Method for the processing of the reception signal of a deramp type synthetic aperture radar

A Deramp type radar used in synthetic aperture radar for radar imaging transmits coherently repeated linear frequency-modulated pulses and carries out a sort of pulse compression in reception by demodulation of the echo signals received by means of a frequency ramp that reproduces all or part of a transmitted pulse, and by a Fourier transform performed in range. The application to a Deramp type radar signal of a standard SAR processing is disturbed by the fact that, in this signal, the effectively demodulated part of an echo signal due to a target has a position with respect to this echo signal and a duration that are variable as a function of the distance from the target to the radar. The proposed method makes it possible to eliminate this disturbance by means of a particular choice of a common temporal support used for the demodulation of the signals of all the targets of the useful swath and a phase correction applied to the level of the pulse response of the image focusing filter of the SAR processing. Secondarily, a second phase correction can be applied to the complex reflection coefficients obtained for the dots of the image at the end of the SAR processing.

BACKGROUND OF THE INVENTION 
Synthetic aperture radars (SARs) are used to obtain high-resolution images 
of the ground in all weather conditions from a moving carrier. They 
illuminate the ground in a direction lateral to the route of the carrier, 
by a coherently repeated recurrent electromagnetic waveform, receive the 
sum of the echoes sent back by the ground and analyze this sum of echoes 
received throughout the route of the carrier for the deduction therefrom 
of the reflection coefficients of the different points of the zone 
examined in order to form a radar image. The phase coherence from one 
recurrence to another enables analysis of the Doppler history of the 
different targets and therefore makes it possible to separate them in 
azimuth, i.e. separate them along the direction of motion of the carrier 
identified on an axis known as an azimuth axis. The resolution in the 
direction perpendicular to that of the movement of the carrier identified 
by an axis called a range axis is done as a function of the echo return 
time. It is improved conventionally by the pulse compression technique. 
The data elements delivered at reception by an SAR type radar are 
constituted by a sampling of the unprocessed video signal received at a 
double rate: a fast rate corresponding to the sequence of range gates that 
subdivide the listening time of the radar after each transmission of a 
waveform and a slow rate corresponding to the succession of recurrences of 
the transmitted waveform. The result thereof is that, for an image 
processing operation, there is available a data table with two dimensions: 
a range dimension along which the samples of the unprocessed video signal 
received in response to the transmission of a waveform are aligned, these 
samples being taken at the fast sampling rate, and an azimuth dimension 
along which there are arranged the successive sequences of samples of the 
unprocessed video signal received for the various recurrences of the 
transmitted waveform. 
The image processing operation consists of the extraction of an image from 
a 2D table such as this that brings together the unprocessed video samples 
received by the radar during the illumination of a zone. This extraction 
is done by separating the contributions, to the total echo, of each point 
of the zone in taking account of the specific progress of the range and 
azimuth coordinates of the point considered during the illumination of 
this point by the radar moving above the zone. The range coordinate 
progresses as follows: the radar approaches the point until it passes 
through the perpendicular to this point and moves away from this point. 
The phenomenon is known as the range migration phenomenon. The progress of 
the azimuth coordinate consists of an approach at decreasing speed 
implying a Doppler effect at decreasing positive frequency followed by a 
moving away at increasing speed implying a Doppler effect at increasing 
negative frequency with cancellation of the Doppler effect when the radar 
passes through the perpendicular to the point. This phenomenon is known as 
the Doppler history. 
If we set aside the pulse compression, the image processing consists of the 
following steps for each point of the zone illuminated by an SAR type 
radar: selecting the azimuth domain in the table of unprocessed video 
samples corresponding to the period of illumination of the point 
considered by the radar; then, in this azimuth domain, identifying the 
band of samples to which the point considered has contributed by its echo, 
in taking account for this purpose of the range migration; finally, 
filtering the collection of samples of this band to take account of the 
Doppler history of the point considered and extract its complex reflection 
coefficient therefrom. These operations which are necessary for each point 
of an imaged zone are done so as to limit, as far as possible, the 
quantity of calculations performed. 
A theoretical analysis of the nature of the operations, including a 
possible preliminary pulse compression operation, that have to be 
conducted on the 2D range and azimuth table of the unprocessed video 
reception samples delivered by an SAR type radar in order to obtain an 
image, shows that these operations can be interpreted as a 2D filtering of 
the table in the range and azimuth dimensions. This filtering is done by 
means of a so-called image focusing filter whose pulse response 
h(.tau.,t,.tau..sub.i) to two temporal variables, .tau. range and t 
azimuth, is not stationary in range. This justifies the presence of a 
second range parameter .tau..sub.i in its expression. From this, there 
follows a standard method of processing the table of unprocessed video 
reception samples delivered by an SAR type radar. This standard method 
consists in: 
determining the pulse response h(.tau.,t,.tau..sub.i) to two temporal 
variables, namely .tau. range and t azimuth, locally valid in the range 
bands .tau..sub.i where this response is assumed to remain stationary, 
that define the image focusing 2D filter carrying out the following at the 
same time: the pulse compression, the correction of range migration and 
the extraction of the frequency components corresponding to the Doppler 
history. This determining of the pulse response is done on the basis of 
the properties of the waveform transmitted and the geometrical parameters 
of the image being taken, 
convoluting this image focusing filter pulse response with the table of 
unprocessed video signal samples received, subdivided beforehand into 
range bands through direct and reverse 2D Fourier transform operations, to 
return to the spaces of the range and azimuth frequencies and avoid 
integration computations inherent in a convolution, and 
combining the image bands obtained to reconstitute a full image. 
It is shown that a certain economy of computation can be achieved by 
performing the pulse compression operations prior to the image processing 
operations. 
The pulse compression which makes it possible to improve the range 
resolution can be done on several waveforms emitted, and especially on a 
waveform constituted by a coherently repeated linear frequency-modulated 
pulse. In theory, the pulse compression consists in filtering the signal 
received with a filter matched to the waveform transmitted or again in 
correlating the received signal with the transmitted waveform. This 
operation is difficult to achieve, and is often costly in terms of 
computation. This is why it is often sought to replace it with simpler 
processing operations giving similar results. One example thereof is 
provided by a type of radar known as the Deramp radar which transmits 
coherently repeated linear frequency-modulated microwave pulses and 
carries out a sort of pulse compression in reception, not by matched 
filtering but by a demodulation of the signal received by the transmitted 
pulses and a frequency analysis of the demodulated signal received. For 
details of the Deramp radar, reference may be made to: [1] J. P. HARDANGE, 
P. LACOMME and J. C. MARCHAIS, "Radars aeroportes et spatiaux" (Airborne 
and space radars), Masson 1995, pp. 168-170. 
In a Deramp radar, advantageous use is made of the fact that with a linear 
frequency-modulated pulse, there is a relationship of proportionality 
between the time that elapses and the instantaneous frequency of the 
pulse. Through this relationship, the mixing of two pulses that are 
staggered in time such as a transmitted pulse and the echo pulse that is 
returned by a point target gives a signal whose frequency is constant with 
respect to time and depends on the relative delay between the two pulses. 
