Steven Noel, Harold Szu, in Progress in Unsupervised Learning of Artificial Neural Networks and Real-World Applications, Russian Academy of Nonlinear Sciences, ISBN 0-620-23629-9, 1998.

Chapter 4 Wavelets and Neural Networks for Radar

In this chapter we apply wavelet and neural network techniques to the processing of radar signals. As a case study we consider the problem of radar proximity fuzing. However, the techniques we develop also apply to other types of radar and sonar systems. We introduce the signals found in radar systems, and how the information they contain about targets can be used for fuzing. We then describe a new wavelet-based model for the random radar reflections from the ocean surface, which are a major source of noise in naval operations. Next we show ways of improving the performance of wavelet denoising when the corrupting noise is fractal. We then describe a new method of pulse radar target detection using a recently developed complex-valued Hermitian wavelet. Finally, we examine a new wavelet video method of processing signals for continuous-wave radar fuzes. 4.1 Radar Signals for Proximity Fuzing

Radar is an electromagnetic sensor used for detecting, locating, tracking, and identifying objects, often at considerable distances. Radar operates by transmitting electromagnetic energy toward targets and observing the resulting echoes. Radar can not only determine the presence, location, and velocity of objects, but in some cases their size and shape as well. Radar is distinguished from optical and infrared sensors by its ability to detect distant objects under all weather conditions, and to determine their distance with precision.

Radar is called an “active” sensor because it has its own source of illumination (a transmitter) for locating targets. Radar typically operates in the microwave region of the electromagnetic spectrum, in particular at frequencies extending from about 400 MHz to 40 GHz. However, it has been used at lower frequencies for longer-range applications, and at optical and infrared frequencies (laser radar).

Radar underwent rapid development during the 1930s and 1940s in order to meet the needs of the military. Indeed, radar is still widely used in the military, and the military continues to subsidize advances in radar technology. Still, radar has been employed in an increasing number of important civilian applications. Notable examples include air traffic control, remote sensing of the environment, aircraft and ship navigation, speed measurement for industrial applications and for law enforcement, space surveillance, and planetary observation [70].

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The circuit components and other hardware of radar systems vary significantly with the operating characteristics of the system. As a consequence, radar systems range in size from fitting in the palm of the hand to taking up several football fields. However, the basic principles of operation of all radar systems remain the same. Figure 4-1 shows a basic block diagram for a radar system. An electromagnetic wave is generated, reflected from an object, and received. The wave may be either a pulse-type signal of duration τ , or a continuously oscillating signal of frequency f . Once the electromagnetic wave is received, the resulting signal is processed, and information about the object is extracted. Some action is then taken based on the resulting information.

Transmitter

TransmittingAntenna

ReceivingAntenna

Receiver

Power Supply

Signal Processor

Control, DataProcessing,and Storage

Display orDecision

Figure 4-1: Basic radar system block diagram. The next two sections describe more fully the types of signals found in radar systems, and how the information they contain can be applied to proximity fuzing. We cover the two main types of radar systems, pulse radar (Section 4.1.1) and continuous wave radar (Section 4.1.2). Then, in Section 4.1.3 we describe how the Doppler effect in radar can be applied to proximity fuzing. 4.1.1 Pulse Radar Radar systems that operate with pulse-type signals are called pulse radars. The basic design of pulse radars is quite simple, as illustrated by Figure 4-2. The system consists of a single antenna, a transmitter, a receiver, a switch connecting the transmitter and receiver to the antenna, and a computer for controlling the system and gathering data. The transmitter is usually relatively powerful, and the receiver is usually relatively

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sensitive. Only a single (switched) antenna is necessary, because the transmitted and received signals occur at different times. The system first transmits a pulse of electromagnetic energy. To do this, the controlling computer must set the transmit/receive switch so that the transmitter is connected to the antenna. The transmitted pulse is of relatively short duration, for example on the order of a tenth of a second. Because the signal energy of the pulse is contained within in a short time span, the resulting power is relatively large. After the pulse is transmitted, the system tries to receive any echoes from the pulse, setting the switch so that the receiver and antenna are connected. The computer samples the received echo and stores the resulting digital signal. If there are no objects nearby to cause a significant echo, the received signal is just mostly background noise. However, if such an object is present, an echo pulse signal will result. The echo signal is generally relatively weak, because signal power decreases inversely to the fourth power of the distance to the object [3].

Transmit/ReceiveSwitch

Antenna

Transmitter

Receiver

SwitchControl Radar

DataRadar Pulses

ReflectedPulses

TransmitterControl

Figure 4-2: Basic pulse radar system. One of the most fundamental and important pieces of information provided by radar is the distance (or range) to the target object. We can calculate the range R as

Rct

=2

, (4.1)

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where t is the elapsed time and c ≈ ×3 108 meters second/ is the velocity of radiation propagation. Pulse radars can then directly determine range by simply measuring the elapsed time between transmitted and received pulses and then applying Eq. (4.1). The time between radar pulses is called the pulse repetition interval (PRI). For a radar pulse of duration τ , the physical distance the pulse occupies in free space is cτ . The resulting maximum practical range maxR (called “first time around” or

“unambiguous” range) is then given by 2/PRImax ⋅= cR . (4.2)

This is the distance to which a pulse can make a round trip before the next pulse is sent. Since a pulse radar’s receiver is switched off while the transmitter is switched on, no targets can be detected during this interval.

Transmitted Received

t = 0 t R c= 2 /

MuchWeakerPulse

BackgroundNoise

Figure 4-3: Transmitted and received signals for pulse radar. In theory, pulse radars could also calculate the target’s relative velocity rv as the rate of change of range R over time t , that is,

dt

dRvr = . (4.3)

Here θcosvvr

= , where v

is the target velocity and θ is the direction between v

and

the line between the target and the radar antenna, with the origin fixed at the antenna. However, this method is known to have relatively poor accuracy and resolution [3].

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4.1.2 Continuous Wave Radar Rather than emitting pulses of radiation, a radar system can also emit radiation at all times, leading to so-called continuous radar systems. The simplest form of continuous wave radar emits a sinusoidal signal of constant amplitude, frequency, and phase. However, for such a simple system it is difficult to separate the relatively weak echo signal from the transmitted one. There is no such problem for pulse radar because transmission and reception occur separately. Also, such a simple continuous wave radar system is unable to measure range to the target, because there is no time delay associated with the transmitted signal. Recall that for pulse radar the time delay between transmitted and received signals determined target range. However, a simple continuous wave radar system can directly obtain the relative velocity of the target, which can be difficult for pulse radar [3].

Transmitted SignalOf Certain Frequency

Sensor

Reflected Signal WithDoppler Frequency Shift

Object

Figure 4-4: Doppler frequency shift between transmitted and reflected signals. Continuous-wave radar determines range by measuring the so-called Doppler frequency shift of the reflected signal. The Doppler shift arises from relative motion between the radar and the target. This relative motion causes the wavelength of the reflected electromagnetic wave to be decreased (or increased) by a fraction that is proportional to its velocity in the direction of propagation, as shown in Figure 4-5. Since the velocity of propagation is constant by Einstein’s special theory of relativity, the corresponding frequency of the wave must increase (or decrease) inversely to the wavelength. The Doppler shift is positive if the radar and target are approaching, and is negative if they are receding. A familiar example of the Doppler shift in acoustics is the noticeable change in whistle pitch as a train passes by.

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λ

λ′

( )cvr /1−=′ λλ

tvr∆c

ft

/

/1

λ==∆

rv

Transmitted

Reflected

Figure 4-5: Change in reflected wavelength caused by target motion.

Figure 4-6: Familiar example of Doppler shift is change in pitch as train passes by.

Let us derive the Doppler frequency shift and the corresponding relative velocity between the radar and the target. Assume that the range between the radar and target is R . The total number of radiation wavelengths λ over the transmitted and reflected paths is then λ/2R . Since one wavelength λ corresponds to an angular phase of π2 , the total phase φ is λπ /4 R . The rate of change in φ with time t is the angular Doppler

frequency dω , which is then

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λπ

λπφπω r

dd

v

dt

dR

dt

df

442 ==== . (4.4)

Here fd is the Doppler frequency shift and vr is the relative velocity of the target with respect to the target. The Doppler frequency shift fd then becomes

c

fvvf rr

d022 ==

λ, (4.5)

where f0 is the frequency of the transmitted signal and c is the velocity of radiation

propagation. The relative velocity rv corresponding to this frequency shift is then

02 f

fcv d

r

= . (4.6)

Thus we can calculate the relative velocity rv between the sensor and an object from

measurements of the Doppler frequency shift df , given the known transmitted frequency

0f .

