[1] In recent years, bistatic pulsed high-frequency ground wave radar models of the ocean clutter have been developed. Several new features, distinct from earlier monostatic developments, appear as products of those analyses. One question that needs to be addressed is, “What characteristics of the theoretical clutter models are likely to be visible in experimental data collected from the ocean surface?” A major consideration in answering this question is the development of an appropriate noise model. Such a model along with an analogous clutter model is derived. This allows a simulation of time series data for both clutter and noise which may be treated using standard Fourier transform techniques to provide a periodogram for the typical combined noise/clutter spectrum of scattering from the ocean surface. The analysis proceeds on the assumption of an externally noise-limited system, with the noise being characterized as a white Gaussian zero-mean process. The aliasing due to noise undersampling is seen to be an integral part of the model. Statistical stationarity is assumed throughout. Both infinite and finite pulse trains are considered.

[2] On the basis of a theory presented by Walsh et al. [1990], Walsh and Dawe [1994] developed a first-order result for the high-frequency bistatic cross-section of the ocean surface. The analysis also appears in the open literature in the work of Walsh and Gill [2000] and with extension to second-order in scatter for the bistatic case in the work of Gill and Walsh [2000, 2001]. Some of the features of the results for the bistatic case are completely new while others are shown to be generalizations of earlier monostatic derivations. To examine whether or not these features are likely to be observed in field data, it is necessary to model the received power due to scatter from a remote surface patch. From the well-known radar range equation, it is clear that this quantity will depend upon the distances of the patch from the radar components, the frequency of transmission, the gains of the antennas, the transmitted power, the rough spherical earth attenuation functions and the scattering cross-section. However, even if these quantities are known, there is no assurance that all of the features described in detail by Gill and Walsh [2001] will always be visible in the radar power spectrum. Obviously, one of the major reasons for this is the fact that any received signal is contaminated, to some extent, by external noise.

[3] Here, a suitable combined noise and ocean clutter signal model for a pulsed radar is developed. Section 2 begins by characterizing the noise signal and, subsequently, its spectral density as observed in a pulsed radar system. An identical technique allows for the development of a realistic model for the ocean clutter. The results for the noise and clutter signals may then be added in the time domain. Aliasing which arises because of noise undersampling is seen to flow directly from the analysis. The models are first developed assuming the availability of a large, ideally infinite, number of pulses, but in Appendix A it is shown that the results are essentially the same for a limited number of pulses. The radar system is assumed to be externally noise-limited and the noise is assumed to be white, Gaussian, zero-mean and statistically stationary. In section 3, results are depicted for a particular pulsed-radar configuration. Section 4 contains a summary and indicates the utility of the technique for modeling high-frequency pulsed radar signals before system deployment.

2. Characterization of the Noise and Clutter Voltages and Their Spectral Densities for Pulsed Radar

2.1. Noise Signal

[4] In the HF band, the external noise power may arise from a combination of atmospheric, galactic and man-made sources [see International Telecommunications Union (ITU), 1994]. It shall be assumed, as in those Recommendations, that the noise is a stationary Gaussian white process and is understood to have zero mean. As a model for such a stationary Gaussian process in one variable, we consider a mathematical construct presented by Pierson [1955] in his development of a theory to describe wind-generated gravity waves. Using Pierson's model, for which he admittedly gave credit to private correspondence from James W. Tukey, the ambient noise voltage, n_{a}(t), may be cast, using the complex exponential, as

where

with the h(·) being the Heaviside function introduced to account for the fact that any receiving system will have a limited noise bandwidth, B, t is time, ω′ is radian frequency (=2πf′), S_{N}(ω′) is the power spectral density of the noise, and ε(ω′) is the random phase uniformly distributed on the interval 0 to 2π, and . The integral limits may be understood to be over the entire set of real numbers, but only −(B/2) < ω′ < (B/2) actually contribute to a non-zero integrand. It is immediately obvious that the construct in (1), containing the variable of integration under the square root, is not an integral in the usual Riemann sense. It does, however, represent the ensemble of, in this case, all possible noise regimes whose power spectral densities are embodied in S_{N}(ω′), the process being stationary and Gaussian. The legitimacy of this representation of a Gaussian process involving the differential under the square root is discussed in meticulous detail by Pierson and its historical significance and links to other important work are elaborated on in Chapter 8 of Kinsman [2002], the latter also containing a complete reproduction of the Pierson's original manuscript.