Thus, the correlation between the transmitted and received waveforms that 
results in the pulse compression can be replaced by a simple demodulation 
of the signal received by the transmitted signal followed by a Fourier 
transform with minor differences in the result. 
The operational advantage of the processing done in a Deramp radar with a 
view to pulse compression is that it enables very high range resolution 
with a narrow instantaneous band receiver. In the case of use in imaging, 
this is done to the detriment of the width of the processed swath, i.e. 
the useful duration of reception between each waveform transmitted for the 
useful instantaneous band in the receiver is proportional to the duration 
of the swath processed. This mode of pulse compression is therefore 
particularly suited to very high resolution radar imaging systems on 
regions that are not extensive in distance. 
The use of a Deramp type radar as a synthetic aperture radar to carry out 
radar imaging raises difficulties owing to the particular properties of 
its demodulated reception signal. Indeed, this signal has the property, in 
relation to the target echo signal from which it originates, of having a 
delay that is variable as a function of the position in range of the 
target in the swath, namely in the zone illuminated by the transmitted 
pulse and also of having a variable duration, smaller than that of the 
target echo signal from which it originates, depending also on the 
position in range of the target in the swath. Its delay, which is variable 
with respect to the echo signal from which it originates, generates a 
parasitic phase that disturbs the subsequent operation of image 
construction processing and adversely affects the sharpness of the image 
while its variable duration gives the Deramp type radar a resolution that 
is variable in range and also affects the sharpness of the image. 
The present invention is aimed at resolving these difficulties in order to 
obtain a radar image that is as sharp as possible. 
SUMMARY OF THE INVENTION 
An object of the invention is a method for the processing of the reception 
signal of a Deramp type SAR to obtain a radar image. The Deramp type SAR 
is placed on board a carrier moving above a region of terrain to be imaged 
that it illuminates. It sends out coherently repeated linear 
frequency-modulated pulses with a duration T and a modulation slope 
.alpha., demodulates the echo signal received in return between each 
transmitted pulse by means of a demodulation ramp that is centered in 
range on the middle of the useful illuminated zone or useful swath, 
resumes the form of all or part of the transmitted pulse and has a 
duration T.sub.d smaller than or equal to the duration T of a transmitted 
pulse, sufficient to cover the reception time interval where the echo 
signals of all the targets of the useful swath overlap and, after 
demodulation, delivers an unprocessed video reception signal available in 
the form of successive samples or reception data elements that are taken 
at a double rate: a fast rate corresponding to the sequence of range gates 
and defining a temporal dimension of range .tau. along a range axis 
oriented laterally to the path of the carrier of the SAR and a slow rate 
corresponding to the succession of recurrences and defining a temporal 
dimension of azimuth t along an azimuth axis oriented in the direction of 
shift of the carrier of the SAR, these samples or data elements taking the 
form of a table of reception data with two dimensions, range and azimuth. 
his method comprises the following successive processing steps: 
the passage of the data elements from the reception data table into a 
dispersive delay line with a pulse response h.sub.l (t) such that its 
instantaneous frequency is a linear function of the time; 
##EQU1## 
with a linear modulation slope having a frequency K chosen to be equal to: 
##EQU2## 
T.sub.a being the useful duration of demodulation or duration of a common 
temporal support chosen for the demodulation of the echo signals coming 
from all the targets of the useful swath, this useful duration T.sub.a 
being smaller than or equal to the duration T.sub.d of the demodulation 
ramp and greater than or equal to the period of time beginning before the 
start of reception of an echo sent by a target placed in distance at the 
far end of the useful swath and ending after the end of reception of an 
echo sent by a target placed in distance at the near end of the useful 
swath while at the same time being centered in distance on the middle of 
the useful swath, 
the selection, from among the data elements of the 2D table, of reception 
data elements, consisting of the withdrawal of the reception data elements 
arriving along the range axis outside the useful duration chosen T.sub.a 
and their replacement by zero values, 
the replacement of the data elements of the table of reception data, 
considered in the range dimension, by their Fourier transform, which is 
one-dimensional in range, for the obtaining, after demodulation and 
passage through the dispersive line, of a type of pulse compression to 
which there is assigned a parasitic phase term of pulse compression, 
the rearrangement of the data elements of the table in the range dimension 
in order to have available data corresponding to an order of moving away 
that increases in range, 
the subdivision of the table into overlapping bands, parallel to the 
azimuth axis, so as to have bands corresponding to narrow zones of range 
.tau..sub.i where it is possible, as a function of the geometrical 
parameters of the image taken, to locally determine a image focusing 2D 
filter having a pulse response h(.tau.t,t,.tau..sub.i) with two temporal 
variables, namely .tau. range and t azimuth, and a function of correction 
of the parasitic phase due to the pulse compression, that are stationary 
in the range band .tau..sub.i, 
the filtering of the table bands by the image focusing 2D filter whose 
pulse response has been modified by the parasitic phase correction 
function, 
the juxtaposition of the table bands resulting from the filtering to obtain 
a table of complex reflection coefficients of the points of the 
illuminated region of ground, and 
the construction of an image of the illuminated region from the moduli of 
the complex reflection coefficients represented in the table obtained in 
the previous step. 
Advantageously, the duration T.sub.d of the demodulation ramp is taken to 
be equal to the duration T of a transmitted linear frequency-modulated 
pulse. 
Advantageously, the useful duration T.sub.a of demodulation is taken to be 
equal to the duration T of a transmitted linear frequency-modulated pulse, 
reduced by half of the duration T.sub.f of the useful swath. 
Advantageously, the 2D filtering of the table bands by the image focusing 
2D filter whose pulse response has been modified by the parasitic phase 
correction function is obtained by going into the spaces of the range and 
azimuth frequencies, by other one-dimensional direct and reverse Fourier 
transform operations in the two dimensions of range and azimuth of the 
data tables so as to avoid computations of convolution.

MORE DETAILED DESCRIPTION 
FIG. 1 is a block diagram of a monostatic synthetic aperture radar with 
pulse compression. This radar is known by the terms "Deramp Radar, 
Deramping Radar, Dechirping Radar etc.". The radar, mounted on a carrier 
moving above a region of the ground to be imaged, consists of an antenna 1 
connected by a circulator 2 to a transmission part 3 and a reception part 
4 with an inverter switch 5 that is placed between the transmission part 3 
and the circulator 2 and enables the extraction, from the transmission 
part 3, of a demodulation signal for the reception part 4, a time base 6 
organizing the sequencing of the various signals and, finally, an image 
processing circuit 7 that is often kept at a distance on the ground in a 
telemetry station. This image processing circuit 7 constructs an image 
from the unprocessed video data delivered by the reception part 4. 