It is also possible to continuous wave radar to measure range to the target though the process of frequency modulation, which is the controlled variation of the frequency of the transmitted signal. This frequency variation adds time information to the transmitted signal. By measuring the time-varying frequency of the reflected signal, the time delay between transmission and reception can be inferred so as to determine range. For this to be effective, the amount of frequency modulation must be sufficiently greater than the expected Doppler shift. The simplest form of frequency modulation is linear, that is, the transmitted frequency changes at a constant rate over time. This is shown in Figure 4-7. The frequency modulated radar system measures the instantaneous frequency difference f∆ between transmitted and received signals. This frequency difference is proportional to the time delay τ for round-trip signal propagation to the target. The time delay is then

12 ff

fT

−∆=τ , (4.7)

where 1f is the minimum modulation frequency, 2f is the maximum modulation frequency, and T is the modulation period. We then obtain range R in the usual fashion, that is 2/τcR = .

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T

f∆

Time

Frequency

1f

2f

TransmittedFrequency

ReceivedFrequency

Time

TransmittedSignal

f∆

Figure 4-7: Linear frequency modulation.

Note that when the frequency modulation sweep resets, the frequency difference becomes negative, which is a problem for the radar system. The system therefore must use a discriminator to clip off the negative portion of the difference signal. The maximum unambiguous range maxR for frequency modulated continuous

wave radar is determined by the modulation period T , that is

2max

cTR = . (4.8)

These radar systems do not have a minimum range as do pulse radar systems. However, they are unsuitable for longer-range detection because the continuous power of their transmitted signals must be considerable lower than the peak power of pulse radar signals. That is, for continuous wave systems the peak power is the same as the average power, but for pulse systems the peak power is many times greater than the average. The transmitted radar signal has a very high frequency, usually somewhere between 400 MHz to 40 GHz. At these frequencies, it is difficult to process signals, either with analog circuits or digitally. Frequency modulated continuous wave radar systems therefore employ the processing shown in Figure 4-8.

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FrequencyModulation

Mixer nth HarmonicFilter

Low-PassFilter

Mixer

Transmitter n Frequency

Multiplier

fd

Figure 4-8: Extraction of signal at the Doppler shift frequency fd.

The received echo signal is mixed with the transmitted signal, resulting in a signal whose frequency spectrum is shown in Figure 4-9. The mixer output has frequency components that are integer multiples of the instantaneous frequency of the transmitter, the so-called harmonics. A filter then extracts a predetermined harmonic, which is usually selected based its particular range-dependent attenuation [3].

fd

fm 2 fm 3 fm 4 fm0

fd

Rel

ativ

e A

mpl

itud

e

Frequency

Figure 4-9: Frequency spectrum of mixer output signal.

The nth harmonic filter output is then mixed with the corresponding integer

multiple of the instantaneous modulation frequency. The output of this second mixer has signal components at both the sum and difference of the transmitter harmonic and its Doppler shifted echo signal. A low-pass filter extracts only the difference, yielding a signal whose frequency is equivalent to the Doppler shift. This signal has a relatively low frequency, as seen from Eq. (4.5), since targets of interest have velocities much lower than the speed of light. The Doppler extracted shift signal is therefore much easier to process with practical hardware.

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One of the primary sources of noise in radar systems operating over the ocean is random reflection of radar from the irregular ocean surface, called sea clutter. The Doppler frequency shift for such sea clutter signals is fairly small, being dictated by the velocity of the wind-blown water waves. For targets whose velocities are significantly larger than that of the waves, the radar system could effectively filter the sea clutter signal from the target signal, as shown in Figure 4-10.

fd

(n-1) fmn fm (n+1) fm

fm fclutter ftarget

SeaClutterFilter

Harmonics ofmodulation frequency

Doppler shift

Figure 4-10: Filtering sea clutter noise for relatively fast targets.

However, many military targets, in both the air and water, have velocities that are not significantly greater than wave velocities. In such cases the sea clutter cannot be easily filtered out. It therefore becomes the dominant noise source, and hampers the ability to effectively measure the signal. 4.1.3 Use of Doppler Effect for Fuzing

We have just seen how measuring the Doppler frequency shift of a reflected signal allows us to calculate the velocity of a radar target. By measuring the Doppler shift as it changes over time, we can also calculate the direction to the target. As we will show, the direction to the target is the critical information needed for optimal proximity fuzing of gun-launched projectiles.

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θ

α

x

y

Radar proximity fuze at origin

Target

Projectile

x vt= −v

vv

r=

cosθ

Figure 4-11: Geometry for radar proximity fuze and target. Figure 4-11 shows a radar proximity fuze and a target. We assume that they are

approaching at a constant velocity v , with the origin fixed at the fuze. For the very short times characteristic of fuzing engagements, this is an appropriate assumption. The relative velocity rv can be written as

θθ coscos vvvr ==

, (4.9)

where θ is the angle toward the target. The angle θ is then calculated as

=

−= −−

v

vvt r11 coscotα

θ . (4.10)

We are unlikely to have a precise value of the closest approach distance α .

However, we can estimate the magnitude vv = of the target velocity v by measuring

the relative velocity rv at a distance sufficiently larger than the closest approach distance

α . We can then determine the instantaneous angle to the target ( )rvθθ = by calculating

the relative velocity rv from the instantaneous Doppler shift. Also, if we have reliable measurements of both v and the time of closest approach, we can calculate α as

( )θα

cotvt−= , (4.11)

with 0=t at the time of closest approach. While the kinematics we just described are somewhat idealized, the behavior does hold true in general. At sufficiently large distances between the fuze and target, both the

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relative velocity and the corresponding Doppler frequency shift asymptotically approach a constant. As the target later passes near the near, the relative velocity decreases, with the frequency shift decreasing proportionately. The nearer the target passes the fuze, the more nonlinear is the change in Doppler frequency shift over time. This general behavior is shown in Figure 4-12.

Point of ClosestApproach Between

Target and Fuze

Time

DopplerShift

More nonlinear for moreclosely passing target

Less nonlinear for lessclosely passing target

Figure 4-12: Change in Doppler frequency shift over time as target passes fuze.

Radar proximity fuzes can take advantage of this behavior in order to gain information about targets. For example, the approach of Doppler shift to zero indicates that the target is at its closest distance from the fuze.

f ffuzing opt where= ∞ cos ,θ θopt

fragment=

−

∞tan 1

v

v

θθθθ

MissDistance

ProjectileTrajectory

Range

DopplerProfile

f∞

α

Figure 4-13: The radar proximity fuzing problem for gun-launched projectiles.

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For gun-launched projectiles the optimal value of optθ for fuzing is known to be

( )vv /tan frag

1opt

−=θ , (4.12)

where fragv is the velocity of the projectile warhead fragments. Note that this is

independent of the closest approach distance (so-called miss distance) α . The fuzing problem is this case is then to estimate the Doppler shift over time, and to detonate when the shift reaches its predetermined optimal value. This is shown in Figure 4-13. In the figure, ∞f is the Doppler shift associated with the relative target velocity rv measured at a distance that is sufficiently larger than the miss distance α . 4.2 Wavelet Model for Sea Clutter Noise In radar proximity fuzing for naval operations, one of the major sources of noise is random reflection of the radar signal from the ocean’s surface. This noise signal is called sea clutter. While sea clutter noise is fractal, it differs considerably from 1/f fractal noise. Wornell’s wavelet-based model for fractal signals was developed for the special case of 1/f noise. We generalize his model so as to be more consistent with the known characteristics of sea clutter. We will use this new model to synthesize sea clutter for corrupting radar signals in our simulations. This will allow a more meaningful assessment of our target detection and velocity estimation algorithms. In Section 4.2.1, we verify the wavelet noise model using real data. In Section 4.2.2, we then extend the model so as to be more consistent with sea clutter noise. 4.2.1 Verifying Fractal Nature of Real Signal The irregularity of the ocean surface arises through the interaction of wind and water. The system is open because of the external forcing of the sun’s heat, and is dissipative because of the heat loss due to friction. Also, the interaction of wind and water is nonlinear. Signals resulting from open dissipative systems with nonlinear dynamics are known to be fractal [54]. Thus we expect sea clutter noise to be fractal. It would be of interest to verify with real data our wavelet-based techniques for analyzing fractal signals. However, we cannot show sea clutter data for actual Navy proximity fuzes because of security issues. The reason is that given real data, and by considering the typical sizes of ocean waves, one could estimate the transmitted frequency of the fuze device. Such transmitted frequency may not be publicly known. However, we can verify the fractal nature of signals from other open dissipative nonlinear systems, to help validate our wavelet-based techniques. One such system is the

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energy consumption over time at an electrical power plant. This system is open because energy is continuously being added, and is dissipative because of loss through consumer usage. It is also nonlinear, for example as consumers flip switches on appliances. Figure 4-14 shows a signal for the consumption over time at a large electrical power plant. We will now test the hypothesis that the wavelet transform coefficients for this signal are consistent with Wornell’s model for 1/f fractal processes. In particular, we will determine if the scale-dependent variances of the wavelet coefficients agree with those predicted by the model.

0 200 400 600 800 1000 1200300

350

400

450

500

550

Figure 4-14: Electrical power consumption over time. Figure 4-15 shows the multiresolution detail signals for the power consumption signal, all plotted with the same vertical scale. Here we applied Daubechies’ 8th-order wavelet basis. From the figure we indeed see the decay in variance of the wavelet coefficients as the scale decreases, which is roughly consistent with Wornell’s model.

