A time-frequency method for detecting VHF underdense meteor signals



[1] Underdense meteor echoes observed using very high frequency (VHF) radar can be accurately modeled as a single complex damped sinusoid in additive white Gaussian noise. The normalized damping coefficient is expected to be between 0 and 0.3 for a VHF meteor return based on the modeled ambipolar diffusion rates near 90 km and a radar operating between 30 and 50 MHz. Current meteor echo detection routines operate either in the frequency domain, where it is difficult to detect highly damped signals, or in the time domain, where it is difficult to detect narrow band signals. An added difficulty is that typical approaches require a priori knowledge of the noise variance which can bias the performance of these estimators. In this paper, a time-frequency waveform detector is proposed to address these problems. By normalizing the signal power and power spectral density in the time and frequency domains, respectively, a detector that is invariant to the noise variance can be implemented. The characteristics of a damped sinusoid in the time and frequency domains are exploited to construct the detector. The threshold for the proposed detector is only a function of of the time series length, and is stable with respect to a range of damping coefficients and noise levels. The time-frequency waveform detector exhibits superior performance to the conventional energy or power detectors when the signal of interest is highly damped or the noise variance is unknown. A derivation of the time-frequency waveform detector, comparison with the energy and power detectors and numerical results demonstrating the effectiveness of this detector are presented.

1. Introduction

[2] Every day millions of small particles enter the Earth's atmosphere from space. These small particles or meteoroids are traveling at speeds ranging from 10 to 70 km/s and ablate near 100 km. This ablation process is a result of the frictional heating that occurs when these meteors impact the increasingly dense atmosphere. During the ablation process, the meteoroid collides with the neutral gas molecules creating a long plasma trail in the wake of the meteoroid. This plasma trail is orders of magnitude more dense than the background ionosphere, and will typically persist for less than half of a second. During the short time while this trail exists, it can be probed using a very high frequency (VHF) radar. Atmospheric parameters, such as the vector wind field, temperature and density, affect the characteristics of the returned radar signal. The Doppler frequency of the returned signal is related to the radial wind field, and the damping coefficient is related to the temperature and pressure fields. Additional information about meteor radar systems can be found in [Avery et al., 1990; Palo, 1994; Valentic, 1996; Hocking et al., 1997; Hocking, 1999; Cervera et al., 1997, and references therein].

[3] While there are some differences in the specific implementations of meteor radar systems the basic components of the signal processing systems are common. All of these systems include a real-time detection algorithm, an offline discrimination algorithm and a final parameter estimation algorithm. In many systems the real-time detection algorithm is a basic energy detector, where the signal-to-noise ratio (SNR) is computed and compared with a predefined threshold. If the signal exceeds this threshold then this echo is saved for further processing. The discriminator is an intermediate step between online detection and offline parameter estimation that typically occurs offline. The goal of the discriminator is to discard as many signals as possible that don't appear as classic underdense meteors before the parameter estimation process. In some cases this discriminator is a second more sophisticated detector, such as the one described in this paper, or a set of heuristics to separate signals of interest from clutter and interference. The final step in this process is to estimate the parameters of interest (eg. frequency and damping) from the detected signal. This step may also have a discrimination component whereby signals with parameters that are outside a predefined range are rejected as not being from underdense meteor echoes.

[4] The discriminator plays an important role in meteor radar systems as it is used to detect underdense meteor echoes while discriminating against clutter and interference derived from other signals, such as overdense meteor echoes, noise spikes, ionospheric E-region echoes, aircraft and lightning interference [e.g., Holdsworth et al., 2004], which can account for more than 50% of the total signals detected [Valentic, 1996] with a simple energy thresholding algorithm. After passing the discrimination stage, the meteor echo can be saved for later offline processing to extract the direction of arrival (DOA) [Jones et al., 1998], range, decay time [Hocking et al., 1997; Hocking, 1999; Dyrud et al., 2001], and the Doppler frequency [Cervera et al., 1997]. The overall meteor radar system performance is directly dependent upon the successful identification of underdense meteor echoes in the background noise and clutter environment.

[5] While the role of the detector and discriminator play an important function in meteor radar systems, very few details can be found in the literature about how these algorithms are implemented. In fact, it is often the case that the detector and discriminator are developed using an ad-hoc, heuristic trial and error approach. One case where details about the detector and discriminator can be found is in Hocking [2001]. The performance of the detector and discriminator is flexible and can be adjusted providing a range of performance from “strict” to “loose”. However, the detector performance is not evaluated in terms of measurable metrics, such as the probability of false alarm (Pfa) and the probability of detection (Pd). As a result, the impact of adjusting the detector and discriminator performance cannot be objectively assessed.

