using direct integration and ray tracing. In his analysis the troposphere is divided into spherical 30.48 m layers of constant refractive index up to 3.048 km and 3.048 m layers up to 30.48 km; the refractive index above 30.48 km is unity. The refractive bending of the propagation path gives rise to antenna pointing errors, range delay errors, and Doppler frequency errors. An encounter between a ground terminal and an aircraft is depicted in Figure 19.13. The refraction angle and range errors are shown in Figures 19.14 and 19.15 respectively. These errors increase with increasing height and decreasing antenna grazing or apparent elevation angle. For example, with receiver heights above 1853.2 km and zero degree grazing angle the maximum refraction angular errors are 0.799° and 0.049° for the wet and dry conditions respectively. For the wet and dry conditions, the corresponding maximum range errors are about 381 and 290 ft (116 and 88.4 m).

The Doppler frequency is dependent on the angle ψ between the receiver terminal velocity vector and the direct propagation path to the receiver and is expressed as⁴

(19.46)

where V is the magnitude of the receiver terminal velocity and f_c is the carrier frequency, and the free‐space speed of light c = 3e⁸ m/s. Referring to Figure 19.13, the velocity error is the difference between apparent path velocity (V_o) and the ray path velocity (V_r) and is evaluated as

(19.47)

where , , and . After separating terms and applying small argument approximations, Millman arrives at the following approximation to (19.47):

(19.48)

The angle Δα_t is determined based on the refractive indices n_g and n_t computed using the appropriate equations (19.42) through (19.45). The indices are used in the spherical earth model of the atmosphere that provides for evaluating Δα_t using the geometry of the encounter and Snell’s law expressed as the ratio n_g/n_t. Millman ([14], pp. 326–328, 346) characterizes the evaluation of Δα_t as

(19.49)

The angle α_o is measured at the ground terminal and is expressed in terms of the an angle θ as

(19.50)

The angle α_o is the apparent ground antenna elevation angle observed at the ground terminal; however, the angle θ is unknown so Millman’s profiles in Figure 19.14 must be used to determine refraction angle error Δα_o at the ground terminal.

Based on the velocity error in (19.48), the one‐way Doppler frequency error is expressed as

(19.51)

As an example application in the determination of the Doppler frequency and the corresponding Doppler frequency error using (19.46) and (19.48), consider the following encounter conditions: Earth radius is R_e = 6378.28 km, h = 12 km, α_o = 5°, ψ = ±90°, f_c = 100 MHz, V = 0.3 km/s, and c = 3e⁸ m/s is the free‐space speed of light. The results are also to be calculated with 100 and 0% humidity corresponding wet and dry conditions. To begin the solutions, the Doppler velocity can be computed using the specified parameters and (19.46). The next step is to use Figure 19.14 and verify that the values of Δα_o are: 0.15° and 0.104° respectively for 100 and 0% relative humidity. With these parameters and the specified data, compute the values of n_g and n_t using (19.42) through (19.45) under the appropriate humidity and height conditions. The next step involves evaluating Δα_t using (19.49) from which the velocity error is determined using (19.48). The final evaluation involves the determination of the one‐way Doppler frequency error (Δf_d) using (19.51). The maximum Doppler error occurs when ψ = 90° and, based on this example, the wet and dry conditions correspond to

(19.52)

Because the tropospheric parameters are independent of the carrier frequency, (19.52) can also be formulated as

(19.53)

This example uses a relatively slow airplane and with higher speeds, lower elevation angles, and higher operating frequencies increased Doppler frequency errors will be encountered. The two‐way Doppler frequency error is of interest in radar applications that simply requires doubling the one‐way results in this example. Additional reading on the subject of refraction is also provided by Skolnik [17] and Barton [18].

19.5.2 Ionospheric Refraction

Electromagnetic propagation through the ionosphere is influenced by two key parameters: the electron density, n_e, and the rms fluctuation in the electron concentration defined in terms of the parameter σ_e. Therefore, before the performance of a communication system can be evaluated, it is necessary to quantify these parameters throughout the ionosphere. The characterization of the ionosphere and its impact on communication link performance is discussed in Chapter 20. However, by way of introduction, the electron density profiles in the natural environment are characterized as a function of height using Chapman’s analytic model. In a nuclear‐disturbed environment, the electron densities are determined from experimental measurements and computer simulations. Ray bending in the ionosphere is not explicitly addressed in Chapter 20; however, an analysis is provided by Millman [14] and follows that given for troposphere in the preceding section.

19.6.1 Knife‐Edge Diffraction

When an incident wave is obstructed by a knife‐edge conductor, shown in Figure 19.16, secondary wavelets result in signal diffraction in the shadow region behind the knife‐edge. The transmitted ray that just intersects the top of the knife‐edge and all lower angle rays are blocked from the receiver. However, the higher angle rays result in secondary wavelets, some of which are directed behind the knife‐edge providing coverage in the shadow of the obstruction. The direct path of length d₁ + d₂ is obstructed; however, for path lengths greater than the refraction path D₁ + D₂ an electric field is produced in the shadow region of the obstruction. The height h, shown as the heavy solid line in Figure 19.16, is the effective height of the knife‐edge between the direct and minimum refraction path; the total height of the obstruction is defined as h_o = h + h′.

The electric field strength at a point in the shadow region is the sum of all the secondary wavelets and is computed as [19]

(19.54)

where E_o is the electric field that would be produced at the receiver over a free‐space path equal to the direct path length. The lower integration limit in (19.54) is the Fresnel–Kirchoff diffraction parameter expressed as

(19.55)

where λ is the carrier frequency wavelength and Δ_k is the differences between the direct path and the refraction path and is evaluated as

(19.56)

The approximation in (19.56) is based on the conditions: λ ≪ h ≪ d₁ and d₂.

Evaluation of the integral in (19.54) is performed in Section 20.6.2 involving the Fresnel integrals [20] C(z) and S(z) with the result

(19.57)

The magnitude of (19.57), expressed in decibels, is plotted in Figure 19.17 as a function of the diffraction parameter v. When v ≤ 0 there is a direct path of length d₁ + d₂ between the transmitter and receiver and the electric field variations can be thought of as resulting from knife‐edge reflections. However, when v = 0 the peak of the knife‐edge just intersects the direct path and the received electric field strength is reduced by 6 dB. For v > 0, the signal reception is due to diffraction into the shadow region.

The function plotted in Figure 19.17 is called the Fresnel diffraction pattern and the peaks and troughs of the ripples result from successive constructive and destructive interference from the received secondary wavelets shown in Figure 19.16 and identified for k > 0. These peaks and troughs occur because of path differences given by Δ_k = kλ/2 for k = even and odd respectively. The plane perpendicular to Figure 19.16, that is, through the page and containing the knife‐edge, contains unique secondary wavelets that intersect the plane forming annular rings with radius h_k that identify Fresnel zones. The Longley‐Rice propagation model, discussed in Section 19.7, makes extensive use of reflection and knife‐edge diffraction in the evaluation of signal loss over various terrain conditions.

19.8.1 Okumura Model for Urban Environments

The Okumura model [23] is based on curves that are fit to measured signal losses taken in an urban environment over frequency (f_c) and distance (d) ranges: 150 MHz ≤ f_c ≤ 1920 MHz, 1 km ≤ d ≤ 100 km respectively and apply for transmitter effective antenna heights: 30 m ≤ h_te ≤ 1000 m and effective receiver antenna heights: h_re ≤ 30 m. The resulting median signal loss is expressed as

(19.59)

where L_fs is the free‐space signal loss, A_mu(f_c, d) is the median excess propagation loss determined from Figure 19.20, and G_e is a frequency‐dependent environmental gain factor determined from Figure 19.21. To attach a level of confidence to these measured values, the estimated standard deviation about the median losses varies between 10 and 14 dB.

19.8.2 Hata Model for Urban and Suburban Environments

The Hata model [24] is derived from measured path loss data and applies over the frequency range: 150 MHz ≤ f_c ≤ 1500 MHz and for transmitter effective antenna heights: 30 m ≤ h_te ≤ 200 m and effective receiver antenna heights: 1 m ≤ h_re ≤ 30 m. There is no specified limit on the distance d; however, for reasonable agreement with the Okumura model requires that d > 1 km. The median loss for the urban environment is expresses as

(19.60)

The correction factor a(h_re) is a function of the cell area and for small to medium cities is given by

(19.61)

and for large cities

(19.62)

For suburban and open rural areas, the correction factors are

(19.63)

and

(19.64)

19.8.3 Erceg Model for Suburban and Rural Environments

The Erceg model [25] characterizes the loss for wireless mobile communication in suburban and rural areas. The results are based on curve‐fit plots of loss vs. distance derived from 1.9 GHz experimental data collected in 95 macro cells across the United States. In this case, the data was collected using omnidirectional azimuth antennas with transmit and receive gains of 8.14 dB and 2.5 dB respectively. The receiver antenna height was fixed at h_r = 2 m.