The transmission part 3 has a waveform generator GFO 30 generating linear 
frequency ramps at an intermediate frequency, a microwave source OH.sub.yp 
31 generating a microwave carrier and a mixer 32 connected to the outputs 
of the waveform generator GFO 30 and of the microwave source OH.sub.yp 31 
in order to deliver linear frequency-modulated microwave pulses to the 
inverter switch 5 and beyond, to the circulator 2 and the antenna 1 as 
well as to the reception part 4. 
The reception part 4 has a demodulation mixer 40 placed at input and 
receiving a microwave signal coming from the circulator 2 and the antenna 
1, and a demodulation signal coming from the transmission part 3. This 
demodulation signal consists of linear frequency-modulated microwave 
pulses generated by the transmission part 3. These pulses are diverted by 
means of the inverter switch 5 and translated into a higher frequency band 
by means of a mixer 41. This mixer is followed by a highpass filter 42 and 
receives signals from a local oscillator OL 43 at intermediate frequency. 
The demodulation mixer 40 is followed by a bandpass filter 44 enabling the 
selection of the useful frequency zone, namely the range domain observed 
or the useful swath, an amplitude-phase demodulator DAP 45 receiving the 
intermediate frequency carrier from the local oscillator OL 43 and two 
analog-digital converters 46, 47. These two analog-digital converters 46, 
47 carry out the sampling and conversion into digital form of the phase 
and quadrature parts of the unprocessed reception video signal, at a 
double rate: a fast rate corresponding to the succession of range gates 
subdividing the listening time of the radar between each of the 
transmissions of linear frequency-modulated pulses and a slow rate 
corresponding to the succession of linear frequency-modulated pulses 
transmitted. 
The time base 6 carries out the sequencing of the linear 
frequency-modulated pulses generated by the transmission part 3 in order 
that they may be repeated coherently with respect to the microwave signal 
from the microwave source SF 31. Secondly, the time base 6 operates the 
inverter switch 5 so that, between each linear frequency-modulated pulse 
transmitted to the antenna 1 and the next pulse, there is a linear 
frequency-modulated pulse diverted to the reception part 4 for the 
demodulation. Thirdly, the time base sets the double rate of sampling of 
the unprocessed reception video signal performed by the analog-digital 
converters 46, 47 of the reception part 4. To accomplish this task, the 
time base 6 is synchronized with the signal of the microwave source 31 of 
the transmission part. 
Owing to this double sampling rate, it is usual to present the samples of 
the unprocessed reception video signal delivered by the reception part 4 
in the form of a table with two dimensions: a range dimension along which 
there are aligned the samples of the unprocessed video signal received in 
response to the transmission of a linear frequency-modulated pulse and an 
azimuth dimension along which there are arranged the successive sequences 
of samples of unprocessed video signals received for the various linear 
frequency-modulated pulses transmitted. A table such as this, containing 
samples of unprocessed video reception signals, is then transmitted to an 
image processing device 7 in order to be processed therein in the two 
dimensions of range and azimuth in order to get an image therefrom. 
FIG. 2 shows the sequencing of the different signals of a Deramp type pulse 
compression SAR. This figure shows the transmitted pulses 10 with a 
duration T. They are linear frequency-modulated pulses that succeed one 
another at the end of a period T.sub.R leaving a duration T.sub.R -T 
between them for reception. The duration T.sub.R -T between the 
transmitted pulses is taken up by a reception period 11 that starts with 
the beginning of an echo due to a near target located at the beginning of 
the useful swath and ends with the end of an echo due to a distant target 
located at the end of a useful swath. Sometime during the period of 
reception, there appears a demodulation pulse 12 that is the copy of a 
part of the transmitted pulse with a duration T.sub.d. This demodulation 
pulse 12 appears with the delay such that it is centered on the middle of 
the useful swath. Two interesting periods of time appear in this FIG. 2, 
the period T.sub.f which corresponds to the duration of the useful swath 
and is the time interval between the reception of the echoes from the 
targets located at the beginning and end of the swath and the period 
T.sub.a which is the usual duration of analysis, in this case the time 
during which the echoes sent back by the targets located in the useful 
swath are received simultaneously. 
FIG. 3 illustrates the principle of demodulation of a linear 
frequency-modulated pulse. 
In the time-frequency domain, a linear frequency-modulated pulse may be 
likened to a straight line with a slope a such that: 
##EQU3## 
where B is the frequency band of the pulse and T is its duration. It may 
be assumed here that we are dealing with linear frequency-modulated pulses 
in the increasing order of frequencies and hence that the straight lines 
which represent them are ramps with a positive slope .alpha.. Furthermore, 
for convenience's sake, the figure shows a pulse used for the demodulation 
or the demodulation ramp with a negative slope -.alpha. for, in this case, 
the demodulation process which consists of a subtractive mixture of two 
instantaneous frequencies is expressed in the figure as the point-to-point 
sum of the useful signal and of the demodulation signal. With this 
convention which shall remain valid hereinafter in the description, and 
taking the center of the demodulation pulse as the starting point of the 
coordinates, the demodulation ramp appears as a straight line segment 20 
with a negative slope -.alpha.. This straight line segment 20 is 
represented by dashes and passes through the starting point. It starts at 
an instant .tau..sub.d, ends at an instant .tau..sub.d +T.sub.d, T.sub.d 
being the duration of the demodulation ramp, and occupies a frequency band 
[+B.sub.a /2, -B.sub.a /2]. An echo coming back from a target appears as a 
straight line segment 21 with a slope .alpha. that is represented by a 
solid line. This straight line segment 21 starts at an instant .tau..sub.c 
at the bottom -B/2 of the frequency band of a transmitted pulse and ends 
at an instant .tau..sub.c +T at the peak B/2 of the frequency band of a 
transmitted pulse. The result thereof is that, after demodulation, it 
gives an unprocessed video signal constituted by a pure frequency: 
EQU f.sub.c/d =-.alpha.(.tau..sub.c -.tau..sub.d) 
represented in FIG. 3 by a horizontal straight line segment 22 in a solid 
line that starts at whichever is the latest of the instants .tau..sub.c or 
.tau..sub.d, in this case the instant .tau..sub.d and ends at whichever is 
the earliest of the instants .tau..sub.c +T or .tau..sub.d +T.sub.d, in 
this case the instant .tau..sub.c +T. It can be seen therefore that the 
unprocessed video reception signal resulting from the demodulation of the 
echo due to a target is a sine signal whose frequency is proportional to 
the relative delay of the echo of the target with respect to the instant 
at which the demodulation ramp is started. 
More specifically, it is possible to express the transmitted signal, the 
received signal received associated with a target corresponding to a 
propagation time .tau..sub.c and the demodulation signal started at the 
instant .tau..sub.d respectively by: 
##EQU4## 
where f.sub.c is the center frequency of a pulse, .alpha. the ratio of the 
frequency band B of a pulse transmitted on its duration T and T.sub.d the 
duration of the demodulation ramp. At the end of the demodulation, the 
received signal associated with this target is therefore given by: 
##EQU5## 
The second cosine term disappears after the bandpass filtering 44 which 
comes into play just after the demodulation. The role of the bandpass 
filtering 44 is to eliminate the image frequencies generated by the 
demodulation mixer 40 and also, as specified here above, to select the 
useful swath and prevent the targets in the vicinity of the useful swath 
from acting as parasites with respect to the useful swath after sampling. 
