VOLUME 20 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY NOVEMBER 2003
Whitening in Range to Improve Weather Radar Spectral Moment Estimates. Part I: Formulation and Simulation
SEBASTIAÂ N M. TORRES Cooperative Institute for Mesoscale Meteorological Studies, University of Oklahoma, and National Severe Storms Laboratory, Norman, Oklahoma
DUSÏAN S. ZRNICÂ National Severe Storms Laboratory, Norman, Oklahoma
(Manuscript received 29 May 2002, in ®nal form 14 March 2003)
ABSTRACT A method for estimation of spectral moments on pulsed weather radars is presented. This scheme operates on oversampled echoes in range; that is, samples of in-phase and quadrature-phase components are collected at a rate several times larger than the reciprocal of the transmitted pulse length. The spectral moments are estimated by suitably combining weighted averages of these oversampled signals in range with usual processing of samples (spaced at the pulse repetition time) at a ®xed range location. The weights in range are derived from a whitening transformation; hence, the oversampled signals become uncorrelated and, consequently, the variance of the estimates decreases signi®cantly. Because the estimate errors are inversely proportional to the volume scanning times, it follows that storms can be surveyed much faster than is possible with current processing methods, or equivalently, for the current volume scanning time, accuracy of the estimates can be greatly improved. This signi®cant improvement is achievable at large signal-to-noise ratios.
1. Introduction and identi®cation of weather phenomena. On the other hand, the need for faster updates between volume scans Doppler weather surveillance radars probe the at- calls for faster antenna rotation rates, which limit the mosphere and retrieve spectral moments for each res- number of samples available for each resolution volume. olution volume in the surrounding space. In computing As mentioned before, the number of samples is inversely these moments, it is customary to average signals from related to the variance of estimates. many pulses to reduce the statistical uncertainty of the Several solutions have been proposed to reduce spec- estimates. With such processing, the variance reduction tral moment errors in weather surveillance radars. In the of averaged estimates is inversely proportional to the quest for ®nding better estimators of spectral moments, equivalent number of independent samples, which de- Zrnic (1979) showed that maximum likelihood (ML) pends on the correlation between samples and the total estimators yield errors one order of magnitude less than number of averaged samples (or pulses) M (Walker et those obtained with conventional autocovariance meth- al. 1980). The number of samples available for aver- ods. Later, Frehlich (1993) improved ZrnicÂ's results and aging is determined by the pulse repetition time T and s derived simpli®ed expressions to test new estimators the dwell time, which is usually controlled by the re- based on the ML approach. Due to the complexity of quired azimuthal resolution. If averaging along sample ML estimators, researchers focused on ways to simplify time is not enough to keep estimation errors below ac- spectral moment estimators by assuming knowledge of ceptable limits, it is possible to trade range resolution some of the underlying parameters of the weather signal. for estimate accuracy by averaging a few samples along Bamler (1991) computed the Cramer±Rao lower bound range time. From an operational point of view, we are (CRLB) for Doppler frequency estimates assuming both faced with con¯icting requirements. On the one hand, the correlation (or spectrum) of samples and signal-to- large estimation errors restrict the applicability of weather surveillance radars for precise quanti®cation noise ratio (SNR) are known. Later, Chornoboy (1993) obtained an optimal estimator for Doppler velocity that is simpler than ML formulations, but again, the SNR Corresponding author address: Dr. SebastiaÂn M. Torres, NSSL, and the spectrum width were assumed to be known. 1313 Halley Circle, Norman, OK 73069. Summarizing, ML estimators provide better accuracy E-mail: [email protected] compared with classic estimators (Zrnic 1979) and are
᭧ 2003 American Meteorological Society 1433
Unauthenticated | Downloaded 10/02/21 09:25 PM UTC 1434 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 20 only moderately complex if the spectrum width is pression due to the required larger transmission band- known a priori. However, this last assumption restricts widths. their applicability because the correlation coef®cient of Range oversampling and whitening can be used to the weather echo along sample time is not known and increase the equivalent number of independent samples must be estimated. That is, to estimate spectral mo- without increasing the transmission bandwidth. The ments, the joint distribution of signal power, mean main advantage of this technique is that the whitening Doppler velocity, and spectrum width would need to be transformation is derived from a known correlation calculated, which turns out to be computationally very function. In a rather short but important study, Dias and intensive. Dias and LeitaÄo (2000) proposed ML esti- LeitaÄo (1993) derived an iterative technique to obtain mators that do not exhibit the characteristic computa- ML estimates of spectral moments. They consider both tional burden. They derived a nonparametric method to the time and space variables and assume a known cor- obtain ML estimates of spectral moments from weighted relation in range. Implicit in their solution is whitening sums of autocovariance estimates; the only assumption of the signals in range; thus, it is likely the earliest is that the power spectral density of the underlying pro- application of this technique to radar remote sensing. cess is bandlimited. More recently, Fjùrtoft and LopeÁs (2001) proposed a Schulz and Kostinski (1997) suggested that knowl- method for estimating the re¯ectivity in synthetic ap- edge of the correlation coef®cient of weather-echo time erture radar (SAR) images with correlated samples (pix- series could improve the variance of spectral moment els). The method is based on a modi®ed whitening trans- estimates. By means of the whitening transformation formation that exhibits low computational complexity (i.e., a linear operation that decorrelates echo samples), and is suitable for oversampled data. they devised estimators that theoretically achieve the This paper describes an application of the whitening CRLB. Further, Koivunen and Kostinski (1999) ex- transformation in range that increases the equivalent plored practical aspects of the whitening transformation number of independent samples while keeping the dwell on estimation of signal power. A principal obstacle to time constant with no signi®cant degradation of the the application of the whitening transformation on the range resolution. Obtaining more independent samples time series is the need to know (or estimate) the cor- reduces the estimate errors at the same antenna rotation relation coef®cient of the signal. These authors suggest rate or speeds up volume scans while keeping the errors ways to overcome this dif®culty and call for an exper- at previous levels. Our work has been inspired by Schulz imental study to verify the technique. Using the whit- and Kostinski (1997) but shares some common elements ening approach, Frehlich (1999) investigated the per- with the work of Dias and LeitaÄo (1993). In simulation formance of ML estimators of spectral moments under studies we consider a receiver with a large bandwidth the assumption of a known spectrum width. Although and a perfect rectangular pulse. We use autocovariance Frehlich concluded that the estimator derived by Schultz processing in sample time and examine in detail