This model is directed toward applications, like personal communication services (PCS) that involve smaller cells, lower transmit antenna heights and higher frequencies. The path loss applies for transmitter effective antenna heights: 10 m ≤ h_te ≤ 80 m and distances in the range: 0.1 m ≤ d ≤ 8 km. The three environments include various terrain hill conditions and tree densities. The Erceg model loss is expressed as

(19.65)

where d_o = 100 m is the minimum close‐in distance. The exponent γ is a Gaussian random variable expressed as⁵

(19.66)

where, the term in brackets, and σ_γ are the mean and standard deviation of γ respectively. The parameter x is a normalized Gaussian random variable characterized as N(0,1).⁶ The s term in (19.65) is a zero‐mean Gaussian random shadow fading term characterized as N(0,σ) and expressed as

(19.67)

where y = N(0,1), z = N(0,1), and σ is a Gaussian random variable characterized by N(μ_σ,σ_σ). The variables x, y, and z are independent random variables. The constants a, b, c, σ_γ, μ_σ, and σ_σ are listed in Table 19.5 for each terrain category.

19.9.1 Modified Exponential Decay Model Link Loss

The MED model applies to suburban and rural environments consisting primarily of trees and vegetation. The total loss through the foliage is evaluated as

(19.68)

where a_n is the specific attenuation expressed as

(19.69)

The losses are shown in Figure 19.22 for the indicated frequencies.

19.9.2 CCIR Link Loss Model

The CCIR model is a modification of the MED model with the total loss given by

(19.70)

and the specific attenuation expressed as

(19.71)

The loss using the CCIR model is shown in Figure 19.23 and is somewhat higher than that predicted by the MED model.

19.9.3 Barts and Stutzman Link Loss Model

The Barts and Stutzman model is also a modification of the MED model for distances ≤14 m; otherwise, the loss predictions are identical to the MED model. The total loss through the foliage is evaluated as

(19.72)

and the specific attenuations for the different ranges are expressed as

(19.73)

and

(19.74)

The loss for the Barts and Stutzman model is shown in Figure 19.24.

19.9.4 CCIR Link Margin Model

The CCIR link margin model [29] data were collected in urban, semi‐urban, suburban, and rural areas at 860 MHz and 1.55 GHz for elevation angles [31] ranging from 19° to 43°. In the following relationships, the angles are entered as degrees and the frequency as gigahertz. The link margin applies for a percentage availability of P_a = 90%, that is, the received signal power exceeds the detection threshold 90% of the time; however, other percentages of availability can be evaluated by subtracting the loss L = 0.1(90 − P_a) dB from the derived link margins.⁸ The range of P_a is 50–90%. Furthermore, a factor K is included that relates the percentage of the surrounding locations or area (area%) for which the received power is expected to exceed the detection threshold; the values of K for area% are tabulated in Table 19.6.

The link margin, in dB, for the urban, suburban, and rural models are expressed as

(19.75)

and

(19.76)

Equations (19.75) and (19.76) are plotted in Figure 19.25 as a function of θ for various carrier frequencies in gigahertz with the indicated conditions of: link availability, percent of area, and L = 0 dB. With P_a = 90%, area% = 50%, and K = 0, the link is available 90% of the time and in 50% of the surrounding area the received power is expected to exceed the detection threshold. If the availability is decreased to 50% then L = −4 dB and the link margin can be decreased by 4 dB. On the other hand, if the detection threshold is to be exceeded in over 99% of the surrounding area then K = 2.35 and the link margin must be increased by the additive term involving K in (19.75) and (19.76). Note that the range of the elevation angle in Figure 19.25 exceeds the stated range of the model by about 5° on each end of the abscissa. The frequency translations method of Goldhirsh and Vogel [32] is recommended to evaluate the link margin requirements at other frequencies.

19.10.1 Introduction

Impulsive noise occurs from thunder storm activity around the world and is present in most regions as the energy from lightning flashes or strikes propagates through the natural wave guide between the earth surface and the ionosphere. The severity of the storm activity varies with geographic regions and seasons; however, the worldwide average rate of lightning strikes is on the order of hundreds‐per‐second. In regions near active storm centers the noise spikes are most pronounced, characterized as short high‐energy pulses that disrupt communications. As the impulsive energy propagates farther from the storm center the wide bandwidth pulses undergo attenuation and dispersion and combine with similarly filtered impulses from storm centers in other regions of the globe. The global effects of lightning strikes resulting from storm activity are most evident at the lower frequencies, typically in the low frequency (LF) region and below. These effects become less troublesome at frequencies in the high frequency (HF); however, the HF region has unique issues [33] to contend with including time‐varying multipath, ducting, and Faraday rotation.

The impulsivity measure V_d is introduced in Chapter 14 as the parameter that characterizes the severity of the impulsive noise and is defined as the ratio of the rms noise envelope to the average noise envelope. The V_d measure is expressed in decibels with the minimum value of 1.049 dB corresponding to minimum storm activity; larger values indicate increased storm activity. As the storm energy from around the globe is combined, the noise addition is subject to the central limit theorem and the impulsive noise approaches white Gaussian noise with the corresponding V_d = 1.049 dB. This condition is observed during periods of relatively calm worldwide storm activity and will change suddenly as a result of a distance storm. The impulsivity is also characterized by the amplitude probability distribution (APD), defined as the probability that the noise envelope exceeds the abscissa [34]. The worldwide characterization of impulsive noise due to storm activity is published by the International Telecommunication Union (ITU) through the CCIR Report 322 [35] and associated reports [36]. These reports characterize the APD based on impulsive noise measurements corresponding to the V_d measure. The impulsive noise from lightning strikes is characterized as a nonstationary random process and the APD data are based on the ensemble average of recorded time sequences.

19.10.2 Lognormal Impulse Noise Model

To the casual observer, a lightning strike appears as a single flash of light; however, in many events each flash is actually composed of multiple strokes separated typically by 50–100 ms. The multiple strokes following the initial lightning strike are referred to as return strokes. The number and interval between the return strokes is modeled statistically based on observations [37]. Uman and Krider [38] have summarized the phenomenon of lightning strikes and, based on the studies of Mackerras [39], conclude that the number of return strokes is typically distributed between 2 and 8 resulting in a mean value of 5 return strokes for each lightning flash. Figure 19.26 shows Mackerras’ results in terms of the probability that the number of return stokes exceeds the abscissa; the dashed curve represents the piece‐wise linear approximation to the data expressed in (19.77) and is used in the computer simulations. Beach and George [40] and Uman [41] report on the time between return strokes based on the data collected by Schonland in South Africa. Figure 19.27 characterizes Schonland’s data in terms of the probability that the time interval between return strokes exceeds the abscissa; the dashed curve corresponds to a piece‐wide linear approximation in (19.78) used for computer simulations. Beach and George observed that the time between stokes corresponding to Schonland’s data can be approximated using the Gamma pdf with α = 2 and mean value α/β = 55. This approximation is also applied to the data of Kitagawa, Brook, and Workman [42] for cloud‐to‐ground lightning using a mean value of α/β = 35. Although these Gamma function pdf approximations are good fits to the data over regions about the mean values, over regions several standard deviations removed from the mean they are not as accurate, so the piece‐wise linear approximations are used in the computer simulations.

The piece‐wise linear approximations for the simulated probabilities are

(19.77)

and

(19.78)

where j and n are integers and the time is in milliseconds.

The impulse noise for lightning strikes is modeled as shot noise [43] and expressed as the summation

(19.79)

where N_p(t) represents the main lightning stroke with the associated return strokes, , and is expressed as

(19.80)

with

(19.81)

The functions N_p(t) and represent complex impulse noise processes with lognormal distributed amplitudes A_p(t), and uniformly distributed phases, φ(t), φ′(t), respectively.

