Frequency domain non-linear signal processing apparatus and method for discrimination against non-Gaussian interference

Apparatus and method for discriminating against non-Gaussian noise. Analog signals from an array of sensors are converted to real and imaginary digital representations and processed such that non-Gaussian noise is separated from signals of interest. The processor uses estimates of Kurtosis and quantiles from either past or adjacent frequency components to construct non-linear elements, which are then used to process remaining signal data to improve the signal-to-noise ratio thereof by removing non-Gaussian noise therefrom.

BACKGROUND OF THE INVENTION 
(1) Field of the Invention 
The present invention relates to signal processing and more particularly to 
an apparatus and method for discriminating against non-Gaussian noise 
contamination of acoustic signals or the like thereby improving detection 
and estimation thereof. 
(2) Description of the Prior Art 
Signal processors generally must separate the signal from the broadband 
noise in which it is additively embedded. Traditional and currently used 
methods for detection and estimation of such noise contaminated signals 
either assume that the underlying noise environment is Gaussian or else 
the noise is not considered in the technique. For example, the estimated 
noise spectrum may be used to detect narrow-band signals, which technique 
is near optimum if the noise is Gaussian. However, if the noise is 
non-Gaussian then such a technique is not optimum and performance is 
significantly degraded. What is needed is a signal processor which can 
discrimate against non-Gaussian noise thereby increasing the 
signal-to-noise ratio. 
A previous processing technique that is important to this invention uses a 
discrete Fourier transform (DFT) or a fast Fourier transform (FFT) to 
extract narrowband frequency domain signal components. Such a technique is 
also employed in the present invention. 
SUMMARY OF THE INVENTION 
Accordingly, it is a general purpose and object of the present invention to 
provide a method and apparatus for removing non-Gaussian noise effects 
from contaminated signals of interest. It is a further object that the 
apparatus and method make use of non-linear elements constructed from past 
or adjacent frequency components. Another object is that the non-linear 
elements be constructed using estimates of Kurtosis and quantiles from 
past or adjacent frequency components. A still further object is to 
operate in the frequency domain by employing Fourier transform techniques. 
These objects are accomplished with the present invention by providing a 
frequency domain non-linear signal processing apparatus and method for 
discrimination against non-Gaussian interference comprising means for 
converting analog signals from an array of sensors such as hydrophones to 
real and imaginary digital representations, and means for converting 
digital components into the frequency domain so as to process the real and 
imaginary signal components in such a way as to separate out ambient 
non-Gaussian noise from signals of interest. The processing means uses 
estimates of Kurtosis and quantiles from either past or adjacent frequency 
components to construct non-linear elements. These non-linear elements are 
then used to process remaining signal data to improve the signal-to-noise 
ratio thereof by removing non-Gaussian noise therefrom.

DESCRIPTION OF THE PREFERRED EMBODIMENT 
Referring now to FIG. 1 there is shown a graphic illustration of the 
increase in peakedness of probability density functions corresponding to 
an increase in Kurtosis K. The Kurtosis value of interest for this 
invention is the K=3 value associated with the normal or Gaussian 
distribution. Values of K greater than 3 indicate a non-Gaussian 
distribution such as is generally exhibited by noise. 
FIG. 2 shows a typical cumulative distribution curve such as can be 
generated from any Gaussian or non-Gaussian distribution. At preselected 
probability levels, such as .lambda..sub.1 and .lambda..sub.2, 
corresponding quantiles a.sub.1 and a.sub.2 respectively may be 
determined. Additional choices of values of .lambda. will yield additional 
quantiles if desired. 
FIG. 3 shows a typical non-linear element which is constructed using 
quantiles a.sub.1 and a.sub.2 from FIG. 2 to yield the non-linearity 
illustrated by the solid line. Additional non-linearities may be 
constructed by using additional quantiles such as a.sub.0 and a.sub.3 
producing a form like the dashed lines of FIG. 3 where a more refined 
element is desired. 
Referring now to FIG. 4 there is shown a system 10 comprising an array of 
sensors 12 which sense impinging energy signals such as acoustic pressure 
and convert the signals to proportional analog electrical signals. The 
input data consists of broadband or narrowband noise in which a signal is 
additively embedded. The electrical signals are converted to a spatial 
beam direction by beam steering electronics 14. The analog, spatial 
domain, output of beam steering electronics 14 is transmitted to 
analog-to-digital(A/D) converter 16 where the analog signal is converted 
to a discrete time signal by sampling in time and quantizing the signal 
thus producing digital data points representative of the analog signal. 