the and Kostinski cannot be applied to the case of ®nite effects of white noise. At the end we brie¯y describe SNR, Kostinski and Koivunen (2000) showed that the some other practical aspects and trade-offs. problem is not in the estimator but in the Gaussian as- sumption for the Doppler spectrum. In their work, they 2. Why whitening? suggested simple numerical recipes intended to avoid the so-called Gaussian anomaly. Current implementations of spectral moment esti- Range oversampling to improve the estimates of per- mators use a simple method of averaging samples in iodograms was explored by Urkowitz and Katz (1996). range at the expense of degradation in range resolution. They acknowledged that the periodograms for each Simple averaging, however, does not yield the best per- range location are correlated; they computed the equiv- formance when the observations are correlated. Schulz alent number of uncorrelated range samples and showed and Kostinski (1997) computed the variance bounds for that the variance reduction is not optimal. Strauch and re¯ectivity estimates and demonstrated that they do not Frehlich (1998) considered oversampled signals in depend on the correlation structure of the observations. range to estimate Doppler velocity within one pulse. Hence, it can be inferred that it is not the correlation This approach fails because the phase shift within a between observations that limits the accuracy of a given pulse is smaller than the uncertainty of the estimates; estimator but the way those observations are used to the authors point out that simple averaging is not enough compute the estimates. Therefore, it is reasonable to to achieve the required equivalent number of indepen- think that knowledge of the correlation coef®cient dent samples. Acquisition and processing of samples (mTs) could be used to formulate estimators that attain over ®ner range scales was investigated also in the con- the CRLB. In their work, Schulz and Kostinski proposed text of pulse compression. Pulse compression can be whitening the time series data along sample time to applied to increase the equivalent number of indepen- produce uncorrelated samples. The main drawback for dent samples by averaging high-resolution estimates in this technique is that the whitening transformation de- range (Mudukutore et al. 1998). However, most ground- pends on other meteorological parameters such as the based weather surveillance radars do not use pulse com- spectrum width. Dias and LeitaÄo (1993) derived man-
Unauthenticated | Downloaded 10/02/21 09:25 PM UTC NOVEMBER 2003 TORRES AND ZRNIC 1435 ageable approximate solutions to the ML estimators of spectral moments for signals that are oversampled in range. Their solution, although not immediately trans- parent to readers, amounts to whitening the samples in range and applying Fourier transforms for estimating spectral moments. We suggest combining the whitening transformation of samples in range with autocovariance processing in sample time and thus improving the spectral moment estimates. The proposed processing increases the equiv- alent number of independent samples in a simple manner while the sacri®ce in range resolution is minimal and the transmission bandwidth is not broadened. While the correlation of samples separated by Ts needs to be es- timated for each particular case (it depends on the me- teorological conditions being observed), samples spaced in range exhibit a correlation coef®cient that can be exactly computed a priori; the underlying assumption here is that the mean echo power changes very little over the averaging interval in range. We elaborate more on this fundamental constraint later. By exactly knowing the correlation coef®cient, it is possible to apply the whitening transformation without worrying about the FIG. 1. Depiction of sampling/oversampling in range and process- pitfalls originating from an estimated quantity. As a re- ing of the signals. (a) Samples in range with spacing equal to the sult, the equivalent number of independent samples be- pulse length ; standard processing to obtain correlation estimates is comes equal to the number of available samples, and indicated; (b) oversampling in range; (c) zoomed presentation of ov- ersampled range locations where range samples to be whitened with the variance reduction through averaging is maximized. matrix W are indicated; (d) processing of whitened samples to obtain Maximization of the equivalent number of independent estimates of correlations in range and average of these estimates in samples leads to the following. range to reduce the statistical errors. • For the same uncertainty as that obtained with cor- related samples, faster scan rates are possible, as the the pulse duration (i.e., oversampling by a factor of total number of samples M for a resolution volume is L). Assume that a range of depth c (where c is the determined by the pulse repetition time and the dwell speed of light) is uniformly ®lled with scatterers, which time. Rapid acquisition of volumetric radar data has is a common occurrence for relatively short pulses. For signi®cant scienti®c and practical rami®cations. For convenience, the contribution from the resolution vol- example, observations at minute intervals are required ume to the received sampled complex voltage V(nT ) to understand the details of vortex formation and de- s ϭ I(nT ) ϩ jQ(nT ) at a ®xed time delay nT can be mise near the ground. Even faster rates of volumetric s s s decomposed into subcontributions s(l , nT ) from L data are required to determine the presence of trans- o s contiguous elemental shells or ``slabs,'' each c/2L verse winds. Fast update rates would also yield more thick. Each of these is an equivalent scattering center. timely warnings of impending severe weather phe- For simplicity and T are dropped hereafter so the nomena such as tornadoes and strong winds. o s indices l and n indicate range-time increments (sam- • For the same scanning rates, lower uncertainties can o pling time) and sample-time increments T (pulse rep- be obtained, making the use of polarimetric variables s etition time), respectively. The voltages s(l, n) are iden- feasible for accurate rainfall estimation and hydro- tically distributed, complex, Gaussian random variables, meteor identi®cation. where the real and imaginary parts have variances 2, With the advent of digital receivers (Brunkow 1999), 2 2 and the average power of s(l, n)is s ϭ 2 . A pulse oversampling is indeed feasible (Ivic 2001). Therefore, with an arbitrary envelope shape p(l) induces weighting it is possible to maintain the same current radar capa- to the contributions from contiguous slabs such that the bilities (as with a digital matched ®lter in classical pro- composite voltage after synchronous detection is cessing) while adding, in parallel, a set of more reliable estimates obtained from whitened oversampled range V(l, n) ϭ I(l, n) ϩ jQ(l, n) data. LϪ1 ϭ s(l ϩ i, n)p(L Ϫ 1 Ϫ i) ૺ h(l), (1) 3. The whitening transformation []iϭ0 The procedure (as depicted in Fig. 1) begins with where the ૺ denotes convolution and h(l) is the impulse oversampling in range so that there are L samples during response of the receiver ®lter. The summation in (1) is
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(R) T 1/2 1/2 T a correct representation for the composite voltage V(l, be obtained as CV ϭ H*H ϭ (U*⌳ )*(U*⌳ ) , where n) given that we consider ``scattering centers,'' and each ⌳1/2 is a diagonal matrix with the square roots of the of these implicitly admits integration. Then, as a gen- eigenvalues on the diagonal. With this decomposition, H eralization of (4.40) in Doviak and Zrnic (1993), the ϭ U*⌳1/2, and W is obtained as W ϭ HϪ1 ϭ ⌳Ϫ1/2UT, correlation of range samples is which is the Mahalanobis transformation (Tong 1995). In