From the discussions in Section 14.3.6, the lognormal amplitude is A_p = e^x, where x is a normally distributed random variable with mean m₀, variance , and phase φ uniformly distributed between −π and π. A zero mean Gaussian background noise term, n_g(t), with variance is added to the impulsive noise; the background noise results from quiescent worldwide thunder storm activity. Therefore, the total atmospheric noise at the input to the receiver antenna is described as

(19.82)

Substituting (19.80) and (19.81) into (19.79) the total atmospheric noise at the receiver antenna input, characterized by (19.82), becomes⁹

(19.83)

The most commonly observed lightning strike or flashes occur as intra‐cloud, cloud‐to‐cloud, and cloud‐to‐ground.¹⁰ To the casual observers the cloud‐to‐ground lightning is the most spectacular. In cloud to ground flashes the first or main strike is preceded by a stepped leader that is followed by dart leader that propagates from the cloud to ground. The dart leader is immediately followed by a return stroke that propagates from ground to cloud and results in the visible lightning flash. Depending on the remaining charge and the electric field intensity, additional dart leaders followed by return strokes may occur resulting in a multiple‐stroke flash. In the communication performance simulation program, the parameter λ is input to establish the mean lightning flash‐rate. The time intervals Δt_i = t_i − t_i−1 between multiple return strokes are randomly distributed according to (19.78). As discussed in Section 19.10.3, the number of strokes and flashes over a recorded ensemble of atmospheric noise is adjusted to match the APD corresponding to the selected V_d (dB).

Two points are noteworthy regarding the noise description in (19.83). The implied bandwidth of the noise impulses is infinite and in areas of intense storm activity the parameter λ may be sufficiently high so that return strokes from several main strokes overlap. The bandwidth issue is handled by passing the impulse noise through the receiver intermediate frequency (IF) filter with one‐sided bandwidth denoted by B. The resulting received atmospheric noise at the output of the IF filter is then evaluated as

(19.84)

where h(t) is the impulse response of the IF filter where the asterisk (*) denotes convolution. Regarding the second point, parameter λ is selected to conform to measured data to result is the prescribed APD.

19.10.3 Fitting the Noise Model to Measured Data

In a simulation model the required sampling frequency f_s is chosen to conform to the receiver Nyquist sampling criterion. For example, when simulating the performance of a communication system using analytic signal representations, the sampling frequency is selected such that f_s ≥ 2B, where B is the bandwidth of the received signal. In this regard, the sampling frequency is selected to result in an acceptably low loss resulting from the detected symbol energy and aliasing distortion. The received samples are processed in the demodulator for waveform acquisition and subsequently symbol and carrier tracking and data detection. In this context t = kT_s: T_s = 1/f_s and (19.84) is the narrowband analytic representation of the received atmospheric noise. For evaluating the communication performance, the modulation symbol rate and the sampling frequency are related to the anti‐aliasing filter bandwidth B as shown in Figure 19.28.

In the following evaluation of the impulse noise model, a minimum shift keying (MSK)‐modulated waveform is used with a symbol rate of R_s = 25 sps, B = 800 Hz, f_s = 2B, and N_s = 32 samples‐per‐symbol. Because λ has units of impulses per second, the impulse rate is restricted to so the parameter N_s can be adjusted as necessary to fit the noise model to the measured data. These relationships are scaled to accommodate specific waveform modulations and data rates as discussed in the case study in Section 19.10.5.

Lightning strikes are characterized by the impulsivity measure V_d defined in (14.67) as

(19.85)

where represents the narrowband time‐sampled receiver analytic noise. The system performance is typically characterized for a specified value of V_d; however, the APD for each V_d must conform to the corresponding measured APD in the ITU publication CCIR 322 [35]. To this end, the measured APD results are shown in Figure 19.29 for several values of V_d; these results are adapted from Gamble [44] and plotted using Rayleigh coordinates that result in a linear APD curve with slope −1/2 for Gaussian noise. The CCIR APD results are measured at the output of a receive antenna modeled as a single‐pole filter with a noise bandwidth of B_n = 243 Hz. Therefore, when specifying V_d in (19.85) the parameters , , m₀, and λ of the filtered noise process characterized by (19.83) and (19.84) must be chosen to conform to the corresponding APD curve. The parameter λ is implicit in (19.83) through the random distribution of the time between the lognormal impulses. The parameters m₀ and are implicit in the normally distributed random variable x denoted as x = N(m₀,σ₀).

An analytic closed‐form solution for V_d in terms of the model parameters is intractable because of the denominator term in (19.85) involving the expectation of the magnitude of the received noise. Therefore, V_d is evaluated numerically in terms of , , m₀, and λ with the background noise power based on a specified receiver signal‐to‐noise ratio¹¹ γ_b = E_b/N_o as

(19.86)

where V_r is the peak voltage of the carrier‐modulated received signal. Upon specifying , λ, and m₀, (19.85) is evaluated by indexing : n = 1, …, N until the computed value of V_d just exceeds the specified value. Each evaluation involves a time series of one million samples to compute the expectations required in (19.85).¹² The starting value of , denoted as , is chosen to minimize the search time and was found to provide sufficient coarse resolution to match the desired APD curve. However, to improve the estimation accuracy, a fine resolution interval of is used with the indexing restarted at the previous value and continuing with : until the computed value of V_d again exceeds the specified value. Upon completion, the value is linearly interpolated between the ending and previous values and the simulation is run once again to verify the evaluation of the desired APD curve using the four parameter values.

The final simulation run uses the interpolated value of and the numerically computed error in V_d in decibels is typically less than 0.1%. The final parameter sets for several V_d values are summarized in Table 19.7. The results of the V_d and APD evaluations are shown as the data points in Figure 19.30 for V_d values of 1.049, 2, 6, and 14 dB; the solid curves are taken from Figure 19.29 and represent the corresponding measured APD curves.

Vd (dB)	λ	m0	γb				a
1.049	—	—	Any	Any	—	—	—
1.5	1150	0	36.5	1.120(−4)	0.2074	0.0163	0.0035
2.0	300	0	28.0	7.920(−4)	0.7625	0.1000	0.0212
6.0	3000	0	15.0	0.01581	2.218	2.0500	0.4400
14.0	1500	3	−30.0	500.00	4.726	5.483(3)	1.354(4)

^aApplies to MSK matched filter.

Figures 19.31, 19.32, and 19.33 show typical recordings of the sampled magnitude, , of the lognormal impulse noise for V_d = 1.049, 2.0, and 14.0 dB, respectively. These recordings are representative of the filtered samples at the output of the 243 Hz noise bandwidth receive antenna used to collect the CCIR‐322 noise data.

The value of V_d will change as the filter bandwidth is changed but the CCIR‐322 results apply only to the 243 Hz noise bandwidth filter [45]. Spaulding, Roubique, and Crichlow [46] have evaluated the conversion of V_d with bandwidth and their results are shown in Figures 19.34 and 19.35 for decreasing and increasing bandwidths respectively. B_i = B_n = 243 Hz is the noise bandwidth in which V_di is measured and B_o is the noise bandwidth of the desired filter. The case study in Section 19.10.5 uses B_i = B_n = 243 Hz with B_o = B_n so no bandwidth conversion is necessary.¹³

19.10.4 Impulsive Noise Mitigation Techniques

Modem performance improvements can be achieved through the use of various impulsive noise mitigation techniques like: clipping [47], limiting [48], excision, and hole‐punching [49]. These techniques are applied in the receiver or demodulator signal path prior to the matched filter detection; however, at the outset of the modem design, the proper selection of the waveform modulation and forward error correction (FEC) coding will result in significant performance advantages. When FEC is applied to the waveform, the use of interleaving is also an effective mitigation technique for impulsive noise. Gamble [44] provides a review of these subjects and summaries the performance results of various authors for constant amplitude waveform modulations: PSK, MSK, and continuous phase frequency shift keying (CPFSK). In Section 19.10.5, the performance improvement using clipping is examined using the MSK‐modulated waveform. Time domain clipping is most effective when applied at a high IF frequency before significant pulse dispersion occurs; this can be achieved for constant envelope waveforms with bandpass limiting followed by narrowband filtering.

19.10.5 Case Study: Minimum Shift Keying Performance with Lognormal Impulse Noise

This case study involves simulating the performance of MSK operating as a bit‐rate of R_b = 50 bps in lognormal noise representative of lightning strikes from worldwide storm centers. In the preceding sections, the parameter V_d is used to characterize the receiver noise described by (19.84). The application of (19.84) involves generating noise samples , where T_s is the sampling interval, and combining the noise samples with the sampled signal (s_i) as shown in Figure 19.36. The sampling frequency f_s = 1/T_s must be chosen to satisfy the Nyquist sampling condition.

