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FIELD
The present invention relates to a method for calibrating an antenna diagram of a MIMO radar sensor having N
_{tx }transmitting antenna elements and N
_{rx }receiving antenna elements.

BACKGROUND INFORMATION
Radar sensors are often used in driver assistance systems for vehicles, for example in systems for automatic spacing control or in collision warning systems, to detect the traffic environment. In addition to the distance and relative speed, the azimuth angle of localized objects is generally also important since, for example when localizing preceding vehicles, the azimuth angle makes possible a lane allocation. The elevation angle of the localized objects can also be important, since it allows a conclusion as to the relevance of the target, for example whether the target can be driven over or under or is an obstacle representing a potential collision hazard.

The azimuth angles and elevation angles of the targets can be ascertained from amplitudes and/or phase differences of transmitting and/or receiving antennas of an antenna array. in order to improve the accuracy and separation capability of the angle estimate, it is possible to use radar sensors that operate on the multiple inputmultiple output (MIMO) principle. Unlike with conventional single inputmultiple output (SIMO) radar sensors—which work with one transmitting antenna and multiple receiving antennas—multiple transmitting antenna elements and multiple receiving antenna elements are used. In order to allow the signals of the transmitting antenna elements to be separated at the receiving antenna elements, the transmitted signals must be uncorrelated (orthogonal). This can be achieved by time multiplexing, frequency multiplexing, or code multiplexing.

In the context of angle estimation, the received signals are compared with a previously established angledependent antenna diagram. For the case in which only a single target is being localized (or multiple targets that can nevertheless be clearly distinguished from one another based on spacing and relative speed), the estimated angle is obtained as the position of the best agreement between the received signal and the antenna diagram. For the general case of multipletarget estimation, special estimating algorithms are known which supply estimated values for the localization angles of all the targets involved.

It has previously been usual to establish the antenna diagrams for each individual sensor at the factory prior to commissioning of the sensor. When the radar sensor is installed in a motor vehicle, for example behind a bumper or behind a relief structure such as an emblem of the motor vehicle brand, distortions of the antenna diagram can occur and can cause systematic errors in the angle estimate. This applies in particular to the transmitting antenna diagrams for MIMO radar sensors.

SUMMARY
An object of the present invention is to enable more accurate angle estimation with MIMO radar sensors.

This object is achieved according to the present invention by a method for calibrating an antenna diagram of a MIMO radar sensor having N
_{tx }transmitting antenna elements and N
_{rx }receiving antenna elements, which method encompasses the following steps:

before commissioning of the radar sensor:
storing an antenna diagram that associates with each of several angles θ a respective control vector a(θ) that is made up of a transmitting control vector a_{tx}(θ) and a receiving control vector a_{rx}(θ);
after commissioning:
performing a radar measurement to localize an object;
checking whether the localized object is a single target or a multiple target;
if it is a single target:
performing a SIMO measurement with each of the transmitting antenna elements;
estimating the angle θ of the object based on the measurement results;
calculating a first comparison variable, dependent on the components of the transmitting control vector a_{tx}(θ), for each transmitting antenna element;
calculating a second comparison variable, dependent on the results of the SIMO measurements, for each transmitting antenna element; and
correcting the transmitting control vector a_{tx}(θ) on the basis of a known relationship between the first and second comparison variables for each transmitting antenna element.
storing an antenna diagram that associates with each of several angles θ a respective control vector a(θ) that is made up of a transmitting control vector a_{tx}(θ) and a receiving control vector a_{rx}(θ);
after commissioning:
performing a radar measurement to localize an object;
checking whether the localized object is a single target or a multiple target;
if it is a single target:
performing a SIMO measurement with each of the transmitting antenna elements;
estimating the angle θ of the object based on the measurement results;
calculating a first comparison variable, dependent on the components of the transmitting control vector a_{tx}(θ), for each transmitting antenna element;
calculating a second comparison variable, dependent on the results of the SIMO measurements, for each transmitting antenna element; and
correcting the transmitting control vector a_{tx}(θ) on the basis of a known relationship between the first and second comparison variables for each transmitting antenna element.
This method allows calibration or postcalibration of the transmitting antenna diagram even after commissioning of the radar sensor, so that interfering influences that occur only after installation of the radar sensor in the vehicle, and therefore cannot be captured in the context of calibration at the factory, can be compensated for later on. For example, when a single object is localized at a specific azimuth angle θ while the motor vehicle in which the radar sensor is installed is being driven, then according to the method described above, the transmitting antenna diagram can be postcalibrated for that specific azimuth angle. Because single objects are localized at different azimuth angles θ over time as the motor vehicle is operated, a corrected (calibrated) transmitting antenna diagram for the entire azimuth angle range is thus gradually obtained. The calibration phase can then be completed. Optionally, however, it is also possible to proceed continuously with postcalibration, or to repeat it at certain intervals, so that agerelated changes in the antenna diagram can also be taken into account.