[5] The desire is to apply the expression in equation (1) to a pulse radar system. It is considered that the pulse repetition period is T_{L} while the temporal width of the pulse is τ_{0} as depicted in Figure 1. At first, to later aid in the analysis for the ocean clutter signal, we consider that the pulse train is large, ideally unbounded. It will be seen in Appendix A that this does not essentially affect the results. The noise, n(t), at any time sampled by the receiving system may be characterized, using (1), as

it being understood here and in the remainder of the article the general “rectangular” function in terms of the Heaviside functions has the form

A power spectral density for n(t) may now be sought. Keeping in mind that the autocorrelation, (t_{1}, t_{2}), for n(t) is given in the usual sense by

where 〈·〉 denotes ensemble averaging, N(·) is the Fourier transform of n(·) and * denotes complex conjugation, we first seek N(ω) – i.e., the Fourier transform of n(t) in (1). Formally, because of the Heaviside functions in the summations of (1),

The time integral in the transform, on changing the variable to t′ = t − mT_{L}, becomes

where Sa[·] = . Substituting the result of (5) into (4) leads to

where δ(·) is the Dirac delta function, T_{P} is a fixed period and t_{d} is time. Examining the summation in (6) in view of (7) and identifying t_{d} and T_{p} as being analogous to (ω′ − ω) and 2π/T_{L}, respectively, the form of the sum is

The Fourier transform in equation (6) is then given by

The randomness here appears only in the phase exponential and given that the ensemble average,

the autocorrelation in the integrand of equation (3), in view of (8), becomes

The delta function constraint immediately requires that

so that, using one of the delta functions to evaluate the ω′ integral, equation (9) may be written as

Using the inverse Fourier transform expression given in equation (3), the autocorrelation function is

where the ω_{2} integral has been executed via the delta function of (10). Defining

and setting ω_{1} = ω, this autocorrelation finally appears as

The noise power spectral density, _{N}(ω, t), clearly follows from the Fourier transform of _{N}(τ + t_{2}, t_{2}) as

where t = t_{2}.

[6] To reduce the complexity of (13), an alternate form is sought for the second summation. To this end, it is noted that a periodic train of impulses, like that in Figure 1, may be written as a Fourier series in the form

where c_{n} are the Fourier coefficients. These are given by

Equation (16) represents the general form of the time-dependent Doppler noise power spectral density for a pulsed radar assuming a noise which is a stationary zero-mean Gaussian process. This time dependency may be considered in terms of the “position” at which the pulses are sampled. Here we choose to sample at the pulse centers since, as is easy to show, this maximizes the noise power spectral density.

[7] To impose sampling at the pulse centers, it is required that t = pT_{L} where p belongs to the integers. Of course, m is also an integer so that the exponential in equation (16) becomes

It has been assumed at the outset that the ambient noise spectrum is flat (i.e., white noise) so that S_{N} (ω + (m2π/T_{L})) may be removed from the summations and designated as S_{N}(ω′). The noise spectral density for the pulsed radar then becomes, from equations (16) and (17),

it being understood that t = pT_{L}. Next, it may be noted that since τ_{0}/2 < T_{L}