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( ) D x t0

( ) D x t1

( ) D x t2

( ) D x t3

( ) D x t4

( ) D x t5

( ) D x t6

Figure 4-15: Multiresolution detail signals for power consumption signal. We can see the scale dependent decay in wavelet coefficients more precisely in Figure 4-16. It shows the logarithm of the wavelet coefficient variances as a function of scale. The plot is nearly linear, which is in good agreement with the model. From the plot we determine that the energy consumption signal is well modeled as a 1/f process with spectral parameter γ ≈ 0 6. .

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 70

0.5

1

1.5

2

2.5

3

3.5

Scale m

log 2v

ar X

nm

Figure 4-16: Scale-dependent variances for power consumption signal.

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Figure 4-17 shows the average normalized correlation ρn nm m,,

′′ between wavelet

coefficients Xnm and X n

m′′ . The plot is a function of integer lag l ≥ 0 between

coefficients, so that ′ = −n n l . Since correlations are taken for lag along a single scale, we have m m= ′ . These lag-dependent along-scale correlations are then averaged over all scales.

0 200 400 60 0 800 1000

0

0 .2

0 .4

0 .6

0 .8

1

ρn n lm m,,−

Lag l

Figure 4-17: Average along-scale lag correlation for power consumption signal.

From the plot we see that the correlation drops off sharply as the lag increases. Again, this is in good agreement with the model. In particular, if we perform any wavelet-based analysis on this signal it would be reasonable to neglect inter-coefficient correlation. This reinforces the interpretation of the wavelet transform as a Karhunen−Loeve type expansion for 1/f processes, that is, an expansion in which the coefficients are uncorrelated. 4.2.2 Wavelet-Based Sea Clutter Model We now develop a new wavelet-based model noise model that is more consistent with sea clutter characteristics than is Wornell’s model for 1/f noise. Let us begin by assuming that at spatial frequencies above some limit, the ocean surface has a 1/f spectrum. In other words, 1/f-type behavior could be expected for higher frequencies, at least to the resolution limit of our measuring instruments. This seems reasonable because full open dissipative nonlinear behavior could be expected at the smaller scales.

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Figure 4-18: The ocean surface. However, consider Figure 4-18. It is obvious that at lower spatial frequencies the ocean surface does not have a 1/f spectrum. In other words, on larger scales the surface seems more flat. Taken in the extreme, the ocean surface appears completely flat when viewed from space. For lower spatial frequencies (longer spatial wavelengths) the wind is not powerful enough to generate waves which are tall enough to exhibit 1/f behavior. In other words, gravity limits the largest possible heights of waves. Thus at the largest scales, the assumption breaks down that the wind-water interaction is an open dissipative nonlinear system. In the case of radar signals, we assume that the measurement bandwidth is wide enough to actually observe large-scale deviations from 1/f behavior. Recall that according to Wornell’s model, a nearly-1/f noise is constructed from a wavelet expansion when the wavelet coefficients Xn

m are uncorrelated zero-mean

Gaussian random variables whose variances depend on their scale level m as var Xn

m m= −σ γ2 2 , (4.13) for noise variance σ 2 and spectral parameter γ . In constructing our new sea clutter

model, we then apply the 2−γm variance condition in Eq. (4.13) to the wavelet coefficients at the smaller scales, as consistent with 1/f behavior.

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For the larger scales, rather than applying the 2−γm variance condition, we assume that variances become smaller rather than larger as the scale increases. In particular, we assume that they vary inversely to Eq. (4.13), that is, they vary as 2γm . Thus we choose the scale-dependent variances varm n

mX for the detail wavelet coefficients Xnm as

( )var.m n

mm

mXm

m=

≤ ≤≤ ≤

−

−

2 1 4

2 5 68

γ

γ (4.14)

Here we assume a 6-level discrete wavelet transform, though this variance structure can be easily generalized to other numbers of wavelet transform levels. For the approximation wavelet coefficients αn

6 we choose variance varα γ

n6 2= . (4.15)

This is consistent with the variance structure for the detail coefficients. For all wavelet coefficient variances the spectral parameter γ is constant. Accurate values of γ could be estimated from measurements of actual sea clutter under particular sea states.

1/f NoiseModel

Sea ClutterModel

RayleighDistribution

Scale m

log

2va

r Xnm

Figure 4-19: Scale-dependent wavelet coefficient variances for 1/f noise model, sea clutter model, and Rayleigh distribution.

When the scale-dependent wavelet coefficient variances in Wornell’s 1/f noise model are plotted on a logarithmic scale, the plot is linear, the slope being the spectral parameter γ . In our new model for sea clutter noise, the plot is piecewise linear. For the

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smaller half of the wavelet scales, the slope is still the spectral parameter γ . However, for the larger half the slope is γ− , so that the coefficient variances become smaller at the larger scales. This is shown in Figure 4-19. Figure 4-19 also shows the scale-dependent variance for the Rayleigh distribution, which is a classical model for sea clutter [3]. The frequency components ( )fR for the Rayleigh distribution follow a Gaussian distribution, that is

( )

−−=2

00 exp

σff

RfR . (4.16)

Taking the base-2 logarithm as consistent with the plot, we have

( )[ ]

( )

( ) ( ) .2ln

1log

loglog

exploglog

20202

2

2

002

2

0022

ffR

eff

R

ffRfR

−−=

−−=

−−=

σ

σ

σ

(4.17)

Here we apply the property of logarithms 2ln/12ln/lnlog2 == ee . The plot is given in terms of scale rather than frequency, which is equivalent to a reflection about the center scale. Because of the symmetry of Eq. (4.17) about 0f , the plot in scale looks the same

as would a plot in frequency. From Figure 4-19 we see that our model is certainly more consistent with the classical Rayleigh distribution than is the model for 1/f noise. While our model does not completely agree with the Rayleigh distribution, it has close agreement near the center scale. This corresponds to the relatively narrow bandwidth radar systems to which the Rayleigh distribution has been traditionally applied. More recent measurements using higher bandwidth radar systems show high-frequency behavior that is more consistent with 1/f noise [55]. These measurements are therefore also more consistent with our model. Also, our new model provides more flexibility than the Rayleigh distribution for the analysis and synthesis of signals. In particular, our model (along with Wornell’s model) allows the modeling of not only fractal noise signals, but also multifractal noise signals. A multifractal is a generalization of a fractal in which the fractal dimension is no longer globally constant. Recall that the fractal dimension determines the degree of roughness of the signal. The multifractal formulation is therefore convenient in modeling sea clutter whose roughness varies because of changing sea states.

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m = 1

m = 2

m = 3

m = 4

m = 5

m = 6

WaveletCoefficients

ScalingCoefficients m = 6

SynthesizedSignal

Figure 4-20: Wavelet synthesis of sea clutter signal for spectral parameter γγγγ = 1.

m = 1

m = 2

m = 3

m = 4

m = 5

m = 6

WaveletCoefficients

ScalingCoefficients m = 6

SynthesizedSignal

Figure 4-21: Wavelet synthesis of sea clutter signal for spectral parameter γγγγ = 1.5.

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m = 1

m = 2

m = 3

m = 4

m = 5

m = 6

WaveletCoefficients

ScalingCoefficients m = 6

SynthesizedSignal

Figure 4-22: Wavelet synthesis of sea clutter signal for spectral parameter γγγγ = 2. In both our model and Wornell’s model, the fractal dimension is determined by the value of the spectral parameter γ . We could therefore model multifractal noises by

allowing ( )tγγ = to vary over time t . However, for the specific case of proximity

fuzing, ( )tγ is expected to be essentially constant, because of the characteristically short engagement times. Figures 4-20 through 4-21 show the synthesis of fractal sea clutter noise signals through our wavelet-based model. Here we apply Daubechies 8th-order nearly symmetric wavelet basis [17]. In the figures the spectral parameter γ varies from 1 to 1.5 to 2. We reset the random number generator seed before each signal was synthesized to allow a more direct comparison of the three signals. The spectral parameter γ controls the degree to which wavelet coefficient variance changes between scale levels m . As we see in the figures, for larger γ the decay in variance is stronger for the smaller scales. The spectral parameter γ also determines the fractal dimension of the signals, which is a measure of signal roughness. We see this more clearly in Figure 4-23, which directly compares wavelet-synthesized sea clutter for 2and,5.1,1=γ . In this figure we see that the signal clearly becomes rougher as γ decreases. The various degrees of roughness then correspond to different levels of sea state activity, making our model more generally applicable.

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1=γ

5.1=γ

2=γ

Figure 4-23: Spectral parameter γγγγ determines sea clutter roughness, which corresponds to different sea states.