The first cosine term which is the only one to remain after demodulation 
and filtering shows that the unprocessed video reception signal coming 
from a target is a sine signal. The frequency -.alpha.(.tau..sub.c 
-.tau..sub.d) of this sine signal is proportional to the relative delay of 
the target with respect to the instant at which the demodulation ramp is 
started, as was already seen in the time-frequency graph of FIG. 3, and 
its phase shift is a function of the echo propagation time .tau..sub.c and 
of the instant .tau..sub.d at which the demodulation signal is started. 
The echo signal .nu..sub.0 (.tau.,t) received before decompression at an 
instant corresponding to range .tau. and an instant corresponding to 
azimuth t results from a 2D range and azimuth convolution between the 
complex reflectivity of the illuminated ground .gamma.(.tau.,t), this 
complex reflectivity being sought in order to plot the image of the 
ground, and the product of multiplication between the type of pulse 
transmitted p(.tau.) and the pulse response h.sub.SAR (.tau.,t) of the 
image focusing filter, this response being determined without taking 
account of the range pulse compression, solely as a function of the 
geometrical parameters of the image being taken, and this response being 
assumed to be stationary for a certain range band so that it is possible, 
for this range band, to write: 
##EQU6## 
where: .tau..sub.0 designates the instant from which the samples of 
unprocessed video reception signals begin to be recorded, 
.tau..sub.c is the time of propagation of the echo signal from a target, 
T.sub.e is the period of illumination of a target by the radar moving past 
in azimuth, 
p(.tau.) designates the type of pulse transmitted, 
h.sub.SAR (t) is the pulse response of the image focusing filter where the 
pulse compression is not taken into account. This pulse response depends 
implicitly on the range variable .tau., which is not mentioned for the 
sake of simplicity, and is such that: 
##EQU7## 
the constant phase term exp.left brkt-bot.i2.pi.f.sub.c .tau..sub.d .right 
brkt-bot. being omitted with a view to simplification, f.sub.c designing 
the center frequency of a transmitted pulse and g.sub.az (t) the azimuth 
aperture pattern of the radar antenna. 
Hereinafter in the development of the description, in order to make the 
mathematical expressions less cumbersome, the following convention will be 
adopted: 
##EQU8## 
The transmitted pulse being a linear frequency-modulated pulse in the B 
frequency band with a duration T and a modulation slope 
.vertline..alpha..vertline.=B/T, it can be admitted that: 
##EQU9## 
The demodulation ramp is characterized by the instant .tau..sub.d at which 
it is started and its duration T.sub.d. The expression of its complex 
envelope is given by: 
##EQU10## 
It is deduced therefrom that the expression of the signal received after 
demodulation is written as follows: 
##EQU11## 
The analysis of this expression shows that in order to be capable of 
assessing the complex reflectivity .gamma.(.tau.,t) of the ground 
illuminated by the radar in order to construct a radar image thereof, it 
is not enough to know the pulse response h.sub.SAR (t) of the image 
focusing filter defined in the absence of pulse compression but that is 
necessary, in addition, to be capable of assessing the products of 
multiplication between the pulse echoes returned by the targets p() on the 
one hand and the demodulation ramp p.sub.d.sup.* () on the other. This 
product, once it is developed, contains two terms: 
a term defining a temporal support: 
##EQU12## 
a phase term associated with the demodulation: 
##EQU13## 
which can be written again as follows: 
##EQU14## 
The temporal support defines the time interval during which the echo signal 
sent back by a target exists and is demodulated. This time interval has a 
position on the range axis and a duration that are variable as a function 
of the position in range of the target considered in the useful swath and 
of the position in range of the demodulation ramp which is usually the 
middle of the useful swath. A constant range resolution on the useful 
swath dictates the adoption of temporal supports of the same duration for 
the echo signals of all the targets, whatever their positions in range in 
the useful swath. This leads to arrangements being made in order that the 
temporal supports actually used for the signals of all the targets of the 
useful swath will have the same duration. 
The phase-shift term associated with the demodulation comes from the fact 
that the temporal supports of the echo signals of the targets have a 
position with respect to the echo signals of the targets themselves, that 
varies as a function of the position in range of the targets. 
To construct an image out of a unprocessed video reception signal delivered 
by a Deramp type radar, it is therefore necessary, in addition to the 
usual operations of image processing, to overcome the dependency of the 
duration of the temporal support used for the demodulation of the echo 
signal sent back by a target on the position in range of this target in 
the useful swath. It is furthermore necessary to correct the effects of 
any phase term that may be associated with the demodulation due to a 
variation in position of the temporal support used for the demodulation of 
the echo signal sent back by a target with reference to the echo signal 
itself as a function of the position in range of the target. 
It is proposed, in order to have a maximum of demodulated signals and 
therefore a range resolution that is both uniform and maximum on the 
useful swath, to make the best possible use of a demodulation ramp with a 
maximum duration equal to the duration T of a transmitted pulse to obtain 
a temporal demodulation support that is unique for all the echo signals of 
the useful swath and has a maximum duration. This leads to the adoption of 
a useful duration T.sub.a common to all the demodulated signals of a 
useful swath equal to the minimum duration of demodulation encountered for 
the echo signals from targets located in distance at the ends of the 
useful swath. Since this minimum duration is equal to the duration T of a 
transmitted pulse minus half of the duration T.sub.f of the useful swath, 
it is sought to obtain a useful duration T.sub.a equal to: 
EQU T.sub.a =T-T.sub.f /2 
This useful duration T.sub.a, which must be common to all the echo signals 
whatever the position in range, in the useful swath, of the targets from 
which they come, is centered on the demodulation ramp. Since this useful 
duration T.sub.a is far too lengthy to be occupied simultaneously by all 
the echo signals from the targets of the useful swath, it is necessary to 
delay the demodulation signals that are in advance in relation to it, 
namely those coming from the targets located in range in the first quarter 
of the useful swath, and anticipate the demodulation signals that are 
delayed in relation to it, namely those coming from the targets located in 
range in the last quarter of the useful swath. To do this, profitable use 
is made of the fact that the demodulated echo signals are pure frequencies 
depending on the position in range of the targets from which they come, to 
reposition them with respect to one another in time by means of a 
dispersive line. As we shall see here below, this repositioning which 
enables all the demodulated echo signals of the useful swath to overlap 
the useful duration T.sub.a do not however make it possible cause the 
useful duration T.sub.a to be occupied by the same partition of each of 
the echo signals. The result thereof is the persistence after demodulation 
of a phase term that is variable as a function of the position in range, 
in the useful swath, of the target causing the demodulated echo signal 
which has to be compensated for in a subsequent SAR processing operation 
for the construction of radar images. 