[6] In this paper a time-frequency detection and discrimination method is proposed. This method could be implemented either as the real-time detection algorithm, or as the off-line discriminator following a simple energy detector with a very low detection threshold. The result of the simple detector with a low-threshold is to provide a very high probability of detection but at a cost of a high probability of false alarm. The off-line discrimination will significantly decrease the probability of false alarm without significantly impacting the detection performance. Given current data storage capabilities, such an approach could be implemented either on-line or off-line.

[7] Underdense meteor echoes can be modeled as a single complex damped sinusoid in complex additive white Gaussian noise [McKinley, 1961]. The problem of detecting sinusoidal signals in additive Gaussian noise has historically been approached either from the time or the frequency domain. We will first examine these methods to provide a context for a joint time-frequency approach.

[8] An energy detector compares the energy of the received echo to the background noise level using a ratiometric approach. One advantage of the energy detector is that a detailed signal model is not required, but knowledge of the noise variance is required. As a result the performance of the noise variance estimation procedure will directly impact the detector performance.

[9] The power detector, also known as Fisher's g-test, is implemented by computing a ratio between the peak power in the power density spectrum and the total power in the signal. This ratio is independent of the noise variance and as a result the detector performance does not rely on estimating the noise variance. This technique has been extended to the multiple sinusoid case by Siegel [1980], but because the probability of observing two meteors simultaneously is quite small, this approach is not considered herein. Although the power detector is invariant to knowledge of the noise variance, it performs poorly when highly damped sinusoidal signals are present, and results in missed detection of such signals [Percival and Walden, 1993]. Therefore, it is important to understand the limitations and biases of the detection and discrimination methods that are used so that improved methods can be developed and any resulting scientific analysis can account for these known biases.

[10] A new time-frequency waveform detector is presented in this paper to detect damped sinusoidal signals in white Gaussian noise with an unknown variance. Rather than looking at the power spectrum or the signal energy, a test is developed that utilizes information both in the time and frequency domains. This is accomplished by interpreting the power spectrum and signal power as probability density functions. The statistical properties of these probability density functions in the time and frequency domains form the basic components of this detector. Monte-Carlo simulations show this detector works well in the presence of noise, where the noise level is unknown and in the presence of highly damped sinusoidal signals, where other detectors fail.

[11] In the next section, a signal model for an underdense meteor echo is developed. By using known diffusion and meteor height statistics, a range of damping coefficients are determined that can be used in evaluating the detector performance. In section 3, the time-frequency waveform detector is derived, and Monte-Carlo simulations are used to determine the performance of the detector under a range of expected conditions. In section 4, the energy and power detectors are explored and compared with the proposed time-frequency waveform detector. Monte-Carlo simulations are also used to estimate the performance of the time-frequency waveform detector.

2. Signal Model of the Echo From Meteor Scattering Radar

[12] The radar equation representing the power received from an underdense meteor trail in a backscatter configuration is [Sugar, 1965]

equation image



  • PR and PT are the transmitted and received power (W),
  • GT and GR are the power gains of the transmitting and receiving antennas,
  • λ is radar operating wavelength (m),
  • R is the distance between the transmitter and the meteor trail (m),
  • re = 2.8178e−15m is the classical radius of an electron,
  • q is the electron line density of the meteor trail (em−1),
  • r0 is the initial radius of the trail (m), and
  • D is the ambipolar diffusion coefficient m2s−1 [Kaiser, 1954].

[13] An empirical relationship between the ambipolar diffusion coefficient D and altitude in the region between H = 80 km and H = 110 km is [McKinley, 1961]

equation image

[14] Equation 1 shows that the received signal power, in the absence of noise, is exponentially decaying with time, and this expression can be simplified to

equation image

where A2 is the maximum received power, measured at t = 0, and α = 16π2D/λ2 is the decay rate.

[15] In the presence of additive noise the received signal can be expressed as y(t) = sR(t) + n(t), where sR(t) is the received signal with PR(t) = ∣sR(t)∣2 and n(t) is the added noise. The received signal can be Doppler shifted due the the radial motion of the meteor trail which gives rise to the following baseband signal model

equation image

where ϕ is the phase of the received signal and fd = 2Vr/λ is the Doppler shift induced by the radial motion Vr of the trail relative to the radar.