4.3 Wavelet Denoising for Fractal Noise The standard wavelet denoising algorithm was derived under the assumption of white Gaussian noise, that is, noise in which the power spectrum is constant. However this assumption does not generally hold for noise in radar systems. In particular, radar noises are often fractal. When a target reflects very little radar energy (a so-called low observable), a dominant form of noise is the 1/f noise in the electronic circuits, particularly at lower frequencies. An example of such a target is a sea-skimming anti-ship cruise missile, which is usually designed to be a low observable. When the target velocity is on the order of velocity of the ocean waves, for example that of a helicopter, the dominant noise is sea clutter. We therefore investigate methods for improving the performance of wavelet denoising when the assumption of white noise no longer holds. From our wavelet-based models of fractal noise, we know that this corresponds to wavelet transform coefficient variances that differ over scale rather than remain fixed. We use this knowledge to improve denoising. In particular, Section 4.3.1 describes various rules for estimating

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noise scale from the wavelet coefficients. Section 4.3.2 then tests signal denoising performance using these rules.

Sea clutter more dominantfor lower velocity targets

(helicopter)

Electronic 1/f noise moredominant for lower observable

targets (cruise missile)

Figure 4-24: Dominant noise sources for different target characteristics. 4.3.1 Noise Scale Estimation Rules We will now examine techniques for improving the wavelet denoising estimate when the corrupting noise is fractal. It involves using different rules for estimating the noise scale parameter σ that appears in the wavelet coefficient shrinkage function

( )δλσ x . One rule is to use only the smallest scale wavelet coefficients F1 to estimate a single parameter σ for all scale levels m . In other words, for this rule the parameter σ is independent of scale. More precisely, for this rule the noise scale parameter σ is

( )σ σ= ~ F1 , (4.18)

where ~σ is a standard function for estimating the noise scale. One such function ~σ is the median absolute deviation [60], a highly robust estimate of scale. It is important to use a robust estimator to prevent the actual signal from leaking into the estimate. This first rule could be considered the safest or most conservative choice, since in general the smallest scale coefficients consist almost entirely of noise. A second rule for estimating the noise scale σ is to form a single estimate from the detail coefficients for all scales in the wavelet transform. Like for the first rule, the noise scale for this rule is independent of scale. For this second rule we can write the noise scale parameter σ as

( )σ σ= ~ , , ,F F F M1 1 . (4.19)

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This rule can give a more accurate estimate of noise scale since it uses all detail wavelet coefficients rather than just the smallest scale ones. However, it is more likely to be influenced by the wavelet coefficients of the signal itself. A third rule is to estimate a separate noise scale parameter σ m for each scale level m . Thus in this case the noise scale parameter is scale dependent. We can then write the noise scale parameter σ m as ( )σ σm

mF= ~ . (4.20)

This third rule is important if the scale of the noise varies from level to level. As we have seen, for 1/f-type noise the scale of the noise does indeed vary with scale level.

Figure 4-25: Signal with sharp peaks corrupted by fractal noise. 4.3.2 Performance For Different Rules We will now compare the three rules for estimating the noise scale parameter σ , using the signal shown in Figure 4-25. The signal is corrupted by additive 1/f noise with spectral parameter γ = 2 . We do this to create an example in which level-dependent

128

estimates of noise scale are needed. In particular, for this example the wavelet coefficients of the noise have the form var

ε σ σγnm m m= =− −2 2 22 2 , (4.21)

where

εnm are the wavelet coefficients of the noise vector

ε . The next three figures show exploratory data analysis plots for the denoising of the signal in Figure 4-25. In these figures we apply hard shrinkage with the universal threshold. However, the figures each use a different rule for estimating the noise scale σ . Figure 4-26 uses the smallest scale wavelet coefficients to estimate a single noise scale σ for all scale levels. In Figure 4-27, we form a single estimate of σ from the wavelet detail coefficients for all scales. Finally, in Figure 4-28 we estimate a separate noise scale σ m for each scale level m . One important thing to point out is the boxplots of wavelet transform coefficients. The superimposed horizontal lines on these plots are proportional to the estimates of noise scale σ (the constant of proportionality is the shrinkage function threshold λ ).

From the horizontal lines we see that when we use only the smallest scale wavelet coefficients to estimate σ , the resulting noise scale estimate is relatively small. This is a consequence of the underlying decay in wavelet coefficient magnitudes for 1/f noise as the wavelet scale decreases. When we estimate σ from the wavelet detail coefficients for all scales (4 scales in this case), the resulting estimate is larger. When we then estimate a separate noise scale σ m for each level m , σ m increases as m increases. In particular, at the larger scales the estimates are significantly larger than for the other two scale estimation rules. From the figures we see that the scale-dependent noise estimates greatly improve the resulting denoised signal. In particular, the denoised signal is significantly smoother. In fact, from the energy plots it is apparent that for the scale-independent noise estimates too little of the noise energy is removed from the noisy signal, particularly at the larger scales. However, from the residual plots it appears that for the scale-dependent noise estimates some of the signal peaks have been slightly oversmoothed. The residual resulting from applying scale-dependent noise estimates have the scale-dependent decay we expect for 1/f noise. In contrast, this scale-dependent decay of the residual is lacking when we apply scale-independent estimates. Also, the quantile-quantile plots show that applying scale-independent estimates results in residuals that are nearly white Gaussian, while the actual noise corrupting this signal is 1/f and therefore non-white.

129

NoisySignal

Estimate

Residual

F1 F 2 F3 F 4 F 5 F 6 α 6

F1F 2F3F 4F5F 6α 6

F1

F 2

F 3

F 4

F 6

F 5

α 6

DWT of Estimate

F1

F 2

F3

F 4

F 5

F 6

α 6

Residual

DWT of Residual Autocorrelation of Residual

Quantiles of Standard Gaussian

Res

idua

l

Histogram and Density EstimateQuantile-Quantile Plot

Figure 4-26: Denoising using smallest scale coefficients to estimate scale of noise.

130

NoisySignal

Estimate

Residual

F1 F 2 F3 F 4 F 5 F 6 α 6

F1F 2F3F 4F5F 6α 6

F1

F 2

F 3

F 4

F 6

F 5

α 6

DWT of Estimate

F1

F 2

F3

F 4

F5

F6

α 6

ResidualDWT of Residual Autocorrelation of Residual

Quantiles of Standard Gaussian

Res

idua

l

Histogram and Density EstimateQuantile-Quantile Plot

Figure 4-27: Denoising using all coefficients to estimate single scale of noise.

131

NoisySignal

Estimate

Residual

F1 F 2 F3 F 4 F 5 F 6 α 6

F1F 2F3F 4F5F 6α 6

F1

F 2

F 3

F 4

F 6

F 5

α 6

DWT of Estimate

F1

F 2

F3

F 4

F5

F6

α 6

ResidualDWT of Residual Autocorrelation of Residual

Quantiles of Standard Gaussian

Res

idua

l

Histogram and Density EstimateQuantile-Quantile Plot

Figure 4-28: Denoising using scale-dependent estimates of noise scale.

132

Single estimate from smallest scale coefficients

Single estimate from all coefficients

Separate estimate for each scale

Figure 4-29: Signal denoised using 3 different rules for estimating noise scale. We also see rather weak autocorrelations for the residuals resulting from scale-independent noise estimates. This is in contrast to the strong autocorrelations for the scale-dependent estimates. Because 1/f noises are strongly autocorrelated, we should also expect strong autocorrelation in the residuals. However, some of the autocorrelation for

133

scale-dependent noise estimates is likely caused by wavelet coefficient leakage from the actual signal at larger scales. That is, the relatively large thresholds at larger scales result in excessive coefficient shrinking. This would appear to be an unavoidable consequence of the relatively large energies at large scales that is characteristic of 1/f processes. 4.4 Complex Hermitian Wavelet for Pulse Radar In pulse radar systems, an abrupt pulse of electromagnetic energy is transmitted toward a target object. The electromagnetic pulse reflected from the target generates a sharp spike in the radar receiver output signal. Mathematically, this spike is known as a singularity, that is, a point at which the signal derivative is undefined (non-unique). We will now develop a generalized form of radar signal processing for detecting target singularities in pulse radar systems. The method is based on the continuous wavelet transform using the recently developed Hermitian hat wavelet [71]. This wavelet is a complex-valued generalization of the Mexican hat wavelet [10]. The Hermitian hat wavelet has an even function as its real part and an odd function as its imaginary part. These functions are analogous to the cosine and sine (complex exponential) applied in traditional quadrate processing [3]. Just like the complex exponential, the Fourier transform of the Hermitian hat wavelet has no complex phase, so that the phase of the wavelet filter does not affect the phase of the signal. However, unlike the complex exponential, the Hermitian hat wavelet is localized in time. This is shown in Figure 4-30. With this generalized quadrature processing we wish to detect singularities representing pulse radar returns from targets. Singularities are defined in terms of non-unique derivatives, but direct calculation of derivatives is inaccurate in the presence of noise. This is because the derivatives are undefined throughout the signal. Techniques involving the complex phase of the Morlet wavelet have been successful in detecting singularities in signals [72]. The singularities are revealed through bifurcations in the phase as the scale of the wavelet decreases. Unfortunately, the Morlet wavelet has certain disadvantages in terms of digital signal processing. The Hermitian hat wavelet exhibits the same singularity detection properties as the Morlet wavelet, but is more advantageous for processing signals digitally. In Section 4.4.1, we derive the Hermitian hat wavelet as a filter that detects target singularities through its complex phase. Specifically, the wavelet is constructed so that its phase is the ratio of multiple-scale derivatives of differing order. In Section 4.4.2, we compare the Hermitian hat wavelet to the Morlet wavelet in terms of its digital signal processing properties. Then, in Section 4.4.3 we show how the Hermitian hat wavelet can be applied to target singularity detection for radar proximity fuzing.