The remaining Heaviside functions ensure that for non-zero results,

where ω is the Doppler radian frequency of the noise. In a typical pulse radar, the receiver bandwidth, B, is on the order of kilohertz (or krad/s in terms of radian frequency) while ω is on the order of rad/s (i.e., Doppler frequency is on the order of hertz). Therefore, to a very good approximation, (20) becomes

and the asymmetry in the summation of (19) is removed. Furthermore, if match filter conditions are assumed to exist [see, for example, Barton, 1987], B = 2π/τ_{0}. The limits on m in (19) are then given by

With these stipulations, for sampling at the pulse centers, the Doppler noise power spectral density in (19) simplifies to

where the radar duty cycle, d, has replaced τ_{0}/T_{L}, and it is understood that m takes integer values. The argument pT_{L} has been explicitly retained only to emphasize sampling at the pulse center. Since S_{N}(ω′) is assumed constant, _{N}(ω, pT_{L}) is also constant for all ω. Equation (23) illustrates the very important property that the aliasing appearing by virtue of the summation is “buffered” by the duty cycle multiplier. That is, a low-duty cycle, d, will increase aliasing as it increases the range of m, but at the same time this small d also multiplies the sum to mitigate the aliasing effect. Conversely, a higher-duty cycle will reduce the aliasing since there will be fewer m's in the sum, but this reduction is subject to multiplication by a larger external factor.

[8] It may be verified numerically that for typical duty cycles in a pulse radar, _{N} (ω, pT_{L}) in (23) does not vary significantly from the ambient noise power spectral density, S_{N}(ω′). This is illustrated in Figure 2. When d > 0.5, there is, logically, only the m = 0 pulse in the summation of equation (23). For these cases, the total multiplier on S_{N}(ω′) is simply d and the slope of the curve is unity. Appendix A establishes a similar result for a finite number of pulses. Indeed, the effect of using only a few pulses is not significantly different from the case given here. To illustrate this fact, such a case is also depicted in Figure 2 where it can be seen that the discrepancy between the two results is always less than 3 dB for all duty cycles.

[9] We next seek a practical form of equation (23) for use with a system which is externally noise-limited. We have noted that in the HF band the external noise of significance falls into three categories: (1) atmospheric; (2) galactic and (3) man-made [see ITU, 1994]. The relative importance of each type of noise depends on location, the frequency of operation, the time of day and even the season of the year, the latter dependencies arising because of variable solar activity. While significant fluctuations may occur as these parameters vary, median noise values will suffice for illustrative purposes.

[10] In the ITU-R Recommendations, hereafter referred to as the Recommendations, the external noise factor, f_{a}, is defined as

where _{n} is the available noise power from an equivalent lossless antenna, k = 1.38 × 10^{−23} J/K is Boltzmann's constant, T_{0} is the reference temperature which is taken as 290 K and B_{n} is the noise power bandwidth of the receiving system. For a matched filter system, B_{n} may be taken as the reciprocal of the transmit pulse width, τ_{0}. It is common to define an external noise figure, F_{a}, as

a median value of which may be designated as F_{am}. This is the quantity which is readily available from the Recommendations. For a white noise process, as is assumed here, the power spectral density, S_{N}(ω′), in equation (23) is

Then, equation (23) for the external Doppler noise power spectral density, _{N}(ω), becomes

Here, the pT_{L} argument has been dropped from the power spectral density as it does not enter the calculation for infinitely many (or a large number of) pulses. As noted in Appendix A, this is also true for even a small number of pulses. Finally, while here consideration is being given only to external noise, if the noise factor of the receiving system is known, then applying (27) to the system itself allows for a simple combination of external and internal noise.

2.2. Ocean Clutter Signal

[11] As is seen later in this subsection it is necessary to invoke the radar range equation (see equation (34)) to calculate the power spectral density for the sea echo received from a particular patch of ocean surface. Implicit in that equation as it appears in the cross-section derivation found in the work of Walsh et al. [1990], and subsequent work based on that analysis, is the fact that the transmitter power P_{t} is the peak pulse power which, by definition, is not the maximum instantaneous power of a single sinusoid but is the power averaged over the pulse length, τ_{0}. This translates into the peak pulse power being one half the peak instantaneous power. The question to be addressed now is “What is the proper form of the Doppler power spectral density for the ocean clutter which is obtained for a particular patch of ocean by gating the received signal on and off repeatedly?” With this in mind, we repeat the general procedure of section 2.1 for the ocean clutter signal.

