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  • View in gallery

    (left) Power spectrum showing noise power spectral density (dashed line). The unambiguous velocity is denoted as . (right) Power spectrum following noise subtraction showing signal-to-threshold ratio (STR). Noise spikes exceeding the threshold level constitute false alarms.

  • View in gallery

    Spreading loss , computed from (14), as a function of N for various normalized spectral widths, assuming a Hanning time-domain window.

  • View in gallery

    Power spectrum with , M = 4, and N = 256 showing multiple peaks in the main lobe region. The correction factor is shown (dB) relative to the expected value of the spectrum (red).

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    Simulated and modeled as a function of DFT length for M = 2 with Hanning weights applied to the time-domain signal: modeled values (solid line), simulated values for (triangles), and simulated values for (diamonds).

  • View in gallery

    From (24), (solid trace) when applying a Hanning window, and DFT length N that maximizes signal processing gain (20) vs normalized spectral width for two false alarm rates: (dashed red trace) and (dotted red trace).

  • View in gallery

    Relative gain at optimal DFT length GR (26) as a function of normalized spectral width, with total number of samples M as a parameter (M = 102–105), for four false alarm probabilities.

  • View in gallery

    Normalized spectral width as a function of pulse repetition frequency for various operating frequencies for spectral widths (left) m s−1 and (right) m s−1.

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    Cumulative distribution function CDF of the average of 32 noise samples for mean noise power (solid curve) and CDF after mean noise (dashed curve) for false alarm probability , showing the prenoise subtraction threshold and the threshold after noise subtraction for noncoherent processing .

  • View in gallery

    Maximum probability of false alarm vs sample length M for noncoherent processing.

  • View in gallery

    False alarm factors: (left) and (right) as a function of total sample length M for spectral plot, showing N = 64 (black traces) and N = 512 (red traces).

  • View in gallery

    Combined DFT-based (20) and noncoherent (3) signal processing gain vs DFT length with spectral width as a parameter using a Hanning time-domain window. (top) Total sample length NM = 256 and (bottom) NM = 20 000, showing noncoherent integration gain [(3); horizontal dashed red trace], DFT-based processing gain [(20); black traces], , (vertical red lines), and (vertical green line). Simulated values shown with symbols represent the average of 2500 trials for NM = 256 and 50 trials for NM = 20 000.

  • View in gallery

    Probability of detection for noncoherent power averaging with noise subtraction as a function of SNR. Theoretical given by (42) assuming all signal samples independent (red trace); (42) with the modified CDF substituted for (black trace); = 0.02, and M = 192. Simulated values shown with symbols representing the average of 10 000 samples.

  • View in gallery

    Simulated and theoretical probability of detection (53) for NM = 20 000 samples and SNR = −24 dB, Hanning window applied. (top) Term = 0.01; (bottom) = 0.25. Noncoherent processing [(42); horizontal red lines] and simulated values represent the average of 1000 trials (symbols), and (vertical red lines mark).

  • View in gallery

    Term (red trace) and (black traces and symbols) as a function of normalized spectral width; is plotted for five different detection probabilities, 0.1 (square), 0.3 (triangle), 0.5 (*), 0.7 (+), and 0.9 (diamond).

  • View in gallery

    Optimal DFT length , assuming Hanning time-domain weights, as computed from measured spectral width in a high SNR region of cloud layer measured at X band, 6 May 2015 (range gates 360–386). The average value of 312 (dashed red line) corresponds with ( m s−1). Data were processed with zero padding to avoid scalloping loss: .

  • View in gallery

    Range profiles of cloud STR with = 0.01 using (top left) noncoherent averaging and mean noise subtraction for sample length M = 24 576; (top right) peak STR with spectral processing with DFT length N = 32, and number of spectral averages ; (bottom left) N = 312, ; and (bottom right) N = 2048, . Data processed with .

  • View in gallery

    Range profiles of cloud STR with = 0.25 using (top left) noncoherent averaging and mean noise subtraction for sample length M = 24 576; (top right) peak STR with spectral processing with DFT length N = 32 and number of spectral averages ; (bottom left) N = 312, ; and (bottom right) N = 2048, . Data processed with .

  • View in gallery

    Probability of detection for noncoherent processing (red traces) for M = 24 576 and for spectrally processed data (black traces) for DFT lengths N = 32, and number of spectral averages (solid traces); N = 312, (dashed traces); and N = 2048, (dotted–dashed traces). (left) Term = 0.01 with dashed lines showing and ; (right) = 0.25 with dashed lines showing and . The theoretical processing gains are as follows: dB; dB for N = 32; dB for N = 312; and dB for N = 2048.

  • View in gallery

    Measured (black trace with asterisks) at X band, 6 May 2015, for the average of range gates 360–386 (range = 7.944–8.594 km) and simulated (red trace) assuming for (left) and (right) = 0.25, showing (vertical red line) and [vertical green line in (right)] for = 0.25. Term is greater than M for . Data processed with .

  • View in gallery

    (left) Measured at X band, 6 May 2015, for the average of range gates 360–386 (range = 7.944–8.594 km) processed using uniform weights with (solid black trace with asterisks), and (dashed black trace) showing deleterious effects of scalloping loss. Simulated (red trace), with uniform weights (vertical red line), and . (right) As in (left), but for Hanning time-domain weights and .

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    Doppler power spectrum (red trace) with spectral width σ showing relative energy of discrete frequency samples (blue rectangles) for spectral samples marked with asterisks.

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Comparison of Meteorological Radar Signal Detectability with Noncoherent and Spectral-Based Processing

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Abstract

Detection of meteorological radar signals is often carried out using power averaging with noise subtraction either in the time domain or the spectral domain. This paper considers the relative signal processing gain of these two methods, showing a clear advantage for spectral-domain processing when normalized spectral width is less than ~0.1. A simple expression for the optimal discrete Fourier transform (DFT) length to maximize signal processing gain is presented that depends only on the normalized spectral width and the time-domain weighting function. The relative signal processing gain between noncoherent power averaging and spectral processing is found to depend on a variety of parameters, including the radar wavelength, spectral width, available observation time, and the false alarm rate. Expressions presented for the probability of detection for noncoherent and spectral-based processing also depend on these same parameters. Results of this analysis show that DFT-based processing can provide a substantial advantage in signal processing gain and probability of detection, especially when the normalized spectral width is small and when a large number of samples are available. Noncoherent power estimation can provide superior probability of detection when the normalized spectral width is greater than ~0.1, especially when the desired false alarm rate exceeds 10%.