In the method described above, only the transmitting section of the antenna diagram is calibrated. Also a subject of the invention, however, is a method that proceeds analogously and in which the receiving section of the antenna diagram is calibrated with the aid of several multiple inputsingle output (MISO) measurements.

Also, one aspect of the present invention is a radar sensor for motor vehicles, in which sensor one of the abovedescribed methods is implemented.

An exemplifying embodiment is explained in further detail below with reference to the figures.

BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a diagram of a MIMO radar sensor having two transmitting antenna elements and four receiving antenna elements.

FIG. 2 is a diagram of the radar sensor according to
FIG. 1 after installation in a motor vehicle and during localization of an object.

FIG. 3 is a flow chart of a method for calibrating the radar sensor.

DETAILED DESCRIPTION OF EXAMPLE EMBODIMENTS
FIG. 1 is a diagram of a very simple example of a (MIMO) radar sensor
10 that, in this example, has only two transmitting antenna elements
12 and four receiving antenna elements
14. Transmitting antenna elements
12 are powered by a control and evaluation unit
16, and emit radar signals that are reflected at an object
18 and are received by each of the receiving antenna elements
14. The received signals are evaluated separately in control and evaluation unit
16.

What has been depicted here, simply for reasons of clarity, is a bistatic antenna system in which transmitting antenna elements
12 are different from receiving antenna elements
14. A monostatic antenna concept, in which the same antenna elements are used for transmission and for reception, can also be used in practice. The number of transmitting and receiving antenna elements can also be appreciably greater in practice than in the example shown here.

In this example receiving antenna elements
14 are disposed at regular intervals along a line (uniform linear array, ULA). The same is also true of transmitting antenna elements
12; the transmitting and receiving antenna elements do not necessarily need to be disposed on the same line. If the radar sensor is intended to be used to measure azimuth angles of the objects, the lines on which the antenna elements are disposed extend horizontally. In the case of a sensor for measuring elevation angles, conversely, the antenna elements would be disposed on vertical lines. Also possible is a twodimensional antenna array with which both azimuth angles and elevation angles can be measured.

In the example shown, radar sensor
10 is operated using the time multiplexing method. In other words, only one of the N
_{tx }(=2) transmitting antenna elements
12 is active at each point in time. The activity phases of the individual antenna elements alternate cyclically with one another.
FIG. 1 symbolically depicts the case in which only the lower of the two transmitting antenna elements
12 is active.

Alternatively, the radar sensor could also be operated using the frequency multiplexing method. All the transmitting antenna elements
12 would then be active simultaneously, but would operate at slightly different frequencies so that the signals of the various transmitting antenna elements can be separated again on the receiving side.

A further possibility would be the code multiplexing method. Here a specific code is modulated onto the signal transmitted by each of the transmitting antenna elements
12, and the signals are separated from one another on the receiving side by codeselective filtering.

The time multiplexing method will be considered hereinafter for explanatory purposes. In a simple signal model it can be assumed that object
18 is a pointlike scattering center at which the signal emitted from the active transmitting antenna element
12 is scattered as a spherical wave that then reaches the various receiving antenna elements
14. The distance between radar sensor
10 and object
18 is, however, unrealistically short in
FIG. 1. In practice, this distance is so great that the dimensions of radar sensor
10 are negligibly small compared with the object distance. Somewhat more realistic conditions are shown in
FIG. 2, where object
18 is located at a greater distance in front of the front portion of a motor vehicle
20 on which radar sensor
10 is disposed. The radar waves arriving at the location of the radar sensor can then be regarded, to a good approximation, as planar waves that reach all the receiving antenna elements
14 at practically the same angle of incidence—the (azimuth) angle θ of object
18.