134

( ) ( ) ( )[ ]θ t q t p t= −tan /1

Low-PassFilter

Low-PassFilter

... ...

... ...

( )p t

( )q t

( )s t

sinω0t

cosω0t

×

×

Traditional Processing

*

*

heven

hodd

( ) ( ) ( )p t h t s t= ∗even

( ) ( ) ( )q t h t s t= ∗odd

Hermitian Hat Processing

( ) ( ) ( )[ ]θ t q t p t= −tan /1

( )s t

Figure 4-30: Traditional vs. Hermitian hat radar signal processing. 4.4.1 Construction of Hermitian Wavelet We will now construct the complex-valued Hermitian hat wavelet. The underlying principle behind the Hermitian hat is that it is a filter that detects signal singularities through its complex phase. More precisely, its phase θ is chosen to be the ratio of the first and second derivatives of the input signal ( )x x= τ . Here τ ω= 0t is

135

dimensionless time, with scale frequency ω π0 2= / a . We can therefore write the

wavelet filter output phase ( )θ θ τ= as

θττ

=−

−tan

/

/1

2 2

dx d

d x d. (4.22)

The Hermitian hat wavelet is also based on the principle that in the presence of noise, differentiation should be done through filtering rather than directly. To accomplish this, it employs dummy convolutions with the Dirac delta function ( )δ δ τ= . In particular, we can write

dx

d

dx

d

d

dx

τδ

τδτ

= ∗ = − ∗ (4.23)

and

d x

d

d x

d

d

dx

2

2

2

2

2

2τδ

τδ

τ= ∗ = ∗ . (4.24)

If we then substitute Eqs. (4.23) and (4.24) into Eq. (4.22), we have

θδ τδ τ=

− ∗− ∗

−tan

( / )( / )

12 2

d d xd d x

. (4.25)

We can then write Eq. (4.25) in terms of filters h d d1 = − δ τ/ and h d d2

2 2= − δ τ/ , which becomes

θ =∗∗

−tan 1 1

2

h x

h x. (4.26)

We now need to determine the form of the filters h1 and h2 . We begin with the

Schwartz definition of ( )δ τ as a unit-area Gaussian function of vanishing width:

δπ

ωω

ω( ) lim ( ) /t e t=→∞

−

0

021

2 02 . (4.27)

We then choose filters h1 and h2 to be the first and second derivatives of this ( )δ τ , yielding

hd

dtte t

1 03 21

20

2

= − = −δπ

ω ω( ) / (4.28)

136

and

hd

dtt e t

2

2

2 02

02 21

21 0

2

= − = − −δπ

ω ω ω[ ( ) ] ( ) / . (4.29)

The constant term ( )1 2

1 2

02/

/π ω divides out when we substitute the filters back into Eq. (4.26), so that h1 and h2 become dimensionless. Since h1 is odd, we call it

( )h hodd odd= τ . Likewise, since h2 is even we call it h heven even= ( )τ . We can then write

h h eodd = = −1

22

τ τ / (4.30) and

h h eeven = = − −2

2 212

[ ] /τ τ . (4.31) These 2 functions are shown in Figure 4-31. Note that the even function heven is the familiar Mexican hat wavelet, which is the reason for the name “hat” in Hermitian hat.

Even Filter Odd Filter

Figure 4-31: Hermitian hat odd and even wavelet filters. If we then replace h1 and h2 with hodd and heven in Eq. (4.26), we have

θ =∗∗

−tan 1 h x

h xodd

even

. (4.32)

Thus our desired complex-valued filter is h h jh( ) ( ) ( )τ τ τ= +even odd , (4.33)

137

and our construction of the Hermitian hat wavelet ( )h τ is complete. Szu has shown that the Hermitian hat is an admissible wavelet [71]. Moreover, it is a Hermitian operator, since it has the property of time reversal under complex conjugacy. More formally, the Hermitian hat obeys the relation h h*( ) ( )τ τ= − . This is the reason for the term “Hermitian” in Hermitian hat. It is worth noting that the Hermitian hat is an analytical continuation of the real-valued Mexican hat to the complex domain. This is the reason for the name “hat” in Hermitian hat. The analytical continuation to the complex domain is precisely what gives the Hermitian hat the ability to detect singularities in signals. 4.4.2 Comparison to Morlet Wavelet It is interesting to analytically compare the Hermitian hat wavelet with another complex-valued wavelet, the Morlet wavelet. Consider multiplying hodd by some real number α , so that we have

h h j h j e( ) ( ) ( ) ( ) /τ τ α τ ατ τ τ= + = + − −even odd 1 2 22

(4.34) Also recall that e jjατ ατ α τ≈ + −1 22 2 / . (4.35)

If it were the case that α = 2 , then we could write

h e ej( ) /τ τ τ≈ −2 22

, (4.36) which is the form of the Morlet wavelet. However, for the Morlet wavelet we would

need α > 6 rather than α = 2 , so that its Fourier transform would nearly vanish at zero. Thus we see that while the Hermitian hat is in some respect similar to the Morlet, the two wavelets are still quite different. The Fourier transform ( )H Ω of the Hermitian hat wavelet h( )τ can be written as

H e( ) ( ) /Ω Ω Ω Ω= − −2 22

, (4.37) where Ω = ω ω/ 0 is dimensionless frequency [71]. The Fourier transform ( )H Ω is real,

which is another property of Hermitian operators. Since ( )H Ω has no complex phase,

the Hermitian hat wavelet does not affect the phase of the signal. Also, ( )H 0 0= , so that h( )τ has no bias.

138

-1 -0.5 0 0.5 1-2

-1

0

1

2

3Real Part

Imaginary Part

-0.1 -0.05 0 0.05 0.1-1

-0.5

0

0.5

1 Real Part

ImaginaryPart

Hermitian Hat Morlet

Figure 4-32: Hermitian hat wavelet vs. Morlet wavelet (single scale).

Hermitian Imaginary

Hermitian Real

Morlet Imaginary

Morlet Real

Time Time

TimeTime

Scale

Scale

Scale

Scale

Figure 4-33: Hermitian hat wavelet vs. Morlet wavelet (multiple scales).

139

Unfortunately, the Morlet wavelet has certain weaknesses in terms of digital signal processing. To be an admissible wavelet, its Fourier transform must vanish at zero. The Morlet wavelet must oscillate at least 6 times within its Gaussian window to adequately approximate this. This implies by the Nyquist theorem a relatively large number of necessary sample points. Also, being a relatively wide finite impulse response filter, it tends to smear the original singularity. The Hermitian hat wavelet has the same phase bifurcation for singularity detection as the Morlet wavelet. However, it requires only about one and a half oscillations to have its Fourier transform nearly vanish at zero. It also exhibits less smearing than the Morlet. It is therefore more attractive for processing signals digitally. 4.4.3 Target Singularity Detection for Fuzing We now apply the Hermitian hat wavelet to the detection of a signal singularity corrupted by fractal noise. The uncorrupted signal is [ ]x n x e nn n= ≤ ≤− −

00 1 256| |/ ,ϖ , (4.38)

for singularity height x0 10= , shift n0 128= , and e-folding length ϖ = 10 . This signal is shown in Figure 4-34, after having been corrupted by simulated sea clutter noise that was synthesized with our wavelet-based model.

Figure 4-34: Singularity signal corrupted by simulated sea clutter noise. Figure 4-35 shows the real and imaginary parts of the Hermitian hat wavelet transform for the signal shown in Figure 4-34. Figure 4-36 then shows the corresponding phase of the transform. We see that the phase plot is very irregular. The Hermitian hat has detected the singularities in the noise as well as in the signal. A threshold of 0.1 was

140

therefore applied to the phase to separate the signal singularity from the noise. The result is shown in Figure 4-37.

Imaginary Part Real Part

Time

ScaleScale

Time

Figure 4-35: Real and imaginary parts of Hermitian hat wavelet transform.

Figure 4-36: Phase of Hermitian hat wavelet transform.

Time

Scal

e

Figure 4-37: Phase of Hermitian hat transform after threshold.