Denotes Open Access content.

Corresponding author address: James B. Mead, ProSensing Inc., 107 Sunderland Road, Amherst, MA 01002. E-mail: mead@prosensing.com

Abstract

Detection of meteorological radar signals is often carried out using power averaging with noise subtraction either in the time domain or the spectral domain. This paper considers the relative signal processing gain of these two methods, showing a clear advantage for spectral-domain processing when normalized spectral width is less than ~0.1. A simple expression for the optimal discrete Fourier transform (DFT) length to maximize signal processing gain is presented that depends only on the normalized spectral width and the time-domain weighting function. The relative signal processing gain between noncoherent power averaging and spectral processing is found to depend on a variety of parameters, including the radar wavelength, spectral width, available observation time, and the false alarm rate. Expressions presented for the probability of detection for noncoherent and spectral-based processing also depend on these same parameters. Results of this analysis show that DFT-based processing can provide a substantial advantage in signal processing gain and probability of detection, especially when the normalized spectral width is small and when a large number of samples are available. Noncoherent power estimation can provide superior probability of detection when the normalized spectral width is greater than ~0.1, especially when the desired false alarm rate exceeds 10%.

Denotes Open Access content.

Corresponding author address: James B. Mead, ProSensing Inc., 107 Sunderland Road, Amherst, MA 01002. E-mail: mead@prosensing.com

1. Introduction

Detection of meteorological radar signals in the presence of noise has been studied extensively (cf. Keeler and Passarelli 1990; Doviak and Zrnić 1993). In the context of noncoherent power averaging, as is used by the WSR-88D for reflectivity estimation (Doviak and Zrnić 1993), it is well known that the signal detection threshold reduces as the square root of the number of samples averaged (Marshall and Hitschfeld 1953; Clothiaux et al. 1995). Spectral-based processing, through the use of the discrete Fourier transform (DFT) to compute the power spectrum, provides an alternative method for estimating reflectivity and other spectral moments (Gage and Balsley 1978; Farley 1985; Kollias et al. 2005). Signal detection in spectral processing has been shown to improve linearly with DFT length, since the DFT implements coherent integration (Farley 1985; Lyons 2004). However, this linear improvement in detectability is strictly true only for zero bandwidth signals in the presence of noise. In addition, averaging multiple DFT-derived power spectra to estimate the final power spectrum results in a reduction in the signal detection threshold by the square root of the number of power spectra averaged (Farley 1985). More recently, dual-polarization weather radars using simultaneous transmission and reception (STAR) processing benefit from asymptotic reduction in noise power when cross correlating the signal from orthogonal polarization channels (Keränen and Chandrasekar 2014; Ivić et al. 2012, 2014).

Given the limited number of radar pulses available for signal estimation, it is useful to consider the optimal combination of coherent integration—through DFT processing—and spectral power averaging that maximizes the probability of detection for a given false detection (or false alarm) rate. In this paper, we show that a simple expression for the optimal DFT length required to maximize the signal-to-threshold ratio is a good predictor of the DFT length yielding maximum probability of detection. The optimal DFT length is dependent on only two parameters, a constant that depends on the time-domain windowing function used in computing the DFT and the normalized Doppler spectral width . The Doppler spectral width prior to normalization is determined by turbulence within the radar pulse volume as well as other factors, such as beam broadening and shear broadening (Kollias et al. 2011; Gage 1990). Cloud and precipitation spectral width as measured using Doppler radars vary from ~0.1 to several meters per second [see Gage (1990); Doviak and Zrnić (1993) for examples of data spanning this range], which results in in the range of ~10−5 to ~0.5 depending on radar wavelength and pulse repetition frequency.

This paper is strictly concerned with optimizing detection probability and does not consider other issues, such as measurement bias or variance in the context of spectral moment analysis for estimating reflectivity, velocity, and velocity spectral width. It is noted in passing that noncoherent methods of moment estimation have well-known errors (Doviak and Zrnić 1993), although they are unbiased for reflectivity and velocity. Reflectivity estimated from the zeroth moment of the noise-subtracted power spectrum is biased at low signal-to-noise ratio if the region of summation is limited to contiguous spectral points of positive value near the spectral peak (Mead 2010). These issues are beyond the scope of this paper, which is primarily concerned with the detection of weak signals in the presence of noise.

The paper is organized as follows: Section 2 describes the well-known problem of signal detection with thresholding using noncoherent power averaging. Section 3 addresses spectral-based processing, deriving expressions for signal processing gain and optimal DFT length. False alarm threshold factors are derived as a function of false alarm rate for both methods in section 4. In section 5, the formulas for signal processing gain are compared to simulated values showing excellent agreement. Expressions for probability of detection are derived in section 6. The relative signal processing gain between noncoherent processing and spectral processing at the optimal DFT length is shown to be approximately equal to the relative signal-to-noise ratio needed to achieve the same probability of detection. Measured X-band cloud radar data processed using both methods are presented in section 7, which demonstrates that the analytical models based on Gaussian signal statistics are predictive for real data.

A list of symbols is provided for convenience in appendix A.