FIGS. 4 and 5 are time-frequency graphs that illustrate this choice of 
demodulation temporal support. They show a system of time-frequency axes 
whose temporal point of origin is the echo propagation time corresponding 
to the middle of the useful swath and whose frequency point of origin is 
the middle of the frequency band [-B/2, B/2] of a transmitted pulse. In 
this system of time-frequency axes, the following are plotted: 
a straight line segment CD 50 with a slope .alpha. represented by a solid 
line, passing through the point of origin and corresponding to the 
response to a transmitted pulse, from a target located in range, in the 
middle of the useful swath. This response is centered on the point of 
origin and lasts for the period T of a transmitted pulse, 
two straight line segments EF and GH 51, 52 with a slope .alpha. shown in 
dashes on either side of the straight line segment 50 and corresponding to 
the responses, having a duration T, of two targets to a transmitted pulse: 
one segment 51 is located in range at the beginning of the useful swath 
and the other segment 52 is located in range at the end of the useful 
swath. These two parallel straight line segments 51 and 52 are separated 
in time by a duration T.sub.f corresponding to the width in range of the 
useful swath. Their ends HGEF form the corners of a parallelogram 53 
which, in the time-frequency domain, demarcates the surface occupied by 
the echoes sent back by all the targets of the useful swath, 
a straight line segment 54 with a slope -.alpha. shown in dashes, passing 
through the starting point and corresponding to the demodulation ramp 
used, with a duration T. This ramp begins in time at an instant 
.tau..sub.d placed at the start of reception of an echo 51 from a target 
located in range at the far end of the useful swath, ends in time at an 
instant .tau..sub.d +T.sub.d or .tau..sub.d +T and occupies a frequency 
band [-B/2, B/2] corresponding to that of the transmitted pulse. 
At the instant .tau..sub.d at which the demodulation ramp starts, the 
linear frequency-modulated pulse sent back by a target placed in range at 
the near end of the useful swath has already begun, so much so that only 
its rear part occupying a frequency band [-B.sub.a /2, B/2] is 
demodulated. Similarly, at this same instant .tau..sub.d, the linear 
frequency-modulated pulse sent back by a target placed in range at the far 
end of the useful swath has not begun, so much so that only its front part 
occupying a frequency band [-B/2, B.sub.a /2] is demodulated. Thus, the 
only part demodulated is the part of the echo signals received from the 
useful swath occupying, in the time-frequency graphs of FIGS. 4 and 5, the 
surface of the hexagon whose corners are the point C, the projection I of 
the point C parallel to the axis of the frequencies on the straight line 
segment EF 51, the points F and D, and the projection J of the point D 
parallel to the axis of the frequencies on the straight line segment GH 
52. 
After demodulation, the target echo signals are converted into pure sine 
signals appearing on the time-frequency graphs of FIGS. 4 and 5 in the 
form of horizontal straight line segments as was explained with reference 
to FIG. 3. Among these straight line segments, it is possible to 
distinguish the following. Firstly, there is the horizontal straight line 
segment 56, drawn in a bold line on the time axis. This segment 56 
corresponds to the demodulation of the echo signal 51 from a target 
located in range at the middle of the useful swath. Then there is the 
straight line segment 57 drawn in dashes corresponding to the demodulation 
of the echo signal 51 from a target located in range at the beginning of 
the useful swath. Finally, there is the straight line segment 58, shown in 
dashes, corresponding to the demodulation of the useful signal 52 from a 
target located in range at the end of the useful swath. 
FIG. 4 shows the respective positions of the echo signals coming from the 
useful swath after their demodulation by the demodulation ramp 54. They 
occupy the surface of a truncated parallelogram LMOPQR. The choice of a 
common demodulation temporal support amounts to the inscribing of a 
rectangle in the surface of this truncated parallelogram. It can be seen 
then that without, any particular processing, the temporal width of the 
greatest rectangle that can be inscribed is limited to T-T.sub.f, which 
amounts to using only the part of the echo signals of the useful swath 
that belongs to the surface of the rectangle GQFM. 
To increase the used part of the echo signals of the useful swath, it is 
then proposed to deform the truncated parallelogram LMOPQR of FIG. 4 
occupied by the demodulated echo signals so as to inscribe a rectangle of 
greater temporal width therein. This is obtained, as shown in FIG. 5, by 
the application to the demodulated echo signals of a delay as a function 
of their frequencies such that the truncated parallelogram is converted 
into a hexagon L'M'OP'Q'R that is symmetrical with respect to the time 
axis. It is then possible to inscribe a rectangle L'M'P'Q' within it, 
having a temporal width T-T.sub.f /2. This amounts to using the part of 
the echo signals of the useful swath belonging to the rectangular surface 
IFJG. Thus, half of the surface of the echo signals that was neglected 
earlier is recovered. It is observed however that, as was the case 
earlier, while a common temporal support is truly obtained for the 
demodulation of all the echo signals of the useful swath, the parts of the 
echo signals taken into account have variable delays with respect to the 
echo signals themselves which give rise to a parasitic phase modulation. 
To delay the demodulated echo signals as a function of their respective 
frequencies, a filter is used with a pulse response such that its 
instantaneous frequency is a linear function of time as in the case of a 
linear frequency-modulated pulse. This type of filter is also called a 
dispersive line because of the technological means used to manufacture or 
generate linear frequency-modulated waveforms (for example surface 
acoustic wave or SAW devices). 
To understand the effect of the passage of the demodulated echo signals 
into a dispersive line, attention is paid to the effect of a dispersive 
line on a purely sine signal with a frequency f.sub.0 and a duration T 
such that: 
##EQU15## 
The application of this signal to the input of a dispersive line having a 
pulse response: 
##EQU16## 
prompts the appearance, at output of the dispersive line, of a signal 
having the form: 
##EQU17## 
Now: 
##EQU18## 
The assessment of the straight-line term of the preceding relationship can 
be made in the basis of the assessment of the function: 
##EQU19## 
Indeed, by assuming: 
##EQU20## 
where C and S designate the Fresnel integrals, we obtain: 
##EQU21## 
when we are in the conditions where: 
##EQU22## 
it is possible to write: 
##EQU23## 
It is deduced therefrom that the output signal of the dispersive line can 
be written as follows: 
##EQU24## 
This shows that the passage of a pure sine signal into a dispersive line 
results in a delay .DELTA.t equal to: 
##EQU25## 
which is a function of the frequency f.sub.0 of the sine and of the slope 
K of the linear frequency modulation of the dispersive line. The passage 
also results in a parasitic phase shift: 
##EQU26## 
also dependent on the frequency f.sub.0 and the slope K of linear 
frequency modulation of the dispersive line. 
In the above computation, it has been assumed that the duration of the 
dispersive line is infinite. In practice, the result obtained remains 
valid for a finite duration so long as the frequency of the input sine 
wave belongs to the band of the dispersive line. 