(R) 2 the case of Cholesky (or triangular) decomposition, the RVsmm(l) ϭ [p (l) ૺ p*(Ϫl)], (2) (R) T correlation matrix is factored as CV ϭ TT* , where the (R) T where the modi®ed pulse envelope pm is given by pm(l) matrix T is a lower triangular matrix. Here, CV ϭ H*H ϭ p(l) ૺ h(l). Hence, the correlation coef®cient of range ϭ (T*)*(T*)T, and H ϭ T*; hence, the whitening matrix (R) Ϫ1 Ϫ1 samplesV is W ϭ H ϭ (T*) is also lower triangular. A possible advantage of lower triangular W matrices is that whitening R(R)(l) p (l) ૺ p*(Ϫl) (R)(l) ϭϭVmm. (3) can proceed in a pipeline manner; that is, computations V R(R)(0) LϪ1 V p2 (lЈ) can start as soon as the ®rst sample is taken and progress m lЈϭ0 through subsequent samples. Non-lower-triangular W ma- trices require the presence of all data before a whitened In practice, (R) can be evaluated by attenuating the V sample can be computed. transmitted pulse, injecting it directly into the receiver, In the following sections, we denote with X(l, n) the and oversampling the result to obtain the modi®ed pulse sequence of whitened samples obtained from V(l, n) for envelope p . Introducing p into (3) produces (R) ; this m m V a ®xed sample time nT as needs to be done only once for a given pulse shape and s receiver bandwidth. LϪ1 X(l, n) ϭ wV( j, n); l ϭ 0,1,...,L Ϫ 1, (7) The procedure for implementing the whitening trans- l,j jϭ0 formation follows Koivunen and Kostinski (1999) and is listed here for completeness (and reader convenience). where wl,j are the entries of the whitening matrix. Al- De®ne the Toeplitz±Hermitian normalized correlation ternatively, the previous equation can be written using (R) matrixCV as matrix notation as
(R)(R) Xnnϭ WV , (8) 1 VV(1) ´ ´ ´ (L Ϫ 1) (R)(R) T (1)* 1 ´ ´ ´ (L Ϫ 2) where Vn ϭ [V(0, n), V(1, n),...,V(L Ϫ 1, n)] and (R) VV CV ϭ . X ϭ [X(0, n), X(1, n),...,X(L Ϫ 1, n)]T. It is important __5_n (R)(R) to note that regardless of the method used to decompose VV(LϪ1)* (L Ϫ 2)* ´ ´ ´ 1 (R) CV , the whitening procedure is given by (8), and even (4) though the whitening matrices may be different, the results in terms of data decorrelation are statistically Because this matrix is positive semide®nite (Therrien equivalent. 1992), it can be decomposed as
(R) T CV ϭ H*H , (5) 4. The noise enhancement effect where the superscript T indicates matrix transpose and The presence of noise is inherent in every radar sys- * is the usual complex conjugation operation. Any H tem; therefore, it is necessary to analyze the perfor- (R) that satis®es (5) is called a square root ofCV (Faddeev mance of the whitening transformation, under noisy and Faddeeva 1963) and is the inverse of a whitening conditions. Let V ϭ VS ϩ VN, where the subscripts S transformation matrix, and N stand for signal and noise components, respec- W ϭ HϪ1, (6) tively. When applying the whitening transformation, both signal and noise are similarly affected: which, if applied to the range samples, produces L un- correlated random variables with identical power (Kay X ϭ WV ϭ WVSϩ WV NSNϭ X ϩ X . (9) 1993). For simplicity, we dropped the subscript ``n'' that is In general, the decomposition of the correlation matrix used to indicate sample time. From (9), we can see that is not unique and many well-known methods can be ap- the signal is whitened and the noise, which was white plied to generate different whitening transformations. Two prior to the whitening transformation, becomes colored. such methods are the eigenvalue decomposition (Therrien Let us apply the transformation matrix to the data (8) (R) 1992) and Cholesky decomposition, which is equivalent and compute the range-time correlationRX for the ran- to Gram±Schmidt orthogonalization (Therrien 1992; Pa- dom vector X using the expectation operation E[.] as poulis 1984). In the eigenvalue decomposition method, (R) TTT (R)(R) T RX ϭ E[X*X ] ϭ W*E[V*V ]W . (10) CCVVis represented as ϭ U⌳U* , where ⌳ is a diagonal (R) matrix of eigenvalues ofCV , and U is the unitary matrix The correlation matrix of V (signal plus noise) is given (R) (R)(R) whose columns are the eigenvectors ofCV . With this by SVCCVVSSϩ NVI, where is given in (4); I is the L- decomposition, an expression of the same form as (5) can by-L identity matrix; SV is the signal power; and NV is
Unauthenticated | Downloaded 10/02/21 09:25 PM UTC NOVEMBER 2003 TORRES AND ZRNIC 1437 the noise power. Then, substituting the expression above minimum mean-square error (MMSE) criterion accom- for the correlation matrix of V in (10), using (6), dis- plishes the desired goal but requires a priori knowledge tributing the matrix product, and using (5), of the SNR at every range location (Ebbini et al. 1993). Alternatively, we can look at the same problem in terms R(R)(ϭ (HϪ1)*[S C R) ϩ N I](HϪ1)T XVVVS of the eigenvalues of the range-time correlation matrix. ϭ S I ϩ N (HHT *)Ϫ1 . (11) The ability to limit the gain of the whitening transfor- VV mation to reduce the noise enhancement effect arises It is thus evident that the range-time correlation of X from the relation between the eigenvalues of a corre- RR(R)(R) is the sum of a signal (XXSN ) and a noise ( ) component, lation matrix and the corresponding power spectral den- RR(R)(I R) HTH Ϫ1 where XXSNϭ SV and ϭ NV( *) . By de®nition, sity. That is, the range spanned by the power spectral XS is white because its correlation is a diagonal matrix. density matches closely the range of eigenvalues (John- Additionally, all components have identical power SV son and Dudgeon 1993). Accordingly, by limiting the R(R) becauseXS is a scalar multiple of the identity matrix. span of eigenvalues, it is possible to place a bound on On the other hand, the noise becomes colored, and its the gain of the transformation (Torres 2001). The anal- mean power after whitening can be computed by av- ysis of these and other suboptimal techniques is a subject eraging the powers of the individual components of XN, for further study. (R) which correspond to the diagonal elements ofRXN . Then, 1 N N ϭ tr[R(R)] ϭ V tr[(HHT *)Ϫ1 ], (12) 5. Spectral moment estimators X LLXN The estimation of spectral moments using a whitening where tr(.) is the matrix trace operation. Therefore, the transformation on oversampled data is performed in noise enhancement factor (NEF) de®ned as NEF ϭ NX/ three steps. First, oversampled data in range are whit- NV can be obtained from (12) as ened as discussed in section 3. Then, the sample-time autocorrelation at lags zero and one are estimated for NEF ϭ LϪ1 tr{[C(R)]},Ϫ1 (13) VS each range location, and these estimates are averaged where we used the cyclic property of the trace and the (in range) to reduce the standard errors. Finally, these matrix decomposition in (5). For an ideal system (i.e., improved correlation estimates are used to compute a system with a rectangular transmitted pulse and radar power, Doppler velocity, and Doppler spectrum width bandwidth much larger than the reciprocal of pulse with the usual algorithms (Doviak and Zrnic 1993). (R) width)V (l) ϭ 1 Ϫ | l |/L for | l | Ͻ L, and the trace Whitening-transformation-based (WTB) estimators for in (13) can be computed using (A32) to obtain NEF ϭ the spectral moments and the theoretically derived equa- L 2(L ϩ 1)Ϫ1. Note that an extra L factor should be added tions for their variances (appendix A) are presented next. if comparing with the noise power in the classical pro- These theoretical equations are used to compare and cessing. To effectively oversample by a factor of L, an validate simulation results. Further, they clearly show L-times larger bandwidth than the reciprocal of the pulse the interplay of the various variables in the reduction width is needed; this increases the noise power by the of variances. same factor. Under these considerations, the effective noise increase over the matched ®lter case (for an ideal a. Signal power estimator system) becomes L3(L ϩ 