2. Signal detection with noncoherent power averaging

To compare noncoherent power averaging to spectral-based processing, it is useful to define a detection threshold set according to an allowable probability of false alarm. The detection threshold T determines if a particular power average is declared a detection; that is,
e1
where is the power average of M samples of the signal-plus-noise vector s and is the mean noise power for the averaging period. For simple power averaging with mean noise power subtraction [cf. (6.28) in Doviak and Zrnić 1993], the detection threshold as a function of false alarm rate may be expressed as [cf. (11) in Clothiaux et al. 1995]
e2
where M is the total number of samples and is the false alarm factor for noncoherent power averaging that is a function of M and the false alarm rate. The false alarm factor raises or lowers the threshold based on the acceptable level of false detections. As a reference point, a baseline detection threshold is established for single sample detection (M = 1), which corresponds to a probability of false alarm of 0.135. Given this reference threshold, (2) exhibits a reduction in the threshold by a factor of , which is typically less than one for practical values of and reasonable false alarm rates ( for probability of false alarm > 0.001). An expression for the false alarm factor as a function of false alarm rate is derived in section 4.
The reduction in threshold achieved by power averaging can be thought of as signal processing gain or noncoherent gain in minimum detectable signal as compared to single sample detection, and is given as
e3
The signal-to-threshold ratio for noncoherent processing STRn is given as
e4
where SNR is the signal-to-noise ratio.

The formulas presented above for noncoherent processing and those derived in the following section for spectral processing, assume that some method is established for accurately estimating noise power. Noise power can fluctuate due to a variety of causes, for example, changes in atmospheric brightness temperature as a function of antenna direction or changes in receiver gain and noise figure. Thus, it is critical to implement methods that continuously estimate noise power, such that each averaging period has a noise estimate specific to the data gathered during that averaging period.

3. Signal detection with spectral-based processing

The power spectrum of a given random signal can be estimated from a finite sample of length M as follows. The M samples are divided into equal segments of length N. Each segment is processed by a DFT of length N (Doviak and Zrnić 1993) and then power averaged to form the power spectrum estimate
e5
where ; is the pth N-length segment of s; and is an N-element vector of time-domain weights, that is, the time-domain window function. The time-domain weights are applied to reduce spectral artifacts resulting from the finite sample length (Lyons 2004). The issue is to find the value of N to optimize signal detectability.

An N-point DFT is a coherent integration algorithm that for zero bandwidth signals results in a gain in SNR that scales approximately linearly with N (Lyons 2004). The signal processing gain is evident when comparing the SNR in the frequency domain to that of the original time-domain signal (Lyons 2004). The deviation from N is due to scalloping loss (Prahbu 2014), which can be essentially eliminated by use of specialized windowing functions (Lyons 2011) or by oversampling in the frequency domain by zero padding the time-domain input (Prahbu 2014). Zero padding results in a small correction to the false alarm threshold, which is addressed in section 4c.

For a nonuniform window, the gain in SNR for a zero bandwidth signal is given by the following expression:
e6
where the numerator is the signal power gain and the denominator is the noise power gain. This expression is derived in appendix D. For the following, it is convenient to define a coherent integration loss factor ,
e7
which gives the loss in processing gain relative to uniform weighting. For example, for N > 2 when using a Hanning window. Loss factor is given for various time-domain window functions in Table 1.
Table 1.

Coherent integration loss factor , spectral broadening scale factor , optimal DFT length scale factor , and spreading loss at optimal DFT length for commonly used windows.

Table 1.
For spectral processing, STR is defined as the ratio of peak spectral power to detection threshold following noise subtraction, as shown in Fig. 1. The detection threshold for spectral processing is expressed in terms of the peak signal in the power spectrum and the noise power spectral density :
e8
where is the false alarm factor for spectral-based processing.
Fig. 1.
Fig. 1.

(left) Power spectrum showing noise power spectral density (dashed line). The unambiguous velocity is denoted as . (right) Power spectrum following noise subtraction showing signal-to-threshold ratio (STR). Noise spikes exceeding the threshold level constitute false alarms.

Citation: Journal of Atmospheric and Oceanic Technology 33, 4; 10.1175/JTECH-D-14-00198.1

Power spectrum–based signal processing gain is defined as the ratio of postprocessing STR to single-pulse SNR and combines both coherent and noncoherent integration gain. Coherent integration gain increases linearly with DFT length N for small N but asymptotically approaches a saturation level when the signal’s spectral peak lobe becomes distributed over multiple DFT bins for finite bandwidth signals. To account for this saturation effect, a spreading loss factor is defined as the ratio of power in the signal peak to the total spectral power, such that coherent integration gain is equal to . For this analysis, a Gaussian power spectrum of the form
e9
is assumed, where is velocity, is mean velocity, and the spectrum width is defined as the half-power full width of the spectral peak1 (m s−1). For discrete Fourier transforms using common windowing functions that do not significantly distort the Gaussian shape of the underlying power spectrum, spreading loss is given by the following expression, derived in appendix C:
e10
where erf is the error function, is the full width at half maximum spectral width broadening arising from the discrete Fourier transform for a particular windowing function, and is the spectral resolution of the DFT, that is,
e11
where is the radar wavelength (m), is the pulse repetition frequency (Hz), and
e12
is the unambiguous velocity (m s−1).
Spectral broadening due to the DFT is linearly related to spectral resolution when applying a suitable windowing function, that is,
e13
where is a window-dependent scale factor, ranging in value from 1.025 to 1.752 for common window functions, or zero for a uniform window.
Substituting (11) and (13) into (10) yields a normalized form for spreading loss:
e14
where the normalized spectral width is given by
e15
To find for a particular time-domain window, we note that Parseval’s theorem can be used to compute directly when and . By definition, the spreading loss for zero spectral width and zero velocity is given by
e16
where is the power in the zero-velocity bin of the power spectrum. When and Parseval’s theorem is
e17
Combining (16) and (17) demonstrates that :
e18
Equating (14) to (18) for yields the desired solution:
e19
The term is plotted as a function of N in Fig. 2. For small N, then begins to drops in magnitude for .
Fig. 2.
Fig. 2.

Spreading loss , computed from (14), as a function of N for various normalized spectral widths, assuming a Hanning time-domain window.

Citation: Journal of Atmospheric and Oceanic Technology 33, 4; 10.1175/JTECH-D-14-00198.1

To posit an expression for spectral signal processing gain, the effects of coherent integration gain (), noncoherent integration gain ( due to averaging of individual power spectra), spectral false alarm threshold, (derived in section 4b), and a correction factor for finite averaging, (described below), are combined. Power spectrum–based signal processing gain is equal to the product of these four factors:
e20
and the spectral signal-to-threshold ratio is
e21
Note that in the limit of M = 1.

