The research that led to the development of the present invention was sponsored by the National Oceanic and Atmospheric Administration's (NOAA's) Earth Systems Research Laboratory (ESRL). NOAA is a part of the U.S. Department of Commerce, a component of the U.S. Federal government. The United States Government has certain rights in this invention.FIELD OF THE INVENTION
The present invention relates to the art of signal processing. In particular, the present invention is directed toward a statistical technique for detection of transient signals in power spectra, particularly for Doppler spectra used in radar and related applications.BACKGROUND OF THE INVENTION
Many remote sensing instruments, (i.e. radars, lidars, sodars), calculate Doppler spectra (power spectra scaled to represent the Doppler shift) as part of their signal processing. Transforming signals from a time-series representation to a power spectrum is very useful since it concentrates the signal energy into a small region of the power spectrum while spreading out the noise energy. Typically, the desired signal appears as a peak in the power spectrum in the presence of noise. If the signal-to-noise ratio (SNR) is large enough, it is easy to detect the signal peak amongst the noise peaks. Usually, the largest peak in the spectrum is chosen as the signal. However, when the SNR is low, it is difficult to select the desired signal since the noise peaks can be larger than the signal. Optimum detectability occurs when the signal exists at least as long as the duration of the sampling time. However, many real-world signals are of shorter duration. These transient signals are difficult to detect in the presence of noise. Under these conditions, the signal energy is contained in many spectral points and difficult to detect.
Most instrument designers will choose the sampling time of an instrument equal to the time the scattering target is in the beam to optimize its detectability. Since the Fourier transform of a continuous signal has narrow bandwidth, its corresponding signal will have a narrow peak in the power spectrum. This narrow peak has the best chance of detection. In addition, the sampled time series is often divided into short blocks whose power spectra are averaged together. This averaging reduces the variance of the noise making signal peaks more detectable. However, often the detection of a transient signal is required. The Fourier transform of a transient signal will have a broad peak in the power spectrum. Therefore, the signal energy will appear in many frequency bins and be difficult to detect. Also, it is not possible to average short power spectra together when trying to detect transient signals, since averaging would reduce the amplitude of a transient peak.
Detecting signals in the presence of noise is a problem when signal amplitude is small compared with the amplitude of the noise signal. Transforming from a time-series representation to a power spectrum is very useful since it concentrates the signal energy into a small region of the power spectrum while spreading out the noise energy. Typically, the amplitude of a signal peak must be larger than one to two standard deviations of the noise peaks to be detected. The Heisenberg-Gabor Uncertainty Principal states that long duration time signals have narrow spectral peaks and short transient signals have broad spectral peaks. This makes detection of long duration signals much easier, since the signal energy will be contained in very few spectral bins. Transient signals with short time duration will spread the signal energy over a larger part of the spectrum, making detection based on the peak exceeding the noise deviations difficult.
This effect is demonstrated in
The present invention comprises a statistical method for detecting small amplitude, short duration signals in the presence of noise. If the noise in the sampled time series has Gaussian voltage statistics for example (usually the case in remote sensing instruments), the resulting power spectrum will have exponential noise statistics. Exponential statistics imply that there is a high probability of large, narrow noise peaks with large amplitudes. Broad spectral width signals that are not taller than the noise peaks are difficult to detect. The present invention selects the peak with the largest area (signal power) then uses a statistical test to determine if the peak is due to a transient signal or noise.
The first step of the processing method breaks the sampled time series of voltages into many short blocks and calculates the power spectrum of each block. It then finds the peak in each spectrum with the largest signal power. A peak is defined as a region where the spectral values are completely above the mean noise level. To find the peak with the largest signal power, the first three moments (signal power, frequency, and spectral width) are calculated for every peak in each spectrum. From all these moments, the peak with the largest signal power is chosen. This effect is demonstrated in
The second step of the processing method uses the low probability of occurrence of wide noise peaks to detect transient signals. This second step is accomplished by comparing the probability density function (PDF) of the measured signal spectral widths to a pre-determined PDF to detect the transient signals. The measured signal PDF is developed by generating a histogram of all the spectral widths calculated from all the largest power peaks from measured power spectra. Enough short spectra have to be processed to generate a histogram of spectral widths. The measured spectral width histogram is then normalized by its area to derive the measured spectral width probability density function (PDF). A similar technique is used to develop a pre-determined PDF. This pre-determined PDF of spectral widths can be calculated from existing theoretical functions, derived from numerical simulations of the system, or measured instrument noise that does not contain any signal.