The receiver filter in Figure 19.36 represents the cascade of receiver and demodulator IF filters with a composite noise bandwidth of B Hz. The bandwidth of the single‐pole filter following the impulse noise generator establishes the impulsivity measure and the corresponding APD for the system evaluation. For example, suppose that the receiver being evaluated has an antenna noise bandwidth of 486 Hz, referring to Figure 19.35 with B_o/B_i = 2, a value of V_d = 6 dB in 243 Hz corresponds to  = 8 dB in the receiver noise bandwidth. Therefore, the system performance is evaluated using  = 8 dB with the corresponding APD response. However, in this case study, B_i = B_n = 243 Hz with B_o = B_n, so the measured V_d values can be used without applying bandwidth conversion. This avoids the uncertainty introduced in the bandwidth conversion processing [46].

The signal must be passed through the equivalent receiver filters and the signal and intersymbol interference (ISI) distortion losses must be considered. Although the simulated signal is characterized as an analytic signal, the modulated signal at the filter output has an equivalent carrier‐modulated peak level of V_r volts corresponding to the signal power P_s = /2. Similarly, the power of the zero mean sampled noise is computed as

(19.87)

This power is measured in the bandwidth of the sampling frequency and the signal‐to‐noise ratio in the sampling bandwidth is N_s = f_s/R_s times higher than E_b/N_o. In an environment involving white noise the channel power can be scaled by the bandwidth ratio and adjusted to correspond to a specified E_b/N_o ratio. However, because the impulse noise does not have a constant power spectral density (PSD) the signal will necessarily involve the channel noise power () at the output of the symbol matched filter. For MSK modulation, the matched filter noise bandwidth is

(19.88)

The noise powers and are listed in Table 19.7 for the indicated values of V_d. The performance simulation evaluates the bit‐error probability as a function of the signal‐to‐noise ratio, γ_b = E_b/N_o measured in the bandwidth to the data rate R_b, that is,

(19.89)

Upon expressing (19.89) in terms of the noise power measured in the bandwidth B_mf and solving for results in

(19.90)

Using (19.90) the voltage gain required to bring up to the level for a specified γ_b = E_b/N_o is

(19.91)

This gain is applied to the simulated impulse noise as shown in Figure 19.36. As stated above, in the following simulations the input filter is the single‐pole filter with a noise bandwidth of B_i = 243 Hz used to characterize the CCIR‐322 data collection hardware.

Following the impulse noise gain adjustment, the received signal, receiver thermal noise, and impulsive channel noise are added and passed through the receive filter, the optional nonlinearity, and the MSK matched filter. The demodulator data estimates are determined from the optimally sampled matched filter and the errors are counted to determine the bit‐error probability. The hard limiting (HL) is ideal, in that, it normalizes the complex baseband signal plus noise (s_rk) as

(19.92)

When clipping is used, the clipping threshold is defined relative to the channel noise power and, because the received signal power is held constant, the clipping level (CL) relative to the received signal voltage V_r is also constant. The matched filter is a complex inphase and quadrature (I/Q) filter with quadrature cosine weighed symbol integrators offset by the bit duration T_b = T/2, where T is the MSK symbol duration. Ideal symbol timing is used, however, the performance using ideal frequency and phase tracking is compared to the performance using a second‐order Costas phaselock loop with various time‐bandwidth products.

The following figures characterize the performance of the MSK‐modulated waveform for various V_d values as described above. The modulation symbol rate of R_s = 25 sps has negligible distortion when passed through the single‐pole equivalent receiver antenna filter. The receiver bandwidth of B = 800 Hz is sufficiently wide to provide for effective clipping of the impulse noise peaks and the associated reduction of the V_d through clipping, HL, or bandwidth reduction is reflected in the performance results.

Gamble [44] has evaluated the theoretical performance of coherently detected binary phase shift keying (BPSK) with a linear receiver for various values of V_d. Gamble also provides comparisons of the theoretical performance with published experimental results for PSK [50] with V_d = 3.75 dB measured in 33.3 Hz and for MSK [51] with V_d = 5 dB measured in 40 Hz. These comparisons indicate the Gambles theoretical results provide a lower‐bound to the experimental results by 2–3 dB; however, the differences may be accounted for by measurement accuracies, tracking losses and other hardware anomalies. The simulation results in the following section focus on MSK modulation and compare reasonable well with Gambles results.

19.10.5.1 Simulated MSK Performance with V_d = 1.5 dB

The simulated performance of MSK in atmospheric noise with V_d = 1.5 dB and ideal symbol and phase tracking is shown in Figure 19.37. The performance using a linear receiver is compared to a HL receiver and a receiver with various CLs ranging from 0 to 21 dB. The Monte Carlo simulations use 1 M bits (500K symbols) for each signal‐to‐noise ratio and, because the channel noise does not have a constant power spectral density, the abscissa is defined more generally as .¹⁴ The dotted curve is the theoretical performance of MSK in additive white Gaussian noise (AWGN) and the circled data points (see Figure 19.38) result from the corresponding simulated performance. These ideal tracking performance results are provided as a baseline for comparison. The modem performance with impulsive noise is degraded from the AWGN performance by about 3.5 dB at P_be = 10⁻⁵ with the linear receiver and an additional 0.2 dB with the HL receiver. The performance with CLs ranging from 0 to 21 dB fall between HL and linear performance.

The CL is defined in terms of the clipping threshold T_cl and the channel noise power as

(19.93)

The clipping threshold is related to the received signal power as

(19.94)

The signal power in the simulation is constant with P_s = 0.5 and, using the computed noise power from Table 19.7,  = 0.0163, the clipping threshold relative to P_s in decibels is

(19.95)

Figure 19.37 also shows the performance with phaselock loop tracking using the time‐bandwidth product B_LT_b and lock detector B_LDT_b products of 0.025 and an initial phase error of 10° with zero frequency error. The critical phaselock loop signal‐to‐noise ratio of the linear receiver is about 8 dB and for the HL receiver it is about 14 dB. Clipping levels ranging from 0 to 21 dB were examined and the phaselock loop never achieved phase‐lock for CLs ≤ 18 dB. With CL = 21 dB phase‐lock was achieve with a critical signal‐to‐noise ratio of about 6 dB. Considering the CL relative to the signal power, as given by (19.95), the 21 dB CL corresponds to a clipping threshold relative to the signal power of 6.13 dB, whereas, CLs ≤ 18 dB correspond to clipping thresholds ≤3.13 dB relative to the signal power. These results indicate that an optimum clipping threshold exits with phase tracking. In all cases, once phase‐lock is achieved the performance loss is less than about 0.25 dB with tracking.

19.10.5.2 Simulated MSK Performance with V_d = 6 dB

The simulated MSK results for V_d = 6 dB are shown in Figures 19.38, 19.39, and 19.40. In these figures, the dotted curve is the theoretical performance of MSK in AWGN and the circled data point corresponded to the simulated MSK performance. The Monte Carlo simulations use 1 M bits (500K symbols) for each signal‐to‐noise ratio. The results in Figure 19.38 compare the performance of a linear receiver with one using HL and various CLs; all of these results use ideal symbol time and phase tracking. As predicted by Gamble the performance of the linear receiver is severely degraded; however, HL and CLs ≤ −6 dB provide significant improvements.

Referring to (19.94) and Table 19.7,  = 2.05 and the clipping threshold relative to the signal power, P_s = 0.5, is expressed in decibels as

(19.96)

Therefore, a CL of −6 dB corresponds to a clipping threshold of relative to the signal power level of 0.13 dB. The CLs of −3 and 0 dB, shown in Figure 19.38, also exhibit considerable improvement relative to the linear receiver. These CLs correspond to the respective clipping thresholds of 3.13 and 6.13 dB relative to the signal power.

The simulation results in Figure 19.39 show the performance of the linear receiver and the HL receiver with phase tracking indicated by the dash‐dot‐dot curves. The phaselock loop (PLL) B_LT_b and lock detector B_LDT_b products are 0.025 and the initial phase error is 10° with zero frequency error. The critical signal‐to‐noise ratio for phase‐lock is about 14 dB for the linear system and 6 dB for the HL receiver.