The correction factor is required when the number of spectral averages is small. This issue is shown graphically in Fig. 3, where several peaks appear in the main lobe region. Each bin of the power spectrum is statistically independent of its neighbor, with chi-square distribution of order Furthermore, any one of the peaks in the region near the spectral peak has the potential of being the peak value in the spectrum. Thus, the expected value of the peak exceeds the mean value of the central peak by the factor .

Fig. 3.
Fig. 3.

Power spectrum with , M = 4, and N = 256 showing multiple peaks in the main lobe region. The correction factor is shown (dB) relative to the expected value of the spectrum (red).

Citation: Journal of Atmospheric and Oceanic Technology 33, 4; 10.1175/JTECH-D-14-00198.1

Through simulation, an approximate expression for was found:
e22
where is an approximation of the number of DFT bins in the peak signal region of a Gaussian spectrum:
eq1
e23
where can take on noninteger values. The term is plotted in Fig. 4 as a function of DFT length for two values of assuming . This figure shows that statistical fluctuations due to finite averaging in the power spectrum provides enhanced detection, equivalent to a processing gain of several decibels under conditions of large N and small . Details of the model for are presented in appendix B.
Fig. 4.
Fig. 4.

Simulated and modeled as a function of DFT length for M = 2 with Hanning weights applied to the time-domain signal: modeled values (solid line), simulated values for (triangles), and simulated values for (diamonds).

Citation: Journal of Atmospheric and Oceanic Technology 33, 4; 10.1175/JTECH-D-14-00198.1

The optimal DFT length can be found by finding N that maximizes the signal processing gain (20). However, (20) is a nontrivial function of N and cannot be readily differentiated. An approximate formula for the optimal DFT length is given by
e24
where is a constant that depends on the particular window function applied, for example, for the Hanning window.

Figure 5 plots the approximation of optimal DFT length (24) and the actual DFT length that maximizes signal processing gain (20) determined numerically, as a function of spectral width and two false alarm probabilities. The approximate and exact values are seen to be in close agreement, demonstrating that (24) is a good predictor of the optimal DFT length for maximizing the signal processing gain.

Fig. 5.
Fig. 5.

From (24), (solid trace) when applying a Hanning window, and DFT length N that maximizes signal processing gain (20) vs normalized spectral width for two false alarm rates: (dashed red trace) and (dotted red trace).

Citation: Journal of Atmospheric and Oceanic Technology 33, 4; 10.1175/JTECH-D-14-00198.1

Optimal signal processing gain occurs when :
e25
where is the spreading loss computed at , is the total observation time, and is the false alarm factor computed at . Note that = 1 when . For all of the weighting functions studied, it was found that . The formula for shows that optimal DFT-based processing gain scales as the square root of the radar wavelength and observation time, and scales linearly with pulse repetition frequency. Furthermore, optimal DFT-based processing gain relative to noncoherent gain (3) is given by
e26
This expression shows that signal processing gain of spectral-based processing as compared to noncoherent processing scales as the square root of wavelength and pulse repetition frequency, and is inversely proportional to the square root of the spectral width. Furthermore, is found to increase monotonically with , since the ratio / increases as the total number of samples increases.

In Fig. 6 is plotted as a function of normalized spectral width for false alarm probabilities between 0.001 and 0.25 with the total number of samples M as a parameter. For a given normalized spectral width, increases as the false alarm rate is decreased. In addition, the ratio / in (26) increases with M, asymptotically approaching a constant value that depends on the false alarm rate and the optimal DFT length. Thus, larger sample lengths M improve the relative gain of DFT-based processing to noncoherent processing due to a more favorable spectral detection threshold.

Fig. 6.
Fig. 6.

Relative gain at optimal DFT length GR (26) as a function of normalized spectral width, with total number of samples M as a parameter (M = 102–105), for four false alarm probabilities.

Citation: Journal of Atmospheric and Oceanic Technology 33, 4; 10.1175/JTECH-D-14-00198.1

As an aid to interpreting Fig. 6, normalized spectral width is plotted in Fig. 7 as a function of pulse repetition frequency for two spectral widths, m s−1 and 1.0 m s−1 with radar frequency as a parameter. Clouds above the turbulent boundary layer can exhibit spectral widths on the order of m s−1, for which is greater than 1 for all of the radar frequencies plotted assuming pulse repetition frequencies above a few kilohertz and a false alarm probability below 0.1. Clouds and clear-air turbulence within the atmospheric boundary layer often exhibit spectral widths on the order of m s−1. In this case, shorter wavelength radars show little sensitivity improvement from spectral processing.

Fig. 7.
Fig. 7.

Normalized spectral width as a function of pulse repetition frequency for various operating frequencies for spectral widths (left) m s−1 and (right) m s−1.

Citation: Journal of Atmospheric and Oceanic Technology 33, 4; 10.1175/JTECH-D-14-00198.1

Assuming, as was found empirically, that , the following formula can be used to solve for :
e27
However, this formula cannot be used for a uniform window, since the expression for spreading loss (14) is only accurate for tapered weighting functions. Therefore, was determined empirically for the uniform window.
By equating DFT-based signal processing gain (20) to the noncoherent integration gain (3), the critical DFT length is found, above which the DFT-based processing gain falls below the pure noncoherent integration gain:
e28
This equation is most easily solved numerically, since are nontrivial functions of N.

Table 1 summarizes the various constants in the formulas given above for commonly used window functions. While the spreading loss (14) has significant errors for the uniform window, the equations for optimal DFT length (24), optimal signal processing gain (25), and relative gain (26) are accurate for the uniform window using the constants in Table 1.