[12] For the temporal periods, typically several minutes, used in examining HF sea echo, Barrick and Snider [1977] have argued that the ocean surface wave field may be modeled as a stationary Gaussian process. This results in the echo signal being likewise approximately stationary and Gaussian for the same time frames. With these assumptions the ocean clutter signal, c_{a}(t), may be cast analogously to the noise voltage of equation (1) as

Here, B_{c} is the clutter bandwidth, ω′_{c} the radian frequency, ε(·) is the random phase and S_{c}(·) is the peak power spectral density of the continuous clutter signal. Gating this signal, under the assumption of infinitely many pulses (in reality, a large number of pulses) (29) may be modified to give

The form of the gated clutter, c(t), is obviously identical to that for the gated noise in equation (2). Therefore the spectral density for c(t) may be written down immediately from that given for the noise in equation (16). Using subscript c to indicate “clutter” quantities, the Doppler power spectral density of the gated signal is

If, next, it is assumed that the sampling occurs at the pulse centers–i.e., t = pT_{L} with p integer –, proceeding as for the noise analysis, equation (31) may be shown to reduce to

Typically, the Doppler clutter bandwidth is on the order of hertz while the pulse repetition period, T_{L}, is in the microsecond range. This means that

and since ∣ω∣ ≤ ,

as well. Thus the limits on the sum in equation (32) which trivially may be written as

indicate that m = 0 is the only surviving term. Therefore, from equation (32), and using the fact that _{c}(·) is clearly independent of p, we may write

Unlike the Doppler noise power spectral density, _{N}(ω), there is no aliasing apparent in _{c}(ω). Of course, this is not surprising as adequate sampling of the clutter signal was implicitly imposed by the assumptions on the summation indices following equation (32). Thus, while the noise is a broadband signal and is folded into the narrow Doppler bandwidth, the ocean clutter is already narrow band and is therefore not affected in this way. The other important feature in equation (33) is the factor (τ_{0}/T_{L}) multiplying the peak spectral density. It is easily shown that this combination is, in fact, the average power spectral density. This means that, when considering the gated clutter signal, the average power, rather than the peak power, should be used in the radar range equation. That equation, modified from the work of Barton [1987] for a narrow beam receiving system, may be written for the Doppler clutter power spectral density as

It should be remarked that (τ_{0}P_{t}/T_{L}) is the average transmitted power as dictated by equation (33); λ_{0} is the operating wavelength; G_{t} and G_{r} are the gains of the transmitting and receiving antennas, respectively; the F's are the usual Sommerfeld attenuation functions; ρ_{01} and ρ_{02} are the distances from the scattering patch to the transmitter and receiver, respectively (for monostatic operation they are identical); A is the scattering patch area and σ is scattering cross-section normalized to the patch area. Differentials strictly associated with received power spectral density and area have been removed under the assumption that the receiver beam width is narrow enough to ensure constancy of the various parameters over the area A being interrogated.

[13] The clutter analysis in conjunction with that of section 2.1 provides a basis for defining the ocean clutter signal to noise ratio for a pulse Doppler radar as

[14] Results of this and the preceding sections, which may be used to examine the relative importance of the various portions of the HF cross-sections of the ocean surface when the radar system is externally noise-limited, are depicted in section 3.