Figure 19.40 shows the phase tracking performance with clipping. With CL = 0 dB phase‐lock is achieved for signal‐to‐noise ratios less than 0 dB and with CL = −3 dB the critical signal‐to‐noise ratio is about 8 dB. For signal‐to‐noise ratios below about 6 dB the PLL never achieved phase‐lock. With CLs ≤ −6 dB phase‐lock was never achieved for any signal‐to‐noise ratio; this in contrasted with HL that did achieve phase‐lock as mentioned above. The loop bandwidth is directly related to the signal level into the PLL and, because the receiver gain following the limiting and clipping is not adjusted, the low signal level associated with low CLs may be attributed to the failure to achieve phase‐lock. In any event, the performance at lower signal‐to‐noise ratios is an important consideration with the use of FEC coding. In all cases, once phase‐lock is achieved the performance loss is less than about 0.25 dB with tracking. The phaselock performance can be improved by reducing the time‐bandwidth products at the expense of less tolerance to the dynamics of the encounter.

19.11.1 Introduction

When communicating with submarines in deep ocean environments, the distortion introduced by surface waves driven by wind is a major factor that degrades the system performance. The impact of wind waves on the signal amplitude and phase is severe at very low frequency (VLF) and LF frequencies and in this section the signal distortion is analyzed and quantified in terms of the sea‐state conditions. The performance degradation is examined in a case study in Section 19.11.5 and signal processing techniques to mitigate the impact of the wave distortion are discussed.

In the real‐world environment, the received signal is corrupted by additive atmospheric noise as discussed in the preceding sections; however, the impulsivity of the atmospheric noise is decreased because of the narrowband filtering through the sea water. As the water depth increases, the V_d measure of the atmospheric noise approaches 1.049 dB corresponding to Gaussian noise. In addition to the filtered atmospheric noise, the noise at the receiver consists of additive thermal noise and man‐made interference noise from equipment that falls within the bandwidth of the receiver IF stages. In these cases, the performance loss is mitigated through the use of robust symbol modulations, FEC coding, interleaving, and adaptive interference cancellation techniques [52].

A theoretical characterization of wind waves is given by Kinsman [53] where, in Chapter 8, the Pierson‐Neumann theory is discussed leading to the Neumann wave‐height energy spectral density (EDS) characterization. An alternate expression of the wave‐height spectral density is based on the Pierson‐Moskowitz theory and the two spectrums are compared. These energy spectrums characterize the wave height for a fully developed sea after a period of sustained winds and that is not influenced by the ocean floor or surrounding shore lines. In other words, the sustained winds are over deep oceans with a large surface area or fetch. In addition to the introduction given by Kinsman, the U.S. Navy [54] provides a description of the growth and decay of waves. The wind wave characteristics are identified in terms of a sea‐state number [55, 56] as given in Table 19.8. In the following sections, the Neumann wave‐height energy EDS is used to compute random wave‐height temporal records; these records are seamlessly linked in a computer simulation program and used to distort the amplitude and phase of the communication waveform for the purpose of evaluating the communication link performance under various sea‐state conditions.

19.11.2 Neumann Wave‐Height Energy Spectrum

The EDS of wave height in a fully developed sea of unlimited depth, fetch, and time is characterized by Neumann [57] as

(19.97)

where g is the gravitational acceleration at sea level, found from experimental data to be 9.8106 m/s², u is the wind velocity in meter per second, and ϖ is the angular frequency of the energy spectrum in radian per second. The constant 4.791 is determined from experimental data and has dimension of m²/s⁵. S(ϖ) has dimensions of square meter‐second and is plotted in Figure 19.41 in terms of the frequency f = ϖ/2π. The severity of the sea condition is designated by the sea‐state number and the corresponding wind velocity as indicated in Table 19.8. Figure 19.41 clearly demonstrates the impact of the wind velocity with an increase of five decades in the peak energy spectrum between sea‐states 1 and 8. The alternate wind wave spectral density is based on the Pierson‐Moskowitz wave theory [58] and expressed by (19.98) and shown in Figure 19.42.¹⁵ The Neumann characterization is less intense at the higher frequencies with a net shift in energy to the lower frequencies.

(19.98)

The integral of the wave‐height EDS is the wave energy and, for the Neumann spectral density, is evaluated as

(19.99)

Commonly used statistical measures of the wave height are [59]: the average height , the average of the 1/3 highest wave , and the average of the 1/10 highest wave . Expressed in terms of the wave energy, these average wave‐height measures are given by

(19.100)

(19.101)

and

(19.102)

The constant 4.791 in (19.99) applies for wave heights between and . The wave energy and corresponding wave heights are listed in Table 19.9 for the sea‐state conditions.

Sea‐State	uk (knots)a	E (m2)	b	b	b
1	5	0.0007	0.047	0.075	0.095
2	12	0.056	0.419	0.668	0.849
3	16	0.235	0.858	1.373	1.746
4	18	0.423	1.151	1.840	2.340
5	24	1.783	2.363	3.778	4.806
6	28	3.854	3.475	5.550	7.070
7	34	10.173	5.645	9.025	11.480
8	42	29.262	9.575	15.307	19.472

^aIn (19.99) wind velocity u is meter per second so u = 0.5148 u_k.

^bMeters.

19.11.3 Wave‐Height Generation Using Neumann’s Energy Density Spectrum

In this section, the received signal phase and amplitude characteristics are examined at a point beneath the ocean’s surface under various sea‐state conditions using Neumann’s wave‐height EDS. The approach is similar to that described in Chapter 20 for the generation of Rayleigh fading signals and involves the uniform sampling of the wave‐height EDS and then using the inverse Fourier transform (IFT) to generate discrete‐time sampled wave data. The resulting record of finite wave data samples is extended to an arbitrary length by joining independently generated records in a seamless manner.¹⁶

The sampling conditions are depicted in Figure 19.43 using an N_fft − point radix‐2 fast Fourier transform (FFT). The wave EDS is bandlimited to B Hz and sampled at the sampling frequency f_s = N_fftΔf Hz with N samples over the EDS response, that is, B = NΔf. The temporal response of the sampled EDS is shown as the sampled wave record in Figure 19.43b. T_B = 1/B is defined as the time‐constant of the wave temporal response and is related to the correlation time of the wave; there are N time‐constants for each record.

The parameter N is chosen such that there are about four samples within the 3‐dB bandwidth (B₃) of the EDS so the frequency increment is Δf ≅ B₃/4. The parameters B₃ and B are determined from Figure 19.41 and the value of Δf is chosen to satisfy the condition

(19.103)

where B is chosen so that integrated alias distortion is more than 25 dB below the EDS in the bandwidth B. Using these results, the record duration T_rec is computed as T_rec = 1/Δf.

Referring to Figure 19.54a, the sampling frequency is selected to satisfy the Nyquist criteria f_s = kB with k ≥ 2. Using this result and (19.103), the size of the FFT is determined as

(19.104)

where k = 2ⁿ: n ≥ 1. The minimum value of k results in the smallest FFT size of 2^m+1 samples and selecting larger values of k increases the size of the FFT by decreasing size sample interval Δt, however, Δf and T_rec remain unchanged. Therefore, for a given number of samples‐per‐symbol, increasing k results allows for more user symbols‐per‐record with fewer records required for the Monte Carlo performance simulations.¹⁷ These relationships are dependent on the modulated symbol duration T = 1/R_s that is typically less than T_B. Table 19.10 identifies the selected parameters that are used for the various sea‐state conditions in the case study of Section 19.11.5 and Table 19.13 identifies the common symbol rate of 32 symbols‐per‐second used for each sea‐state condition.

The complex samples of S(f) are generated using identically distributed zero‐mean independent complex Gaussian random variables b(nΔf): n = 0, …, N − 1 with variance . Although S(f) extents over all positive frequencies, the maximum frequency (N − 1)Δf is selected to result in negligible distortion due to aliasing. The sampling frequency increment Δf determines the record length given by T_rec = 1/Δf. To simplify the notation, the complex samples b(nΔf) are defined in terms of b_n as

(19.105)

and, letting i = {I,Q}, the pdf of the independent quadrature samples b_ni are characterized by N(0,σ_s). The discrete‐time samples of the wave‐height data h(t), t = mΔt, are obtained using the radix‐2 inverse fast Fourier transform (IFFT) with M = 1/(ΔtΔf) and are evaluated as

(19.106)

Because the wave function is real‐valued, the negative frequency response, that is, the response from f_s/2 to f_s = MΔf, is the complex conjugate of the positive frequency response; this property simplifies the evaluation of (19.106). Figure 19.44 shows a typical wave‐height record. The M‐sample record describing the wave heights h_m is concatenated with other independently generated records to create arbitrarily long temporal wave‐height conditions for Monte Carlo performance simulations. The record concatenation uses a third‐degree polynomial in t, such that, the amplitudes and slopes are equal at the record boundary points. The boundary points are separated by the wave‐height correlation interval τ_c. The normalized correlation response is evaluated as

(19.107)

and the correlation time is defined as the time at which

(19.108)

With and the discrete‐time correlation function is evaluated as

(19.109)

Using (19.107) through (19.109) to determine the correlation time, the number of interpolation samples required to concatenate the wave‐height records is . Figure 19.45 shows a typical correlation response for sea‐state 6.