1)Ϫ1. The trade-off between noise enhancement (radar sen- The WTB power estimator for oversampled signals sitivity) and variance reduction makes the whitening in noise is given by transformation useful in cases of relatively large SNR. 1 LϪ1 MϪ1 Ãà 2 For weather surveillance radars, the SNR of signals S ϭ SX ϭ |X(l, m)| Ϫ N(NEF), (14) from storms is large and the effects of noise when using (WTB) LM lϭ0 mϭ0 the whitening transformation are negligible. For ex- where L is the oversampling factor, M is the number of ample, a 3 mm hϪ1 rain in the Weather Surveillance pulses, N is the noise power, NEF is the noise enhance- Radar-1988 Doppler (WSR-88D) produces an SNR of ment factor in (13), and X(l, n) is the whitened over- 37 dB at 50 km. Clearly, for measuring light rain, the sampled weather signal as in (7). Using the results in noise enhancement by whitening would not be a prob- the appendix for an ideal system, the normalized stan- lem to the full 230 km of required coverage. However, dard deviation of WTB signal power estimates is ob- the threshold for display is set at an SNR of 6 dB mainly tained from (A34) as to observe snow, which typically has smaller re¯ectiv- à ity. This SNR falls in the range where noise enhance- SD{S} (WTB) 1112LN ment dominates and may preclude the use of whitening. ϭϩ SLLϩ 1 S A solution to the noise enhancement problem is to ͙M [2vn͙ relax the whitening requirements and select a transfor- 1/2 L(3L2 ϩ 2L Ϫ 3) N 2 mation such that the output noise power is also mini- ϩ , (15) mized. A transformation that is optimized based on the 2(L ϩ 1)2 S ]
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where vn, the normalized spectrum width, is de®ned the lag-one sample-time autocorrelation function for the as /2 a, and a is the maximum unambiguous veloc- oversampled whitened signal X(l, n) is estimated as ity. 1 LϪ1 RÃÃ(T)((1) ϭ R T)(1) XXL l b. Mean Doppler velocity estimator lϭ0 1 LϪ1 MϪ2 The WTB mean Doppler velocity estimator for ov- ϭ X*(l, m)X(l, m ϩ 1), (17) ersampled signals is given by L(M Ϫ 1) lϭ0 mϭ0 where X(l, n) is the whitened oversampled weather sig- a à (T) à ϭϪ arg[RX (1)], (16) nal. From (A35) in the appendix, the standard deviation (WTB) of WTB mean Doppler velocity estimates for the ideal where a is the maximum unambiguous velocity, and case is
1/2 SD{Ã } (2 )(22 )22 2 (WTB) 1 e vn Ϫ 11 LN evn L(3L ϩ 2L Ϫ 3) N ϭϩ[2 sinh(2 )]2 ϩ . (18) 2 LLvn 1 S 22(L 1)2 S a 2͙M Ϫ 14Ά[]vn͙ ϩ []ϩ · c. Doppler spectrum width estimator component level). Alternatively, oversampling-and-av- erage-based (OAB) estimators operate on oversampled The WTB Doppler spectrum width estimator for ov- data but use incoherent averaging; that is, averaging in ersampled signals is given by range is performed at the correlation level. Throughout ÃÃ1/2 this work we stress the comparison of WTB estimates, a͙2 SSXX Ã ϭ ln(T)( sgn lnT) , (19) (the proposed implementation) with MFB estimates, the (WTB) |RÃÃ(1)| |R (1)| ΗΗΆ·[]XX [] current implementation in the WSR-88D. OAB esti- mators are only included for illustrative purposes; for where SÃ andRÃ (T) (1) are given in (14) and (17), re- X X more details the reader is referred to Torres (2001). spectively. From (A36) in the appendix, the standard Figure 2 shows the normalized standard deviation of deviation of WTB Doppler spectrum width estimates WTB, MFB, and OAB power (Fig. 2a), Doppler velocity for the ideal case is (Fig. 2b), and spectrum width estimators (Fig. 2c) as a
SD{Ã } 2 function of the SNR for the ideal case and a normalized e(2vn) 1 (WTB) ϭ spectrum width of 0.08. The oversampling factor L ϭ 2 42 a []vn ͙M Ϫ 1 8 is a realistic value that can be achieved on weather surveillance radars. The SNR in these ®gures refers to 22 e(2vn)(Ϫ 4e vn) ϩ 31 the output of the digital receiver, before the digital ϫ L matched ®lter for the case of MFB estimates. With this Ά[]4vn͙ assumption one can make a fair comparison of estimates LN because the common signal path ends at the digital re- ϩ 2[cosh(2 )2 Ϫ 1] vn L ϩ 1 S ceiver. After the digital receiver, either whitening or a matched ®lter (or both) can be applied. 2 2 1/2 e(2vn)2ϩ 2 L(3L ϩ 2L Ϫ 3) N When compared with MFB (or OAB) estimates, WTB ϩ . estimates exhibit a superior performance for large SNR. 22(L 1)2 S []ϩ · Nonetheless, it is evident from these plots that the per- (20) formance of the WTB estimator worsens as the SNR decreases due to the noise-enhancement effect inherent to the whitening transformation. The performance of d. Results OAB estimators is not optimum in terms of variance The performance of WTB estimators is compared reduction as the average is done on correlated variables. with that achieved by the classical matched-®lter-based However, since there is no noise enhancement, OAB (MFB) estimators and the estimators obtained from ov- estimates exhibit lower errors than MFB estimates for ersampled data and regular averaging. MFB estimators a broader range of SNR than the WTB estimates. The are derived from oversampled data using coherent range variance reduction factor (VRF) for the three WTB es- averaging, that is, the type of conventional processing timators compared with the conventional matched-®lter that uses a digital matched ®lter at the receiver's front approach can be computed as the ratio of the variance end (averaging in range is performed at the I and Q of MFB estimates to the variance of WTB estimates.
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FIG. 2. Standard error of (a) signal power, (b) mean Doppler velocity, and (c) Doppler spectrum width estimates vs the SNR for the ideal case. The three sets of curves in each ®gure correspond to WTB, MFB, and OAB estimators, respectively. Solid lines show the results from simulations (averaging 1000 realizations), and dashed lines show the theoretical predictions.
For the ideal receiver case and a large SNR, the variance timates in the matched-®lter implementation. This is reduction factor is equal to the oversampling factor L because WTB estimators should be utilized only for [cf. (A34)±(A36) with (A39)±(A41), respectively]. Note SNRs greater than or equal to the crossover signal-to- that theoretical predictions and simulation results, both noise ratio (SNRc). By de®nition, SNRc is found by plotted in Fig. 2, are in remarkable agreement. equating the VRF to 1 and is plotted in Fig. 4 versus Figure 3 shows the bias of WTB, MFB, and OAB the normalized spectrum width for power (Fig. 4a), signal power (Fig. 3a), mean Doppler velocity (Fig. 3b), Doppler velocity (Fig. 4b), and spectrum width esti- and spectrum width estimators (Fig. 3c) as a function mates (Fig. 4c). For completeness, the SNRc with re- of the SNR under the same conditions as in Fig. 2. spect to OAB estimates is also shown. These plots were Estimators of S and are unbiased even for relatively obtained directly from the theoretical results derived in low SNR. On the other hand, estimates exhibit a appendix A and veri®ed by simulation studies (see ap- small bias even for large SNR with WTB estimates be- pendix B). ing the least biased of the three. At low SNR, WTB estimates of the spectrum width are heavily biased due 6. Discussion to the noise-enhancement effect. It is of practical signi®cance to determine the value Section 5 discussed the application of the whitening of SNR for which the variance of estimates obtained transformation to the estimation of spectral moments. from whitened samples is equal to the variance of es- Estimators operating on whitened signals were termed
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FIG. 3. Bias of (a) signal power, (b) mean Doppler velocity, and (c) Doppler spectrum width estimates vs the SNR for the ideal case. The three curves in each ®gure correspond to WTB, MFB, and OAB estimators, respectively. The results were obtained from simulations by averaging 1000 realizations.