The input signal can be narrowband or broadband. The noise is composed of 
narrowband non-Gaussian components or broadband measurable in the 
frequency domain distributed over the band. All the frequency components 
need not be non-Gaussian. Some or all can be Gaussian, since the invention 
disciminates against this type of noise. Buffer 18 receives the digital 
data from A/D converter 16 and consecutively stores N digital data 
samples. The N samples then are transmitted to fast Fourier transform 
(FFT) 20 which converts the N temporal digital samples to N discrete 
complex Fourier coefficients representing the digital signal in the 
frequency domain. The N samples stored in buffer 18 are chosen to match 
the size FFT selected. Non-linear processors 22 are connected to FFT 20 
one each for receiving the real and imaginary part of each discrete 
frequency component. Each processor 22 separately operates on a real or 
imaginary frequency component thereby eliminating impulsive and/or 
Non-Gaussian interference which also may be characterized as fluctuating 
or frequency modulated interference. Memory 24, connected to each 
processor 22, receives, stores and transmits to/from each processor 22 
data necessary to control processor functioning. Counter 26 counts the 
number of FFT 20 blocks of N samples in order to establish the convergence 
rate for the quantile estimation algorithm. Counter 28 counts the number 
of FFT blocks in order to bound the number of samples used in the Kurtosis 
estimator. 
The input data to processors 20 is described mathematically as the time 
series 
##EQU1## 
where X.sub.lN+i represents the data at the (lN+i).sup.th time sample. 
Further, X.sub.lN+i =S.sub.lN+i +n.sub.lN+i, where S.sub.lN+i is the 
signal component and n.sub.lN+i is the noise component at the 
(lN+i).sup.th time sample, respectively. N identifies the FFT size, and 
l(=0, 1, . . . M) represents data blocks of N samples each, corresponding 
to consecutive FFT outputs for a total of M+1 outputs. 
After each M+1 block of N samples each has been FFT'd, the real and 
imaginary parts of each N/2 non-redundant frequency component are 
accumulated for M.sub.k .ltoreq.M+1 blocks. Mathematically the frequency 
components are described as follows: 
EQU F.sub.l (k)=F.sub.l.sup.R (k)+jF.sub.l.sup.I (k)=0, 1, . . . M; k=1, 2, . . 
.N/2, 
where j=.sqroot.-1 and I represent the real and imaginary parts of each 
frequency component respectively. 
Then 
##EQU2## 
for l=0, 1, . . . M.sub.k are accumulated and stored. The number M.sub.k 
is chosen based on the desired confidence bound and fixed. Each output 30 
of each processor 22 of the N complex frequency domain data samples may be 
further selectively processed as desired, e.g., frequency domain output 
30a may be further processed by inverse fast Fourier transform 32 (IFFT) 
which converts the output back to the discrete time domain. Output 30 may 
also be further processed in the frequency domain by coherent processor 34 
or incoherent processor 36 depending upon whether phase information is 
desired in addition to magnitude information. 
FIG. 5 shows a typical non-linear processor 22 further comprising a 
Kurtosis estimator 50 in parallel with a dynamic range estimator 52. 
Estimators 50 and 52 feed into a quantile estimator 54 which in turn 
controls the functioning of non-linear element 56. Each estimator 50, 52 
and 54 has associated therewith separate memory modules 50a, 52a and 54a, 
respectively, to store each iterative estimate of Kurtosis, dynamic range 
and quantiles respectively. In operation input data values X.sub.l are 
used by estimator 50 to determine the Kurtosis for that particular real or 
imaginary data frequency Fourier coefficient. For each non-redundant 
frequency component for both the real and imaginary parts the Kurtosis is 
estimated and stored, the Kurtosis of the k.sup.th frequency component of 
the real part for the M.sub.k +1 FFT outputs being defined as: 
##EQU3## 
where F.sub.l.sup.R (k) is the mean of the real part of the k.sup.th 
frequency component over the M.sub.k +1 FFT outputs, and VAR.sup.R (k) is 
the corresponding variance of the real part of the k.sup.th frequency 
component over the M.sub.k +1 FFT outputs. The Kurtosis of the imaginary 
part of each non-redundant frequency component is estimated in a similar 
way. 
