Direction-of-arrival estimation apparatus, method, and non-transitory medium

Abstract

A direction of arrival estimation apparatus includes at least first and second sub-arrays to receive a reflected wave of a transmission waveform from a target; first and second phasing parts that perform phasing of reception signals at the first and second sub-arrays to generate first and second sub-array beams; an arrival time difference calculation part that calculates first and second correlations of the reception signals of the first and second sub-array beams at first and second time points to find an arrival time difference between times of the reflected wave arriving at the first and second sub-arrays, based on a result of a predetermined operation on the first and second correlations and a time difference between the first time point and the second time point; and a direction of arrival calculation part that finds a direction of arrival of the target based on the arrival time difference.

Claims

1. A direction of arrival estimation apparatus comprising: first and second sub-arrays, each including a plurality of receivers configured to receive a wave including a reflected wave of a transmission wave from a target, which is transmitted from the direction of arrival estimation apparatus and reflected by the target; first and second phasing parts configured to perform phasing of a plurality of reception signals received at the first and second sub-arrays to generate first and second sub-array beams, respectively; an arrival time difference calculation part configured to calculate a first correlation {S.sub.r(t.sub.1)S.sub.l*(t.sub.1)} (where * is a complex conjugate operator) of reception signals S.sub.r(t.sub.1) and S.sub.1(t.sub.1) of the first and second sub-array beams at a first time point t.sub.1 and a second correlation {S.sub.r(t.sub.2)S.sub.1*(t.sub.2)} of reception signals S.sub.r(t.sub.2) and S.sub.l(t.sub.2) of the first and second sub-array beams at a second time point t.sub.2, wherein the arrival time difference calculation part is configured to calculate a first complex conjugate product {S.sub.r(t.sub.1)S.sub.l*(t.sub.1)}*{S.sub.r(t.sub.2)S.sub.l*(t.sub.2)} of the first and second correlations and a second complex conjugate product {S.sub.r(t.sub.1)S.sub.l*(t.sub.1)} {S.sub.r(t.sub.2)S.sub.1*(t.sub.2)}* of the first and second correlations, and wherein the arrival time difference calculation part is configured to find an arrival time difference between arrival times of the reflected wave from the target at the first sub-array and at the second sub-array, using a result of an inverse trigonometric operation on a value obtained by dividing a difference between the first and second complex conjugate products by a sum of the first and second complex conjugate products, a distance between the first and second sub-arrays beams, a frequency change rate of the transmission wave, and a time difference between the first time point and the second time point; and a direction of arrival calculation part configured to find a direction of arrival of the target, using the arrival time difference.

2. The direction of arrival estimation apparatus according to claim 1, wherein the arrival time difference calculation part is configured to calculate and store first to third correlations of the reception signals of the first and second sub-array beams at time points including at least first to third time points, and wherein the direction of arrival estimation apparatus further includes: a direction averaging part configured to average at least two directions of arrivals (DOAs) calculated using combinations of two pairs of the correlations selected from the first to third correlations at the time points including at least the first to third time points.

3. The direction of arrival estimation apparatus according to claim 1, further comprising: a fitting part configured to perform fitting of a straight line or a curve to a plurality of directions of arrivals (DOAs) calculated using the first to third correlations of the reception signals of the first and second sub-array beams at the time points including at least the first to third time points.

4. The direction of arrival estimation apparatus according to claim 1, further comprising: at least a third sub-array, wherein the arrival time difference calculation part and the direction of arrival calculation part are configured to calculate at least two directions of arrivals (DOAs), for respective different combinations of two sub-arrays among the first to third sub-arrays, and wherein the direction of arrival estimation apparatus comprises a sub-array averaging part configured to calculate an average of the at least two directions of arrivals (DOAs).

5. The direction of arrival estimation apparatus according to claim 1, wherein the direction of arrival calculation part is configured to calculate the direction of arrival of the target, using the arrival time difference, a distance between the first and second sub-array beams, and a reflected wave velocity.