The space-distance resolution is inversely proportional to the frequency 
band B.sub.a occupied by the used demodulated part of the echo signals of 
the useful swath: 
EQU B.sub.a =.vertline..alpha..vertline.T.sub.a 
To study the expression of the useful temporal support after demodulation 
for any target of the temporal position .tau.'+.tau..sub.c, we begin by 
paying attention to the characteristics of the first and last targets of 
the useful swatch. The temporal position of the first target verifies the 
relationship: 
EQU .tau..sub.df +T=.tau..sub.d -T.sub.a 
The temporal position of the last target ascertains for its part the 
following relationship: 
EQU .tau..sub.ff =.tau..sub.d +T.sub.d -T.sub.a 
These last two expressions which reflect the use of the maximum duration of 
analysis T.sub.a makes it possible to find the width of the useful swath: 
EQU T.sub.f =T+T.sub.d -2T.sub.a 
as well as the temporal position of the middle of the useful swath: 
##EQU27## 
It is observed that, in the case of FIGS. 4 or 5 where T.sub.d =T has been 
taken, we get: 
EQU T.sub.f =2(T-T.sub.a) 
The width of the swath is herein greater than the difference between the 
duration T of the transmitted pulse and the duration of the temporal 
support T.sub.a used for the demodulation. It is furthermore observed that 
the spectral band occupied by the echo signals coming from the useful 
swath is given by: 
##EQU28## 
The result thereof is that the sampling frequency F.sub.e of the 
analog-digital converters 46, 47 placed on the output channels of the 
amplitude-phase demodulator DAP 45 of the reception part of the Deramp 
radar, which corresponds to the fast rate of sampling in range, must be at 
least equal to this band B.sub.FI to meet the Nyquist theorem. In 
practice, it is necessary to take an over-sampling margin that depends on 
the equivalent filtering template of the receiver. This over-sampling is 
generally greater than it is for a mode of pulse compression by 
correlation for, in this case, the parasitic signals liable to undergo 
aliasing in the useful band include the targets neighboring the useful 
swath in addition to thermal noise. 
If we take position at the level of the demodulation by the linear 
frequency-modulated ramp, the starting and ending instants of the useful 
temporal support with a duration T.sub.a during which the echo signal of 
the first target of the useful swath is actually demodulated are 
respectively given by: 
EQU .tau..sub.d and .tau..sub.d +T.sub.a 
Those of the last target of the useful swath are given by: 
EQU .tau..sub.d +T.sub.d -T.sub.a and .tau..sub.d +T.sub.d 
By proportionality, it is deduced therefrom that the starting and ending 
instants of the useful temporal support for an unspecified target at the 
temporal position .tau.'+.tau..sub.c are given respectively by: 
##EQU29## 
Consequently, the middle of the temporal support of this target is equal 
to: 
##EQU30## 
and the expression of the useful temporal support is: 
##EQU31## 
in taking account of the instant .tau..sub.0 from which there begins the 
storage of the digital samples of the demodulated unprocessed video 
signal. 
The unprocessed video signal .nu..sub.1 (.tau.,t) delivered by the Deramp 
radar after demodulation performed in such conditions may then take the 
form: 
##EQU32## 
It was seen, with reference to FIGS. 4 and 5, that in order to make the 
temporal support used common for the demodulation of the echo signals of 
all the targets of the useful swath, the demodulation signal was made to 
pass into a dispersive line. The problem therefore arises of defining the 
slope K of linear frequency modulation of this dispersive line. To resolve 
this, it is observed that the middle of the useful swath is at the 
temporal position: 
##EQU33## 
that, at the end of the demodulation, since the signal with a point of 
origin that is a target in the middle of the useful swath has a zero 
frequency, it is therefore not delayed by a dispersive line and has a 
useful temporal support whose midpoint is at the temporal position: 
EQU .tau..sub.d +T.sub.d /2 
Furthermore, the demodulated signal coming from a target at the temporal 
position .tau.'+.tau..sub.c has a frequency equal to: 
-.alpha.(.tau.'+.tau..sub.c -.tau..sub.d +(T-T.sub.d)/2) 
It is deduced therefrom that the slope K of linear frequency modulation of 
the dispersive line must verify the relationship: 
##EQU34## 
In the borderline case where T.sub.d =T.sub.a, the slope of linear 
frequency modulation of the dispersive line becomes infinite. This means 
that no delay must be applied whatever the frequency and therefore that 
the dispersive line becomes unnecessary. In this borderline case, the 
duration of the demodulation ramp is necessarily smaller than T-T.sub.f, 
so that all the echo signals of the useful swath entirely overlap the 
demodulation ramp and so that it is therefore no longer necessary to 
stagger them in time to have a common useful temporal support. 
With all the computations made, the expression of the demodulated echo 
signal on the useful duration T.sub.a, after passage into a dispersive 
line with a linear frequency-modulated slope complying with the 
relationship (2), is given by: 
##EQU35## 
The rectangular window function rect() which no longer depends on the 
integration variables .tau.' and t', may be taken out of the double 
integral and no longer constitutes an obstacle to its computation. What 
remains to be done then is to take account of the phase terms associated 
with the demodulation. 
After demodulation and passage into the dispersive line, the unprocessed 
video signal delivered by a Deramp radar is subjected to a Fourier 
transform in range with possible weighting in order to discern the echoes 
from the targets of the useful swath, each of which has a frequency 
depending on its delay with respect to the instant at which the 
demodulation ramp is started. Let: 
##EQU36## 
where W is a weighting window used to limit the level of the minor lobes. 
Now, we have: 
##EQU37## 
where wsinc is the core of the pulse response and represents a weighted 
cardinal sine value. 
We also have: 
##EQU38## 
Finally, after compensation for the constant terms and simplification of 
the phase terms, we obtain: 
##EQU39## 
The weighting window W () used may be a Hamming window such that: 
##EQU40## 
In this case, the core of the pulse response is given by: 
##EQU41## 
where x represents the reduced variable x=FT.sub.a and where the 
coefficient k assumes a value smaller than 1, for example 0.58. 
In reality, it is worthwhile to take the discrete Fourier transform in 
range solely for the samples of the table of unprocessed video reception 
data associated with the temporal support used for the demodulation. For 
it is reasonable not to introduce the parasitic samples outside this 
support into the computations. Mathematically, this amounts to taking the 
Fourier transform in range of the signal .nu..sub.1 (.tau.,t) advanced in 
range by .tau..sub.d +T.sub.d /2-.tau..sub.0 -T.sub.a /2. Under these 
conditions, we obtain a new signal .nu..sub.3 (F,t) such that: 
##EQU42## 
By adopting the same approach as here above, the same expression is 
obtained for .nu..sub.4 (F,t) as in the relationship (3), after 
compensation for a new constant term. 