141

4.5 Wavelet-Generated Video for Continuous Wave Radar

Radar proximity fuzing involves detecting the presence of target objects near the fuze. We now propose a novel wavelet-video based method of processing signals for fuzes that employ continuous wave radar. In proximity fuzing with continuous wave radar, interesting signal structures are localized in time. Wavelet representations are therefore ideal, since they have temporal localization. The radar proximity fuzing problem could then be seen as the recognition of any patterns among the time varying wavelet transform coefficients of the fuze signal that may indicate the presence of targets. In our scheme we place a window of fixed width over the incoming signal, so as to localize the processing near the present time. We then perform a continuous wavelet transform on the signal within the window, resulting in an image of the transform. As the signal window then moves forward in time, the corresponding sequence of transform images forms a video. From this wavelet transform video, we then extract features as input to pattern recognition algorithms such as artificial neural networks. This is shown in Figure 4-38.

SensorSignal Time

One-DimensionalWindow

Time-VaryingSignal in Window

Two-DimensionalWindow

ContinuousWavelet

Transform

Scal

e

Shift

Video FromTime-Varying

Image in Window

Moving In Time

Moving In Time

Wavelet VideoFeatures

PatternRecognitionNeural Net

Figure 4-38: Generation of wavelet transform video from time varying radar proximity fuze signal. Temporary expansion of dimensionality allows us to extract

salient features, leading to reduced computational complexity.

142

Many systems other than radar proximity fuzes have the need to detect the

existence of objects in their environment. For example, proximity sensing is widely applied in manufacturing automation and robotics. More recently, there has also been a strong interest in proximity sensing for automobile collision avoidance. Our proposed method is also applicable to processing signals in sonar sensors.

(a) (b)

Figure 4-39: Other applications of proximity sensing: (a) manufacturing automation and robotics and (b) automobile collision avoidance.

The next six sections describe our new method of continuous wave radar

proximity fuzing based on wavelet-generated video. In Section 4.5.1, we discuss the various advantages of our approach. Then, in Section 4.5.2 we show how fast wavelet denoising algorithms can dramatically improve signal quality, which leads to enhanced recognition performance. In Section 4.5.3, we show the continuous wavelet transform representation of continuous wave radar proximity signals, and contrast the representation with other wavelet and Fourier ones. Section 4.5.4 then introduces a type of pattern recognition neural network that can form an arbitrary mapping of inputs to outputs, and Section 4.5.5 describes how signal features extracted from wavelet-generated video can be used along with the neural networks to improve proximity fuzing. Section 4.5.6 then shows the performance of our new signal processing method for various degrees of Doppler signal nonlinearity and levels of corrupting noise. 4.5.1 Advantages of Our Approach

The processing of video data in real time is considered to be somewhat impractical given the current state of technology. The utility of such processing in real-world applications would therefore seem to be limited. However, recent developments at Trident Systems, Incorporated have made available real-time wavelet processing of

143

video, in the form of the WaveNet technology [73]. Also, in the future a variety of fast architectures for computing wavelet transforms will surely be developed.

Besides, in our proposed scheme, wavelet video processing is not a particular

computational hindrance, but rather allows salient features to be extracted via the wavelet coefficients. Because of the quality of the wavelet video features, it is likely that fewer numbers of inputs will be needed for pattern recognition. In this sense our scheme could be considered to be a form a data compression. In particular, it seems to be a form of data compression that is ideal for pattern recognition.

The multiresolution nature of wavelets also allows us to explore the tolerance of

imprecision in the processing of signals. This provides the freedom to tailor the design of the sensor to the resolution requirements of the signals being processed. This tolerance of imprecision is in the spirit of fuzzy logic, but in this case the imprecision is in the scale of the signal structures rather than in the membership of sets.

The important idea is that useful information in signals is generally found at the

larger scales (lower frequencies). The less useful, smaller scale signal structures can therefore be disregarded. Neglecting unnecessary details allows a reduction in the amount of data to be processed. This in turn reduces the complexity of the processing, leading to improvements in processing time, system size, and system cost.

This reduction of data through the explicit use of scale is a powerful form of data

compression. While there are several other strategies for data compression, this one has the advantage of being based on the extraction of signal features. Through wavelet transform time integration, a single coefficient provides the correlation between the signal and a wavelet at a particular scale and time shift. Wavelets are known to provide good signal features for pattern recognition algorithms such as artificial neural networks. Indeed, natural sensors such as eyes and ears carry out wavelet-type processing.

The continuous wavelet transform effectively increases the dimensionality of the

signal representation from one to two. While this might cause some concern at first glance, it is really not a problem. The reason is that the wavelet representation will be used to extract signal features only. Thus the pattern recognition neural networks need not suffer from the “curse of dimensionality.” After all, the extracted features are of a single dimension only, so that the increase in dimensionality is only temporary. Indeed, because of the high quality of wavelet features, it is quite possible that fewer features will be needed, and that recognition performance will be improved.

Mallat’s multiresolution analysis [20] leads to discrete orthogonal wavelets at dyadic scales and shifts, implemented via the efficient pyramid algorithm. These discrete wavelets have been successful in many applications, particularly data compression. However, discrete wavelets have limited utility for pattern recognition problems. This is because interesting signal structures are not constrained to follow such power-of-two patterns. In particular, discrete wavelet transform coefficients are shift-variant, which in general causes problems for pattern recognition.

144

In contrast, the continuous wavelet transform has coefficients at all scales and

shifts, not just dyadic ones. The continuous transform therefore has the desirable property of shift invariance. Another advantage of continuous wavelets is that they have less stringent requirements for admissibility, which allows a wider choice of basis functions. They also have the possibility of being basis functions for adaptive wavelet networks.

Through the inclusion of all scales and shifts, the continuous wavelet transform

effectively increases the dimensionality of the signal representation. That is, the representation is made to be a function of two variables rather than one. We note that the discrete wavelet transform introduces no such increase in dimensionality, since the number of transform coefficients is the same as the number of signal sample points. This is because the discrete wavelet transform employs an orthonormal basis rather than an overcomplete frame.

However, the fact that we are using the continuous wavelet transform coefficients

merely for feature extraction means that we need not be plagued by the curse of increased dimensionality. In particular, the goal is to use only the relatively few coefficients that provide the best features. In fact, the use of such high quality features may well mean that fewer numbers of inputs will be needed for the pattern recognition neural networks. Of course, these high quality features are also likely to improve the performance of the neural networks.

Our scheme could therefore be considered a form a data compression. The

temporary increase in dimensionality could then improve compression quality, at least when measured with respect to pattern recognition performance.

If we disregard the issue of dimensionality, it might still be argued that

computation of the discrete wavelet transform is faster, which has complexity ( )nO . However, a continuous wavelet transform implemented via the fast Fourier transform has complexity ( )nnO log , which is still quite acceptable for many applications. Also, a continuous wavelet transform has the potential for massive parallelism, and allows the possibility of adaptive wavelet bases [74]. 4.5.2 Preprocessing With Wavelet Denoising To improve performance for noisy Doppler signals, we apply Donoho’s ( )nO wavelet denoising algorithm [56]. The algorithm first does the discrete wavelet transform with Mallat’s pyramid algorithm [20]. The pyramid algorithm computes the transform for some J dyadic levels of scale, resulting in vectors of detail and smooth wavelet coefficients JJJ sdddd ,,,,, 121 − .

145

The algorithm then shrinks the detail coefficients for scales 1−≤ Jj to obtain

121

~,,

~,

~−Jddd . Here the jd

~ are

)(~

jj jjdd σλδ= , (4.39)

where )(xλσδ is a nonlinear threshold shrinkage function given by

>−≤

=.||if)|)(|(sign

||if0)(

λσλσλσ

δ λσ xxx

xx (4.40)

This threshold shrinkage function is shown in Figure 4-40.

−−−−λσλσλσλσ

λσλσλσλσ x

( )xλσδ

Figure 4-40: Nonlinear threshold shrinkage function for wavelet denoising.

The threshold shrinkage function )(xλσδ is parameterized by a threshold λ and

an estimate of the standard deviation of the noise σ . We use a universal threshold

)log(2 Nj =λ , where N is the number of data samples [56]. For σ we use the

median absolute deviation, which is a robust estimation of standard deviation. Finally, the wavelet denoising algorithm computes the inverse discrete wavelet

transform using the new coefficients JJJ sddd ,,~

,,~

11 − . This results in a non-parametric estimate of the signal without the noise. The entire wavelet denoising algorithm is shown in Figure 4-41.

For the discrete wavelet transform in the denoising algorithm, we apply a super-Haar wavelet, which is a linear superposition of shifted Haar wavelets [75]. The super-Haar scaling function ( )tφ is given by

( ) ( )

−=k

Hk ktst φφ , (4.41)

146

where ks are integer coefficients and ( )tHφ is the Haar scaling function [22], given by

( ) [ )[ )

∉∈

=.1,0,0

1,0,1

t

ttHφ (4.42)

We apply the particular super-Haar in which [ ]1,2,2,1=ks .

d1

d2

d3

d4

s 4

DWTInverseDWT

NoisySignal

DenoisedSignal

WaveletCoefficients

CoefficientShrinkage

Figure 4-41: Wavelet denoising algorithm.