[16] Nearly all atmospheric meteor radar systems operate as pulsed radars which gives rise to a discretization of the received signal. Since the signal power decay rates and Doppler frequencies are small relative to a typical radar pulse width (10–100 μsec) the received signal is only sampled once every pulse repetition period (TIPP). Letting t = n · TIPP results in the following discretized received signal model

equation image
equation image

where N is the number of radar samples collected for the echo while αN = 16π2D · TIPP/λ2 and fN = fd · TIPP are the sampling normalized decay rate and frequency.

[17] The received noise, w(n), is parameterized as a complex white Gaussian noise process,

equation image

where E[·] means the statistical expectation and the power SNR of the echo is defined as

equation image

[18] We have plotted the altitude distribution of meteor echoes based on that typically obtained for an all sky meteor radar at 46.3 MHz in Figure 1. This vertical profile is approached using a Gaussian curve with mean altitude μ = 93 km and variance σ2 = 25 km [Janches et al., 2004; Nakamura et al., 1997]. Estimates for the normalized diffusion value using TIPP = 3.3 ms are also shown in Figure 1. It is clear that within 2σ above and below the average expected altitude (83–103 km), where more than 95% of meteors are located, the normalized damping coefficient is 0.3 or less at a frequency of 50 MHz (λ = 6 m), which corresponds to a diffusion coefficient of 20 m2s−1. This value will decrease for longer operating wavelengths. Because the detector performance is dependent upon the damping rate, more highly damped signals (larger αN) are more difficult to detect, the analysis presented herein will be limited to a normalized damping coefficient between 0 and 0.3.

Figure 1.

Normalized damping coefficient as a function of altitude (80–110 km) and frequency (30–50 MHz).

[19] Figure 2 shows how the damping rate affects the structure of the received meteor echo signal in both in the time and frequency domains. A narrowband sinusoidal signal with a small damping coefficient, will be broad in time domain while having a sharp peak in the frequency domain. Conversely a highly damped sinusoidal signal is localized in the time domain but broad in the frequency domain. As a result simply picking the highest power or highest energy signal will result in a failure to detect either highly damped signals in the frequency domain or narrow band signals in the time domain. To address this issue, a time-frequency waveform detector is proposed that makes use of information in both the time and frequency domain.

Figure 2.

Time (top) and frequency (bottom) domain representations of two damped sinusoids with normalized frequency fN = 0.1 and normalized damping cooefficients αN = 0.1 (dashed) and αN = 0.3 (solid). Also shown is white Gaussian noise (dotted).

3. Time-Frequency Waveform Detector

[20] In this section a time-frequency waveform detector is described that can be used to detect a single damped sinusoidal signal in an unknown additive white Gaussian noise background. This time-frequency waveform detector has better performance than either a simple power or energy detector.

[21] In the absence of a signal the received waveform possesses a Gaussian distribution in the time domain and a chi-squared distribution in the frequency domain. Additionally the statistical distribution of the signal energy in the time domain also possesses a chi-square distribution. The mean and variance of these distributions can be utilized in the time and frequency domains to develop an effective detector. The derivation of this time-frequency waveform detector is outlined below.

[22] The power of an echo can be written as

equation image

and the power spectral density of the echo is

equation image

where Y(k) is the Fourier transform of the echo at the normalized frequency k/N. Notice that Equation (9) and (10) have exactly the same form, therefore a generalized expression for these terms can be written as

equation image

where z(m) is either y(n) or Y(k). Next normalize (11) by the sum over all N points.

equation image

Assume purely for mathematical convenience, that the length of the realization N is odd (i.e. we can write N = 2L + 1 for some positive integer L). The coordinate m = 0,⋯, N − 1 can be written as a series of symmetric samples m = −L, ⋯, L. Accordingly P(m) can be rewritten as

equation image

[23] Notice that P(m) is invariant to scaling of z(m). In this case, scaling ∣z(m)∣2 by the noise variance σ2, as is done for Fisher's g-test, results in no change for P(m). From the definition of P(m) in (13), it can be shown that equation imageP(m) = 1, and therefore P(m) can be interpreted as a probability density function. The mean and variance of m can be written as

equation image
equation image

where μm and δm2 are the basic statistical components that are used to form the hypothesis test described herein.