To demonstrate the fidelity of the wave‐height generation, the Fourier transform of the sea‐state 6 wave correlation response in Figure 19.45 is compared to the theoretical Neumann EDS. The results are shown in Figure 19.46 where the triangular data points represent the re‐constructed EDS for an average of 1000 ensembles or records. Figure 19.47 shows the reconstructed EDS with different sampling conditions for an average of 20 ensembles. Although a single ensemble will differ distinctly from the theoretical EDS, this is the nature of statistical sampling ensemble records as required for communications system performance evaluation.

19.11.4 Signal Generation using Temporal Wave‐Height Records

Having characterized the wave height for different sea‐state conditions, it is now necessary to solve for the sampled received signal with amplitude and phase temporal fluctuations [60] influenced by the changing height of the wind‐driven waves. The solution is found in the characterization of an electric field traversing the surface of a conducting medium expressed as [61]

(19.110)

where μ is the permeability and σ is the conductivity of the medium and ω_c = 2πf_c, where f_c is the carrier frequency of the incident electric field E_o(t,ω_c) in hertz. Typical values of μ and σ for sea water are: 4π10⁻⁷ H/m and 4 mho/m respectively; values of sea water conductivity range between 2 and 8 mho/m. Equation (19.110) is used to evaluate the fluctuations in the received signal with changes in the water depth due to the wave‐height variations h(t), described by (19.106). By noting that , the exponent in (19.110) is expressed as

(19.111)

where¹⁸

(19.112)

The last equality in (19.112) is based on the above typical parameter values with f_c expressed in hertz. Using (19.110) through (19.112), the temporal expression of a CW signal is

(19.113)

where A(t) and θ(t) are the wind wave‐dependent amplitude and phase functions given by

(19.114)

and

(19.115)

The attenuation A(t) results from the variation in the signal level due to the wave‐height temporal variations, however, (19.114) also expresses the static loss as the water depth (d) increases beneath the surface of the sea. This static attenuation or loss, expressed in decibel per meter, is given by

(19.116)

Equation (19.115) is plotted in Figure 19.48 as a function of the carrier frequency for various sea water conductivities.

The theoretical characteristics of the signal phase function are evaluated in terms of the derivatives of θ(t) using (19.115) with the result

(19.117)

The phase derivatives represent stochastic processes with correlation functions (τ) and corresponding phase energy EDSs are expressed as

(19.118)

where

(19.119)

The phase variance is defined as

(19.120)

where E⁽ⁿ⁾ is the energy of the derivatives of the wave height. Using (19.119) E⁽ⁿ⁾ is evaluated as

(19.121)

Performing the integration in (19.121) using the Neumann EDS and substituting the result into (19.120) gives the standard deviation of the phase derivatives expressed as

(19.122)

The theoretical standard deviations expressed by (19.122) are listed in Table 19.11 for n = 0, 1, and 2 and Table 19.12 lists some corresponding results based on Monte Carlo simulation of the phase function as expressed by (19.115). The simulation results are based on eight wave‐height samples within the wave time‐constant T_B depicted in Figure 19.43 with samples taken over 5 N time‐constants. Using a first‐order polynomial to link the records, the simulated results for resulted in discontinuities at the record boundaries. These discontinuities degrade the performance of PLL tracking with coherent demodulation; however, they can be eliminated by using a higher order polynomial for linking the records.¹⁹

Sea‐State Number
Parameter	1	2	3	4	5	6	7	8
(degrees)	1.04	9.31	19.11	25.65	52.66	77.42	125.8	213.4
(deg/s)	4.59	17.07	26.28	31.36	48.28	60.84	81.42	111.8
(deg/s2)	35.00	54.22	62.61	66.40	76.68	82.83	91.27	101.4

^aNeumann spectrum corresponding to f_c = 30 kHz.

Parameter	Sea‐State Number
	2	4	6
(degrees)	11.26	20.54	37.98
(deg/s)	16.87	14.10	31.85

^aNeumann spectrum corresponding to f_c = 30 kHz.

The application of the wave amplitude and phase fluctuations, expressed in (19.114) and (19.115), respectively, to the received signal in a communication link is expressed in terms of the analytic signal representation as

(19.123)

where the transmitted analytic signal is expressed as

(19.124)

with

(19.125)

In these expressions, and are the modulation‐dependent amplitude and phase functions and P_s is the transmitted signal power.

The correlation response of a typical record of g(t) for sea‐state 6 is shown in Figure 19.49. Polar plots of the amplitude and phase fluctuations, in decibels and degrees, are shown in Figures 19.50, 19.51, and 19.52 for sea‐states 6, 4, and 2 respectively. Along the polar trajectory there are numerous phase reversals that are not apparent, for example, the occurrence of the first four phase reversals is indicated by the labeled arrows in Figure 19.50; the label 0 represents the record starting position. These phase reversals occur in the first quarter of the record and the entire record has 12‐phase reversals.

19.11.5 Case Study: Differentially Coherent BPSK Performance with Sea‐State

This case study examines the performance of differentially coherent BPSK (DCBPSK) waveform modulation with sea‐state distortion resulting from wind waves. The transmitted message is preceded by a single phase reference symbol that is used to detect the information bits. The ideal performance of DCBPSK in an AWGN channel is shown in Figure 19.53 and compared to the performance of coherent BPSK (CBPSK). The circled data points correspond to Monte Carlo simulated performance using 1 M bits for each signal‐to‐noise ratio. The following performance simulations with sea‐state examine the probability of receiving a 16‐bit word correctly so the word‐error probability is also plotted in Figure 19.53 under the ideal conditions involving AWGN. FEC coding is not used. These performance plots using the ideal AWGN channel are used for subsequent comparisons with the performance under various sea‐state conditions. The sea‐state simulations correspond to a carrier frequency of f_c = 30 kHz.

The criteria for the sampling of the temporal wave‐height records are outlined in Section 19.11.3 and the results are summarized in Table 19.10. Implicit in these requirements is the sampling of the communication waveform as determined by the selection of the FFT size. The most important consideration is that the selected sampling frequency results in an acceptable aliasing distortion loss. This condition requires that f_s ≥ f_N = 2(ρR_s + |f_e|) where f_N is the Nyquist frequency, R_s is the transmitted symbol rate, f_e is the maximum received carrier frequency error, and ρ is the number of spectral sidelobes permitted in the principal bandwidth |f| < f_N/2. The aliased spectrum is depicted in Figure 19.54 as the dashed curves for the signal spectrum S_s(fT) = sinc(fT) with ρ = N_s = 4 and f_e = 0.²⁰ Although the FFT size is identified in Table 19.10 for the generation of the wave‐height samples, the selection criteria involve the symbol rate and the number of samples‐per‐symbol as characterized by

(19.126)

where the record length T_rec = 1/Δf is determined in Section 19.11.3. Table 19.13 summarizes the selected FFT sizes for the various sea‐state conditions and the corresponding symbol rates with N_s = 4 samples‐per‐symbol. With these sampling conditions, the following bit‐error performance comparisons with sea‐state use identical symbol rates. Using 4 samples‐per‐symbol, as in this case study, the symbol rate can be increased by increasing the FFT size; however, this requires generating a new set of data records. On the other hand, the symbol rate can be decreased by simply increasing the samples‐per‐symbol without having to generate new sea‐state data records.²¹ The use of an integer number of samples‐per‐symbol is simply for convenience and avoids the necessity of rate conversion in the demodulator processing.

Figures 19.55 and 19.56 show the error performance for a DCBPSK‐modulated waveform with sea‐states 2, 4, and 6. Figure 19.55 corresponds to bit‐error probability using 1 M bits for each signal‐to‐noise ratio. The performance of antipodal and ideal DCBPSK operating in the AWGN channel are included for reference. When the waveform amplitude distortion, given by (19.114), is removed in the simulation, it is found that the performance is essentially unchanged indicating that the differentially coherent detection is predominantly impacted by the sea‐state phase distortion. The simulation conditions result in R_sT_B = 16, 32, and 64 symbols for each sea‐state time‐constant so increasing the symbol rate by about 8 : 1 or (32 : 1) should not result in a significant performance loss due to the phase distortion. The use of FEC coding and bit interleaving, with an interleaver span comparable to T_B, will mitigate the performance loss resulting from wave fluctuations. Figure 19.56 shows the performance of DCBPSK‐coded message words with 8 and 16 bits‐per‐word.