4. False alarm factors

a. Derivation of false alarm factor for noncoherent processing

The sum of M noise power samples of a bivariate Gaussian voltage distribution representing the noise-signal envelope has a chi-square distribution with 2M degrees of freedom, (Ulaby et al. 1982, 486–487). The cumulative distribution function (CDF) for a chi-square distribution with 2M degrees of freedom is denoted by D [Abramowitz and Stegun 1964, their (26.4.1); cf. Wolfram 2012]:
e29
which gives the probability that a particular average of M power samples is less than a threshold power for a process having mean noise power . For example, if the threshold , then for large M. The inverse function is written as
e30
which gives the threshold power as a function of .
The probability of false alarm (Skolnik 1990) is equal to the probability that a noise sample in the absence of a signal exceeds , thus . Prior to mean noise subtraction, the threshold for a given probability of false alarm is therefore
e31
After noise subtraction, this threshold drops by the mean noise power:
e32
where and are shown graphically in Fig. 8 along with the cumulative distribution functions for the average of 32 noise samples before and after the mean noise subtraction. For this analysis, negative powers after the noise subtraction are discarded. The maximum probability of false alarm for noncoherent processing occurs when (i.e., . This maximum is plotted in Fig. 9 as a function of the sample length.
Fig. 8.
Fig. 8.

Cumulative distribution function CDF of the average of 32 noise samples for mean noise power (solid curve) and CDF after mean noise (dashed curve) for false alarm probability , showing the prenoise subtraction threshold and the threshold after noise subtraction for noncoherent processing .

Citation: Journal of Atmospheric and Oceanic Technology 33, 4; 10.1175/JTECH-D-14-00198.1

Fig. 9.
Fig. 9.

Maximum probability of false alarm vs sample length M for noncoherent processing.

Citation: Journal of Atmospheric and Oceanic Technology 33, 4; 10.1175/JTECH-D-14-00198.1

Equating (32) to (2) gives the false alarm factor for noncoherent processing:
e33

b. Derivation of false alarm factor for spectral processing

For spectral processing, the probability of false alarm for each bin in the power spectrum is denoted by . The probability that all points in the power spectrum fall below the threshold is given by
e34
Thus, the false detection rate for the entire power spectrum is
e35
Solving for the per-bin false alarm rate,
e36
The threshold prior to noise subtraction is found by substituting (36) in (31):
e37
where is the noise power spectral density. The spectral processing detection threshold is
e38
The maximum false alarm rate for spectral-based processing is very close to 1.0, since there is a high probability that at least one point in the power spectrum will exceed the estimate of the mean noise power spectral density.
Equating (38) to (8) gives the false alarm factor for spectral processing:
e39
The false alarm factors and are plotted in Fig. 10 for various false alarm rates as a function of the total number of samples. When comparing the processing gain of spectral and noncoherent processing, the false alarm rates and should be equal. Note that the false alarm factor computed at , , is equal to evaluated at .
Fig. 10.
Fig. 10.

False alarm factors: (left) and (right) as a function of total sample length M for spectral plot, showing N = 64 (black traces) and N = 512 (red traces).

Citation: Journal of Atmospheric and Oceanic Technology 33, 4; 10.1175/JTECH-D-14-00198.1

c. Impact of zero padding on spectral false alarm threshold

To avoid reduction in signal power due to scalloping loss, the time-domain signal and the windowing function may be zero padded prior to computing the power spectrum; that is, in (5) the DFT is taken over with zeros appended to the vector of signal-plus-noise voltages and the time-domain weights vector . For example, if , then the scalloping loss reduces from 1.42 to 0.35 dB when using a Hanning time-domain window function. With zero padding, the false alarm factor is not accurately modeled by replacing N with in (39), since adjacent spectral values of noise become increasingly correlated as increases. Through simulation, it was found that rises by no more than 6% relative to the nonzero padded value, with little change as increases beyond 3N. If this small change in the threshold level is ignored, then the false alarm rate will increase by a few percent.

5. Simulation of signal processing gain

An Interactive Data Language (IDL) program was written to simulate signals of various spectral widths in the presence of noise and to simulate spectral-based and noncoherent signal processing gain. The simulated spectra were generated using the method described in Sirmans and Bumgarner (1975). Noncoherent gain (3) and DFT-based signal processing gain (20) are plotted in Fig. 11 for total sample lengths M = 256 and M = 20 000. These figures show excellent agreement between (3) and (20) and the simulated gain values. The ratio of peak spectral processing gain to noncoherent gain is seen to be greater for the larger total sample length as predicted by (26).

Fig. 11.
Fig. 11.

Combined DFT-based (20) and noncoherent (3) signal processing gain vs DFT length with spectral width as a parameter using a Hanning time-domain window. (top) Total sample length NM = 256 and (bottom) NM = 20 000, showing noncoherent integration gain [(3); horizontal dashed red trace], DFT-based processing gain [(20); black traces], , (vertical red lines), and (vertical green line). Simulated values shown with symbols represent the average of 2500 trials for NM = 256 and 50 trials for NM = 20 000.

Citation: Journal of Atmospheric and Oceanic Technology 33, 4; 10.1175/JTECH-D-14-00198.1

6. Probability of detection analysis

a. Probability of detection for noncoherent processing

First, consider a simplified case where all signal-plus-noise samples are independent, bivariate Gaussian distributed random variables. The probability of detection, (Skolnik 1990) is the probability that a given realization of the signal-plus-noise exceeds the false alarm threshold :
e40
which can be reexpressed in terms of the signal-to-threshold ratio (4):
e41
Eliminating by use of (32):
e42
This formula gives the probability of detection for simple noncoherent integration of M uncorrelated samples of the signal and noise. Equation (42) is strictly valid only when each signal sample is independent. However, signal samples are often partially correlated, thus the distribution of the sum of M signal-plus-noise samples is not always accurately described by chi-square statistics.

The distribution function of the average signal-plus-noise power can be obtained from the convolution of two independent distributions, the first distribution being pure noise and the second distribution consisting of independent signal samples in the presence of noise. Although this is an artificial construct, the resultant distribution function is correct, provided the power of the independent signal samples is scaled to yield the original signal-to-noise ratio.