(b) (c)(a)

Figure 4-42: Doppler signal denoising: (a) pure, (b) noisy, and (c) denoised.

147

We also apply the scale-dependent rule for estimating the scale of the noise that we introduced in Section 4.3. This rule handles the more general case of fractal rather than white noise. In particular, it performs better for fractal sea clutter noise.

Figure 4-42 shows pure, noisy, and denoised versions of a simulated Doppler signal. The closest-approach distance α is such that the change in frequency is nearly linear over time. We assume that the sinusoid amplitude is constant over time, which is appropriate over the short distances applicable to proximity sensing. 4.5.3 Wavelet Representation for Feature Extraction The Fourier transform is the cornerstone of signal processing. However, since it lacks time localization, it is less suited to the processing of signals whose frequencies change over time. The time-dependent (or windowed) Fourier transform localizes time by doing the transform over a window, which shifts in time. Unfortunately, the width of the window is fixed over the entire transform, which causes problems in the high-frequency limit [76].

In contrast, a wavelet transform has a window whose bandwidth varies in proportion to the center frequency of the wavelet. This is the so-called constant-Q property from electrical engineering. The result is that the wavelet transform performs time-scale processing rather than time-frequency processing. Also, wavelet transforms allow more freedom in the choice of basis, so that the basis functions can be better matched to the shape of the signal.

The wavelet transform provides the local scale of the signal over time, which for

Doppler signals is the local period or inverse of frequency. Wavelet representations of the Doppler signal are particularly necessary in the case of closely passing targets, for which the change in frequency is more abrupt. These representations are also convenient when the signal is embedded in nonstationary noise. The continuous wavelet transform [10] ( )baFw , of a signal ( )tf is given by

( ) ( ) 0,, 2/1 > −= ∞

∞−

− adta

bttfabaFw ψ . (4.43)

Here a and b are scale and shift parameters, respectively. A necessary and sufficient condition for Eq. (4.43) to be invertible is that ( )tψ satisfies the wavelet admissibility condition

( ) ∞<Ψ ∞

∞−

− ωωω d12

, (4.44)

148

where ( )ωΨ is the Fourier transform of ( )tψ . If ( )tψ has reasonable smoothness and decay at infinity, which is usually the case, the admissibility condition can be written as

( ) 0= ∞

∞−dttψ . (4.45)

Under certain conditions, it is possible to reconstruct ( )tf from samples of ( )baFw ,

taken on a hyperbolic lattice. The collection of wavelet functions ( )abt −ψ over this lattice

is then said to constitute a frame. A frame, in contrast to a basis, is an overcomplete set. This redundant representation allows more flexibility in the choice of inputs to pattern recognition neural networks. In particular, we are not constrained to the power-of-two scales characteristic of the discrete wavelet transform.

We choose for ( )tψ the real part of the Morlet wavelet [77], which is

( ) ( ) ( ) 2/0

2/ 220 cosRe ttti eteet −−− == ωψ ω , (4.46)

with 336.52ln/20 == πω , which is a standard value. The real Morlet wavelet is a

Gaussian-modulated sinusoid, which is well suited to processing sinusoidal Doppler signals. The wavelet transform with the real Morlet is similar to the time-dependent cosine Fourier transform with a Gabor [77] (Gaussian-shaped) window. This type of Fourier transform is also called the Gabor transform, and is given by

( ) ( ) ( ) ( ) ∞

∞−

−−= dtettfbF bt 2/2

cos, ωω . (4.47)

For comparison, we can write the wavelet transform in Eq. (4.43) as

( ) ( ) ( )[ ] ( )dtebttfbF

bt

w ∞

∞−

− ′−

−′=′2/

2

/0cos, ωωωω , (4.48)

where a/0ωω =′ . For the Gabor transform in Eq. (4.47), the width of the window GW ,

given by

( ) −−= 2

2

1exp btWG (4.49)

remains fixed. However, for the wavelet transform in Eq. (4.48) the window width WW ,

given by

′−−=

2

0 /2

1exp

ωωbt

WW (4.50)

149

varies inversely with the frequency a/0ωω =′ . Thus the frequency bandwidth of the

wavelet window varies in proportion to ω′ , through the inverse scaling property of Fourier conjugate variables.

Also, the cosine term ( )[ ]bt −′ωcos for the wavelet transform shifts in time along with the window, through the shift parameter b . In contrast, for the Gabor transform only the window shifts in time, and the cosine term remains fixed. A comparison of the kernels for the Gabor and Morlet transforms is shown in Figure 4-43.

e j t− ω0 e j t− 2 0ω e j t− 4 0ω

( )ψ t ( )ψ t / 2 ( )ψ t / 4

Morlet

Gabor

Figure 4-43: Kernels for Gabor and Morlet wavelet transforms.

Figure 4-44 shows the continuous Morlet wavelet transforms of the three signals in Figure 4-42. Figure 4-45 shows the same three transforms, using a surface plot rather than a grayscale image. The wavelet transforms show the increase in local signal scale over time. In this case the increasing signal scale is the increasing period of the frequency-modulated sinusoid.

To some extent the time-scale structure of the Doppler signal is visually apparent in the transform of the noisy signal. However, if samples of the noisy transform are used as neural network inputs for proximity sensing, the high-frequency fluctuations result in poor performance. These fluctuations are largely removed by the wavelet denoising, which results in much improved performance.

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Shift b

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e a

(a) (b) (c)

Sca

le a

Shift b

Figure 4-44: Grayscale images for continuous wavelet transforms of Doppler signals: (a) pure, (b) noisy, and (c) denoised.

Shift b

Scale aScale a

Scale a

Shift bShift b

(a) (b) (c)

Figure 4-45: Surface plots for continuous wavelet transforms of Doppler signals: (a) pure, (b) noisy, and (c) denoised. 4.5.4 Artificial Neural Networks for Pattern Recognition The continuous wavelet transform correlates a Doppler signal with time-localized wavelets at various scales and shifts. It gives the change in local signal scale over time, which in this case is the Doppler period or inverse frequency. When a moving window is placed on the incoming Doppler signal and the windowed signal is wavelet transformed, the corresponding time-varying transform imagery constitutes video. Samples of this wavelet-generated video over time then form signal features for pattern recognition neural networks. These networks are then trained to extract the Doppler frequency shift over time. This frequency shift is critical information for proximity sensing.

The continuous wavelet transform constitutes a frame rather than a basis. Such a redundant representation allows more flexibility in the selection of signal features. In

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terms of the most efficient signal representation, these features should be orthogonal. However, such a representation in which the features are completely independent is less robust with respect to noise immunity and fault tolerance. The search for the best representation is therefore a tradeoff between redundancy and robustness [78].

We extract the Doppler shift with feedforward multilayer neural networks, known as multilayer perceptrons [79]. After computing the continuous wavelet transform of the denoised Doppler signal, we sample the transform coefficients to provide inputs for the multilayer perceptrons. The networks are trained with the Levenberg-Marquardt [80][81] rule to provide the Doppler shift at a given time. This rule is a powerful generalization of gradient descent that employs an approximation of Newton’s method. It is much faster than standard gradient descent algorithms such as backpropagation, although it does require more memory.

Figure 4-46 shows the neural network architecture we employ for Doppler

frequency estimation. The network is comprised of 3 layers of artificial neurons: an input layer, a middle or hidden layer, and an output layer. Signals flow forward through the network, that is, from input layer to hidden layer to output layer. This architecture is known as a multilayer feedforward network, or multilayer perceptron.

ΣΣΣΣ

σσσσ

σσσσ

σσσσ

Weights forhidden neurons

Inputs

Weights foroutput neuron

Hiddenneurons

(sigmoidal)

Outputneuron(linear)

Figure 4-46: Neural network architecture for proximity sensing pattern recognition.

The input neuron layer in Figure 4-46 performs no processing; it merely provides means for coupling the input vectors to the hidden layer. The neurons in the middle layer sum the weighted network inputs, along with an internal bias for each neuron, then apply the nonlinear sigmoidal activation function

( ) ( )( )j

j

v

vj

je

evv −

−

+−=

=1

1

2tanhσ , (4.51)

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where jv is the weighted sum for neuron j . This sigmoidal nonlinearity limits the

neuron outputs to (-1,1). The single output neuron computes the weighted sum of the outputs of the hidden neurons, along with its internal bias, without applying the sigmoidal function.