The Kurtosis estimate for the real and imaginary parts of each frequency 
component is compared with a fixed number K.sub.k. This number depends 
upon M.sub.k and the desired confidence bound in the estimate. For the 
real data evaluation case of the invention, K.sub.k was set to 4 and 
M.sub.k was 100 and greater. Then for each Kurtosis estimate which exceeds 
K.sub.k the quantiles, at some determined probability level, which are 
estimated concurrently for real and/or imaginary parts at the 
corresponding frequency, are used to construct the appropriate 
non-linearity. Concurrently, data values X.sub.l are used by estimator 52 
to determine the maximum and minimum values of X.sub.l determining the 
difference between C.sub.max and C.sub.min which by definition is the 
dynamic range. Estimator 54, once initiated, receives ongoing estimates 
from estimator 50 and 52 and continually re-estimates the quantiles. The 
quantiles are used to construct a non-linearity in which the data at the 
corresponding frequency and appropriate real and/or imaginary parts are 
processed. The data at the output of the non-linearity has the property 
that its distribution is Gaussian or nearly Gaussian. 
The quantiles are defined as: 
EQU a.sub.q =F.sup.-1 (.lambda..sub.q), q=1,2, . . . , m-1 
where 
##EQU4## 
and f(x) is the underlying density function. 
The {.lambda..sub.q }q=1,2, . . .,m-1 are chosen and fixed, e.g., m=3, 
.lambda..sub.1 =0.25 and .lambda..sub.2 =0.975 etc. 
The quantile recursive estimation algorithm is as follows: 
Let 
##EQU5## 
represent the real or imaginary part of a frequency component. Set 
.lambda..sub.q as appropriate. The initial values are defined as, 
a.sub.q.sup.0 =0, C.sub.max =0 and C.sub.min =0. The functions are defined 
as, C.sub.max =maximum of (C.sub.max, X.sub.l), C.sub.min =minimum of 
(C.sub.min, X.sub.l) and 
##EQU6## 
Then the i-th quantile estimate is defined for each {X.sub.l }l=1,0, . . . 
M.sub.k+1 as follows: 
Compute C=C.sub.max -C.sub.min and A=.mu.(a.sub.q.sup.l -X.sub.l; ). Then 
##EQU7## 
Once the quantiles are estimated from the the data, the non-linearity is 
constructed and the data is processed through the non-linearity. 
Mathematically the operation of the non-linearity is defined as: 
##EQU8## 
where F.sub.l.sup.R (k) represents the data at the output of the 
non-linearity for the real part at the k.sup.th frequency component, and 
a.sub.1.sup.R (k), a.sub.2.sup.R (k) are the quantiles for the real part 
at the k.sup.th frequency component. 
The imaginary part is processed in a similar way. This process is repeated 
for each M.sub.k +1 block making the technique adaptive, which is also 
part of the invention. The quantiles are then transmitted to non-linear 
element 56 which uses the quantiles to establish the linear range for that 
particular X.sub.l data component fed directly to element 56 and which is 
outputted as y.sub.l. 
FIG. 6 shows an example of a circuit diagram for a non-linear processor. 
Signal processing blocks 50, 52, 54 and 56 of FIG. 5 are identified in 
FIG. 6 by broken lines surrounding the appropriate portions of the FIG. 6 
circuit diagram. The input data X.sub.l is fed in parallel to Kurtosis 
estimator 50 and to dynamic range estimator 52. Kurtosis estimator 50 
converts subsequent data blocks of the now discrete frequency data to a 
Kurtosis estimate. The first phase of this conversion squares the current 
input value of X.sub.l in multiplier 60. The inputs of multiplier 60 are 
multiplied to produce the output X.sub.l.sup.2. The output of multiplier 
60 is fed to two parallel paths which eventually produce the Kurtosis 
estimate of equation (2). One portion of the output of multiplier 60 is 
immediately multiplied by the reciprocal of the presently stored value in 
counter 28 and then added to the value previously stored in register 62 by 
adder 64. Register 62 stores the current iterative value of X.sub.l.sup.2 
and resets to zero after N.sub.k iterations. Counter 28 controls the total 
number of iterations, also reseting at N.sub.k. This output of adder 64 is 
now stored in register 62 for the next iterative cycle. The output of 
adder 64 is also squared and inverted by inverter 70 to produce the 
denominator of equation (2). The other portion of the output of multiplier 
60 is squared again then multiplied by the reciprocal of the presently 
stored value in counter 28 and then added to the value previously stored 
in its associated register, which sum is then stored in the register for 
the next iterative cycle. This operation produces the numerator of the 
Kurtosis estimate of equation (2) which is then multiplied by the output 
of inverter 70 to produce the Kurtosis estimate. 