Consider the power average of M partially correlated signal samples in the presence of noise. The distribution of the sum of two independent distributions is equal to the convolution of the individual distributions (Papoulis 1984). Two distributions are created, length and , consisting of pure noise and signal plus noise:
e43
e44
where . The effective number of independent signal samples is given by an approximate formula valid for normalized spectral widths up to 0.5 (6.12 in Doviak and Zrnić 1993):
e45
The total signal power associated with each of these distributions is chi-square distributed, with and degrees of freedom for noise and signal plus noise, respectively. The mean noise power in both distributions equals . The signal-to-noise ratio of the independent signal samples is scaled to account for the fact that fewer signal samples are employed than the original M samples:
e46
The mean power of each of the signal-plus-noise samples is given by
e47
In this way, the sum of the two distributions maintains the original signal-to-noise ratio. In addition, the sum of the two distributions contains the correct number of independent samples of the signal () and the correct number of independent samples of the noise (M). The cumulative distribution function resulting from the convolution of these two distributions is substituted in (42) to yield the probability of detection. A closed form solution for was derived, but factorial terms involving M were too large to compute for large M, thus was evaluated numerically.

Analytical and simulated results for the probability of detection for noncoherent processing are presented in Fig. 12 for four spectral widths, showing excellent agreement. This figure shows a significant improvement in detection probability for signals with wide spectral width when the signal-to-threshold ratio is greater than one. This trend reverses for low signal-to-threshold ratios (below 0.8).

Fig. 12.
Fig. 12.

Probability of detection for noncoherent power averaging with noise subtraction as a function of SNR. Theoretical given by (42) assuming all signal samples independent (red trace); (42) with the modified CDF substituted for (black trace); = 0.02, and M = 192. Simulated values shown with symbols representing the average of 10 000 samples.

Citation: Journal of Atmospheric and Oceanic Technology 33, 4; 10.1175/JTECH-D-14-00198.1

b. Probability of detection for spectral processing

The probability of detection per DFT bin in the region of the spectral peak lobe, , is given by
e48
which can be reexpressed in terms of the spectral signal-to-threshold ratio (21):
e49
where is a modified version of (generated by convolution as in section 6a) to account for the fact that all spectral values at a particular value of k (prior to spectral averaging) are independent for , while the spectral values prior to power averaging are partially correlated for :
e50
e51
The condition is equivalent to
eq2
which is somewhat less than Note that the multiple peak correction factor is removed from by division in (49), since is computed per DFT bin. Probability of detection for the entire spectrum is given by
e52
where it is assumed that the samples near the spectral peak have equal probability of detection, . For spectral processing, the probability that either noise or signal plus noise will exceed the false alarm threshold and thus be detected is given by
e53
This latter equation is useful when the probability of detection is close to or less than the probability of false alarm.

Simulated and theoretical values of probability of detection are plotted in Fig. 13, showing excellent agreement between (42) and (53) and simulated data. The maximum probability of detection is seen to occur when the DFT length is approximately equal to . This reflects the fact that (53) is near to its maximum value when STRs is maximized. It was found that (28) does not accurately predict the point at which the probability of detection is equal between the two processing methods. This is not surprising, since STRn = STRs when , but the formulas for noncoherent detection probability (42) and spectral-based detection probability (53) have a significantly different dependence on STR.

Fig. 13.
Fig. 13.

Simulated and theoretical probability of detection (53) for NM = 20 000 samples and SNR = −24 dB, Hanning window applied. (top) Term = 0.01; (bottom) = 0.25. Noncoherent processing [(42); horizontal red lines] and simulated values represent the average of 1000 trials (symbols), and (vertical red lines mark).

Citation: Journal of Atmospheric and Oceanic Technology 33, 4; 10.1175/JTECH-D-14-00198.1

A direct comparison between signal processing gain and probability of detection can be made by comparing the required difference in SNR needed to achieve the same detection probability for the two processing methods. In Fig. 14, the relative signal processing gain at the optimal DFT length, (26) in decibels, is compared to the difference in the signal-to-noise ratio, in decibels, required to achieve the same probability of detection for the two signal processing methods when the DFT length is set to . Specifically, , where the subscripts n and s refer to noncoherent and spectral processing, respectively, and the SNR values are selected to equate probability of detection. This comparison was made at five detection probabilities, showing agreement within ±0.4 dB for > 0.3. Thus, , which is computed at , is seen to be a good predictor of the relative signal-to-noise ratio required for the two processing methods to achieve the same detection probability for a given probability of false alarm.

Fig. 14.
Fig. 14.

Term (red trace) and (black traces and symbols) as a function of normalized spectral width; is plotted for five different detection probabilities, 0.1 (square), 0.3 (triangle), 0.5 (*), 0.7 (+), and 0.9 (diamond).

Citation: Journal of Atmospheric and Oceanic Technology 33, 4; 10.1175/JTECH-D-14-00198.1

7. Experimental results

A 2.45-s record (M = 24 576 samples per range gate) of raw in-phase and quadrature (I/Q) data from a cloud layer located between 9.2 and 10.9 km was gathered in Amherst, Massachusetts, at 1401 UTC 6 May 2015 using the DOE ARM program’s X-band scanning ARM Program cloud radar (SACR; DOE 2012). The radar pulse repetition frequency was 10 kHz, the range resolution was 45 m, and the antenna beamwidth was 1.4°.

The optimal DFT length was estimated by computing the normalized spectral width from a power spectrum computed using a 2048-point FFT with Hanning weights with = 12 spectra averaged to form the power spectrum. To compute , was estimated from the measured spectral width using (C2) to remove the spectral broadening term, . Term is plotted as a function of range in Fig. 15, showing an average value of 312 in the ranges between 7.9 and 8.6 km.

Fig. 15.
Fig. 15.

Optimal DFT length , assuming Hanning time-domain weights, as computed from measured spectral width in a high SNR region of cloud layer measured at X band, 6 May 2015 (range gates 360–386). The average value of 312 (dashed red line) corresponds with ( m s−1). Data were processed with zero padding to avoid scalloping loss: .

