polar fourier transform processing of marine radar signals

Polar Fourier Transform Processing of Marine Radar Signals

DAVID R. LYZENGA

Naval Architecture and Marine Engineering Department, University of Michigan, Ann Arbor, Michigan

(Manuscript received 10 August 2016, in final form 18 October 2016)

ABSTRACT

This paper describes a method of processing marine radar signals for the purpose of generating phase-

resolved surface elevationmaps as well as statistical measures of ocean surfacewave fields. Themethod is well

suited to the processing of data collected by marine radars because it allows for the incorporation of effects

dependent on the radar look direction relative to the propagation direction of ocean waves. Applications to

Doppler radar and backscattered power measurements are described, and example results are presented

using simulated radar data.

1. Introduction

Marine radars are mainly used for navigational pur-

poses, such as determining the location of coastlines and

hazards to navigation, and backscattered signals from

the ocean surface itself are undesirable for these pur-

poses. For that reason, horizontally polarized antennas

are used and methods of suppressing ‘‘sea clutter’’ sig-

nals are built into most commercial systems. However,

these signals also contain potentially useful information

about the sea state. The earliest attempts to quantify

and utilize this information involved the use of photo-

graphically recorded radar images, which were then

digitized and processed using rectangular Fourier

transform techniques (Young et al. 1985). Later, the

development of fast analog-to-digital conversion de-

vices allowed ‘‘raw’’ radar signals to be recorded (Nieto

Borge et al. 1999; Dankert and Rosenthal 2004). These

signals consist of the time histories of the backscattered

signals received from each transmitted pulse. The pulses

are transmitted and received from an antenna having a

narrow beamwidth in the horizontal direction, while the

antenna rotates about a vertical axis. Thus, the back-

scattered signals are resolved in the radial direction by

time-of-flight ranging methods and in the azimuthal di-

rection by recording the antenna rotation angle for each

transmitted pulse.

For conventional radars the signal is proportional to

the backscattered power. However, these radars can be

modified to measure the Doppler shift of the back-

scattered signals as well (e.g., Smith et al. 2013). At near

range, the backscattered power is primarily a function of

the radial component of the surface slope, although it

may also depend on other geophysical parameters. The

Doppler shift is a more direct measurement of the radial

component of the surface velocity. In both cases the

signal associated with a given wave component falls off

for look directions that are not parallel to the wave

propagation direction.

A methodology was proposed by Nwogu and Lyzenga

(2010) for obtaining surface elevationmaps fromDoppler

radar measurements by integrating the radial velocity in

the range direction to obtain an estimate of the velocity

potential function. The time derivative of the velocity

potential function can then be computed by frame dif-

ferencing to obtain the surface elevation using the dy-

namic free-surface boundary condition. To the extent

that the radial component of the surface slope can be

estimated from conventional marine radar images, the

same procedure could be used to estimate the surface

elevation by integrating the radial slope. The resultsmust

be interpreted with caution, however, because the qual-

ity of the estimate varies with position within the image.

To illustrate this, consider the case of a surface wave

or Fourier component with amplitude a and wave-

number k, propagating in the y direction. This wave

component can be described by the velocity potential

function

F(x, y, t)5Rehagiv

ei(ky2vt)i, (1)Corresponding author e-mail: David R. Lyzenga, lyzenga@

umich.edu

FEBRUARY 2017 LYZENGA 347

DOI: 10.1175/JTECH-D-16-0158.1

� 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS CopyrightPolicy (www.ametsoc.org/PUBSReuseLicenses).

Unauthenticated | Downloaded 02/15/22 01:02 AM UTC

mailto:[email protected]

mailto:[email protected]

http://www.ametsoc.org/PUBSReuseLicenses



where v is the wave frequency and g is the gravitational

acceleration. The surface elevation is given by the dy-

namic free-surface boundary condition as

h(x, y, t)521

gF

t5Re[aei(ky2vt)] , (2)

and the components of the surface slope in the x and y

directions are hx 5 0 and

hy(x, y, t)5Re[iakei(ky2vt)] , (3)

respectively. The corresponding components of the

surface velocity are u5Fx 5 0 and

y(x, y, t)5Fy5Re

�agk

vei(ky2vt)

�. (4)