WTB estimators, and they exhibit reduced standard er- In the previous equation, is any of the variables dis- rors with respect to MFB estimators if the SNR is rel- cussed in this chapter, namely, S, , or ; and SNRc atively large. A performance comparison of all WTB is the crossover SNR (different for each estimator) de- estimators with MFB estimators for the ideal case ®ned as the SNR that corresponds to a variance reduc- showed that for all the variables the variance reduction tion factor of 1. factor for large SNR is L, where L is the oversampling Throughout this study we dealt with an ideal system factor. That is, approximately L-times fewer samples are and a known range-time correlation coef®cient, which needed for WTB estimators to keep the same errors as derives from an assumption of uniform re¯ectivity. Al- the ones obtained without the aid of a whitening trans- though a rectangular pulse and in®nite bandwidth re- formation. ceiver are not realistic, they closely approximate the For low SNR, the performance of all WTB estimators characteristics of operational systems and serve to il- deteriorates as the noise enhancement effect discussed in lustrate the potential of the method. For example, an section 4 becomes signi®cant. In such cases, the estimates intermediate-frequency (IF) bandwidth of about 10 times on nonwhitened data result in better performance. The the reciprocal of the pulse width will be enough to still rule for selecting the best estimate is given by consider the noise white in relation with the weather signal bandwidth (what really counts is the value of the ˆ if SNR Յ SNRc (MFB) correlation coef®cient of noise at lag 1, which we submit ˆ ϭ (21) is close to zero in our application). This noise model ˆ if SNR Ͼ SNRc . (WTB) differs from the one in Dias and LeitaÄo (1993), where
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FIG.4.SNRc vs the normalized spectrum width for (a) signal power, (b) mean Doppler velocity, and (c) Doppler spectrum width estimators in an ideal system. Solid lines represent the SNR at which the errors of WTB estimates equal the errors of MFB estimates. Dashed lines represent the SNR at which the errors of WTB estimates equal the errors of OAB estimates. WTB estimates are accepted if SNR Ͼ SNRc; otherwise, classical estimates are preferred. the assumption is that the noise and signal components sumption of uniform re¯ectivity has been used in prac- are equally correlated in range and the noise is white tice before. For example, the WSR-88D averages four only in sample time. In the National Weather Service samples in range (over 1 km) to improve the accuracy (NWS) WSR-88D, the IF bandwidth exceeds 8 MHz of estimates. Thus, operational meteorologists ``accept'' and the short pulse is ϭ 1.57 s; therefore, the band- averaging gradients over 1-km volume depths. Although width is over 12 times larger than the reciprocal of . the range extent of the resolution volume with the pro- In addition, although the transmitted pulse is not exactly posed method is smaller (about 500 m versus 1 km), rectangular, rise and fall times are short enough that degradation in regions of nonuniform re¯ectivity will experimental measurements of the range correlation of occur, as shown through simulations by Torres (2001). samples revealed an almost perfectly triangular corre- For example, for a re¯ectivity gradient of 30 dB kmϪ1 lation similar to the one assumed throughout this work. [referred to as ``extreme'' in model 1 of Rogers (1971)], The constraint of uniform re¯ectivity is the principal it can be shown that the variance reduction factor is assumption required to precompute the exact correlation approximately cut in half. This is because the re¯ectivity of oversampled signals in range. However, it is under- pro®le modi®es the correlation of range samples, and stood that these idealized conditions will not be satis®ed whitening based on uniformity assumptions is incom- for all the resolution volumes in an operational envi- plete. However, the same simulation analyses showed ronment, especially at the edge of precipitation cells, that gradients of 10 dB kmϪ1 would create negligible where very sharp gradients could exist. Still, the as- performance deterioration.
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Approximate (or incomplete) whitening reduces the exploits the idea of whitening to obtain independent correlation of samples and hence the variance of esti- samples, such as in the works of Dias and LeitaÄo (1993), mates to a certain degree. Some decorrelation occurs Schulz and Kostinski (1997), Koivunen and Kostinski regardless of re¯ectivity gradients because the range± (1999), Frehlich (1999), and Fjùrtoft and Lopes (2001). time correlation function always has the same compact It operates on oversampled echoes in range, and the support; that is, it is zero outside of a ®nite interval that spectral moments are estimated by suitably combining is solely dictated by the receiver impulse response and weighted averages of these oversampled signals with the transmitter pulse length. Implicit in the compactness usual processing of samples (spaced at pulse repetition is the assumption that the receiver bandwidth is much time) at a ®xed range location. As with the previous larger than the reciprocal of the pulse length. For ex- works, the whitening transformation is used in such a ample, for a ®nite-impulse-response (FIR) receiver ®lter way that the equivalent number of independent samples with F taps, the support of the range-time correlation equals the number of samples available for averaging function is the symmetric interval of length 2(L ϩ F) and, consequently, the variance of the estimates de- about zero, where L is the oversampling factor. Another creases signi®cantly. way to look at this is in the frequency domain by ex- Whitening-transformation-based (WTB) estimators amining the power spectral density of range samples. of spectral moments were explored, and their perfor- The support invariability of the correlation function im- mance was compared with that of classical estimators plies that the nulls of the spectrum always correspond that use a digital matched ®lter. The variance reduction to the same frequencies. Since the whitening transfor- achieved by WTB estimators under ideal conditions as- mation is designed to boost those regions where the ymptotically tends to L for large