The term
xn is intended to designate the fourcomponent vector whose components (x
_{n,1}, x
_{n,2}, x
_{n,3}, x
_{n,4}) indicate the complex amplitudes of the signals that are transmitted by the nth transmitting antenna element
12 and are received by the four receiving antenna elements
14. If d is the distance from one antenna element to another, A. is the wavelength of the radar radiation, and s=x
_{n,1 }is the (timedependent) complex amplitude of the signal that is received by the first of receiving antenna elements
14 (for example, the antenna element located farthest to the right in
FIG. 2), then the differences in path length between the signals that reach the various receiving antenna elements
14 result in the following equation:

The superscripted symbol “T” is intended to denote transposition, since vectors are described here as row vectors but are to be regarded as column vectors. The vector
a
_{rx }is referred to as a “receiving control vector.” This control vector indicates the geometric properties and wave propagation properties of the particular receiving antenna array being considered. A control vector of this kind can be defined not only for ULA antenna arrays but also very generally for other antenna configurations.

It is correspondingly also possible to define, for the array of (in this example, only two) transmitting antenna elements
12, a control vector
a
_{tx }that in this example would substantially indicate the differences in path length of the optical paths from transmitting antenna elements
12 to object
18.

The control vector obtained for the MIMO antenna array as a whole is

The symbol * here denotes the Kronecker product. In the exemplifying case considered here, therefore:

The received signals constitute a vector
x having N
_{tx }N
_{rx }components (eight components in this example), and the following applies:
x(θ)=
s a(θ) (4).

A knowledge of the control vector
a(θ) makes it possible to create an (under suitable conditions, unequivocal) correlation between the angle θ of the object and the received signals
x, and to infer the azimuth angle θ of the object from the amplitude relationships and phase relationships of the received signals. But because the received signals will in practice be more or less affected by noise, the azimuth angle cannot be exactly calculated but can only be estimated, for example using a maximum likelihood estimate.

If this principle is generalized to multipletarget estimates, the individual angle θ becomes a vector
θ whose components indicate the angles of the various targets, the control vector
a becomes a control matrix A, and the relevant equation is
x=Aθ (5).

In principle, for a given radar sensor the antenna diagram, i.e. the totality of all control vectors
a(θ) for all relevant azimuth angles θ, can be established before the sensor is commissioned. In the example shown in
FIG. 2, however, radar sensor
10 is installed behind a bumper
22 of the motor vehicle, and this bumper
22 furthermore has a certain relief
24, for example because an emblem of the motor vehicle brand is impressed or embossed. Because the optical density of the bumper is different from the optical density of air, refraction of the radar waves at and in bumper
22 causes path length differences that have an influence on the antenna diagram. This influence depends on the exact installation location of the radar sensor relative to relief
24, and is therefore difficult to compensate for computationally, especially since effects such as soiling of the bumper can further distort the antenna diagram.

An object of the present invention is thus to find a way to calibrate the antenna diagram at a later time, after the radar sensor is installed on the vehicle, so that such distortions of the antenna diagram do not result in appreciable errors in angle estimation.

Firstly a possibility will be described for postcalibrating the transmitting control vector
a
_{tx}(θ). It is useful in this context to define relative control vectors
a′
_{tx}(θ) and
a′
_{rx}(θ):
a′(θ):=
a′
_{tx}(θ)/
a
_{tx,1}(θ) (6)
and
a′
_{rx}(θ):=
a
_{tx,1}(θ)
a
_{rx}(θ) (7).

Based on equation (2),
a(θ):=
a′
_{tx}(θ)
a′
_{rx}(θ) (8).

since the factor
a′
_{tx}(θ) condenses out of this. Postcalibration of the relative transmitting control vector
a′
_{tx}(θ) is therefore equivalent to a postcalibration of the original transmitting control vector
a
_{tx}(θ).

The quality of an angle estimate is described by the socalled quality function q(
θ). This function is an indicator of the probability that the estimated values ascertained using the estimation algorithm correspond to the actual angular positions of the localized objects. For the general case of a multipletarget estimate, in which multiple targets that are not distinguishable on the basis of their distances and relative speeds are localized at different angles θ
_{j}, the quality function is defined by the equation
q
^{2}(
θ)=
x
^{H}
P
_{A}(θ)
x (9),

in which
x is the vector of the signals obtained with the various combinations of transmitting and receiving antenna elements,
x
^{H }is the vector Hermitianconjugated with
x, and P
_{A}(
θ)=A(A
^{H }A)
^{−1 }A
^{H}, where A is the control matrix that contains the control vectors of all the targets. The control matrix can optionally also contain multiple control vectors for the same target if, as a result of reflections from guard rails or the like, multiple optical paths lead from the same target to the radar sensor.