The radial components of the surface slope and velocity

are given by

hr5h

ycosf and F

r5F

ycosf , (5)

respectively, where f is the radar look direction relative

to the y axis. Measurements of hr and Fr also include a

random component, denoted by «s and «u, respectively,

due to system noise as well as contributions from other

waves. Thus, we can model the radar signals as

~hr(x, y, t)5Re[iakei(ky2vt)] cosf1 «

s(6)

for the ‘‘conventional’’ radar case, and

~Fr(x, y, t)5Re

�agk

vei(ky2vt)

�cosf1 «

u(7)

for the Doppler radar case. It is apparent that the signal-

to-noise ratio goes to zero as f approaches p/2 in both

cases. This is not a problem in principle for the estima-

tion of wave spectra, provided there are no obstructions

in the radar field of view, but it means that there are

locations in the radar image where the wave field cannot

be accurately measured regardless of the method used.

Furthermore, if there are waves propagating over a wide

range of angles, there is no location in the image where

the instantaneous wave field is accurately measured.

The solution, as discussed below, is to use the mea-

surements to predict the wave field some tens of seconds

into the future.

2. Polar Fourier transform processing

Fourier methods are useful not only for computing the

wave spectrum but also for predicting the phase-resolved

wave field, because each Fourier component propagates

at a different velocity. Conventionally, this has been

done by resampling a portion of the data into rectan-

gular coordinates and using a 2D FFT. However, an

alternative method for computing the Fourier transform

turns out to be especially useful for processing marine

radar data. Consider the ordinary two-dimensional

Fourier transform of the function f (x, y),

F(kx, k

y)5

ðLx

2Lx

ðLy

0

w(x, y)f (x, y)e2i(kxx1kyy) dx dy, (8)

where w(x, y) is a weighting or aperture function. The

Fourier transform along the ky axis (kx 5 0) is given by

F(0, ky)5

ðLx

2Lx

ðLy

0

w(x, y)f (x, y)e2ikyy dx dy. (9)

This calculation can be repeated for other coordinate

system orientations, as illustrated in Fig. 1, and the re-

sults can be written as

F(k,f)5

ðLx

2Lx

ðLy

0

w(x, y)f (x, y)e2iky dx dy, (10)

where f represents the orientation of the y axis relative

to north. This integral is referred to in the following as

the polar Fourier transform (PFT). It can be evaluated

numerically by various methods, for example by pro-

jecting the input data onto the y axis and Fourier

transforming in the y direction using an FFT.

FIG. 1. Rotated coordinate system for wave measurements.

348 JOURNAL OF ATMOSPHER IC AND OCEAN IC TECHNOLOGY VOLUME 34


Using the weighting function w(x, y)5xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

(x2 1 y2)p 5

cosf0, the Fourier transform of the y component of the

surface velocity can be written as

Fu(k,f, t)5

ðLx

2Lx

ðLy

0

w(x, y)Fy(x, y, t)e2iky dx dy

5

ðLx

2Lx

ðLy

0

cosf0Fr(x, y, t)e2iky dx dy. (11)

The cosf0 weightingmatches the falloff in the envelope of

the radial velocity signal, as shown in (7), and therefore

minimizes the contribution of noise to the Fourier trans-

form (see, e.g., Turin 1960). Assuming that the weighting

function varies much more slowly than the surface eleva-

tion, the derivative of w can be neglected in comparison

with that ofF, and (11) can be integrated by parts to yield

Fu(k,f)5 ik

ðLx

2Lx

ðLy

0

w(x, y)F(x, y)e2iky dx dy. (12)

Thus, we can estimate the Fourier transform of the ve-

locity potential function as

Fp(k,f)[

ðLx

2Lx

ðLy

0

w(x, y)F(x, y)e2iky dx dy

51

ikFu(k,f, t). (13)

Computing the time derivative of the result then pro-

duces an estimate of the Fourier transform of the surface

elevation. To compute the time derivative, we note that

the amplitude of the Fourier transform varies more

slowly than its phase, so the time derivative ofFu(k, f, t)

can be approximated as

›

›tFu(k,f, t)’ ivF

u(k,f, t), (14)

where v is the frequency of the wave component with

wavenumber (k, f). This frequency can in turn be esti-

mated as

v5 arg[C(k,f, t)]/Dt , (15)

where C(k, f, t) is the conjugate product of successive

frames, that is,

C(k,f, t)5Fu*(k,f, t2Dt)F

u(k,f, t), (16)

and Dt is the time interval between frames, or the an-

tenna rotational period. Frequency ambiguities occur

when the phase of the conjugate product C(k, f, t) is

near p. Two methods have been investigated for

removing these ambiguities. The first method is to

‘‘unwrap’’ the phase of C(k, f, t), by stepping through k

at each f, and adding or subtracting 2p to the phase

whenever the change in phase (from k to k1Dk) is

smaller than2p or larger than p. The second method is

to compute the group velocity,

dv

dk5

1

iDtC

n*(k,f, t)