2.3. A Model for Estimating the Doppler Spectral Density of Sea Echo in the Presence of Noise

[15] To implement the preceding analysis to produce a model for estimating a noise-contaminated HF Doppler spectrum for sea clutter, we note that Pierson's model for a one-dimensional stationary Gaussian process for a time function, f(t), of limited bandwidth, B, found in equation (1) may be written as

Here

and F_{s}(ω) is the power spectral density of f(t). Now,

where n(t) and c(t) are defined in equations (2) and (30), respectively. Similarly,

and sampling at the pulse centers is assumed. _{N}(ω) and _{c}(ω) are then given by equations (28) and (34), respectively. Since _{N}(ω) and _{c}(ω) are developed from ensemble averages using ideal conditions, they represent the “true” (i.e., best possible) power spectral densities of the noise and clutter.

[16] The integral in equation (36) may be represented by the limit of a partial sum [see Pierson, 1955] as given by

This is then used to approximate a time series for the noise, n(t), and the clutter, c(t). The time function, s(t), representing the sum of the clutter and noise signals received by the pulsed radar system is simply

The signal, s(t), may be fast-Fourier-transformed (FFT) using any suitable algorithm (e.g., the nfft(·) function from MATLAB [see The Mathworks Inc., 1994] is used for the plots in Figure 6 of section 3). The magnitude-squared of the FFT algorithm is a normalized estimate of the power spectral density, S(f), and may be divided by the time series “length” to give the proper units of W/Hz. This, of course, is the so-called periodogram or sample spectrum.

3. Calculation and Illustration of Typical Noise and Clutter Power Spectral Densities

[17] In this section, the previous models will be illustrated in two ways: (1) firstly, equations (28) and (34) will be implemented separately, and the noise and clutter spectral densities will be plotted for pulse radar operating frequencies which cover the lower, middle and upper sections of the HF band; (2) secondly, the analysis in section 2.3 along with equations (28) and (34) will be used to illustrate an example of a combined noise and clutter spectrum. The second method mimics the process of producing Doppler spectra from pulsed HF radar sea echo which is invariably contaminated by noise to some extent. It is intended that this model should prove helpful in determining hardware configuration and signal processing parameters for particular deployments of HF pulsed Doppler radar for the purpose of surface parameter estimation.

[18] For the purpose of illustration, we first choose radar system parameters similar to those used in a long-range system operated by Northern Radar Inc. in the 1990s. Subsequently, the noise conditions are addressed.

[19] Initially, the following radar parameters are considered (of course, to illustrate the effect of changing operating frequency, the antenna element spacing must be appropriately altered from that given below to maintain fixed patch areas):

[30] 11. distances from patch to transmitter and receiver: ρ_{01} = ρ_{02} = 50 km

[31] 12. rough spherical earth attenuation functions: F(·)'s are calculated from a FORTRAN routine devised by Dawe [1988]. In monostatic operation, the angle of the wind with respect to the radar beam is an important parameter. In bistatic operation the wind directions with respect to the transmit beam and the receive beam are used in determining F(ρ_{01}) and F(ρ_{02}), respectively. The surface roughness is a function of wind speed. Antenna heights are chosen to be zero. The relative permittivity of seawater is taken as 80, and an average conductivity of 4/m is used. An extensive discussion of transmission losses over the rough sea is provided by Barrick [1971]. In fact, the calculations by Dawe [1988] provided a comparison of Barrick's results with later results by Srivastava [1984]. As an illustration we note that for a 25 MHz signal, a wind velocity of 15 m/s at an angle of 120° relative to the look direction (as in Figure 3), the magnitude of F for a range of 50 km is approximately 0.312.