[24] Under the null hypothesis (H0) when only noise is present, P(m) can be approximately modeled as a uniform distribution. This is because for white Gaussian noise w(n), no specific point m is more probable than any others and hence m is a uniformly distributed random variable. If m is further normalized by dividing by the sample number (N), the mean and variance of m can be expressed as

equation image
equation image

[25] However, the analytical distribution functions for μm and δm2 are difficult to obtain therefore a Monte-Carlo simulation has been utilized. Figures 3 and 4show the Monte-Carlo computer simulation for μm and δm2, where 105 independent white Gaussian noise values were used to generate the distributions for μm and δm2. The distributions are shown for time series of length 11, 51 and 101 points. The distribution for μm is zero mean and symmetric regardless of the time series length, however the width of the distribution does decrease with increasing time series length. Comparatively the distribution for δm2 is nonzero mean and asymmetric for short time series while becoming increasingly narrower and symmetric for larger values of N.

Figure 3.

The probability distribution of μm for different length time series (N = 11, 51, 101) determined using Monte-Carlo simulations.

Figure 4.

The probability distribution of δm2 for different length time series (N = 11, 51 and 101) determined using Monte-Carlo simulations.

[26] Substituting (14) and (15) back into the time and frequency domain, three statistics μk, δk2 and μn are obtained which when combined form a new time-frequency waveform detector which can be expressed as

equation image

where the values a, b and c are related to the detection threshold, and mδ is the mean value of δk2 for a specific value of N. The statistics μk and δk2 are the frequency domain statistics and are used to detect narrow-band signals, while μn is a time-domain statistic and is used to detect highly damped signals.

[27] Under the alternative hypothesis (H1) that a signal plus noise is present, Y(k) has a larger variance (δk2) than under the null hypothesis (H0). Therefore the threshold b can be set to detect a narrow-band signal in noise. However for high Doppler frequencies the effectiveness of only using δk2 is decreased due to the skewness of the power spectrum. To compensate for this effect μk is included in the detector formulation. A detector using only these two statistics will effectively detect narrow-band signals but will miss highly damped signals which are broadband in the frequency domain. To accommodate this problem a third time-domain statistic μn is included in the detector formulation. The combination of these three statistics and the thresholds a, b, and c form an effective time-frequency detector that can be implemented to detect underdense meteor signals in the presence of additive white Gaussian noise. This three-dimensional detector can be seen as a prolate spheroid where values falling outside the prolate spheroid are characterized as detections and the alternative hypothesis is in effect, otherwise the null hypothesis is in effect.

[28] Notice that under the null hypothesis (H0), μk and μn have exactly the same probability distribution. As a result the threshold set for μk can also be applied to μn in the same way, i.e., c can be replaced by a, hence this three-dimensional prolate spheroidal detector can be reduced to a two-dimensional ellipsoidal detector expressed as

equation image

[29] Notice that equation image ≥ 0, therefore a half ellipse is sufficient to represent the threshold curve. Under the null hypothesis (H0) the semi-major axes a and b are defined as follows

equation image
equation image

where equation image is the variance of μn,k, equation image is the variance of equation image, while q is a scale factor determined from Monte-Carlo simulations.

[30] Figure 5 shows the threshold curves for different false alarm probabilities (Pfa). A false alarm occurs when the null hypothesis (noise only) is in effect but the alternative hypothesis (signal present) is chosen because the detection statistic exceeds the detection threshold. The results in Figure 5 are computed for 1000 time series of length N = 15 which only contain noise. Points falling outside of the 1% threshold (11 of 1000) are plotted as circles, between the 1% and 2% threshold as asterisks (9 of 1000) and between the 2% and 5% thresholds as crosses (31 of 1000). The detector threshold can be adjusted using the a and b parameters to provide a desired false alarm rate.

Figure 5.

Threshold curves determined for different false alarm probabilities (1%, 2%, 5%) with N = 51.

[31] Both the distribution of signals under the null hypothesis and the detection threshold will change as a function of the time series length because the relevant detection statistics are a function of the time series length which was shown in Figures 3 and 4. Figure 6 shows the threshold curves for different length time series with a fixed false alarm probability (Pfa = 1%). As expected, this figure shows that the detection threshold becomes smaller as the time series length increases because equation image and equation image decrease with increasing N.

Figure 6.