›Cn

›k, (17)

where Cn(k, f, t) is the normalized conjugate product,

that is,

Cn(k,f, t)5

C(k,f, t)

jC(k,f, t)j5 exp(ivDt) . (18)

The group velocity is not subject to ambiguity prob-

lems, although it is sensitive to noise. The sign of the

group velocity can thus be used to determine the wave

propagation direction for each wavenumber, that is,

to discriminate between approaching and receding

waves.

Combining (13) and (14), and using the dynamic free-

surface boundary condition (2), the PFT of the surface

elevation can be estimated as

Fh(k,u, t)52

v

gkFu(k,f, t). (19)

Wavenumber samples with v, 0 or ›v/›k, 0 are reset

to zero to produce a one-sided PFT containing only

approaching waves.

It should be noted that for typical marine radar an-

tenna heights, Doppler signals become noisy and diffi-

cult to interpret at range distances larger about 1 km due

to wave shadowing effects. At near range, the signals

may be corrupted by returns from the ship itself. To

remove these effects, the observed velocities are set to

zero for range distances less than a few hundred meters

and larger than a kilometer or so. The consequences of

this are discussed below.

3. Surface elevation spectrum

The wave height variance spectrum is conventionally

defined as

S(k,f)51

(2p)2AhjF

h(k,f)j2i, (20)

where A is the effective measurement area. This defi-

nition makes the integral of the spectrum over all

wavenumbers equal to the variance of the surface ele-

vation. In our case, the measurement area is limited by



the minimum and maximum range distances (r1 and r2)

over which useable Doppler signals can be measured

and is modified by the square of the weighting function

w(x, y) discussed above. Since the Fourier transform is

squared to obtain the spectrum, the effective area for

our measurement is

A5

ðr2r1

ðp/22p/2

cos4f0r dr df0 5 3p(r22 2 r21)/16. (21)

In addition, when the wave frequency or group velocity is

used to remove receding waves, the energy or variance in

the spectrum is reduced by a factor of 2. The one-sided

spectrum is therefore doubled in order to make the in-

tegral over all wavenumbers equal to the elevation vari-

ance. Thus, the surface elevation spectrum is calculated as

S(k,f)58

3p3(r22 2 r21)hjF

h(k,f)j2i. (22)

4. Surface reconstruction

The surface elevation is obtained by inverse Fourier

transforming Fh(k, f, t), but the effect of the weighting

function must be taken into account in the interpretation

of the results. The inverse transform can be calculated by

resampling Fh(k, f, t) onto a rectangular coordinate sys-

tem (kx, ky), where kx 5 k sinf and ky 5 k cosf, that is,

h(x, y, t)51

2p2

ððFh(k

x,k

y, t)ei(kxx1kyy) dk

xdk

y. (23)

For simplicity, continuous notation is used here with

unspecified limits of integration but, in fact, the inverse

transform is evaluated as a sum over the available

wavenumber samples. Alternatively, the inverse trans-

form can be computed in polar wavenumber space as

h(r,f0, t)51

2p2

ððFh(k,f, t)eikr cos(f2f0)k dk df. (24)

The former procedure is advantageous if the surface

elevation is to be computed on a rectangular grid, since

FFT methods can be used to evaluate (23), but the sec-

ond procedure may be useful if the elevation is needed

over only a few points, since (24) can be evaluated di-

rectly without resampling the Fourier transform data.

In either case, the surface can be evaluated at times

other the measurement time by appropriately adjusting

the phases of the Fourier coefficients, that is,

h(x, y, t0)51

2p2

ððFh(k

x, k

y, t)ei[kxx1kyy1v(t02t)] dk

xdk

y.

(25)

The Fourier transform (Fh) was calculated using the

effective weighting or aperture function,

w2(r,f0)5 cos2(f2f0)rect(r, r1, r

2), (26a)

where

rect(r, r1, r

2)5

�1 r

1, r, r

2

0 elsewhere. (26b)

This weighting also appears in the inverse transform,

which means that different wave components will appear

in different parts of the image. Thus, for example, if there

is a locally generated wave field propagating eastward

and a swell propagating northward (a typical situation for

coastal California), the wind waves will appear in the

western portion of the reconstructed wave field and the

swell will appear in the southern portion of the image.