6. A direction of arrival estimation apparatus comprising: first and second sub-arrays, each including a plurality of receivers configured to receive a wave including a reflected wave of a transmission wave from a target, which is transmitted from the direction of arrival estimation apparatus and reflected by the target; first and second phasing parts configured to perform phasing of a plurality of reception signals received at the first and second sub-arrays to generate first and second sub-array beams, respectively; an arrival time difference calculation part configured to calculate first and second correlations of reception signals of the first and second sub-array beams at time points including at least first and second time points and perform a predetermined operation on the first and second correlations to find an arrival time difference between arrival times of the reflected wave from the target at the first sub-array and at the second sub-array, using on a result of the predetermined operation on the first and second correlations and a time difference between the first time point and the second time point; and a direction of arrival calculation part configured to find a direction of arrival of the target, using the arrival time difference, wherein the transmission wave has a linear frequency modulation (LFM) waveform with a frequency change rate ξ, wherein a direction of arrival calculation part is configured to derive the direction of arrival of the target θ from the following equation: $\sin \theta =\frac{c\left(t_{l0}-t_{r0}\right)}{d}$ wherein c is the reflected wave velocity, wherein d is a distance between the first and second sub-array beams, wherein t.sub.l0−t.sub.r0 is the arrival time difference between arrival times of the reflected wave from the target at the first sub-array and at the second sub-array and given by: $t_{l0}-t_{r0}=\frac{1}{2\mathrm{\pi \xi }\left(t_1-t_2\right)}{\tan }^{-1}\left[\frac{j\left[{\left\{S_r\left(t_1\right)S_l^{*}\left(t_1\right)\right\}}^{*}\left\{S_r\left(t_2\right)S_l^{*}\left(t_2\right)\right\}-\left\{S_r\left(t_1\right)S_l^{*}\left(t_1\right)\right\}{\left\{S_r\left(t_2\right)S_l^{*}\left(t_2\right)\right\}}^{*}\right]}{\left\{S_r\left(t_1\right)S_l^{*}\left(t_1\right)\right\}{\left\{S_r\left(t_2\right)S_l^{*}\left(t_2\right)\right\}}^{*}+{\left\{S_r\left(t_1\right)S_l^{*}\left(t_1\right)\right\}}^{*}\left\{S_r\left(t_2\right)S_l^{*}\left(t_2\right)\right\}}\right]$ wherein ξ is the frequency change rate, wherein t.sub.1−t.sub.2 is the time difference between the first time point t.sub.1 and the second time point t.sub.2, wherein tan.sup.−1 is an inverse tangent (arctangent) function, wherein j.sup.2=−1, S.sub.r(t.sub.1)S.sub.l*(t.sub.1)} is the first correlations at the first time point t.sub.1, wherein S.sub.r(t.sub.1) and S.sub.l(t.sub.1) are the reception signals of the first and second sub-array beams at the first time point t.sub.1, wherein {S.sub.r(t.sub.2)S.sub.1*(t.sub.2)} is the second correlations at the second time point t.sub.2, wherein S.sub.r(t.sub.2) and S.sub.l(t.sub.2) are the reception signals of the first and second sub-array beams at the second time point t.sub.2, and wherein * is a complex conjugate operator.

7. A direction of arrival estimation method used for an apparatus comprising at least first and second sub-arrays respectively including a plurality of receivers configured to receive reception signals including a reflected wave of a transmission waveform from a target, which is transmitted from the direction of arrival estimation apparatus and reflected by the target, the method comprising: performing phasing of a plurality of the reception signals received at the first and second sub-arrays to generate first and second sub-array beams, respectively; calculating a first correlation {S.sub.r(t.sub.1)S.sub.1*(t.sub.1)} (where * is a complex conjugate operator) of reception signals S.sub.r(t.sub.1) and S.sub.l(t.sub.1) of the first and second sub-array beams at a first time point t.sub.1 and a second correlation {S.sub.r(t.sub.2)S.sub.l*(t.sub.2)} of reception signals S.sub.r(t.sub.2) and S.sub.l(t.sub.2) of the first and second sub-array beams at a second time point t.sub.2; calculating a first complex conjugate product {S.sub.r(t.sub.1)S.sub.l*(t.sub.1)}*{S.sub.r(t.sub.2)S.sub.l*(t.sub.2)} of the first and second correlations and a second complex conjugate product {S.sub.r(t.sub.1)S.sub.1*(t.sub.1)} {S.sub.r(t.sub.2)S.sub.1*(t.sub.2)}*of the first and second correlations; finding an arrival time difference between arrival times of the reflected wave from the target at the first sub-array and at the second sub-array, using a result of an inverse trigonometric operation on a value obtained by dividing a difference between the first and second complex conjugate products by a sum of the first and second complex conjugate products, a distance between the first and second sub-array beams, a frequency change rate of the transmission wave, and a time difference between the first time point and the second time point; and finding a direction of arrival of the target, using the arrival time difference.