In the case of a single target having only one path, this equation is simplified to

The control vector
a(θ) can be measured and normalized before commissioning of the radar sensor. The signal sector
x can likewise be respectively normalized once measurement has occurred. It will therefore be assumed hereinafter that both the control vector and the signal vector are normalized, thereby simplifying the equation even further to

The vector x can be described as
x={xi}, i=1 . . .
N
_{tx} (12),

in which
xi are the vectors that indicate the measured signals that were transmitted by the ith transmitting antenna element and were received by the N
_{rx }receiving antenna elements. For a (singletarget) MIMO angle estimate using all the transmitting antenna elements
12, the quality function is then obtained as:

in which the sum is over all N
_{tx }transmitting antenna elements (summation index i), and a
_{tx,i}*(θ) is the complex conjugate of the individual components a
_{tx,i}(θ) of the transmitting control vector
a
_{tx}(θ). The last transformation in equation (13) follows from equation (2).

If variables y
_{i}(θ) are then defined as
y
_{i}(θ)=
a
_{rx}
^{H}(θ)
xi (14),

it then follows from equation (13) that

For a singletarget SIMO angle estimate that is executed only with the ith transmitting antenna element, the following is obtained (with normalization):

Based on this angle quality it is then possible to decide whether a singletarget situation or multipletarget situation exists with regard to the angle estimate. The quality function will have an appreciably lower value for the multipletarget situation. The decision can therefore be made using the criterion that the quality function for the estimated angle θ is below a suitably selected threshold value.

Alternatively, however, other methods and criteria can be employed in order to decide between a singletarget situation and a multipletarget situation.

If signal noise is ignored, the following (approximately) describes the true angle θ:
xi=sa
_{tx,i}(θ)
a
_{rx}(θ) (17)

where s represents the complex amplitude of the signal emitted from the target, and θ is the actual angle of the target.

Using equation (17) in equation (14), the result is:
y
_{i}(θ)=
a
_{rx}
^{H}(θ)
sa
_{tx,i}(θ)
a
_{rx}(θ) (18).

Because
a
_{rx}(θ) is normalized, this is simplified to:
y
_{i}(θ)=
s a
_{tx,i}(θ) (19).

If SIMO measurements are then carried out successively using each of the transmitting antenna elements
12, N
_{tx }correlations of the kind indicated in equation (19) are then obtained. It is not yet possible, however, to check directly whether and how accurately these equations (19) are satisfied, since the amplitude s is not known. This problem can nevertheless be circumvented by respectively dividing the vectors on the left and the right side of equation (19) by a single one of their components (with no limitation as to generality, by the first component a
_{tx,1}(θ) and y
_{1}(θ), respectively). What is obtained is then:
a
_{tx,n}(θ)/
a
_{tx,1}(θ)=
y
_{n}(θ)/
y
_{1}(θ) (20).

The variables on the left side of equation (20) are the components of the relative transmitting control vector a
_{tx}′(θ). The variables on the right side are obtained, using equation (14), from the measurement results
xi and the known receiving control vector
a
_{rx}(θ) that was originally used.

Postcalibration can therefore easily be accomplished by replacing the previously used relative transmitting control vector
a
_{tx}′(θ) with the vector having the components y
_{n}(θ)/y
_{1}(θ). It is also simple to determine, based on the discrepancy between the old and the new relative transmitting control vector, how severely the previously used antenna diagram was distorted.

With the procedure described above, firstly a coherent summation (y
_{i}(θ)=
a
^{H}
_{rx}(θ)
xi) must be performed, and then the ratio y
_{n}(θ)/y
_{1}(θ) is calculated. In general, however, a tolerably small error occurs if the calculation is simplified by first calculating the ratios x
_{n,k}/x
_{1,k }and then averaging over all k (i.e. over all receiving antenna elements). As an approximation, instead of equation (20) the following relationships can also be used:
a
_{tx,n}(θ)/
a
_{tx,1}(θ)=(1/
N
_{rx})Σ
_{k}(
x
_{n,k}
/x
_{1,k}) (21)

in which x
_{n,k}=(
xn)
_{k}=x
_{j}, where j=(k−1)N
_{tx}+n, the nth component of the partial vector
xn of
x.