Citation: Journal of Atmospheric and Oceanic Technology 33, 4; 10.1175/JTECH-D-14-00198.1

Figure 16 plots the signal-to-threshold ratio processed noncoherently and with spectral processing employing an FFT algorithm with Hanning time-domain weights using three different DFT lengths of N = 32, 312, and 2048 with the probability of false alarm set to 0.01. The signal-to-threshold ratio is superior for all of the spectral plots as compared to the noncoherently processed signal, with the best sensitivity seen for a DFT length of 312. Note that the cloud layer between 9.4 and 9.8 km is poorly detected in the noncoherently processed data and that it has the highest STR in the data processed with N = 312. These figures are repeated in Fig. 17 with the probability of false alarm increased to 0.25.

Fig. 16.
Fig. 16.

Range profiles of cloud STR with = 0.01 using (top left) noncoherent averaging and mean noise subtraction for sample length M = 24 576; (top right) peak STR with spectral processing with DFT length N = 32, and number of spectral averages ; (bottom left) N = 312, ; and (bottom right) N = 2048, . Data processed with .

Citation: Journal of Atmospheric and Oceanic Technology 33, 4; 10.1175/JTECH-D-14-00198.1

Fig. 17.
Fig. 17.

Range profiles of cloud STR with = 0.25 using (top left) noncoherent averaging and mean noise subtraction for sample length M = 24 576; (top right) peak STR with spectral processing with DFT length N = 32 and number of spectral averages ; (bottom left) N = 312, ; and (bottom right) N = 2048, . Data processed with .

Citation: Journal of Atmospheric and Oceanic Technology 33, 4; 10.1175/JTECH-D-14-00198.1

The signal-to-threshold ratios STRn for the noncoherently processed signal and STRs for the spectrally processed data with DFT length = 32 and 2048 are approximately equal when , since all three cases have nearly identical signal processing gain (23.7–23.9 dB). However, the probability of detection is considerably lower for the noncoherently processed signal for the cloud layer between 9.4 and 9.8 km where the STR is less than 5 dB. This behavior is explained by Fig. 18, which shows that for the SNR required to achieve probability of detection of 95% is −21 dB for N = 32, −22 dB for N = 2048, and −16 dB for noncoherent processing. In general, the detection probability curves plotted against SNR are steeper for the spectrally processed data, especially at higher false alarm rates. This is because the probability distribution of the peak noise power after noise subtraction is narrower for spectrally processed data than the distribution of noise subtracted power for the noncoherently processed data. Steeper probability of detection curves are more favorable, since a smaller difference in SNR above the threshold is needed to yield a high detection probability.

Fig. 18.
Fig. 18.

Probability of detection for noncoherent processing (red traces) for M = 24 576 and for spectrally processed data (black traces) for DFT lengths N = 32, and number of spectral averages (solid traces); N = 312, (dashed traces); and N = 2048, (dotted–dashed traces). (left) Term = 0.01 with dashed lines showing and ; (right) = 0.25 with dashed lines showing and . The theoretical processing gains are as follows: dB; dB for N = 32; dB for N = 312; and dB for N = 2048.

Citation: Journal of Atmospheric and Oceanic Technology 33, 4; 10.1175/JTECH-D-14-00198.1

Relative signal processing gain is plotted in Fig. 19 for this same dataset along with simulated values for false alarm probabilities of 0.01 and 0.25 showing excellent agreement. The critical DFT length exceeds M for but was correctly estimated to be 2400 for . The effect of scalloping loss can be seen in Fig. 20, where was computed with uniform and Hanning weights using zero padding and no zero padding . The peak signal power is underestimated by as much as 2.4 dB in the uniform weights case without zero padding due to scalloping loss. The maximum scalloping loss observed for the case of Hanning weights is ≈1 dB. Note that the effects of scalloping loss are only apparent for , as the spectral peak lobe is spread over multiple DFT bins when and is thus not subject to scalloping loss.

Fig. 19.
Fig. 19.

Measured (black trace with asterisks) at X band, 6 May 2015, for the average of range gates 360–386 (range = 7.944–8.594 km) and simulated (red trace) assuming for (left) and (right) = 0.25, showing (vertical red line) and [vertical green line in (right)] for = 0.25. Term is greater than M for . Data processed with .

Citation: Journal of Atmospheric and Oceanic Technology 33, 4; 10.1175/JTECH-D-14-00198.1

Fig. 20.
Fig. 20.

(left) Measured at X band, 6 May 2015, for the average of range gates 360–386 (range = 7.944–8.594 km) processed using uniform weights with (solid black trace with asterisks), and (dashed black trace) showing deleterious effects of scalloping loss. Simulated (red trace), with uniform weights (vertical red line), and . (right) As in (left), but for Hanning time-domain weights and .

Citation: Journal of Atmospheric and Oceanic Technology 33, 4; 10.1175/JTECH-D-14-00198.1

8. Conclusions

The analysis presented above shows that spectral processing has the potential to provide significant sensitivity improvement as compared to noncoherent power averaging. However, it has been shown that spectral processing can result in loss of sensitivity relative to power averaging when the normalized spectral width is greater than ~0.2. The simple expression presented for was shown to accurately predict the DFT length that maximizes the probability of detection. The false alarm rate was found to play an important role in determining the relative detection performance of the two processing methods. Experimental data taken at X-band showed excellent agreement with the analytical expressions for processing gain and the probability of detection. Data processed at using a uniform time-domain window exhibited an 8-dB processing gain advantage as compared to noncoherently processed data.

Acknowlegments

The author thanks the U.S. Department of Energy for providing access to the scanning ARM cloud radar AMF-2 facility used to gather the cloud data presented in this paper. Also, Dr. Mark Goodberlet and Dr. Andrew Pazmany of ProSensing, Inc., are thanked for their helpful discussions and suggestions related to the analyses presented herein.