The architecture in Figure 4-46 is known to be a universal function approximator, that is it can represent an arbitrary function arbitrarily well, given a sufficient number of neurons in the hidden layer [79]. The particular function mapping that the network performs is determined by the values of the weights between neuron layers and the internal neuron biases. Various learning algorithms exist for computing the network weights and biases for a given problem. The most popular learning algorithm is backward error propagation, which attempts to minimize the squared error of the network through gradient descent in weight space [79]. We can define the error signal for neuron j as

( ) ( ) ( )nyndne jjj −= , (4.52)

where n indexes the training vectors, ( )nd j is the desired response for neuron j , and

( )ny j is the actual output of neuron j . The instantaneous value of the sum of squared

errors ( )ne j2

2

1 over all neurons in the output layer of the network can then be written as

( ) ( )

∈

=Cj

j nen 2

2

1E , (4.53)

where the set C includes all neurons in the output layer and N is the number of vectors in the training set. The squared error averaged over all training vectors is then

( )

==

N

n

nN 1

av

1 EE . (4.54)

The average squared error avE constitutes a cost function that is to be minimized. It is

minimized approximately by iteratively reducing ( )nE for each training vector. The

correction ( )nw ji∆ to be applied to weight ( )nw ji is then defined by the delta rule

( ) ( )( )nw

nnw

jiji ∂

∂η E−=∆ . (4.55)

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Here η is a parameter that determines the rate of learning. The minus sign in Eq. (4.55) results in gradient descent in weight space, that is weights are moved in the opposite direction of the error gradient.

We apply a powerful generalization of backward error propagation known as the Levenberg-Marquardt weight update rule [80][81]. This rule can be written in matrix notation as

( ) eJIJJW T1T −+=∆ µ , (4.56)

where W∆ is the matrix of weight updates, e is the error vector, and J is the Jacobian matrix of derivatives of each error to each weight. When the parameter µ is very large, Eq. (4.54) approximates gradient descent, and if µ is small it becomes the Gauss-Newton method.

The Gauss-Newton method is faster and more accurate near an error minimum. The idea is therefore to shift towards Gauss-Newton as quickly as possible. The parameter µ is thus decreased after each successful step, and increased only when a step increases the error. The Levenberg-Marquardt update rule is known to train networks much more quickly than standard backward error propagation. However, it does require more memory, usually a factor of NC * more, where C is the number of output neurons and N is the number of training vectors. 4.5.5 Signal Features from Wavelet-Generated Video We now demonstrate the improvement in pattern recognition performance that wavelet-generated video features can provide. We begin by showing how wavelet transform features outperform both time-domain and frequency-domain ones for classifying signals according to frequency. In particular, we test the ability of pattern recognition neural networks to classify signals as either being either above or below a certain threshold frequency, in the presence of noise. Figure 4-47 shows pattern recognition performance using 3 different signal representations for neural network input: wavelet transform coefficients, time-domain samples, and Fourier transform coefficients. A variety of frequencies were used for the test signals, equally distributed about the threshold frequency. The networks were trained to determine whether the signal frequencies were below (output of zero) or above (output of one) the threshold. Because this problem is binary, the networks were made to have sigmoidal rather than linear activation functions. The noise was white Gaussian, with signal-to-noise ratio of –1 dB.

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AboveThreshold

(Output = 1)

BelowThreshold

(Output = 0)

ThresholdFrequency

ThresholdFrequency

ThresholdFrequency

(a) (b) (c)Frequency

NetworkOutput

Frequency

NetworkOutput

Frequency

NetworkOutput

Figure 4-47: Frequency classification using (a) wavelet, (b) time-domain, and (c) frequency-domain features. We see from Figure 4-47 that classification performance for wavelet features is better than for either time-domain or frequency-domain features. In particular, for the time-domain features there are many misclassifications at the lowest frequencies, and at frequencies just above the threshold. Also, for the frequency-domain features there are many misclassifications near the threshold. If we look more carefully at Figure 4-47, we see that at the highest frequencies, and at frequencies just below the threshold, performance is slightly better for time-domain features than for wavelet features. Also, at the lowest and highest frequencies, performance is slightly better for frequency-domain features than for wavelet features. Interestingly, it appears that the wavelet transform has formed a compromise between the time and frequency domains in which overall classification performance is improved. Now that we have demonstrated the superior frequency classification performance of wavelet features, we can investigate which wavelet transform coefficients might provide the best features for estimating time varying Doppler shift. A fundamental issue is whether to sample from a single time shift of the transform, or to sample over multiple shifts. Sampling at a single time shift makes the signal representation a function of scale and time only. Conversely, sampling over multiple shifts makes the representation a function of scale, shift, and time. This latter representation is true video, that is, time-varying imagery.

Sampling from a single shift completely localizes time, which may be advantageous in some applications. Sampling over multiple shifts gives additional information that may improve estimation performance. Also, sampling over multiple shifts provides a degree of redundancy that will likely improve performance for noisy signals. As a test of single-shift versus multiple-shift features, we used each type of feature as input to a pattern recognition neural network. For single-shift features, we used 32 samples of the continuous wavelet transform of the Doppler signal, taken over

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various scales at a single time shift. For multiple-shift features, we used 16 samples of the transform at the original time shift, and 16 more samples at an additional time shift of 6. Figure 4-48 shows the sampling scheme for the multiple-shift case.

NN input vectors NN input vectors

16 scalesamples

16 scalesamples

from timeshift of 6

Training inputs(pure signal, every

4th time shift)

Testing inputs(distor ted signal,

all time shifts)

Figure 4-48: Samples of wavelet-generated video for training and testing inputs to pattern recognition neural networks.

We trained the neural networks with samples of transforms of pure Doppler signals, sampling only every 4th time shift of the transform. For training outputs, we supplied the known instantaneous frequency of the pure signals for each time shift. Thus the networks were trained to estimate the instantaneous frequency of the Doppler signals, given samples of their wavelet transform.

After training for frequency estimation, we tested the networks with transforms of denoised versions of noisy Doppler signals. The networks were tested for every time shift of the transform, with a signal-to-noise ratio of –2 dB. Figure 4-49 shows the test results. It is obvious that performance is much better for the case of sampling wavelet coefficients over multiple time shifts.

We again point out that sampling over multiple time shifts versus sampling at only a single shift constitutes true image sampling, since both scale and shift variables (two dimensions) are sampled. When the Doppler signal window is then moved forward in time, we have sequence of images over time, that is, we have video. In our simulations, we sample from this wavelet-generated video. In particular, for each wavelet transform image in the sequence that forms video, the samples provide an estimate, via neural networks, of the instantaneous Doppler shift corresponding to the image.

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Time

Dop

pler

Shi

ftE

stim

atio

n

(a) (b)

Time

Dop

pler

Shi

ftE

stim

atio

n

Figure 4-49: Estimation performance for time varying Doppler shift: (a) sampling at single scale of wavelet transform, and (b) sampling from wavelet generated video. Smooth lines show true Doppler shift over time. 4.5.6 Per formance for Time Varying Doppler Shift We have just demonstrated the effectiveness of features extracted from wavelet-generated video. We now test the pattern recognition performance of such features in the estimation of time varying Doppler shift from noisy sensor signals. In particular, we sample the wavelet video as shown in Figure 4-48, use the samples as neural network inputs, and then train the networks to estimate the Doppler shift. We test performance for different degrees of nonlinearity in the Doppler shift over time, as well as for different levels of noise. Figure 4-50 shows network test results for various signal-to-noise ratios, where the Doppler signal frequency decreases nearly linearly over time, corresponding to a relatively large closest-approach distance α . The networks were tested for every time shift of the wavelet transform. Since the networks were trained with only every 4th shift, this shows their ability to generalize to other frequencies.

We see from Figure 4-51 that the frequency estimation performance of the neural networks is overall good. The estimation performance degrades as the signal-to-noise ratio decreases, as would be expected. We point out that the selected signal-to-noise ratios of -0.5 dB, -2 dB, and –4 dB represent very noisy signals. We choose these as representative of the extremely noisy signals that may occur for challenging targets in proximity fuzing.

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Time

Fre

quen

cy

(a)

Time

Fre

quen

cy

(b)

Time

Fre

quen

cy

(c)

Figure 4-50: Estimation of Doppler shift using features from wavelet-generated video and pattern recognition neural networks (nearly linear change in shift over time): (a) signal-to-noise ratio of –0.5 dB, (a) signal-to-noise ratio of –2 dB, (a) signal-to-noise ratio of –4 dB.

Figure 4-51 shows neural network frequency estimation performance for smaller closest-approach distance α . A smaller value of α results in a more pronounced nonlinearity in the frequency shift over time. Network performance is similar to that for the nearly linear case. The difference is that the estimation variance is increased in the region of lowest and highest frequencies. Analytically, for a signal in which the frequency content is constant over time, and assuming white noise, the signal-to-noise enhancement of our proposed method of

processing is a factor of N , where N is the number of video frames. However, in our previous experience, for signals with time varying frequency components, or when the

noise is nonstationary, an improvement exceeding this N can be expected. A more detailed analysis is necessary to further quantify this.

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Time

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cy

(a)

Time

Freq

uenc

y

(b)

Time

Fre

quen

cy

(c)

Figure 4-51: Estimation of Doppler shift using features from wavelet-generated video and pattern recognition neural networks (nonlinear change in shift over time): (a) signal-to-noise ratio of –0.5 dB, (a) signal-to-noise ratio of –2 dB, (a) signal-to-noise ratio of –4 dB.