19.12.1 Introduction

A laser detector using a photomultiplier tube (PMT) [62] is shown in Figure 19.57. The collimated incident light waves (or photons) from the transmitter source are focused on the optical frequency filter and the desired filtered output illuminates the photosensitive surface of the photomultiplier tube releasing elections. The electrons are attracted to the photomultiplier anode by an electric field, created using the photomultiplier prime power, resulting in a current (i_s) through the load resistor R_L. The current fluctuates with the source modulation as does the voltage across R_L which is used to demodulate the source information. The gain of the photomultiplier is related to the number of free electrons at the anode relative to the energy of the incident light wave and gains on the order of 10³ to 10⁴ are not uncommon. The spatial filter eliminates light waves, entering the optical detector from directions other than from the communication transmitter [63, 64].

The equivalent received signal power at the input to the PMT is denoted as P_s (Watts) and the current created by the PMT is determined as

(19.127)

where G is the gain of the PMT and S is the responsivity of the PMT defined as

(19.128)

where q is the electron charge = 1.602 × 10⁻¹⁹ C, η is the efficiency of the PMT, h is Planck’s constant = 6.62 × 10⁻³⁴ J‐s,²² and ν is the frequency in hertz.

The received signal power at the output of the PMT is the power delivered to the load resister and is computed as . Substituting (19.127) for i_s results in the received power

(19.129)

19.12.1.1 Photomultiplier Tube Noise Sources

The noise developed across the load resister results from several noise sources as described in this section.

DC or non‐fluctuating noise is defined in terms of the noise current ; therefore, the DC power into the load resistor is

(19.130)

that behaves like the received signal power.

Thermal noise is determined by the system temperature T_s and is unaffected by the PMT gain; the two‐sided noise power density of the thermal noise is

(19.131)

Signal‐induced noise is shot‐noise resulting from the random release of electrons from the PMT optical surface. The shot‐noise is excited by the incident signal photons and results in the two‐sided noise power density expressed as

(19.132)

Background‐induced noise results from background light energy seen by the optical receiver. It consists of the Sun, Moon, bioluminescence, and other sources that are within the optical field of view (FOV); the two‐sided background noise power density is

(19.133)

Dark current noise results from shot‐noise caused by the random release of photoelectrons from the PMT optical surface due to the surface temperature. Although dark current noise encounters the gain of the PMT and it is usually negligible compared to the background induced noise and is often neglected. The dark noise power density is denoted as N_od.

19.12.1.2 Demodulator Signal‐to‐Noise Ratio

The following description is based on the M‐ary pulse position modulation (PPM) waveform format as shown in Figure 19.58. The symbol frame consists of a synchronization slot, containing the shaded transmitted laser pulse, followed by a dead‐time to provide for laser recharging, shown as the cross‐hatched interval, and M symbol slots that are selected based on the unique (log₂M)‐bit data symbol. The transmitted laser pulses encounter dispersion through the channel and are depicted as the dotted received pulses. Selecting the slot duration of T_s seconds involves a tradeoff between several system design considerations. For example, in consideration of the pulse dispersion, increasing the slot duration increases the average received pulse energy capability at the detriment of the frame efficiency and message throughput. The dead‐time (T_d) determines the pulse repetition frequency (prf) of the laser and is a major contributor to the frame overhead.²³ The PPM parameters are examined in more detail in Section 19.12.3.

The demodulator post‐detection bandwidth is the reciprocal of the slot duration, that is, B_s = 1/T_s. The demodulator signal‐to‐noise ratio is defined in terms of the pulse energy, E_p = P_LT_s, as

(19.134)

The noise density is the sum of the noise sources discussed above, excluding the DC noise that is removed by the post‐detection filter, so the noise density is expressed as

(19.135)

Upon substituting (19.129) for P_L, neglecting the dark and thermal noise sources, and substituting (19.135) with (19.132) and (19.133) into (19.134) results in

(19.136)

The responsivity (S) has units of amperes‐per‐watt, so the units of the product PS are amperes or coulombs‐per‐second. Therefore, since q is the electron charge with units of coulombs‐per‐electron, the quantity PS/q has units of electrons‐per‐second. In the following, the notation λ = PS/q is used and (19.136) simplifies to

(19.137)

where the second equality results by multiplying the numerator and denominator by T_s and recognizing that N = λT_s is the number of photoelectrons in T_s seconds. Equation (19.137) is expressed in terms of the signal‐to‐noise ratio in the bandwidth equal to the bit‐rate, R_b = 1/T_b, using , where

(19.138)

Upon solving (19.137) for N_s and expressing the result in terms of γ_b, the signal photoelectron count is evaluated as

(19.139)

These parameters are exactly those required to characterize the bit‐error performance of the demodulator in terms of the Poisson distribution as discussed in the following case studies [65, 66]. The photoelectron counting is also modeled using the Nakagami m‐distribution [67].

19.12.2 Pulse Dispersion in the Atmosphere

Based on experimental results [68–70], the generally accepted shape of a pulsed laser after emerging from clouds is shown in Figure 19.59 and characterized as

(19.140)

In the following analysis, the PPM slot duration, measured in microsecond, is selected to satisfy two conflicting requirements: increase T_s to capture more received signal energy (E_s) for detection or decrease T_s to provide the shortest PPM frame. The second of these options is dominated by the peak‐to‐average power ratio and the resulting duty cycle [71] of the laser transmitter. The laser pulse dispersion²⁴ in the atmosphere results from the propagation path length (L_cp) through clouds and is characterized principally by the geometric thickness (L_cg) and the mean scattering angle (θ_cs) of the cloud. The delay (T_fd) and width (T_fw) of the integrate‐and‐dump (I&D) detection filter must be selected to maximize the signal‐to‐noise ratio at the optimum sample time of the detection filter. Communication links from the atmosphere to underwater links encounter additional dispersion that is dependent on the geometry of the encounter and the water type [72], for example [73], Jerlov II and III water²⁵ and bioluminescence is a major source of background noise.

Stotts [74] has developed a closed‐form expression for laser pulse dispersion through clouds, for which, the pulse dispersion is expressed as

(19.141)

where c = 3e8 m/s is the free‐space speed of light, ω_o > 0.999 is the cloud albedo²⁶ (in the following analysis ω_o = 1.0), τ_c is the optical thickness of the cloud,²⁷ and L_cp and θ_cs are in meters and radians respectively. The relationship between the geometric cloud thickness (L_cg) and the optical cloud thicknesses is defined as

(19.142)

where σ_c is defined as the average extinction (or attenuation) coefficient of the cloud with dimensions of meters⁻¹. The extinction coefficient is a measure of absorption of light in the cloud. In the following analysis σ_c = 0.04 m⁻¹ which results in L_cg having dimensions of meters. The optical thickness of various types of clouds [75, 76] is listed in Table 19.14.

Figure 19.60 is a plot of (19.141) showing the dependence of the pulse dispersion on the propagation path L_cp for the indicated mean scattering angles and average extinction of the cloud.

The pulse dispersion is measured as the pulse duration between the half‐power points of the pulse relative to the peak power P_pk that occurs at t = T_pk = 1/α. Defining the half‐power points of the pulse as occurring at T₁ and T₂, the pulse dispersion, expressed in (19.141), is T_cd = T₂ − T₁. Referring to Problem 23, T₁ and T₂ are related to the parameter α as T₁ = 0.232/α and T₂ = 2.677/α. Using these relationships, the pulse is completely characterized in terms of the dispersion through the cloud by the parameter α expressed as

(19.143)

Figure 19.61 shows the received pulse responses for the indicated values of the parameter L_cp, in kilometer, corresponding to the mean scattering angles used in Figure 19.60. In these plots, the received pulses are normalized by the total pulse energy E_p = 1/α² resulting in unit energy pulses for each condition. Therefore, the integral

(19.144)

and E(T_cd)/E_p represents the normalized received pulse energy in the interval T_cd. Conversely, 1 − E(T_cd)/E_p represents the loss of energy in decibels.