[32] The assumption of the system being externally noise-limited is still in view and, consequently, the noise analysis in the receiving system as a whole will not be an issue here. Extension to that case is a well understood process and is given in detail, for example, by Collin [1985]. For the 4% duty cycle, (τ_{0}/T_{L}), suggested above, the summation index in the noise equation (28) ranges from −12 to 12. From the Recommendations, the noise figure F_{am} may be determined as 22 dB, 36 dB and 42 dB, for operating frequencies of 25 MHz, 10 MHz and 5.75 MHz, respectively. It may be noted that the “quiet receiving site” category was used for the man-made noise. Also, in the case of 25 MHz, the man-made and atmospheric noise is negligible, while galactic contributions are relevant for all three frequencies. The atmospheric noise figure was chosen halfway between that which is exceeded 0.5% and 99.5% of the time. From Figure 2, the multiplier on S_{N}(ω′) (see equations (23), (27) and (28)) is not significantly different from unity for a 4% duty cycle. Therefore, in the decibel sense, the noise power spectral density of equation (28), to an extremely good approximation, may be given on using frequency, f, in hertz rather than ω in radians/second as

in the simulations here.

[33] For the purpose of cross-section illustration as in Figure 3, the ocean wave surface is modeled as the product of a Pierson and Moskowitz [1964] non-directional spectrum and a cardiod directional pattern with a spreading parameter of 2 [see, for example, Tucker, 1991]. For the bistatic case with a broad beam transmitting antenna, the scattered signal arriving co-temporally at the distant receiving site is from an elliptical patch of ocean. However, under the assumption of a narrow beam receiving array, the observed signal comes from a small portion of this ellipse. The angle between the transmitter and receiver as viewed from this scattering patch is bisected by the ellipse normal at that “point”. Each portion of this bisection is the bistatic angle of which a representative value over the patch is labeled ϕ_{0}, and to illustrate we have used ϕ_{0} = 30°. Figure 3 shows that the basic structure of the bistatic and monostatic results are very similar. In conjunction with Figure 4, it is clear that the observation of various spectral peaks is dependent on the directional nature of the wind-driven sea. In fact, at 25 MHz, if there is a 15 dB increase in the noise floor from sources not included here, at a range of 50 km the increase in the spectral tails due to scatter at the transmitter or receiver [see Gill and Walsh, 2001] will not be visible. However, the peaks near zero Doppler at 25 MHz may still exceed the noise floor and may be visible provided there is no dc contribution from other sources. In passing, it may also be noted as discussed by Walsh et al. [1990], and in subsequent related work referred to throughout this article, that the effect on the dominant portions of the spectra of increasing the modulating pulse width, and thus the scattering patch width, is to cause a decrease in the spectral width of these first-order regions.

[34]Figure 5 clearly shows the improvement in SNR as the operating frequency is lowered. This is largely due to the fact that increasing noise levels are mitigated by a large increase in the spherical earth attenuation functions F(·) appearing in the radar range equation. For the stated parameters, the magnitudes of F for the 25, 10 and 5.75 MHz cases are 0.312, 0.815 and 0.959, respectively. Clearly, the clutter signal, as received from a fixed patch of ocean, will have experienced significantly less attenuation when operation occurs in the lower part of the HF band. The result will be that, for a fixed patch of ocean, the ratio of the clutter power to the noise power will be higher for lower operating frequencies even though the noise figures behave oppositely. It may also be noticed that, in relation to the overall spectrum, the peaks near zero Doppler for 10 MHz and 5.75 MHz contain very little power and may be easily masked by system effects. For example, even small amounts of spectral smearing due to finite system timing could obliterate this phenomenon. However, at these lower frequencies the increase in the spectral tails due to scatter near the transmitting and receiving sites is well above the noise floor.

[35] It must be remarked that the results presented in Figures 3–5 are for an idealized system. Such factors as ground losses and receiver noise, for example, are not included. Furthermore, and perhaps more importantly, the results all find their basis in the ensemble averaging of the received electric field. By definition, this presumes that infinitely many, statistically similar oceans are available for interrogation. Thus the results discussed to this point represent an idealized scenario and no adjustment of system parameters may be effected to improve the (SNR)_{c} beyond that depicted.