signal-to-noise ratios signal spectral components are weak, it follows that a (SNRs). For low SNRs there is a crossover point (SNR ) considerable degree of whitening will be achieved even c for the variances of WTB and classical estimators. An- if there is a mismatch of correlation functions caused alytical expressions that allow the computation of SNR by re¯ectivity gradients. Further study with simulations c as well as with real data is required to quantify this for any variable and different conditions were derived. effect. For SNRs larger than the SNRc, WTB estimates are An alternative technique that increases the equivalent preferred over classical estimates. Below the SNRc, the number of independent samples without the pitfalls of noise enhancement effect dominates and classical esti- noise enhancement and correlation function mismatches mates are favored. is pulse compression. Then, it is natural to compare The application of this technique is possible because pulse compression to oversampling and whitening. As- • the correlation of samples in range is known exactly sume that peak transmitter power, pulse duration, and if the resolution volume is uniformly ®lled with scat- receiver bandwidth are the same for both techniques. In terers (although not optimum, estimate variance re- addition, the compressed pulses are averaged in range, duction is also possible in resolution volumes with and the number of averages is the same as the number re¯ectivity gradients); of averages of whitened pulses in range. Then, at large • the receiver bandwidth is large compared with the SNRs, both achieve the same variance reduction of es- reciprocal of the pulse length; and timates because the oversampling factor (L) is equal to • for most weather phenomena of interest, the SNR is the compression factor. The difference is that pulse com- relatively high (above the SNR ), so the increase of pression requires an L-fold increase in bandwidth for c noise power is not critical and the method works well. the transmitted signal. This increased bandwidth has a payoff at weak SNRs. If noise dominates estimation Realistic simulations (see appendix B) were per- accuracy, pulse compression has approximately an L 2 formed using known statistical properties of signals re- edge in SNR over whitening. This is because there is ¯ected by scatterers in ¯uids. The simulations also take an L-fold increase in signal power and, further, the noise the known properties of the probing pulse and receiver is not enhanced by about a factor of L as occurs with ®lter to reconstruct a composite signal from distributed whitening. In fact, averaging one L-compressed pulse scatterers illuminated by the pulse. This work con®rms is equivalent to a matched ®lter in terms of SNR. The that WTB estimators are indeed viable candidates for increase in bandwidth (noise power) is compensated for future enhancements of the WSR-88D radar network. by an increase in signal power, and averaging in range Summarizing, the use of WTB estimators of spectral L pulse-compressed samples affects signal and noise in moments allows increasing the speed of volume cov- the same way. In summary, pulse compression outper- erage by weather radar so that hazardous features can forms whitening for low SNRs, but its practical imple- be detected more timely. It also leads to better estimates mentation is only affordable in remote sensing devices of precipitation and wind ®elds. that can allocate large transmission bandwidths. Acknowledgments. The authors would like to thank 7. Conclusions Chris Curtis for constructive discussions concerning the A method for estimation of Doppler spectral moments overall structure of this paper and the derivations in on pulsed weather radars was presented. The scheme appendix A. We appreciate references brought to our
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attention by two anonymous reviewers. The Federal where L is the oversampling factor, M is the number of Aviation Administration provided support for part of pulses, and X(l, m) is the whitened signal as in (7). this work through Interagency Agreement DTFA 03-01- Whereas the variance of SÃ can be obtained by direct X9007. Funding for this research was provided under computation, if the distributions of (A4) and (A5) are NOAA-OU Cooperative Agreement NA17RJ1227. smooth and narrow around their mean values, the vari-
ances forà and à can be computed using perturbation APPENDIX A analysis (Zrnic 1977). The results are summarized here: Ãà Derivation of Estimator Variances Var {S} ϭ Var{PX}, (A6) (WTB) à Estimators of the spectral moments S, Ã, and à are à 2 ÃÃ(T)(T) obtained from estimates of total power P and the sample- aXX1|R (1)| R (1) (T) Var {à } ϭ Re Var Ϫ Var , time (T), lag-one autocorrelation function Rà (1) as (T)(T) (WTB) 2|Ά·[][]R (1)| R (1) Ãà S ϭ P Ϫ N, (A1) (A7) a à (T) à ϭϪ arg[R (1)], (A2) Var{à } (WTB) 1/2 2 a͙2 SSÃà 2(T) à ϭ ln sgn ln , (A3) a |R (1)| ÃÃ(T)(T) ϭ ΗΗΆ·[]|R (1)| [] |R (1)| 2 S [] where N is the system noise power and is the max- a PÃÃÃ1|R(T)((1)| R T)(1) imum unambiguous velocity. ϫ Var XXXϩ Re Var ϩ Var Consider ®rst the case of WTB estimators. Estimates Ό S 2|Ά·[][]R(T)((1)| R T)(1) of power and lag-one autocorrelation are given by LϪ1 MϪ1 ÃÃ(T) 1 PRXX(1) Pà ϭ X*(l, m)X(l, m), (A4) Ϫ 2 Re Cov , . (A8) X SR(T)(1) LM lϭ0 mϭ0 Ά·[] 1 LϪ1 MϪ2 Rà (T)(1) ϭ X*(l, m)X(l, m ϩ 1), (A5) To ®nd these variances we must evaluate the following X L(M Ϫ 1) lϭ0 mϭ0 four expressions:
1 LϪ1 LϪ1 MϪ1 MϪ1 E{PÃ 2 } ϭ E[X*(l, m)X(l, m)X*(lЈ, mЈ)X(lЈ, mЈ)], (A9) X 22 LM lϭ0 lЈϭ0 mϭ0 mЈϭ0 1 LϪ1 LϪ1 MϪ2 MϪ2 E{|RÃ (T)(1)|2 } ϭ E[X*(l, m)X(l, m ϩ 1)X*(lЈ, mЈϩ1)X(lЈ, mЈ)], (A10) X 22 L (M Ϫ 1) lϭ0 lЈϭ0 mϭ0 mЈϭ0 1 LϪ1 LϪ1 Mϭ2 MϪ2 E{[RÃ (T)(1)]2 } ϭ E[X*(l, m)X(l, m ϩ 1)X*(lЈ, mЈ)X(lЈ, mЈϩ1)], (A11) X 22 L (M Ϫ 1) lϭ0 lЈϭ0 mϭ0 mЈϭ0 1 LϪ1 LϪ1 MϪ1 MϪ2 E{PRÃÃ(T)(1)} ϭ E[X*(l, m)X(l, m)X*(lЈ, mЈ)X(lЈ, mЈϩ1)]. (A12) XX 2 LM(M Ϫ 1) lϭ0 lЈϭ0 mϭ0 mЈϭ0
The expectation operations E[.] inside these expressions 1 E{PÃ 2 } ϭ {E[X*(l, m)X(l, m)] can be simpli®ed using the identity X 22 LM l,lЈ,m,mЈ ϫ E[X*(lЈ, mЈ)X(lЈ, mЈ)] E[X 1234*XX*X ] ϭ E[X* 12X ]E[X* 34X ] ϩ E[X*(l, m)X(lЈ, mЈ)] ϩ E[X 14*X ]E[X* 32X ], (A13) ϫ E[X*(lЈ, mЈ)X(l, m)]} which is valid for zero-mean, complex, Gaussian ran- dom variables (Reed 1962). After applying this identity 1 ϭ [R (0, 0)]2 to (A9)±(A12), each expectation can be expressed as 22 X LM l,lЈ,m,mЈ the autocorrelation of X at particular lags; that is, E[X*(l, ϩ RXX(lЈϪl, mЈϪm)R (l Ϫ lЈ, m Ϫ mЈ), m)X(k, n)] ϭ Rx(k Ϫ l, n Ϫ m). For example, for (A9) we have (A14)