However, if the reconstructed wave field is propagated

forward in time, these waves will converge in the region

near the radar to produce a more complete and accurate

representation of the wave field in that region than is

contained in a reconstruction at the time of the mea-

surement. This effect is illustrated in the simulation results

presented in the following section.

5. Simulation results

Numerical simulations are a useful supplement to field

testing for validation purposes, since environmental

conditions, radar system parameters, and noise levels

can be arbitrarily changed. Also, estimated variables can

be directly compared with the assumed inputs, so that

measurement errors are not an issue. In fact, there are

no alternative technologies for full-scale spatial and

temporal measurements of ocean surface wave fields, so

empirical validations are limited to point measurements

(as provided by wave buoys) or two-dimensional snap-

shots (as provided by airborne lidar, for example). The

value of simulations depends on the accuracy and

comprehensiveness of the models used for generating

the simulated signals, so field measurements are also

required to fully evaluate the methodology. In the fol-

lowing, the methodology is illustrated for one set of

conditions. Further insights could be gained by system-

atically varying the environmental conditions and radar

system parameters, but a comprehensive study of that

sort is beyond the scope of the present paper.

The simulation model used here contains many approx-

imations but appears to adequately reproduce the statistics

of the observedmarine radar signals at vertical polarization.

First, a time-evolving ocean surface is generated by

selecting a set of Fourier coefficients consistent with a



JONSWAP-type spectrum. The wavenumber magni-

tude and direction of each coefficient are chosen at

random by treating the spectrum as a probability

density function, and the amplitude of each is chosen

from a Rayleigh distribution with a mean equal to the

square root of the spectral density at the chosen

wavenumber. The phases are chosen from a uniform

distribution and are advanced between time samples.

The surface elevation and radial velocity are then

calculated at each output range and azimuth sample

location by summing over the Fourier coefficients.

The effects of unresolved waves are simulated by

adding a random component to the calculated radial

velocity, the random samples being selected from a

Gaussian distribution with a standard deviation

sy 5 (2ktc)21, where k5 2p/l is the electromagnetic

wavenumber and tc is the coherence time.

To model coherent speckle effects, the received

power is calculated as p52p log(z), where z is a

uniformly distributed random variable. The expected

value of the backscattered power is calculated as

p5 SNR(r/ro)23, where SNR is the signal-to-noise ratio

at the range distance ro. The assumed r23 dependence

of the mean backscattered power is roughly in accord

with our (vertically polarized) observations, although

the exact falloff depends on the wind speed and wave

conditions.

Wave shadowing is included by computing the local

depression angle q5 sin21[(h2h)/r] at each range

sample location, where h is the antenna height, h is the

surface elevation, and r is the range distance. Starting at

near range, the depression angle at each sample is

compared with the minimum angle encountered so far.

If q is greater than qmin, then the current point is con-

sidered to be geometrically shadowed, otherwise qmin is

set to the current value. At the shadowed locations, the

received power is set to zero.

Combining these effects, the complex received signal

is calculated as

s(t)5ffiffiffip

pexp(i2kyt)1 «

n, (27)

where y is the radial velocity and «n is a complex nor-

mally distributed random variable with zero mean and

unit variance representing the effects of thermal noise.

The Doppler frequency is defined as the rate of change

of the phase of this signal, that is,

vd5 arg[s*(t2 dt)s(t)]/dt , (28)

where dt5 1/PRF is the time between pulses. Dividing

the Doppler frequency by 2k results in the estimated

radial velocity that is used as the input data for the

methodology described in this paper.

The performance of the processing methodology un-

der bimodal wave conditions is illustrated by combining

two sets of Fourier coefficients: the first set is selected

TABLE 1. Radar parameters used in the simulations.

Radar frequency 9.41GHz

Pulse repetition frequency 2000. Hz

Range resolution 10. m

Range sample spacing 3.75m

No. of range samples 1024

Horizontal beamwidth 1.8Azimuth sample spacing 1.8Antenna rotational period 2.5 s

Antenna height 30. m

Near-range SNR 50. dB

Signal coherence time 0.01 s

FIG. 3. Square root of the estimated wave height spectrum.