The signal .nu..sub.4 (F,t) is the unprocessed video signal after pulse 
compression. To make it compatible with the other processing steps that 
are more specific to the SAR processing, it is desirable to change its 
frequency variable F against a temporal variable s that is more usual in a 
unprocessed video reception signal of an SAR. The following change in 
variable is therefore assumed: 
##EQU43## 
The expression of the unprocessed video reception signal of the Deramp 
radar after pulse compression then becomes: 
##EQU44## 
By making another change in variable, we get: 
##EQU45## 
the sign *.sub..tau. designating an operation of convolution according to 
the temporal range variable .tau.. 
This expression shows that, after pulse compression, the resultant signal 
is a convolution of the intrinsic pulse response of the processing 
(weighted cardinal sine) with the signal that is (the sub-integral term) 
multiplied by a parasitic phase term, delayed by the propagation time and 
referenced with respect to the middle of the useful swath. As indicated 
here above, the parasitic phase term arises out of the fact that the 
demodulated signals, although they all have the same temporal support, 
have variable delays with respect to the echo signals which are their 
source, these delays being a function of the positions in range, in the 
useful swath, of the targets that have sent back the echoes. 
To apply an image processing operation to the table of unprocessed video 
reception data of a Deramp radar obtained after pulse compression by 
demodulation, passage into a dispersive line, one-dimensional Fourier 
transform in range and the replacement, in the range dimension, of the 
frequency variable F by a temporal variable .tau., it is necessary to 
re-order the range data elements so that they succeed one another in 
increasing order of remoteness. Indeed, the center of the useful swath is 
associated for example with the zero frequency. The result of this is 
that, by spectral aliasing, the negative frequencies associated with the 
echoes from the near targets appear in the domain of the positive 
frequencies at the top of the band occupied by the positive frequencies 
associated with the echoes of the distant targets as can be seen in FIGS. 
6 and 7. 
Indeed, let: 
N be the number of samples or data elements of the table on which the 
Fourier transform in range has been performed, 
F.sub.e is the sampling frequency of the radar in range, 
i is the index of the current frequency slot, the relationship between a 
frequency slot and the associated frequency is given by: 
##EQU46## 
Since the spectral band occupied by the useful swath covers the interval 
.left brkt-bot.-B.sub.FI /2,B.sub.FI /2.right brkt-bot., it is deduced 
therefrom that the number N.sub.f of useful points at output of the 
Fourier transform in range is equal to: 
##EQU47## 
where: rint is the function that gives the integer nearest to its 
argument, 
ODD is the function that makes its argument an integer if it is an odd 
number or, if not, adds 1 to it. 
The permutation of the samples at output of the Fourier transform in range 
may therefore be done in a vector with a size N.sub.f. Owing to the 
relationship: 
##EQU48## 
which relates the frequency F to the propagation time .tau. or to the 
range, two cases have to be considered depending on the sign of the slope 
.alpha. of linear frequency modulation of the transmitted pulse. Resuming 
the notation v.sub.3 for the signal or vector obtained directly after the 
Fourier transform in range and using v.sub.4 to note the signal or vector 
obtained after rearrangement of the frequency slots, we get: 
##EQU49## 
These relationships of rearrangement are illustrated in FIG. 6 for the case 
where the sign of the modulation slope .alpha. is positive and FIG. 7 for 
the case where the sign of the modulation slope .alpha. is negative. 
From the mathematical viewpoint, the rearrangement of the frequency slots 
results in a modification of the expression of the signal .nu..sub.4 
(.tau.,t) given by the relationship (4) which becomes: 
##EQU50## 
It is observed that the signal obtained after a pulse compression by this 
mode of processing is the result of the convolution of the intrinsic pulse 
response of the processing (weighted cardinal sine) with the signal that 
is sought multiplied by a parasitic phase term constituted by a quadratic 
phase centered on the middle of the useful swath. 
It can thus be understood why it is unsatisfactory to compensate for the 
parasitic phase by multiplying the compressed signal directly by the 
conjugate of the parasitic phase term. Indeed, the correction is valid 
only for the peak of the pulse response of the pulse compression 
processing (peak of the weighted cardinal sine). This correction is not 
compatible with an SAR image extraction processing operation in the 
spectral 2D domain which alone is capable of giving an accurate pulse 
response on the final image for the very high resolution values. 
Furthermore, it is certain that, with other types of SAR image extraction 
processing operations such as for example the Doppler-range operation, 
this phase correction also leads to a deterioration of the pulse response 
on the image. 
To obtain a more efficient parasitic phase compensation, it is observed 
that the parasitic phase introduced by the demodulation depends on the 
position of the target in the swath. Now this position depends on the 
instant considered during the illumination time. It may be expressed not 
only in range by the variable .tau.+.tau..sub.df as is the case in the 
formula of the parasitic phase shift appearing in the expression of the 
signal received after pulse compression .nu..sub.5 (.tau.,t) developed in 
the relationship (5) but also in azimuth by the variable .tau..sub.c. 
There is therefore an equivalence between the two expressions: 
##EQU51## 
By using this equivalence in the relationship (5), we get: 
##EQU52## 
It is now easier to determine the way in which to compensate for this 
parasitic phase in the rest of the SAR image construction processing 
operation. Indeed, the parasitic phase term is then added to the pulse 
response of the conventional image focusing filter, which amounts to 
adopting a modified image focusing filter whose new pulse response is: 
##EQU53## 
Assuming: 
EQU .tau..sub.c (t)=.tau..sub.c0 +.tau.'.sub.c (t) 
with .tau..sub.co representing the propagation time at the middle of the 
illumination time, we get: 
##EQU54## 
represents a phase term constant in azimuth that may be compensated for at 
the end of the image construction processing operation on the table of 
complex reflection coefficients .gamma.(.tau.,t) in taking account of the 
fact that .tau..sub.c0 then represents the position of the current pixel 
in the useful swath after compression and not in the useful swath of the 
final image. Here, it is possible that a difference might arise owing to 
the number of the negative margin range slots used during the subdivision 
into range blocks that is done during the image construction processing 
operation, these range slots being eliminated at the end of processing. 
This phase term, which has no influence on the radar image constructed out 
of the moduli of the complex reflection coefficients of each point of the 
imaged zone, should be compensated for only if it is necessary to know the 
phase of the complex reflection coefficient, for example when it is 
desired to carry out interferometrical processing operations on radar 
images of one and the same area taken at different angles. 
The second exponential term: 
##EQU55## 
represents an additional phase term that affects only the Doppler phase 
history during the illumination time and not the migration path so that it 
can be added to the traditional pulse response of the image focusing 
filter. Referring only to the Doppler terms up to the third order, this 
second exponential term to be taken into account in the expression of the 
image focusing filter may be placed in the approximate form: 
##EQU56## 
with a mean Doppler f.sub.dc, a Doppler slope f.sub.dr and a Doppler 
acceleration f.sub.dq. It is observed that a part of this term depends on 
the position of the target with respect to the middle of the useful swath. 