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à 22 Quadruple summations can be simpli®ed by letting lЉ E{PX Ϫ P } ϭ lЈϪl, mЉϭmЈϪm, and collecting terms so that 1 LϪ1 (R) 2 LϪ1 LϪ1 MϪ1 MϪ1 ϭ (L Ϫ |l|)|R (l)| 22 XS f (lЈϪl, mЈϪm) LM lϭϪLϩ1 lϭ0 lЈϭ0 mϭ0 mЈϭ0 MϪ1 (T) 2 LϪ1 MϪ1 ϫ (M Ϫ |m|)|R (m)| XS ϭ (L Ϫ |lЉ|)(M Ϫ |mЉ|) f (lЉ, mЉ). mϭϪMϩ1 lЉϭϪLϩ1 mЉϭϪMϩ1 (A15) 2 LϪ1 ϩ Re (L Ϫ |l|)R(R)((l)[R R)(l)]* 22 XXSN Then, LM ΆlϭϪLϩ1 1 LϪ1 MϪ1 MϪ1 E{Pà 22Ϫ P } ϭ ϫ (M Ϫ |m|)R(T)((m)[R T)(m)]* X 22 XXSN LM lϭϪLϩ1 mϭϪMϩ1 mϭϪMϩ1 · 2 LϪ1 ϫ (L Ϫ |l|)(M Ϫ |m|)|RX(l, m)| . (A16) 1 ϩ (L Ϫ |l|)|R(R)(l)|2 22 XN Analogously, the three remaining expressions become LM lϭϪLϩ1 E{|Rà (T)(1)|2(Ϫ |R T)2(1)| } MϪ1 X ϫ (M Ϫ |m|)|R(T)(m)|2 . (A21) XN 1 LϪ1 MϪ2 mϭϪMϩ1 ϭ 22 L (M Ϫ 1) lϭϪLϩ1 mϭϪMϩ2 Recall that the whitening transformation is applied to the samples along the range dimension [see (8)]; the sample± ϫ (L Ϫ |l|)(M Ϫ |m| Ϫ 1)|R (l, m)|2 , (A17) X time correlation of weather signals, which carries the information needed to estimate the spectral moments, is E{[Rà (T)(1)]2(Ϫ [R T)2(1)] } X preserved after whitening. Therefore, for a Gaussian sam- LϪ1 MϪ2 T 1 ple-time correlation function and white noiseRXS (m) ϭ ϭ 2 T 22 S exp[ 2( m) j2 m] andR (m) N (m), Ϫ n ϩ n XN ϭ ␦ L (M Ϫ 1) lϭϪLϩ1 mϭϪMϩ2 where n ϭ /2 a and vn ϭ /2 a, are the normalized ϫ (L Ϫ |l|)(M Ϫ |m| Ϫ 1)RX(l, m ϩ 1) velocity and spectrum width, respectively (Doviak and Zrnic 1993). Summations in (A21) involving these func- ϫ R*(X l, m Ϫ 1), (A18) tions can be approximated as
ÃÃ(T) (T) MϪ1 E{PRXX(1) Ϫ PR (1)} (T) 2 (M Ϫ |m|)|RX (m)| LϪ1 MϪ2 S 1 mϭϪMϩ1 ϭ 2 ϱ 2 LM(M Ϫ 1) lϭϪLϩ1 mϭϪMϩ2 2 MS S 2 (M Ϫ |x|)edxϪ(2vnx) , (A22) ഠ ഠ 1/2 ͵ 2vn ϫ (L Ϫ |l|)(M Ϫ |m| Ϫ 1)RXX(l, m ϩ 1)R*(l, m). Ϫϱ
(A19) if Mvn k 1, Because X(l, m) has uncorrelated signal and noise MϪ1 (M Ϫ |m|)R(T)((m)[R T)(m)]* ϭ MSN, (A23) components, the autocorrelation function can be de- XXSN composed into a sum of the signal (S) and noise (N ) mϭϪMϩ1 MϪ1 autocorrelation functions, that is, R X ϭ RXXϩ R . SN (M Ϫ |m|)|R(T)(m)|22ϭ MN . (A24) Also, because we are dealing with precipitation par- XN ticles and the width of the range-weighting function mϭϪMϩ1 [(4.22) of Doviak and Zrnic 1993] is much smaller Closed-form solutions for the summations involving than the pulse repetition time, the time and space di- range-time correlations can be obtained if we work with mensions are approximately independent [see, e.g., correlation matrices instead of correlation functions. To 1 LϪ1 section 5.5 of Bringi and Chandrasekar (2001)]. make the conversion we can use the identity ⌺lϭϪLϩ1 Therefore, two-dimensional autocorrelation functions (L Ϫ | l |)R1(l)(R*2 l) ϭ tr{C1C2}, where C1 and C2 are are separable (see, e.g., Dias and LeitaÄo 1993) as they the correlation matrices corresponding to the correlation can be decomposed into a product of one-dimensional functions R1 and R 2, respectively, and tr{.} is the matrix sample-time (T ) and range±time (R) autocorrelation functions: 1 This can be proved by expressing the trace as the sum of diagonal (R)(T)(R)(T) RXXXXX(l, m) ϭ R SS(l)R (m) ϩ R NN(l)R (m). (A20) elements of the matrix product, expanding the matrix product using the Hermitian and Toeplitz properties of complex correlation matri- Double summations can now be decoupled and (A16) ces, and ®nally performing a simple substitution of summation in- becomes dexes.
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LϪ1 It can be veri®ed by matrix multiplication that (L Ϫ |l|)|R(R)(l)|22ϭ tr{[C(R)] }, (A25) XS XS L(L ϩ 2) LL lϭϪLϩ1 Ϫ 0 ´´´ 0 2L ϩ 22 2L ϩ 2 LϪ1 (L Ϫ |l|)R(R)((l)[R R)(l)]* ϭ tr{CC(R)(R)}, (A26) LL XXSN XXSN Ϫ L Ϫ 5 0 lϭϪLϩ1 22 LϪ1 (R) 22(R) L (L Ϫ |l|)|R (l)| ϭ tr{[C ] }. (A27) XN XN 0 Ϫ 555 _ lϭϪLϩ1 (R) Ϫ1 2 [CVS ] ϭ . Correlation matrices for the signal and noise com- _55550 ponents of the whitened sequence X can be obtained L from the correlation matrix of the range (correlated) 0 55 L Ϫ samples V by recalling that X ϭ WV (8), where W ϭ 2 H Ϫ1 and C (R) ϭ H*HT . Due to the linear relationship VS LLL(L ϩ 2) between V and X, the correlation matrix of X can be 0 ´´´ 0 Ϫ (R)(R)(T R) 2L ϩ 222L ϩ 2 written as CCXVϭ W* W . Decomposing C V into its signal and noise components and distributing the (A32) matrix products, Then, by summing the diagonal elements of (A32), (R)(R) TT (R) Ϫ1 3 Ϫ1 CXVϭ W*(SC S ϩ NI)W ϭ SI ϩ N(W*W ); (A28) tr{[CVS ] } ϭ L (L ϩ 1) . In addition, for symmetric 2 2 (R)(R) T matrices tr(A ) ϭ⌺i ⌺j (A)ij and from (A32) we ®nd therefore, CCXXSNϭ I and ϭ W*W . Finally, (A25)± (R) Ϫ2 3 2 Ϫ2 that tr{[CVS ] } ϭ (1/2)L (3L ϩ 2L Ϫ 3)(L ϩ 1) . (A27) can be written as Introducing the results of (A22)±(A24) and (A29)± (R) 2 2 (A31) into (A21) for the ideal case, tr{[CXS ]}ϭ tr{I } ϭ L, (A29) S 2 2SN L (R)(R) TT Ã 22 tr{CCXXSN} ϭ tr{W*W } ϭ tr{WW*} E{PX Ϫ P } ϭϩ 2MLvn͙ MLϩ 1 (R) Ϫ1 ϭ tr{[CV ] }, (A30) S NL22(3L ϩ 2L Ϫ 3) ϩ . (A33) tr{[C(R)(]}2 tr{W*WWTT*W } tr{[C R)]Ϫ2 }. (A31) 2 XVNSϭ ϭ M 2(L ϩ 1) For the ideal range-time correlation coef®cient (corre- In a similar way we can derive expressions for (A17)± sponding to a rectangular transmitter pulse and an in- (A19) to use in (A7) and (A8). Doing so we arrive at ®nite receiver bandwidth) the elements of the received the following simpli®ed expressions for the variances (R) Ϫ1 signal correlation matrix are (CVS )ij ϭ 1 Ϫ | j Ϫ i | L of WTB spectral moment estimators:
S 2211 LNL(3L ϩ 2L Ϫ 3) N 2 Var {SÃ} ϭϩ2 ϩ , (A34) 2 (WTB) MLLϩ 1 S 2(L ϩ 1) S []2vn͙
2 2 2 2(2e vn)(2Ϫ 11 LNevn)2 L(3L ϩ 2L Ϫ 3) N Var {Ã } ϭϩa 2 sinh(2 )2 ϩ , 2 vn 2 (WTB) (M Ϫ 1) LLϩ 1 S 22(L ϩ 1) S Ά 4vn͙ [] · (A35)
222 22(2eevn)(2 vn)(Ϫ 4e vn) ϩ 31 LN Var{Ã } ϭϩa 2[cosh(2 )2 Ϫ 1] 42 vn (WTB) 4vn(M Ϫ 1)Ά 4vn͙ LL ϩ 1 S
2 2 e(2vn)2ϩ 2 L(3L ϩ 2L Ϫ 3) N ϩ . (A36) 22([]L ϩ 1)2 S ·
We can repeat the same procedure for the case of a previous case is in the way that the total power and lag- digital matched ®lter where the only difference from the one autocorrelation function are estimated. In this case
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1 MϪ1 digital matched ®lter and ϭ [3(2L 2 ϩ 1)Ϫ1 ]1/2 is a PÃ ϭ |Y(m)|2 , (A37) Y normalization factor that preserves the power of the M mϭ0 input signal under ideal conditions. Following the 1 MϪ2 RÃ (T)(1) ϭ Y*(m)Y(m ϩ 1), (A38) steps outlined in the previous case we obtain the Y M Ϫ 1 mϭ0 variances of the estimators for the matched-®lter LϪ1 where Y(m) ϭ ⌺ lϭ0 V(l, m) is the output of the case as