FIG. 2. Square root of the input wave height spectrum.



from a JONSWAP spectrum with a peak wavelength of

240m and a peak wave direction of 1808T, and the sec-

ond has a peak wavelength of 100m and a peak direc-

tion of 3158T. The coefficients were scaled to produce

a significant wave height of 1m for each mode. The

2D surface elevation spectrum calculated from these

coefficients is shown in Fig. 2 (note this plot indicates

the direction waves are coming from, rather than

propagating toward).

These coefficients were used, along with the radar

parameters shown in Table 1, to generate a simulated

Doppler radar dataset consisting of 200 frames, each

frame corresponding to one antenna rotation or 2.5 s of

data. The radar parameters in Table 1 were chosen to

represent the radar system used in previous field studies

(Lyzenga et al. 2010; Lyzenga andNwogu 2010; Lyzenga

et al. 2015). The simulated dataset was then processed

using the procedures described in sections 2 and 3. The

PFT of the Doppler data, Fu(k, f, t), was calculated

over range samples 16–256 (60–960m). The wave fre-

quency was calculated for each wavenumber sample and

was exponentially filtered using a filter length of four

frames. The PFT of the radial velocity was converted

into the one-sided PFT of the surface elevation,

Fh(k, f, t), using the group velocity criterion ›v/›k, 0

to eliminate receding waves.

The wave height spectrum was calculated from (22)

and averaged over the first 12 frames, with the result

shown in Fig. 3. The estimated spectrum agrees well

with the input spectrum (Fig. 2), making allowances for

the expected spectral blurring due to the finite measure-

ment region. In particular, the correct propagationdirection

FIG. 4. (a) Instantaneous surface elevation calculated from the

first radar frame, at t5 t0. (b) Surface elevation calculated from the

same data at t 5 t0 1 30 s.

FIG. 5. (a) Surface elevation calculated from the first radar

frame, at t 5 t0 1 60 s. (b) Input surface elevation at the same

time as (a).



is selected for both wave systems, and the significant wave

height is reproduced to within a few percent.

The phase-resolved surface elevation was then cal-

culated from (25) at various times, as shown in Figs. 4

and 5 . Figure 4a shows the surface elevation calculated

from the first radar frame, at the measurement time for

this frame (t0). This plot shows the effect of the

weighting function discussed at the end of section 4, with

the longer waves appearing at the bottom of the image

and the shorter wave system appearing on the left side.

Figure 4b shows the surface elevation calculated from

the same data but 30 s later, that is, at t5 t0 1 30 s.

Figure 5a shows the surface elevation at t5 t0 1 60 s, and

Fig. 5b shows the input surface elevation at the same

time. At 30 s both wave systems have propagated closer

to the origin, and at 60 s both systems have converged to

produce a reasonably complete reconstruction of the

wave field near the center of the image (i.e., the radar

location).

A more quantitative comparison of the input and

output surface elevations at the radar location is shown

in Fig. 6. In this case data from all 200 frames are used,

and the surface elevation is calculated at a time corre-

sponding to 60 s after the collection time (this is referred

to as the forecast horizon in the following discussion).

The solid line in this plot is the surface elevation calculated

from the simulated radar data, and the dashed line is the

‘‘actual’’ or input surface elevation at the same time. The

correlation coefficient between the input and estimated

surface elevations is 0.96, and the rms error is 0.10m for

these data.

These calculations were repeated for values of the

forecast horizon from 0 to 120 s, and the correlation

coefficient was calculated for each case, with the results

shown in Fig. 7. Also shown is the ratio of the rms sur-

face elevations for the estimated and input data at the

radar location, and a quantity referred to as the average

weighting function. The latter is obtained by tracing

each wave component (Fourier coefficient) back to its

location at the time of the measurement, computing the

weighting function (26a) at that location, and summing

over all wave components. The value of this average

weighting function tracks the correlation coefficient

fairly well. The significance of this is that the weighting

can be computed independently of any surface truth

data, and it can be used to predict the locations and

times where the wave field is expected to be accurately

reconstructed.

6. Summary and conclusions

Shipboard Doppler radar data can be processed us-

ing the methods described in this paper to produce

estimates of the directional wave spectrum and to re-

construct the wave surface in the vicinity of the ship.