APPENDIX A

List of Symbols

 Gain scale factor to account for signal fading

 Chi-square distribution variable

 Cumulative distribution function for chi-square distribution with 2M degrees of freedom

 Cumulative distribution function resulting from the sum of two independent chi-square distributions

 Modified version of to account for the effect of DFT length on sample independence

 Difference in signal-to-noise ratio between noncoherent and spectral-based processing required to achieve the same probability of detection

 Spectral velocity resolution (m s−1)

 False alarm scale factor regulating the threshold level for noncoherent processing

 False alarm scale factor regulating the threshold level for spectral processing at optimal DFT length

 False alarm scale factor regulating the threshold level for spectral processing

 Pulse repetition frequency (Hz)

 Noncoherent (power averaging) processing gain

 Optimal spectral signal processing gain

 Ratio of optimal signal processing gain to noncoherent processing gain

 Spectral-based processing gain

 SNR gain of weighting function

 Weighting function–dependent constant used to determine optimal DFT length

 Weighting function–dependent scale factor relating spectral broadening factor to spectral resolution

 Radar wavelength

 Spreading loss

 Spreading loss at optimal DFT length

 Loss in the SNR gain of weighting function relative to uniform weights

 Total number of samples, that is,

 Number of spectra averaged together to form the final power spectrum estimate

 Effective number of spectral bins in the signal peak lobe

 DFT length

 DFT length after zero padding

 Effective number of independent signal samples

 Number of noise samples when creating separate noise and signal-plus-noise vectors

 Number of signal-plus-noise samples when creating separate noise and signal-plus-noise vectors

 Critical DFT length above which the noncoherent processing gain exceeds spectral-based processing gain

 Optimal DFT length to maximize the signal processing gain

 Expected value of signal plus noise

 Probability of detection

 Probability of detection per DFT bin

 Probability of false alarm

 Probability of false alarm for each bin of the power spectrum

 Probability of false alarm for the entire power spectrum

 Probability of detection for spectral processing

 Probability of detecting either signal or noise for spectral processing

 Probability that a sample from the chi-square distribution is less than threshold

 Power spectrum estimate for the kth spectral bin

 Mean noise power

 Power spectral density of noise

 Expected value of peak signal power in the power spectrum

 Expected value of power samples in the artificial signal-plus-noise vector

 Signal power spectrum model

 General threshold variable

 False alarm threshold for spectral processing

 False alarm threshold for noncoherent processing

 Normalized spectral width of the meteorological target

 Spectral width of the meteorological target (m s−1)

 Spectral broadening due to the time-domain weighting function (m s−1)

s Signal power

M-length vector of signal-plus-noise samples

 Segment of the signal-plus-noise sample vector used in computing DFT

SNR Signal-to-noise power ratio

 Signal-to-noise power ratio in the artificial signal-plus-noise vector

 Signal-to-noise power ratio for noncoherent processing

 Signal-to-noise power ratio for spectral processing

STR Signal-to-threshold power ratio

 Signal-to-threshold power ratio for noncoherent processing

 Signal-to-threshold power ratio for spectral processing

 Detection threshold

 Detection threshold set by the allowable false alarm rate for noncoherent processing.

 Detection threshold set by the allowable false alarm rate for spectral-based processing

is the reference threshold for M = 1.

 Total observation time (s)

υ Velocity (m s−1)

υ0 Mean velocity (m s−1)

υa Unambiguous Doppler velocity

 Time-domain weight vector

APPENDIX B

Description of Model for

The signal in any of the spectral bins in the spectral peak lobe region has a significant probability of being the highest value. Each bin of the power spectrum is statistically independent of its neighbor, with a chi-square distribution of order 2M. Regardless of the underlying probability density function, the expected value of the maximum of samples of a random variable x is of the form
eb1
where is the mean value of x (Watkins 2012). This form assumes that the mean value of x is the same for all samples. While this is not true for the case of a Gaussian spectrum shape, where the central bin has the highest mean value, (B1) was found to correctly model the expected value of the highest peak in the region near the spectral peak relative to the mean value of the spectral peak.

Data for curve fitting was generated by simulating a Gaussian spectrum using the method described in Sirmans and Bumgarner (1975). The parameter in (B1) was found to depend on the number of individual power spectra averaged used to form the power spectrum and on the normalized spectral width . The condition that = 1 for is enforced when most of the spectral power falls in the DFT bin associated with the spectral peak. Note that = 1 when for all of the windowing functions listed in Table 1.

APPENDIX C

Derivation of Spreading Loss

For a discrete power spectrum computed using (5), the spreading loss is equal to the ratio of the energy contained in the central spectral bin to the total energy in the spectrum. Referring to Fig. C1 and assuming a Gaussian spectrum shape, the spreading loss for the discrete frequency spectrum with spectral resolution can be approximated by the error function
ec1
where
eq3
and is the standard deviation of the spectrum. The scale factor in (C1) was determined by setting when the spectral resolution equals the spectral width, that is, .
Fig. C1.
Fig. C1.

Doppler power spectrum (red trace) with spectral width σ showing relative energy of discrete frequency samples (blue rectangles) for spectral samples marked with asterisks.

Citation: Journal of Atmospheric and Oceanic Technology 33, 4; 10.1175/JTECH-D-14-00198.1

The expression for spreading loss (10) has a spectral width that combines spectral broadening due to the windowing function with the underlying spectral width of the meteorological signal . The resultant spectrum represents the convolution of the meteorological signal power spectrum with the power spectrum time-domain weighting function; thus, in (C1) is replaced by
ec2
The expression for spreading loss (10) was compared to simulated spreading loss and was found to be accurate to within one percent when using the nonuniform time-domain window functions listed in Table 1. Use of a uniform window results in a non-Gaussian power spectrum, which introduces a maximum spreading loss overestimation of approximately 0.7 dB near the region of peak processing gain.

APPENDIX D

Derivation of

Consider a fixed frequency discrete time signal,
eq4
where for some specific frequency index . Using this signal in (5) with M = 1, the resultant spectrogram is given by
eq5
The spectral peak occurs at :
eq6
Next, consider a discrete-time complex-valued noise signal,
eq7
where the amplitude Bi is a Rayleigh distributed random variable with mean value B0 and is a uniformly distributed random variable. The spectrogram of the noise is
eq8
Since is uniformly distributed and all samples are uncorrelated, the expected value of is equal to the power sum:
eq9
Setting the input SNR to 1.0 (A = B0), the signal processing gain is given by the output SNR:
eq10

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1

The relationship between and the mathematical standard deviation is .

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