[36] In practice, the power spectral density is often estimated using the squared magnitude of the Fourier transform of a finite time series. Thus the spectral model suggested in section 2.3 better reflects reality. To illustrate, we choose an operating frequency of 5.75 MHz and a wind direction is taken to be 60° to the radar look direction. Other parameters are as for Figure 5c. The “ideal” Doppler power spectral densities for the noise and clutter as found from equations (28) and (34), respectively, were used in equations (39) and (40) to produce a combined noise and clutter time series. A typical 512-point time series is depicted in Figure 6a. Figures 6b–6d show the effects which differing time series lengths and averaging have on the periodograms. All data were subjected to a Blackman window [see Harris, 1978]. While no dedicated field data are available for comparison, as a point of discussion we include in Figure 6d a similarly windowed and averaged “real” spectrum collected in late fall of 1998 at Northern Radar's Cape Race site. The range for both the simulated and actual data is 50 km. Since the overall system gains are unknown and we are interested in relative effects, the data have been scaled for easy comparison. In the simulation result (Figure 6d), the apparent (SNR)_{c} for the largest Bragg peak is approximately 70 dB. In the real data, this value is greater than 60 dB. Given the fact that the exact system parameters and wind regime for the real data are unknown, this discrepancy between modeled and real data is quite reasonable. Besides additional internal system noise which no doubt is partially responsible for the difference in the SNRs of the field and simulated data, it is possible that there may have been man-made noise not falling into the “quiet receiving site” category of the Recommendations from which the noise figures for the simulation were chosen. Also, it must be remembered that median external noise values were employed in the simulations, and from the Recommendations this could easily account for a 10 dB variation between measured and modeled spectra. When the model is used as a precursor to an actual deployment it will be important to use an F_{a} in equation (28) that is appropriate to the time of day, the season of the year and the position in the sun spot cycle. Comparison with Figure 5c indicates that the apparent noise floor is raised slightly by the side lobes of the Blackman window. Furthermore, it should be pointed out that, as range increases, the spherical earth attenuation functions, F(·), used in the radar range equation decrease significantly [Dawe, 1988]. As a result, in the 5–6 MHz operating frequency interval, even the first-order clutter power will not be significantly above the noise floor when the range exceeds about 300 km [see Hickey et al., 1994].

[37] As a final note, it is obvious that the manifestation of the spectral peaks derived in the works of Walsh et al. [1990], Walsh and Dawe [1994] and Gill and Walsh [2001] will depend on the ocean conditions, the ambient noise regime, the distance to the scattering patch, and the signal processing schemes implemented. Thus the application of the models derived here will be important in assessing the usability of HF surface wave radar under a known set of these parameters, or, on the other hand, the models will be useful in establishing the necessary radar parameters for successful operation under a particular set of environmental constraints.

4. Discussion and Conclusions

[38] As HF radar is becoming increasingly important for sea surface remote sensing, it is equally important to develop models of the received signal, including noise, that can aid in system design and deployment which are subject to specific operating parameters, including those of site geometry. Here a means of modeling the combined clutter and noise signals as a single time series from which simulated spectra may be determined has been conducted. Both signal components have been modeled and combined in the time domain via a representation for a zero-mean, stationary Gaussian process presented by Pierson [1955].

[39] The initial reason for the work was to examine the likelihood of being able to observe certain spectral features predicted in the cross-section work by Walsh and Dawe [1994] and Gill and Walsh [2001]. The main parameters which are incorporated into the model, for both monostatic and bistatic operation, include operating frequency, transmitter power, antenna array gains, wind velocities and ranges from the transmitter and receiver to the scattering patch. Realistic clutter signal to noise values have been examined in view of noise factors available in ITU [1994]. Initially, the results from ensemble averaging were illustrated. Then, using a time model for the sea echo and noise, the customary procedure of implementing the magnitude-squared of a fast Fourier transform of a data time series as an estimate of the power density spectrum was carried out. Encouraging cursory comparison has been made between the model and field data obtained from a 5.75 MHz system operated on the east coast of Canada in the late 1990s.