Because the surface may not be observable at very

near range due to interference from the ship itself,

short-term predictions are frequently more accurate

than instantaneous reconstructions at the ship’s loca-

tion. The spatiotemporal region over which the pre-

dictions are valid can be computed by spatially

translating the PFT weighting function using the wave

group velocity, and averaging over all wave compo-

nents. The maximum prediction interval, or forecast

horizon, depends on the maximum range of the

Doppler measurements, which is ultimately limited by

wave shadowing and azimuthal resolution effects. The

FIG. 6. Input and radar-estimated surface elevation at ship.

FIG. 7. Correlation coefficient, rms ratio of input and estimated

surface elevations, and mean weighting function at ship.



maximum range is on the order of 1 km for typical

antenna heights. The maximum range for back-

scattered power signals is larger, because wave shad-

owing effects produce strong modulations of the

backscattered power and are therefore beneficial

rather than detrimental for wave imaging. However,

the relationship between the wave height and the

backscattered power—that is, the modulation transfer

function—is not always predictable. The solution may

be to use near-range Doppler measurements to de-

termine the statistics of the wave field and use these

statistics to infer the modulation transfer function for

the backscattered power.

Acknowledgments. This work was supported by

ONR contracts N00014-05-1-0537 and N00014-11-D-0370.

The author thanks Prof. Robert Beck for coordinating

these projects and Dr. Okey Nwogu for his insights and

discussions on the subject matter of this paper. Prof. Joel

Johnson and colleagues are also thanked for their experi-

mental contributions to both projects.

REFERENCES

Dankert,H., andW.Rosenthal, 2004:Ocean surface determination

from X-band radar-image sequences. J. Geophys. Res., 109,

C04016, doi:10.1029/2003JC002130.

Lyzenga, D. R., and O. G. Nwogu, 2010: Shipboard radar mea-

surements of ocean waves for real-time prediction of nonlinear

ship motions. 28th Symposium on Naval Hydrodynamics 2010,

Vol. 2, Office of Naval Research, 1164–1177.

——, ——, D. Trizna, and K. Hathaway, 2010: Ocean wave field

measurements using X-band Doppler radars at low grazing

angles. 2010 IEEE International Geoscience and Remote Sens-

ing Symposium: Proceedings, IEEE, 4725–4728, doi:10.1109/

IGARSS.2010.5650065.

——, ——, R. F. Beck, A. O’Brien, J. T. Johnson, A. dePaolo, and

E. Terrill, 2015: Real-time estimation of ocean wave field from

marine radar data. 2015 IEEE International Geoscience and

Remote Sensing Symposium: Proceedings, IEEE, 3622–3625,

doi:10.1109/IGARSS.2015.7326606.

Nieto Borge, J. C., K. Reichert, and J. Dittmer, 1999: Use of nau-

tical radar as a wave monitoring instrument. Coastal Eng., 37,331–342, doi:10.1016/S0378-3839(99)00032-0.

Nwogu, O., and D. Lyzenga, 2010: Surface-wavefield estimation

from coherent marine radars. IEEE Geosci. Remote Sens.

Lett., 7, 631–635, doi:10.1109/LGRS.2010.2043712.

Smith, G., and Coauthors, 2013: High power coherent-on-receive

radar for marine surveillance. Proc. 2013 Int. Conf. on

Radar, Adelaide, SA, Australia, IEEE, 434-439, doi:10.1109/

RADAR.2013.6652028.

Turin, G. L., 1960: Introduction to matched filters. IRE Trans. Inf.

Theory, 6, 311–329, doi:10.1109/TIT.1960.1057571.

Young, I. R., W. Rosenthal, and F. Ziemer, 1985: A three-

dimensional analysis of marine radar images for the de-

termination of ocean wave directionality and surface

currents. J. Geophys. Res., 90, 1049–1059, doi:10.1029/

JC090iC01p01049.



http://dx.doi.org/10.1029/2003JC002130

http://dx.doi.org/10.1109/IGARSS.2010.5650065



http://dx.doi.org/10.1016/S0378-3839(99)00032-0

http://dx.doi.org/10.1109/LGRS.2010.2043712

http://dx.doi.org/10.1109/RADAR.2013.6652028

http://dx.doi.org/10.1109/RADAR.2013.6652028

http://dx.doi.org/10.1109/TIT.1960.1057571

http://dx.doi.org/10.1029/JC090iC01p01049

http://dx.doi.org/10.1029/JC090iC01p01049

polar fourier transform processing of marine radar signals

Documents