While there are many ways to look at the difference between measured and pre-determined PDFs, the simplest scheme looks at the difference between the two PDFs. If the measured probability is larger than the pre-determined probability at some width, a signal was detected. In another embodiment, a scheme may be used that assigns a weight to each of the probabilities in the PDF. The measured PDF is then compared with the pre-determined PDF by calculating the difference between them. This difference is multiplied by its weight and these products summed into a score. Finally, each score is compared against an empirically determined output threshold. Scores greater than this threshold are considered detections.BRIEF DESCRIPTION OF THE DRAWINGS
A dashed, horizontal line shows the mean noise level.
DETAILED DESCRIPTION OF THE INVENTION
No theoretical PDFs for the moments of noise peaks or measured instrument noise exist for the simulated radar data; therefore, a model has been created that generates simulated radar Doppler spectra containing pure noise. This model was run about 6000 times, and moments from each run were used to generate a pre-determined PDF.
In Step 150, a histogram of the largest signal power peaks is generated. In step 160, this histogram is normalized to produce a pre-determined PDF. Histograms of all three moments (signal power, velocity and spectral width) were then plotted are shown in
Once the pre-determined PDF value has been calculated, the process of
In step 270, the difference between the measured PDF from step 260 is compared with the pre-determined PDF from step 150. There are many ways to look at the difference between measured and pre-determined PDFs. One technique is a weighting scheme that allows assigning a weight to each of the bins in the PDF. The measured PDF is then compared with the pre-determined PDF by calculating the difference between them and multiplying the differences by the appropriate weight. The pre-determined PDF values may be stored in memory. As previously noted, the pre-determined PDF value may be measured once for a given data stream and stored and used repeatedly, thus eliminating the need to continually recalculate the pre-determined PDF value.
In step 280 the weighted differences between the measured and pre-determined PDF are summed and compared with a pre-determined threshold value. In this example, the comparison starts at a spectral width of four bins since is it expected that the transient signal to be wider than four bins. If the measured PDF value is larger than the pre-determined PDF value, the difference is weighted by multiplying the difference by its associated weight. The sum of these weighted differences is recorded as the score for each group of data. Finally, each score is compared against an empirically determined output threshold. Scores greater than this threshold are considered detections.
This threshold value may be determined empirically and adjusted accordingly to provide the proper level of signal detection. If the sum of the weighted differences exceeds the threshold value, a signal is detected as indicated in step 295. If the sum of the weighted differences is less than the threshold value, no signal is detected, as indicated in sep 290.
In adder 755, the difference between the measured PDF from block 750 is compared with a pre-determined PDF stored in memory 760. Predetermined PDF 760 may be calculated using the techniques of
In block 775, the weighted differences between the measured and pre-determined PDF are summed and compared with a pre-determined threshold value in block 780. In this example, the comparison starts at a spectral width of four bins since is it expected that the transient signal to be wider than four bins. If the measured PDF value is larger than the pre-determined PDF value, the difference is weighted by multiplying the difference by its associated weight. The sum of these weighted differences is recorded as the score for each group of data. Finally, each score is compared against an empirically determined output threshold. Scores greater than this threshold are considered detections. This threshold value may be determined empirically and adjusted accordingly to provide the proper level of signal detection. Detection reports 785 may be output indicating whether a signal is present based upon this threshold comparison.
While the preferred embodiment and various alternative embodiments of the invention have been disclosed and described in detail herein, it may be apparent to those skilled in the art that various changes in form and detail may be made therein without departing from the spirit and scope thereof.