## 1. Introduction

Detection of meteorological radar signals in the presence of noise has been studied extensively (cf. Keeler and Passarelli 1990; Doviak and ZrniÄ‡ 1993). In the context of noncoherent power averaging, as is used by the WSR-88D for reflectivity estimation (Doviak and ZrniÄ‡ 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 (KerÃ¤nen and Chandrasekar 2014; IviÄ‡ 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 integrationâ€”through DFT processingâ€”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 ^{âˆ’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 ZrniÄ‡ 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

*T*determines if a particular power average is declared a detection; that is,where

*M*samples of the signal-plus-noise vector

**s**and

*M*is the total number of samples and

*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

*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

_{n}is given aswhere 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

*M*as follows. The

*M*samples are divided into

*N.*Each segment is processed by a DFT of length

*N*(Doviak and ZrniÄ‡ 1993) and then power averaged to form the power spectrum estimate

*p*th

*N*-length segment of

**s**; and

*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.

*N*> 2 when using a Hanning window. Loss factor

Coherent integration loss factor

*N*for small

*N*but asymptotically approaches a saturation level when the signalâ€™s spectral peak lobe becomes distributed over multiple DFT bins for finite bandwidth signals. To account for this saturation effect, a spreading loss factor

^{1}(m s

^{âˆ’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:where erf is the error function,

^{âˆ’1}).

*N*in Fig. 2. For small

*N*,

*M*= 1.

The correction factor

*N*and small

*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 bywhere

Figure 5 plots the approximation of optimal DFT length

In Fig. 6 *M* as a parameter. For a given normalized spectral width, *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.

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, ^{âˆ’1} and 1.0 m s^{âˆ’1} with radar frequency as a parameter. Clouds above the turbulent boundary layer can exhibit spectral widths on the order of ^{âˆ’1}, for which ^{âˆ’1}. In this case, shorter wavelength radars show little sensitivity improvement from spectral processing.

*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

*M*noise power samples of a bivariate Gaussian voltage distribution representing the noise-signal envelope has a chi-square distribution with 2

*M*degrees of freedom,

*M*degrees of freedom is denoted by

*D*[Abramowitz and Stegun 1964, their (26.4.1); cf. Wolfram 2012]:which gives the probability

*M*power samples is less than a threshold power

*M*. The inverse function is written aswhich gives the threshold power as a function of

### b. Derivation of false alarm factor for spectral processing

### 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 *N* with *N*. 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).

## 6. Probability of detection analysis

### a. Probability of detection for noncoherent processing

*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.

*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

*M*samples:The mean power of each of the signal-plus-noise samples is given byIn 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 (

*M*). The cumulative distribution function

*M*were too large to compute for large

*M*, thus

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).

### b. Probability of detection for spectral processing

*k*(prior to spectral averaging) are independent for

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 _{s} is maximized. It was found that _{n} = STR_{s} when

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, *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 Â±0.4 dB for

## 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 programâ€™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 1.4Â°.

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

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.

The signal-to-threshold ratios STR_{n} for the noncoherently processed signal and STR_{s} for the spectrally processed data with DFT length = 32 and 2048 are approximately equal when *N* = 32, âˆ’22 dB for *N* = 2048, and âˆ’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.

Relative signal processing gain *M* for

## 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

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

*2M* degrees of freedom

^{âˆ’1})

*k*th spectral bin

^{âˆ’1})

^{âˆ’1})

sâ€ƒSignal power

*M*-length vector of signal-plus-noise samples

SNRâ€ƒSignal-to-noise power ratio

STRâ€ƒSignal-to-threshold power ratio

*M* = 1.

Ï…â€ƒVelocity (m s^{âˆ’1})

Ï…_{0}â€ƒMean velocity (m s^{âˆ’1})

Ï…_{a}â€ƒUnambiguous Doppler velocity

# APPENDIX B

## Description of Model for

*M*. Regardless of the underlying probability density function, the expected value of the maximum of

*x*is of the formwhere

*x*(Watkins 2012). This form assumes that the mean value of

*x*is the same for all

Data for curve fitting was generated by simulating a Gaussian spectrum using the method described in Sirmans and Bumgarner (1975). The parameter

# APPENDIX C

## Derivation of Spreading Loss

# APPENDIX D

## Derivation of

*M*= 1, the resultant spectrogram is given byThe spectral peak occurs at

*B*

_{i}is a Rayleigh distributed random variable with mean value

*B*

_{0}and

*A = B*

_{0}), the signal processing gain is given by the output SNR:

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^{1}

The relationship between