[40] For the sake of completeness, it was established that the basic models yielded results that were not significantly different whether many pulses or a small number of pulses were sampled. The analysis also progressed so as to show how the broad band noise is aliased into the spectrum but is mitigated by duty cycle effects. It was also seen that the model properly reflects the fact that no such aliasing occurs for the narrowband ocean clutter signal. Subsequently, it was shown that, for the pulse radar ocean cross-section models developed by Walsh et al. [1990], Walsh and Dawe [1994], and Gill and Walsh [2001], the average, rather than peak, transmit power should be utilized when implementing the radar range equation to determine clutter spectral density model for a pulsed HF radar.

[41] It is intended that the modeling discussed here will provide useful input into the determination of appropriate operating parameters for HF radar as a remote sensor and will also give an indication of the limitations imposed on the resulting measurements due to the ambient noise environment.

Appendix A:: The Noise Power Spectral Density for a Finite Pulse Train

[42] Consider a finite pulse train of (2q + 1) pulses formed by gating an infinite sequence of pulses as illustrated in Figure A1. With the assumptions on the nature of the noise as given in equation (1), this truncated version may be characterized as

From here, an autocorrelation of n(t) is sought so that, on Fourier transforming the result, a power spectral density may be obtained. The randomness in (A1) appears only in the phase exponential, and given that the ensemble average,

the autocorrelation, _{Nf}(t_{1}, t_{2}), for a finite number of pulses is given by

Putting τ = t_{1} − t_{2} and t_{2} = t and Fourier transforming with respect to τ gives, after some algebra, the noise power spectral density expression, _{Nf}(ω, t), as

Here, ω is the transform variable and is, physically, the radian Doppler noise frequency. A further change of variables using τ_{1} = τ + t produces

The τ_{1} integral in equation (A4) may be easily shown to evaluate to

Now, e^{jmTL(ω′−ω)} is a geometric progression whose first term is e^{−jqTL(ω′−ω)}, whose constant ratio is e^{jTL(ω′−ω)}, and which contains (2q + 1) terms. This summation over m may therefore be reduced to

Equation (A7) is, formally, the Doppler noise power spectral density when a finite number (i.e., 2q + 1) of pulses is sampled. It is clearly time-dependent. If, as for equation (16), we choose to sample at the pulse centers–i.e., at t = pT_{L}, say–where p is an integer and ∣p∣ ≤ q (see Figure A1), then the first Heaviside expression reduces to unity. Furthermore, it may be noted that since (τ_{0}/2) < T_{L},

This immediately implies

When sampling occurs at the pulse center, equation (A7) thus becomes

If matched filter conditions [e.g., Barton, 1987] are assumed to exist, B = 2π/τ_{0}. Then, by defining α = (ω′ − ω)τ_{0}/2 and assuming S_{N}(ω′) to be flat (i.e., white noise), it is easy to show that

where d = τ_{0}/T_{L} is, by definition, the duty cycle of the radar. The fact that B ≫ ω (i.e., the radar receiver bandwidth is very much greater than the Doppler bandwidth of the echo from the ocean) has been used to write the integral in this form. It may be verified numerically that (A11) is essentially independent of p, even for a small number of pulses. This fact is depicted in Figure 2 for q = 10. It was seen in equation (23), that as the number of pulses becomes unbounded (i.e., (2q + 1) → ∞), no explicit p-dependence remains. It is not difficult to show that if in equation (A7) (2q + 1) → ∞, then the spectral density reduces to that given in equation (16).

Acknowledgments

[43] This work was supported by Natural Sciences and Engineering Research Council Discovery Grants (held by the authors). The authors also appreciate the work of Jianjun Zhang in producing the time series simulation and of Dr. Mal Heron for his helpful comments on the original manuscript and Dr. Weimin Huang for his critical reading of the work and latest calculation of the attenuation functions.