S 2 13LN 3LN22 Var {SÃ} ϭϩ2 ϩ , (A39) 22 (MFB) M 2L ϩ 1 S 2L ϩ 1 S []2vn͙
2 2 22 2(2e vn)(2Ϫ 13LNevn) 3LN Var{Ã } ϭϩa 2 sinh(2 )2 ϩ , (A40) 222vn (MFB) (M Ϫ 1) 2L ϩ 1 S 22L ϩ 1 S []4vn͙
222 22(2eevn)(2 vn)(Ϫ 4e vn) ϩ 33LN Var{Ã } ϭϩa 2[cosh(2 )2 Ϫ 1] 42 vn 2 (MFB) 4vn(M Ϫ 1)Ά 4vn͙ 2L ϩ 1 S
2 22 e(2vn) ϩ 23LN ϩ . (A41) 22L2 ϩ 1 S ·
APPENDIX B alent scattering center, which backscatters the complex (I, Q) voltage s(l). This is reasonable because there are Simulation of Oversampled Weather Echoes numerous scatterers in the resolution volume so that The simulation method uses known statistical prop- their contribution causes the voltage backscattered by erties of signals re¯ected by scatterers in ¯uids and is each slab to be a Gaussian complex random variable. based on the well-known procedure for generating sin- Note that implicit is the assumption that the slabs are gle-polarization time series by Zrnic (1975). Further, it large compared with the radar wavelength and there are takes into account the known properties of the probing many uniformly distributed scatterers in each slab. Thus, pulse and receiver ®lter to reconstruct a composite sig- each element of the sequence s(l) is an independent, nal from the distributed scatterers illuminated by the identically distributed (IID) complex Gaussian random pulse; therefore, it is not dif®cult to modify the pro- variable (RV) with zero mean and unit variance. It is cedure to include re¯ectivity gradients and additive assumed that the slab centers are separated in range by noise. The constructed time series exhibits the required c/2L and that the weather signal is sampled at a rate autocorrelation in range and sample time. This proce- L-times faster than the reciprocal of the pulse width, dure is justi®ed by the fact that at the horizontal inci- that is, o ϭ /L s apart. Figure B1 shows a simpli®ed dence common to weather surveillance radars, raindrops scheme of the basic elements involved in the simulation can be regarded as frozen scatterers since the time sep- procedure. aration between echoes from overlapping range inter- vals is very small. Signals received by a Doppler weather surveillance radar at any given time are due to the superposition of the waves backscattered by the hydrometeors that are present in the radar resolution volume. The range lo- cation rs of the resolution volume with respect to the radar depends on the time delay between the transmitted pulse and the sampling time s as given by rs ϭ cs/2, where c is the speed of light. The simulation procedure starts with oversampling in range (along the range-time axis) so that there are L samples during the pulse duration . The contribution of scatterers within the resolution volume is distributed among L slabs, where each slab encompasses a large FIG. B1. Basic elements involved in the simulation of oversampled number of hydrometeors but is represented by its equiv- weather signals.
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For an ideal (in®nite bandwidth) receiver, the impulse LϪ1 response h(n) is given by the unit-sample sequence, that V (l, n) ϭ s(l ϩ i, n)p(L Ϫ 1 Ϫ i) ૺ h(l), 3 is, h(n) ϭ ␦(n). If range time is indexed with l, a range []iϭ0 sample of the weather signal is given by for 0 Յ l Ͻ L ϩ F Ϫ 1 and LϪ1 0 Յ n Ͻ M, (B5) V (l) ϭ s(l ϩ i)p(L Ϫ 1 Ϫ i), 0 Յ l Ͻ L, (B1) 1 iϭ0 where s isa(2L ϩ F Ϫ 2)-by-M matrix of IID, zero- where s is a 2L Ϫ 1 vector of IID Gaussian random mean, unit-variance, Gaussian random variables. Tran- variables with zero mean and unit variance, and p is the sient removal can be accomplished by constructing a transmitted pulse envelope. The autocorrelation of the truncated version of V3 as V 4(l, n) ϭ V3(l ϩ F Ϫ 1, n), signal in (B1) is for 0 Յ l Ͻ L and 0 Յ n Ͻ M. It follows that for a given time (®xed n) the samples V 4(l, n) for 0 Յ l Ͻ LϪ1 L have a correlation given by (B2), and for a given R(R)(m) ϭ p( j)p( j Ϫ m), (B2) V1 range (®xed l) they are IID complex Gaussian RV with j 0 ϭ zero mean and variance G (see previous section). There- which agrees with (4.39) of Doviak and Zrnic (1993). fore, the usual coloring procedure can be applied along A nonideal receiver channel can be modeled by assum- sample time. Start by expressing the power spectrum ing that the receiver is a linear, shift-invariant system (on a discrete Doppler velocity axis) of (B4) as with impulse response h(n) ± ␦(n). Then, the convo- lution operation ૺ can be used to obtain S ( Ϫ )2 ( ) ϭ exp Ϫ k , k 2 2 V (l) ϭ V (l) ૺ h(l) ͙2 G [] 21 2k LϪ1 ϭϪ ϩ a , ϭ s(l ϩ i)p(L Ϫ 1 Ϫ i) ૺ h(l). (B3) kaM []iϭ0 k ϭϪM, ϪM ϩ 1,...,2M Ϫ 1, (B6) The power of this signal is now affected by the re- ceiver's ®lter. The mean signal power after adding where a is the maximum unambiguous velocity. The FϩLϪ2 2 next step is to alias this spectrum into the Nyquist in- correlation in range is G ϭ ⌺ lϭ0 |(h ૺ p)(l )| , where F is the receiver's ®lter impulse response terval (M samples) and then ¯ip it to change the Doppler length. In addition, the length of the input data se- velocity axis to the frequency axis ( ϭϪ f d/2). This ``¯ipped'' sequence is referred to as Ј(k), where 0 Յ quence V1 in (B3) should be adjusted to L ϩ F Ϫ 1 samples so that enough convolution samples are com- k Ͻ M. Summarizing, the time series with sample±time puted in order to bypass transients and obtain a se- correlation is obtained (for a ®xed l) using discrete-time Fourier transforms (F )as quence V 2 with L samples. For Doppler measurements, the radar is pulsed at a V(l, n) ϭ F Ϫ1{F [V (l, n)]͙Ј(k)}, suf®ciently high rate so that the atmospheric phenomena 4 produce correlated signal samples. Samples for every 0 Յ l Յ L Ϫ 1; (B7) range location are taken at intervals of Ts s, giving origin to the ``sample time.'' It is generally assumed that the where V 4 is the truncated version of the time series with sample-time correlation of weather signals is Gaussian only range-time correlation, and V exhibits the required (Doviak and Zrnic 1993) and given by correlation in both range and sample time. Finally, note that because (B6) is normalized, the mean power of V (T) RV (m) ϭ E[V*(l, n)V(l, n ϩ m)] is S, as required.
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