Evaluating (19.144) over the range t = 0 to T_s, the received pulse energy in the PPM slot is determined as E_s and is plotted in Figure 19.62 as a percent of the total received pulse energy. Figures 19.61 and 19.62 or Table 19.15 show that there is a large variation in T_s to cover the conditions; however, with θ_cs = 40° a slot duration of 23.3 µs will capture 95% of the pulse energy with L_cp = 1 km, whereas, a slot duration of 100 µs is required with θ_cs = 40 and L_cp = 2 km capture the same energy. If the transmitter can estimate the cloud conditions then T_s can be minimized and the receiver can form an estimate of the received slot duration during the waveform acquisition processing.

19.12.3 Pulse Position Modulation Waveform

The transmission of M‐ary coded data [77, 78] using a pulsed laser power source is often transmitted using PPM as described in Figure 19.58. The slot synchronization pulses are shown to occur prior to each M‐ary symbol location; however, for covert synchronization, they may occur at predetermined random locations known by the demodulator. Using periodic synchronization pulses simplifies the synchronize processing through the use of a narrowband bandpass filter centered at the synchronization pulse repetition rate; however, the use of FEC coding improves the bit‐error performance at the expense of increased complexity [79, 80]. Example FEC configurations are: a rate r_c convolutional code with a prescribed constraint length, a Reed–Solomon code, for example a concatenated code with a constraint length K convolutional inner code and an M‐ary Reed–Solomon outer code with M = 2^k bits per Reed–Solomon symbol. Data interleavers should also be used with FEC coding; especially between concatenated codes.

The source bit‐rate (R_b) of M‐ary PPM, without FEC coding, is expressed in terms of the bits‐per‐pulse (k) as

(19.145)

where k = log₂(M). Equation (19.145) is plotted in Figure 19.63a and b as a function of k for T_d = 5 and 1 ms respectively with the slot duration (T_s) as the parametric parameter. The laser dead‐time and the slot duration are seen to have a significant impact on the maximum data rate that also dictates the selection of k. The PPM waveform design, including the selection of the slot duration, bits‐per‐slot, and maximum user bit‐rate, are established using Figure 19.63. For example, using the 5 ms dead‐time shown in Figure 19.63a, and selecting k = 7 and T_s = 10 µs results in a 128‐ary PPM waveform with a maximum uncoded user bit‐rate of 1.113 kbps.

Defining the slot duration as in Figure 19.59, the total received pulse energy in the slot and the percentage of pulse energy is determined from Figure 19.62 or Table 19.15. The optimum I&D detection filter delay and duration, shown in Figure 19.59, must be determined in the context of the statistical characterization of the PMT noise and other noise sources as discussed in Sections 19.12.1.1 and 19.12.1.2.

19.12.4 Differential Pulse Interval Modulation Waveform

With differential pulse interval modulation (DPIM) [81], the pulse intervals are synchronized to the transmission of an initial synchronization pulse followed by M‐ary coded pulses based on the differential change in the time slots relative to the preceding pulse. DPIM modulation is depicted in Figure 19.64 using the M‐ary coded waveform with symbol sequence (m, 5, 2, …). In this example, the DPIM‐coded waveform transmits the first information pulse in slot m following the reference pulse and the associated dead‐time, the second pulse is transmitted in slot five following the laser recovery of the first pulse, and the third pulse is transmitted in slot two following the laser recovery time. Therefore, the three information pulses are transmitted in a time equal to (m + 8)T_s + 3T_d µs; whereas, the PPM‐modulated waveform, shown in Figure 19.58, transmits the first three information pulses in (3M + 1)T_s + 4T_d µs. The average time to transmit a frame of M‐ary coded random data using the DPIM‐modulated waveform is

(19.146)

compared to  µs for the PPM‐modulated waveform.

Figure 19.65 shows the bit‐rate of DPIM as a function of the bits‐per‐slot under similar conditions for the PPM modulation shown in Figure 19.63. The average bit‐rate for DPIM is expressed as

(19.147)

19.12.5 Case Study: PMT Demodulator Performance Using OOK Modulation

In this case study, the laser transmitter is either on‐off keying (OOK) resulting in the binary hypothesis in which a binary zero (b₀) is represented by no signal energy being transmitted and a binary one (b₁) is represented by transmitting a signal energy of E_s. Under the maximum‐likelihood (ML) decision rule, the a priori probabilities are represented by P(b₀) = P(b₁) = 1/2 and the decision rule is expressed in terms of the transition probabilities using the likelihood ratio as

(19.148)

The transition probabilities are characterized in terms of the Poisson distributions [82]

(19.149)

where λ_i is the photoelectron rate and T_s is the interval in which the photoelectrons are counted, so N_i is the number of photoelectrons in T_s seconds. Under the conditions b₁ and b₀, the respective notations for the photoelectron counts are and . T_s is the reciprocal of the receiver post‐detection bandwidth and typically T_s ≅ T_b to allow for dispersion of the bit energy through the optical channel. Substituting (19.149) into (19.148) and rearranging the terms of the log‐likelihood ratio (LLR), the log‐likelihood ratio test (LLRT) is expressed, in terms of the threshold k_T, as

(19.150)

This threshold is the optimum threshold for making the decision that minimizes the bit‐error probability, evaluated as

(19.151)

P_be is evaluated in terms of γ_b by specifying the background photoelectron count N_b, and using (19.139) to determine N_s. The resulting bit‐error performance is shown in Figure 19.66 under ideal channel conditions with T_s = T_b and the indicated values of background noise. The bit‐error performance for N_b = 0 results in k_T = 0 so P_be simplifies to . The slight ripple in the N_b = 1 curve results from the integer summations involving the Poisson distributions; for threshold levels exceeding 14 the Gaussian approximation to the Poisson distributions is use to evaluate P_be. The noncoherent OOK performance with AWGN is discussed in Chapter 6.

19.12.6 Case Study: PMT Demodulator Performance of M‐ary PPM Waveform

In this case study, the bit‐error performance is examined for the M‐ary PPM waveform [77] as characterized in Figure 19.58 with k = log₂M bits transmitted in one of M time slots during each symbol frame of duration T_f = MT_s + T_d seconds. The laser pulse contained in the interval T_sync is used for frame synchronization and tracking. In this case, the ML decision rule uses equal a priori probabilities for each symbol with P_m = 1/M: ∀ m ∈ M. The rule that minimizes the symbol‐error probability selects the time slots having the largest post‐detection filter output when the post‐detection filter is matched to the received pulse. An upper bound on the symbol error is given by Bucher [83] as

(19.152)

where the exponent term E_c(N_s/N_b) is equal to

(19.153)

Solving (19.137) for N_s/N_b and substituting the result into (19.153) results in

(19.154)

As in the preceding binary example, the following performance is characterized in terms of the background noise count N_b; however, in this case, the independent variable will be the signal‐to‐noise ratio as measured in the slot bandwidth. Referring to (19.139), the parameter N_s is expressed in terms of N_b with the signal‐to‐noise ratio (γ_sym) measured in the symbol bandwidth equal to 1/T. Therefore, the signal photoelectron count, measured in the slot bandwidth, is computed as

(19.155)

where converts the signal‐to‐noise ratio from the symbol bandwidth to the signal‐to‐noise ratio in slot bandwidth.

The M‐ary PPM performance is shown in Figure 19.67 as a function of γ_s for various values the background noise count N_b. The symbol‐error results are based on the union‐bound [84] which is a loose upper bound on the performance; however, for a fixed M, the union‐bound becomes increasingly tighter as γ_s increases. Figure 19.68 shows the union‐bound symbol‐error performance for the indicated values of k = log₂(M) using a background photoelectron count of N_b = 200.

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ADDITIONAL WIRELESS MOBILE COMMUNICATION REFERENCES

1. K. Allesbrook, J.D. Parsons, “Mobile Radio Propagation in British Cities at Frequencies in the VHF and UHF Bands,” IEEE transaction on Vehicular Technology, Vol. 26, No. 4, pp. 313–323, November 1977.
2. J.J. Egli, “Radio Propagation Above 40 Mc Over Irregular Terrain,” Proceedings of the Institute of Radio Engineers (IRE), Vol. 45, No. 10, pp. 1383–1391, October 1957.
3. V. Erceg, S. Ghassemzadeh, M. Taylor, L. Dong, D.L. Schilling, “Urban/Suburban Out‐of‐Sight Propagation Modeling,” IEEE Communications Magazine, Vol. 30, No. 6, pp. 56–61, June 1992.
4. B. Goldberg, Communications Channels: Characterization and Behavior, IEEE Press, New York, 1976.
5. W.C. Jakes, Editor, Microwave Mobile Communications, IEEE Press, New York, 1994.