8. The direction of arrival estimation method according to claim 7, the method comprising: calculating and storing first to third correlations of the reception signals of the first and second sub-array beams at time points including at least first to third time points; and averaging at least two directions of arrivals (DOAs) calculated using combinations of two pairs of the correlations selected from the first to third correlations at the time points including at least the first to third time points.

9. The direction of arrival estimation method according to claim 7, the method comprising: performing fitting of a straight line or a curve to a plurality of directions of arrivals (DOAs) calculated using the first to third correlations of the reception signals of the first and second sub-array beams at the time points including at least the first to third time points.

10. The direction of arrival estimation method according to claim 7, the method comprising: calculating at least two directions of arrivals (DOAs), for respective different combinations of two sub-arrays among first to third sub-arrays included in the apparatus; and calculating an average of the at least two directions of arrivals (DOAs).

11. The direction of arrival estimation method according to claim 7, the method comprising: calculating the direction of arrival of the target, using the arrival time difference, a distance between the first and second sub-array beams, and a reflected wave velocity.

12. A direction of arrival estimation method used for an apparatus comprising at least first and second sub-arrays respectively including a plurality of receivers configured to receive reception signals including a reflected wave of a transmission waveform from a target, which is transmitted from the direction of arrival estimation apparatus and reflected by the target, the method comprising: performing phasing of a plurality of the reception signals received at the first and second sub-arrays to generate first and second sub-array beams, respectively; calculating first and second correlations of the reception signals of the first and second sub-array beams at time points including at least first and second time points and performing a predetermined operation on the first and second correlations to find an arrival time difference between arrival times of the reflected wave from the target at the first sub-array and at the second sub-array, using a result of the predetermined operation on the first and second correlations and a time difference between the first time point and the second time point; and finding a direction of arrival of the target, using the arrival time difference, wherein the transmission wave has a linear frequency modulation (LFM) waveform with a frequency change rate ξ, wherein the method comprises: deriving the direction of arrival of the target θ from the following equation: $\sin \theta =\frac{c\left(t_{l0}-t_{r0}\right)}{d}$ wherein c is the reflected wave velocity, wherein d is a distance between the first and second sub-array beams, and wherein t.sub.l0−t.sub.r0 is the arrival time difference between arrival times of the reflected wave from the target at the first sub-array and at the second sub-array and given by: $t_{l0}-t_{r0}=\frac{1}{2\mathrm{\pi \xi }\left(t_1-t_2\right)}{\tan }^{-1}\left[\frac{j\left[{\left\{S_r\left(t_1\right)S_l^{*}\left(t_1\right)\right\}}^{*}\left\{S_r\left(t_2\right)S_l^{*}\left(t_2\right)\right\}-\left\{S_r\left(t_1\right)S_l^{*}\left(t_1\right)\right\}{\left\{S_r\left(t_2\right)S_l^{*}\left(t_2\right)\right\}}^{*}\right]}{\left\{S_r\left(t_1\right)S_l^{*}\left(t_1\right)\right\}{\left\{S_r\left(t_2\right)S_l^{*}\left(t_2\right)\right\}}^{*}+{\left\{S_r\left(t_1\right)S_l^{*}\left(t_1\right)\right\}}^{*}\left\{S_r\left(t_2\right)S_l^{*}\left(t_2\right)\right\}}\right]$ wherein ξ is the frequency change rate, wherein t.sub.1−t.sub.2 is the time difference between the first time point t.sub.1 and the second time point t.sub.2, wherein tan.sup.−1 is an inverse tangent (arctangent) function, wherein j.sup.2=−1, {S.sub.r(t.sub.1)S.sub.l*(t.sub.1)} is the first correlations at the first time point t.sub.1, wherein S.sub.r(t.sub.1) and S.sub.l(t.sub.1) are the reception signals of the first and second sub-array beams at the first time point t.sub.1, wherein {S.sub.r(t.sub.2)S.sub.1*(t.sub.2)} is the second correlations at the second time point t.sub.2, wherein S.sub.r(t.sub.2) and S.sub.l(t.sub.2) are the reception signals of the first and second sub-array beams at the second time point t.sub.2, and wherein * is a complex conjugate operator.