Threshold curves determined for different time series lengths (N = 21, 51, 101) with Pfa = 1%.

[32] Figure 7 shows an example of the detector performance. Four experiments were conducted to elucidate the performance of the detector. Three of the experiments contain sinusoids plus noise and the fourth is noise only. For the three cases where a signal is present the signal-to-noise ratio was fixed at 3dB while the frequencies and damping coefficients were varied. In all of the examples the time series length was fixed to N = 51 points, the false alarm probability was set to 1% and 1000 realizations were utilized. It can be seen from Figure 7 that the detector can easily discriminate between noise and signals that are strongly damped (αN = 0.3) at moderate frequencies (fN = 0.2), which lay far from the detection threshold and are denoted by crosses. The undamped signals (αN = 0) for a range of frequencies lay closer to the detection threshold and hence exhibit a higher probability of missed detection than do the highly damped signals. Also indicated in the figure are the detections from noise which lay outside of the detection threshold and are classified as false alarms. The overall performance of the detector will improve with fewer missed detections for stronger echoes (larger signal-to-noise ratio) because the separation between the noise and the signal plus noise distributions increases.

Figure 7.

Performance of the time-waveform detector for four cases: Noise only (dots), fN = 0.2, αN = 0.3 crosses, fN = 0.2, αN = 0 circles, and fN = 0, αN = 0 asterisks. In all cases where a signal is present the SNR is 3dB. The detection threshold was chosen for a Pfa = 1% while the time series are all 51 points long.

[33] To provide some idea about how the time-frequency waveform detector performs we have included results from three selected meteor echoes collected using the South Pole meteor radar system [Lau, 2005] (E. M. Lau et al., Comparison of collocated noninterferometric and interferometric VHF meteor radar systems at the South Pole, submitted to Radio Science, 2006). These results are shown in Figure 8. The South Pole meteor radar uses an energy detector with a very low threshold as the real-time detector where the noise level is computed continuously when a meteor echo is not present. If the SNR of an echo exceeds a predefined threshold, the data is saved starting from the time where the signal exceed the threshold until the time where the power decays to 3dB above the noise floor. Using a low threshold this initial detector misses few significant meteor signals but at the cost of many false alarms. The time-frequency waveform detector runs as an off-line detector or discriminator with the goal of identifying and discarding as many of the false detections as possible while retaining the meteor-like signals for further post-processing. In Figure 8 a noise like (top row), highly damped sinusoid (middle row) and slightly damped sinusoid (bottom row) were selected from the data. The middle column shows the meteor echo time series (both the inphase and quadrature components), the right column shows the frequency content of the echo while the left column shows the detection results. It can be seen that the noise like signal was successfully identified and discarded by the time-frequency waveform detector while both the highly damped and slightly damped sinusoids were retained, which demonstrates the effectiveness of the time-frequency waveform detector.

Figure 8.

Detection of meteor echoes from the South Pole meteor radar system. The location of the meteor echo relative to the detection ellipse is shown in the left column while the time domain signal both inphase (solid) and quadrature signals (dotted) are shown in the center column and the frequency domain signal in the right column. The top row shows a ‘noise like’ signal, while the middle row shows a highly damped signal and the bottom row shows a slightly damped signal.

4. Performance of the Time-Frequency Waveform Detector

[34] The energy and power detectors are two widely used approaches for detecting sinusoids in additive white Gaussian noise. The energy detector operates in the time domain while the power detector operates in the frequency domain. In this section, the performance of the time-frequency waveform detector is compared to both the energy and power detectors. The performance of these detectors, measured as the probability of detection and probability of false alarm, will be examined under a range of signal characteristics such as time series length and signal-to-noise ratio.

[35] The energy detector selects the alternative hypothesis (H1), that a signal is present, if

equation image

where γ is the detection threshold. Provided a probability of false alarm is specified a priori, the detection threshold can be computed analytically as [Kay, 1993]

equation image

where δ2 is the variance of the background complex Gaussian noise distribution. The probability of detection then follows as

equation image

where η = equation images(n)∣2/δ2, and Q is the error function for computing the upper-tail probability of the Gaussian distribution.