The estimate from Kurtosis estimator 50 is next fed to quantiles estimator 
54. The current Kurtosis estimate K from estimator 50 is subtracted in 
subtractor 66 from a value K.sub.0 stored in memory 68 which represents a 
preselected threshold. The difference K.sub.0 -K from subtractor 66 is 
compared in comparator 72. If K.sub.0 -K is greater than zero then the 
output of comparator 72 is one. Otherwise the output of 72 is zero. When 
the output of comparator 72 is one, the current Kurtosis estimate is below 
the threshold of significance and stored value L.sub.N, representing a 
large number within the dynamic range, will be used instead of the 
estimated quantiles. This is done so that the linear region of 56 will 
encompasses the whole available dynamic range of the data. When the output 
of comparator 72 is zero then the estimated quantiles from estimator 54 
will be used to limit the linear region of element 56. In this way a 
non-linear element is construced which depends on the available data 
through the estimation of quantiles and Kurtosis. 
The remaining portions of the circuit of FIG. 6 use adders, subtractors, 
multipliers, counters, comparators, registers, inverters and memory 
arranged in such a fashion as to produce the desired outputs of estimators 
52 and 54 and non-linear element 56. The dynamic range estimator 52 
estimates the maximum and minimum values of the X.sub.l which are then 
used to control the gain in the quantiles estimator and also to limit the 
quantile values to within the estimated dynamic range of the data. 
Therefore quantiles estimator 54 is independent of the amplitude level of 
the data. 
The output of quantiles estimator 54 is fed to non-linear element 56. The 
processor input X.sub.l is passed through non-linear element 56 to produce 
the output y.sub.l. When the processor is first initiated the output 
y.sub.l will equal the input X.sub.l for several iterations until the 
estimators can adjust themselves to reflect the data. When the estimators 
have adapted to the data and the data then changes the estimators will 
readapt themselves to the new data conditions after counters 26 and 28 
have been reset. 
The advantages of this invention are that it: (1) improves performance in 
terms of increased signal-to-noise ratio and gives a high probability of 
detection at a constant false alarm rate in non-Gaussian noise 
environments; (2) removes only the non-Gaussian noise components, leaving 
the Gaussian components unchanged, thereby not degrading performance in 
Gaussian environments; and (3) adapts to changing noise conditions by 
estimating the Kurtosis for each M.sub.k +1 sample and modifying the 
discrimination process appropriately. 
The new features include: 
a. Estimation of Kurtosis for both the real and imaginary parts for each 
frequency component for each M.sub.k +1 sample. 
b. For each block of M.sub.k +1 samples the Kurtosis is used to 
discriminate the data and detect non-Gaussian noise in the frequency 
domain. 
c. When non-Gaussian noise is detected the quantiles which are estimated 
concurrently for the real and/or imaginary parts for the appropriate 
frequency components, are used to construct the appropriate non-linearity. 
d. The quantiles are estimated by a recursive algorithm The recursive 
algorithm used in the invention is a modified version of a known recursive 
algorithm. The application to the frequency domain is new. 
e. Once the quantiles are known the appropriate data is processed through a 
non-linearity which depends upon the data. 
f. At the output of the non-linearity the data is essentially Gaussian. 
This process reduces the variance of the noise and therefore improves 
performance, depending upon the application. 
What has thus been described is a frequency domain non-linear signal 
processing apparatus and method for discrimination against non-Gaussian 
intereference comprising means for converting analog signals from an array 
of sensors such as hydrophones to real and imaginary digital 
representations and means for processing digital components so as to 
process the real and imaginary signal components in such a way as to 
separate out ambient non-Gaussian noise from signals of interest. The 
processing means uses estimates of Kurtosis and quantiles from either past 
or adjacent frequency components to construct non-linear elements. The 
non-linear elements are then used to process remaining acoustic signal 
data to improve the signal-to-noise ratio thereof by removing non-Gaussian 
noise therefrom. 
Obviously many modifications and variations of the present invention may 
become apparent in light of the above teachings. For example: 
a. The estimation of Kurtosis at the output of a FFT for both the real and 
imaginary parts at each frequency component is a new concept. In the test 
and evaluation of the invention using real data the skew was also 
estimated at the same time as the Kurtosis. This skew could also be used 
to supplement the Kurtosis as a discriminate technique. 
b. The Kurtosis at the output of an FFT could be used to detect 
non-Gaussian signals or non-stationanary or transient signals directly 
instead of using this information to construct a non-linearity as in the 
case of non-Gaussian noise. 
c. For non-Gaussian interference with Kurtosis values less than 3, a 
discriminate technique could also be employed. The non-linearity in this 
case would be different. It would be designed to otimize detection under 
this type of interference. 
In light of the above, it is therefore understood that within the scope of 
the appended claims, the invention may be practiced otherwise than as 
specifically described.