It is possible to take account of this non-stationary character in range 
at the same time as that of the pulse response of the image focusing 
filter by the adoption, for the image construction processing operation, 
of range blocks narrow enough to make it possible for the two 
non-stationary characteristics to be overlooked. 
For the phase correction proper, it is necessary to be capable of 
expressing the range position corresponding to the term (.tau..sub.c0 
-.tau..sub.mf) in the range blocks subdivided in the data table obtained 
after pulse compression, namely demodulation, range Fourier transform, 
passage into the dispersive line, the replacing of the range frequency 
variable by a range temporal variable and the rearrangement of the range 
slots of the table in rising order. Looking more closely at the blockwise 
subdivision of the data table which is done with overlappings to take 
account of the range migration, and assuming that: 
N.sub.d is the number of samples in range or range slots of the data table 
corresponding to the extent of the useful swath after pulse compression 
(migrations included), 
N.sub.dbn is the number of range slots of the data table forming the 
negative margin of a range block, 
N.sub.dpb is the number of range slots of the data table forming the 
positive margin of the range block, 
N.sub.db is the total number of range blocks, 
N.sub.dbu is the number of useful range slots for the N.sub.db -1 first 
blocks, it can be written that the total number of range blocks is equal 
to: 
##EQU57## 
where ceil designates a function that makes the integer greater than or 
equal to its argument. The last range block then possesses a number of 
useful range slots equal to: 
EQU N'.sub.dbu =N.sub.d -N.sub.dbn -N.sub.dhp -(N.sub.db -1)N.sub.dbu 
Thus, the temporal position of the middle of the useful swath after pulse 
compression is given by: 
EQU .tau..sub.mf =0.5.times.(N.sub.d -1)T.sub.e 
and that of the center of the useful portion of each range block is given 
by: 
##EQU58## 
It is noted that the absolute temporal position of the first range slot of 
the swath has not been introduced since the only point of interest is the 
relative position of a target with reference to the center of the useful 
swath after pulse compression. 
FIG. 8 illustrates the main steps of the processing undergone by the 
reception signal of a Deramp radar used in a synthetic aperture radar for 
radar imaging. 
The reception signal of the Deramp radar .nu..sub.0 (.tau.,t) is first of 
all demodulated in a demodulator 60 by a demodulation ramp .nu..sub.d 
(.tau.,t) with a duration T.sub.d resuming all or a part of the 
transmitted linear frequency-modulated pulse. It is then converted into 
digital form and sampled by an analog-digital converter 61 in a 2D range 
and azimuth data table. 
The data elements of this table, taken in their order of succession along 
the range dimension, are then subjected to a filtering operation by means 
of a dispersive line 62. This filtering operation is achieved digitally by 
going through the frequency space in range by means of a one-dimensional 
Fourier transform in range 620. In this frequency space, it is got, as is 
well known, by a single complex multiplication, recalled in the figure by 
the multiplier 621, of the data with the conjugate value H.sub.l.sup.* of 
the transfer function of the filter which corresponds by Fourier transform 
to the pulse response. In the present case, the filter is a particular 
dispersive line whose pulse response is: 
##EQU59## 
with a coefficient K verifying the relationship (2): 
##EQU60## 
.alpha. being the modulation slope of the linear frequency-modulated 
pulses transmitted by the radar, T their duration, T.sub.d the duration of 
a demodulation ramp and T.sub.a the useful duration of demodulation or 
duration of analysis. After multiplication, the data elements are 
repositioned in a temporal space in range by a one-dimensional reverse 
Fourier transform in range. 
After they have been filtered by the dispersive line 62, the data elements 
of the table are subjected to a truncation operation 63 in which zero 
values are made to replace the data elements belonging to range slots 
located outside the duration of analysis T.sub.a, for these data elements 
constitute a parasitic noise for the subsequent operations. 
After the truncation operation 63, the data table is subjected to a 
weighting window 64 designed to reduce the minor lobes induced by the 
subsequent image focusing filtering operation whose pulse response is a 
cardinal sine response. 
After the weighting window 64 operation, the data table with two temporal 
dimensions, range and azimuth, is subjected to a one-dimensional range 
Fourier transform 65 aimed at continuing the pulse compression begun with 
the demodulation 60. The result thereof is a data table formed by the 
range-related and azimuth-related samples of a signal .nu..sub.4 (F,t) 
studied earlier. 
The frequency variable in range F resulting from the range Fourier 
transform is then changed into a range temporal variable .tau. during a 
variable changing step 66. Then the range slots of the table are 
rearranged during a rearrangement step 67 so that they succeed one another 
in the table, in a rising order of range. These steps which terminate the 
pulse compression proper lead to a data table formed by the range-related 
samples of a signal .nu..sub.5 (.tau.,t) that was studied earlier, this 
signal being subjected to a slightly modified SAR image construction 
processing operation. 
This SAR processing operation begins, as is usually the case, with a 
subdivision 68 of the data table into range blocks with overlapping to 
take account of the phenomenon of range migration. Each block has a span 
in range that is small enough for the pulse response of the focusing 
filter and the relationship of parasitic phase variation introduced by the 
pulse compression to be capable of being considered as being stationary in 
range. Each range block is then subjected to a one-dimensional range 
Fourier transform 69 followed by a one-dimensional azimuth Fourier 
transform 70. Then, its data elements placed in the frequency domains in 
range and azimuth are multiplied at 71 by the conjugate 
H'.sub.SAR/das.sup.* (t) of the transfer function of an image focusing 
filter modified to take account of a variable phase term introduced by the 
pulse compression. This modified transfer function is the 2D Fourier 
transform in range and azimuth of a pulse response h'.sub.SAR/das (t) 
expressed as a function of the pulse response h.sub.SAR (t) of a standard 
image focusing filter by a relationship having the form: 
##EQU61## 
coming from the expression of the modified pulse response h.sub.SAR/das in 
the relationship (6) from which the constant part of the phase term due to 
the demodulation has been removed. 
Each range block then has its samples replaced in the temporal range and 
azimuth domains by a one-dimensional reverse Fourier transform in azimuth 
72 followed by a one-dimensional reverse Fourier transform in range 73. 
The range blocks are then deprived of their margins of overlapping and 
juxtaposed at 74 to reconstitute a table of complex reflection 
coefficients .gamma.(.tau.,t) of the points of the zone to be imaged, the 
moduli of these reflection coefficients determining the contrast of each 
dot of the image. 
FIG. 8 shows an additional step 75 in which compensation is provided for 
the constant part of the phase term due to the pulse compression in order 
that certain processing operations, such as interferometrical processing 
operations, may be applied to the image. 
The processing that has just been proposed enables the use of the greatest 
part of the signal of a Deramp type radar for the construction of an 
image. At the same time, it ensures the taking into account of the 
parasitic phase term flowing from the particular conditions in which the 
pulse compression is done. Apart from the filtering by a dispersive line, 
it does not significantly increase the quantity of computations needed for 
the construction of a radar image.