2.1. Noise Signal
 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  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, na(t), may be cast, using the complex exponential, as
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′), SN(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 SN(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 , the latter also containing a complete reproduction of the Pierson's original manuscript.
 The desire is to apply the expression in equation (1) to a pulse radar system. It is considered that the pulse repetition period is TL while the temporal width of the pulse is τ0 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, (t1, t2), 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 − mTL, becomes
where Sa[·] = . Substituting the result of (5) into (4) leads to
Lathi  (chapter 1) presents the identity
where δ(·) is the Dirac delta function, TP is a fixed period and td is time. Examining the summation in (6) in view of (7) and identifying td and Tp as being analogous to (ω′ − ω) and 2π/TL, 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 ω2 integral has been executed via the delta function of (10). Defining
and setting ω1 = ω, this autocorrelation finally appears as
The noise power spectral density, N(ω, t), clearly follows from the Fourier transform of N(τ + t2, t2) as
where t = t2.
 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 cn are the Fourier coefficients. These are given by
which easily gives
From equations (14) and (15), the noise power spectral density in (13) simplifies to
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.
 To impose sampling at the pulse centers, it is required that t = pTL 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 SN (ω + (m2π/TL)) may be removed from the summations and designated as SN(ω′). The noise spectral density for the pulsed radar then becomes, from equations (16) and (17),
it being understood that t = pTL. Next, it may be noted that since τ0/2 < TL
which immediately implies
Equation (18) further reduces to
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/τ0. 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 τ0/TL, and it is understood that m takes integer values. The argument pTL has been explicitly retained only to emphasize sampling at the pulse center. Since SN(ω′) is assumed constant, N(ω, pTL) 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.
 It may be verified numerically that for typical duty cycles in a pulse radar, N (ω, pTL) in (23) does not vary significantly from the ambient noise power spectral density, SN(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 SN(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.
Figure 2. The multiplier on the ambient noise power spectral density SN(ω′) compared for a infinite and finite number of pulses.
Download figure to PowerPoint
 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.
 In the ITU-R Recommendations, hereafter referred to as the Recommendations, the external noise factor, fa, 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, T0 is the reference temperature which is taken as 290 K and Bn is the noise power bandwidth of the receiving system. For a matched filter system, Bn may be taken as the reciprocal of the transmit pulse width, τ0. It is common to define an external noise figure, Fa, as
a median value of which may be designated as Fam. This is the quantity which is readily available from the Recommendations. For a white noise process, as is assumed here, the power spectral density, SN(ω′), in equation (23) is
Therefore, from equations (24)–(26),
Then, equation (23) for the external Doppler noise power spectral density, N(ω), becomes
Here, the pTL 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
 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. , and subsequent work based on that analysis, is the fact that the transmitter power Pt 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, τ0. 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.
 For the temporal periods, typically several minutes, used in examining HF sea echo, Barrick and Snider  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, ca(t), may be cast analogously to the noise voltage of equation (1) as
Here, Bc is the clutter bandwidth, ω′c the radian frequency, ε(·) is the random phase and Sc(·) 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 = pTL 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, TL, 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 (τ0/TL) 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  for a narrow beam receiving system, may be written for the Doppler clutter power spectral density as
It should be remarked that (τ0Pt/TL) is the average transmitted power as dictated by equation (33); λ0 is the operating wavelength; Gt and Gr are the gains of the transmitting and receiving antennas, respectively; the F's are the usual Sommerfeld attenuation functions; ρ01 and ρ02 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.
 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
 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
 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
and Fs(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.
 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.