[36] The fundamental disadvantage of this test is that one must know the background noise level δ2 to be able to calculate the threshold effectively. However, δ2 is not typically known a priori and must be also estimated. Any error in estimating δ2 will cause a bias in determining the threshold thereby degrading the performance of the energy detector because the measured false alarm probability (Pfa) will not accurately represent the true Pfa. Figure 9 shows how the true Pfa will deviate from the measured Pfa as a function of estimation error in δ2. In the absence of any estimation errors the measured Pfa is equivalent to the true Pfa. However, when the noise level is under- or over- estimated, the measured Pfa deviates from the true value, indicating the performance of the detector is degraded. For a true Pfa below 10% the measured Pfa can have errors in excess of 10x for a 20% estimation error in δ2.

Figure 9.

Sensitivity of the false alarm probability to errors in estimating the noise variance. Results are shown for 1%, 10%, and 20% errors.

[37] Because the energy detector is commonly used by the meteor radar community we have also investigated methods for estimating δ2. The MEDAC/SC system uses a 5-point sliding window method for estimating the noise level [Valentic et al., 1996], and SKiYMet systems use the pre-t0 data to estimate the noise level [Hocking, 2001]. In both cases a limited number of time domain measurements are used to estimate δ2.

[38] Herein we propose a method that utilizes all of the available data. First the time series is transformed into the frequency domain, the average power is computed (Pavg(0)) and the frequency of maximum power is determined. Next M data points around the frequency of maximum power are removed from the spectrum and then the average power is again computed (Pavg(1)). If (Pavg(0) − Pavg(1))/Pavg(0) < 0.1 then δ2 = Pavg(1) otherwise this process continues recursively. The idea here is that any narrow-band signals are eliminated from the frequency domain and the remaining data are used to compute the noise level. By utilizing as many observations as possible the precision of the noise estimate is improved. The value of M is adjustable and will depend on the expected damping rates, however a value of 3 is a reasonable choice.

[39] Figure 10 shows the relative errors in estimating the noise level for white Gaussian noise using our proposed maximum data set method and the 5-point sliding window method. The results for the pre-t0 method have not be included because the number of points used in this approach is variable, but always less that the proposed maximum data set method. Therefore the results for the pre-t0 method are expected to fall between the two curves shown in Figure 10. It is clear from this figure that the maximum data set method has a much better accuracy and precision than the sliding window particularly for longer time series. However both methods tend to under-estimate the noise level. This error is in excess of 10% for time series less than 30 points.

Figure 10.

Comparison of the percentage error in the estimated noise level between 5-point sliding window method and the maximum data set method.

[40] Figures 11 and 12 both show examples of echoes that were rejected by the time-frequency waveform detector but accepted by the energy detector. In both cases the detector thresholds were computed using Pfa = 1%. Because the noise floor was underestimated the energy detector incorrectly identified these echoes as signals rather than noise. As a result of this underestimation, the energy detector will have far more false alarms than the design value of 1%, a problem that is not encountered by the time-frequency waveform detector. However the time-frequency waveform detector is more computationally intensive than the energy detector. Therefore one prudent approach is to implement a real-time energy detector with a high Pfa followed by an off-line time-frequency waveform based discriminator to eliminate many of the false detections incurred by the energy detector.

Figure 11.

Characteristics of a noise-like echo are shown. This echo was rejected by the time-frequency waveform detector but accepted by the energy detector due to an underestimation of the noise variance. The detector threshold was set for a Pfa 1%.

Figure 12.

Characteristics of a second noise-like echo are shown. This echo was rejected by the time-frequency waveform detector but accepted by the energy detector due to an underestimation of the noise variance. The detector threshold was set for a Pfa = 1%.

[41] A second approach, to avoid the energy detector sensitivity to estimating the background noise level, is to use a power detector, also known as Fisher's g-test. The power detector selects the alternative hypothesis (Ha) if

equation image

where Y(p)(k) is the periodogram of y(n).

[42] The value g can be interpreted as the ratio of the peak power to the total power in the power spectrum. Note that δ2 acts as a proportionality constant in the distribution of both max(Y(p)(k)) and ΣY(p)(k), so that the power detector is invariant to δ2. Fisher showed that the exact distribution of g under the null hypothesis is given by [Percival and Walden, 1993]

equation image

where M is the largest integer satisfying both M < 1/g0 and ML. By using only the first term of the summation, the threshold g0 can be derived

equation image

where α is the significance value. This detector is very effective for detecting narrowband signals which have low damping but fails to detect highly damped signals. The reason for this failure is that highly damped signals are broadband in nature and the spreading of power across the frequency spectrum significantly reduces the effectiveness of the power detector.