The method according to the present invention therefore encompasses in essence the following steps:
calculating a first comparison variable dependent on the components of the transmitting control vector a_{tx}(θ) (e.g. a_{tx,n}(θ)/a_{tx,1}(θ)) for each transmitting antenna element (having the index n);
calculating a second comparison variable dependent on the results of the SIMO measurements (e.g. y_{n}(θ)/y_{1}(θ) or Σ_{k}(x_{n,k}/x_{1,k})) for each transmitting antenna element; and
correcting the transmitting control vector a_{tx}(θ) (or, equivalently thereto, the relative transmitting control vector a′_{tx}(θ)) based on a known relationship (equation (20) or (21)) between the first and second comparison variables for each transmitting antenna element.
calculating a first comparison variable dependent on the components of the transmitting control vector a_{tx}(θ) (e.g. a_{tx,n}(θ)/a_{tx,1}(θ)) for each transmitting antenna element (having the index n);
calculating a second comparison variable dependent on the results of the SIMO measurements (e.g. y_{n}(θ)/y_{1}(θ) or Σ_{k}(x_{n,k}/x_{1,k})) for each transmitting antenna element; and
correcting the transmitting control vector a_{tx}(θ) (or, equivalently thereto, the relative transmitting control vector a′_{tx}(θ)) based on a known relationship (equation (20) or (21)) between the first and second comparison variables for each transmitting antenna element.

An analogous method also allows calibration or postcalibration of the receiving antenna diagram, i.e. of the receiving control vector
a
_{rx}.

What is then obtained instead of equation (13) is:

where
x′(θ)=Σ
_{i}
a
_{tx}*(θ)
xi (23)

and, analogously to equation (12),
x′={xn′}, n=1 . . .
N
_{rx} (24).

For a singletarget MISO (multiple inputsingle output) angle estimate that is performed only with the nth receiving antenna element, the following is obtained (with normalization):

Again, a decision can be made on the basis of the angle quality as to whether a singletarget situation or a multipletarget situation exists in the context of the angle estimate.

If a singletarget situation exists and if signal noise is ignored, the true angle θ is then approximately described by:
xi=sa
_{tx,i}(θ)
a
_{rx}(θ) (26).

A calculation analogous to equations (18) to (20) then yields:

In this case the first comparison variables are therefore the components a
_{rx,n}(θ)/a
_{rx,1}(θ) of a relative receiving control vector
a″
_{rx }(defined analogously to equation (6)). The second comparison variables, which are now dependent on the results of the MISO measurements, are calculated using the variables x′
_{n}(θ)/x′
_{1}(θ) for each receiving antenna element (index n).

In this case as well, the comparison variables and relationships analogous to equation (21) can be used as a good approximation:

FIG. 3 depicts a complete calibration procedure that can be performed while the motor vehicle that has radar sensor
10 is being driven.

In step S
1, firstly normal localization operation is carried out, i.e. objects in the surroundings of the vehicle are localized with the aid of radar sensor
10. For the localized objects, multipletarget angle estimates are carried out on the basis of the antenna diagram that was originally established upon commissioning of the radar sensor or was recalibrated in previous postcalibration procedures.

Step S
2 checks whether, in the current localization cycle, an object was localized at an angle for which a postcalibration had not yet taken place, or for which the most recent postcalibration was already some time in the past.

If so (Y), then in step S
3 a SIMO measurement and an angle estimate using one of the transmitting antenna elements
12 is performed. Otherwise (N) execution jumps back to step S
1, and the loop having steps S
1 and S
2 is cycled through until an object is found at an angle for which a postcalibration needs to take place.

Based on the angle quality of the SIMO angle estimate carried out in step S
3, a decision is made in step S
4 as to whether or not the object localized at the angle θ is a single object.

If it is not a single object (N), execution again jumps back into the loop having steps S
1 and S
2. If it is a single object (Y), then in step S
5 further SIMO measurements are performed with all the transmitting antenna elements
12. A complete set of the variables y
_{i}(θ) for all the transmitting antenna elements is thereby obtained. Based on the comparison variables calculated therefrom, in step S
6 the (relative) transmitting control vector a′
_{tx}(θ) is then corrected (recalibrated).

Optionally a step S
7 can also be added, in which MISO measurements using all the receiving antenna elements are carried out for the same object. In step S
8 the (relative) receiving control vector a″
_{rx}(θ) is then corrected on the basis of those measurements.

If the time multiplexing method is used, the time intervals between the individual SIMO measurements or MISO measurements should not be too long, so that errors caused by any location shifts of the object in the time elapsing between the measurements do not produce a significant error. It is possible, however, to interleave the measurements in time, and/or to combine the measurement results, in such a way that the errors caused by time shifts are averaged out.

An analogous problem also occurs with the frequency multiplexing method, since here the distance to the target, in conjunction with the frequency shift between the transmitting antenna elements, can result in a phase difference between the variables y
_{i}(θ) which may need to be compensated for.