[43] Figure 13 compares the performance of the power detector and the time-frequency waveform detector for normalized damping coefficients ranging from αN = 0.0 to αN = 0, 3. The performance metric is the probability of detection and it is evaluated as a function of SNR. From this figure it is clear that the power and frequency waveform detectors exhibit similar performance for a normalized damping coefficient of 0.1. However for normalized damping coefficients in excess of 0.1 the time-frequency waveform detector is superior and for a normalized damping coefficient of 0.3 the power detector does not exceed a probability of detection of 10% for any SNR. Referring back to Figure 1, it is expected that most meteor returns above 90 km, for a radar operating at 50 MHz, will have a normalized damping coefficient larger than 0.1. Using the power detector would cause many of these signals to go undetected.

Figure 13.

Detection probability as a function of SNR for the power and time-frequency waveform detectors in the presence of a damped sinusoid with a fixed probability of false alarm (Pfa = 5%).

[44] For normalized damping coefficients below 0.1 the power detector shows improved performance over the time-frequency waveform detector for a given SNR. However the probability of detection is nearly 100% for the time-frequency waveform detector and the power detector (for αN ≤ 0.1). Since the performance of the parameter estimation algorithms that are used to estimate the physical parameters from the meteor echoes, such as wind and temperature, degrade significantly for weak signals, any signals with an estimated SNR below 5dB are discarded. Hence the improved performance of the power detector for low SNR is of no benefit. Conversely the inability of the power detector to detect moderately and strongly damped signals will provide a bias towards less damped signals which could bias the atmospheric temperatures that are derived from the detected meteor echoes.

[45] Figures 14 and 15 show examples of meteor echoes collected from the South Pole meteor radar that have a normalized damping coefficient larger than 0.1. A rough estimate of the normalized damping coefficient has been determined by performing a linear least squares fit to the echo power. The resulting normalized damping coefficients are 0.173 and 0.297 respectively. In both cases the time-frequency waveform detector correctly detected these signals while the power detector rejected them, resulting in a missed detection. Because the time-frequency waveform detector utilizes information in both the time and frequency domains it has the ability to detect moderately and highly damped signals that would go undetected by the power detector.

Figure 14.

Characteristics of moderately damped echo (αN = 0.173). This echo was accepted by the time-frequency waveform detector but rejected by the power detector due to a broadening of the signal in the frequency domain.

Figure 15.

Characteristics of a highly damped echo (αN = 0.297). This echo was accepted by the time-frequency waveform detector but rejected by the power detector due to a broadening of the signal in the frequency domain.

5. Conclusion

[46] We have presented a novel time-frequency waveform detector that can be used to detect damped sinusoidal signals in the presence of additive white Gaussian noise. This time-frequency approach shows improved performance over classic power and energy detectors that are implemented in either the time or frequency domains. A fundamental drawback of the time domain energy detector is that this detector is not invariant to the noise power. As has been shown, the background noise is typically underestimated leading to an increase in the false alarm rate. Comparatively the frequency domain power detector is invariant to the the background noise power, but it becomes increasing ineffective in detecting meteors with a normalized damping coefficient in excess of 0.1 and totally fails to detect echoes with a normalized damping coefficient larger than 0.3. Since many of the meteor echoes occurring above 90 km will have a normalized damping coefficient larger than 0.1 for a VHF radar, the power detector will not detect these highly damped signals regardless of the signal to noise ratio. The result will be a bias toward signals with lower damping and hence diffusion rates, which could adversely effect the estimation of atmospheric temperature from these diffusion rates. The proposed time-frequency waveform detector is invariant to the background noise level, therefore it does not suffer the problems associated with the energy detector. Additionally, it has been shown that the time-frequency waveform detector outperforms the power detector for normalized damping coefficients greater than 0.1. Furthermore the time-frequency waveform detector achieves the same level of performance as the power detector for normalized damping coefficients below 0.1 when the signal to noise ratio exceeds 5dB. Since the post-detection parameter estimation algorithms cannot reliably estimate the physical parameters of the meteor echo for signal-to-noise levels below 5dB, the ability to detect these weak signals does not provide an improvement to the system performance.


[47] This material is based upon work supported by the National Science Foundation under grants ATM-0336946, OPP-9981903, and OPP-0538672. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. Useful discussions with Wei Dai are also acknowledged.