13. A non-transitory computer readable medium storing a program causing a computer included in an apparatus including at least first and second sub-arrays respectively including a plurality of receivers configured to receive reception signals including a reflected wave of a transmission wave from a target, which is transmitted from the direction of arrival estimation apparatus and reflected by the target, to execute processing comprising: performing phasing of a plurality of the reception signals received at the first and second sub-arrays to generate first and second sub-array beams, respectively; calculating a first correlation {S.sub.r(t.sub.1)S.sub.1*(t.sub.1)} (where * is a complex conjugate operator) of reception signals S.sub.r(t.sub.1) and S.sub.l(t.sub.1) of the first and second sub-array beams at a first time point t.sub.1 and a second correlation {S.sub.r(t.sub.2)S.sub.l*(t.sub.2)} of reception signals S.sub.r(t.sub.2) and S.sub.l(t.sub.2) of the first and second sub-array beams at a second time point t.sub.2; calculating a first complex conjugate product {S.sub.r(t.sub.1)S.sub.l*(t.sub.1)}*{S.sub.r(t.sub.2)S.sub.l*(t.sub.2)} of the first and second correlations and a second complex conjugate product {S.sub.r(t.sub.1)S.sub.1*(t.sub.1)} {S.sub.r(t.sub.2)S.sub.1*(t.sub.2)}*of the first and second correlations; finding an arrival time difference between arrival times of the reflected wave from the target at the first sub-array and at the second sub-array, using a result of an inverse trigonometric operation on a value obtained by dividing a difference between the first and second complex conjugate products by a sum of the first and second complex conjugate products, a distance between the first and second sub-arrays beams, a frequency change rate of the transmission wave, and a time difference between the first time point and the second time point; and finding a direction of arrival of the target, using the arrival time difference.

14. The non-transitory computer readable medium according to claim 13, storing the program causing the computer to execute processing comprising: calculating and storing first to third correlations of the reception signals of the first and second sub-array beams at time points including at least first to third time points, and averaging at least two directions of arrivals (DOAs) calculated using combinations of two pairs of the correlations selected from the first to third correlations at the time points including at least the first to third time points.

15. The non-transitory computer readable medium according to claim 13, storing the program causing the computer to execute processing comprising: performing fitting of a straight line or a curve to a plurality of directions of arrivals (DOAs) calculated using the first to third correlations of the reception signals of the first and second sub-array beams at the time points including at least the first to third time points.

16. The non-transitory computer readable medium according to claim 13, storing the program causing the computer to execute processing comprising: calculating at least two directions of arrivals (DOAs), for respective different combinations of two sub-arrays among first to third sub-arrays included in the apparatus; and calculating an average of the at least two directions of arrivals (DOAs).

17. The non-transitory computer readable medium according to claim 13, storing the program causing the computer to execute processing comprising: calculating the direction of arrival of the target, using the arrival time difference, a distance between the first and second sub-array beams, and a reflected wave velocity.

18. A non-transitory computer readable medium storing a program causing a computer included in an apparatus including at least first and second sub-arrays respectively including a plurality of receivers configured to receive reception signals including a reflected wave of a transmission wave from a target, which is transmitted from the direction of arrival estimation apparatus and reflected by the target, to execute processing comprising: performing phasing of a plurality of the reception signals received at the first and second sub-arrays to generate first and second sub-array beams, respectively; calculating first and second correlations of the reception signals of the first and second sub-array beams at time points including at least first and second time points and performing a predetermined operation on the first and second correlations to find an arrival time difference between arrival times of the reflected wave from the target at the first sub-array and at the second sub-array, using a result of the predetermined operation on the first and second correlations and a time difference between the first time point and the second time point and finding a direction of arrival of the target, using the arrival time difference, wherein the transmission wave has a linear frequency modulation (LFM) waveform with a frequency change rate ξ, wherein the processing further comprises: deriving the direction of arrival of the target θ from the following equation: $\sin \theta =\frac{c\left(t_{l0}-t_{r0}\right)}{d}$ wherein c is the reflected wave velocity, wherein d is a distance between the first and second sub-array beams, and wherein t.sub.l0−t.sub.r0 is the arrival time difference between arrival times of the reflected wave from the target at the first sub-array and at the second sub-array and given by: $t_{l0}-t_{r0}=\frac{1}{2\mathrm{\pi \xi }\left(t_1-t_2\right)}{\tan }^{-1}\left[\frac{j\left[{\left\{S_r\left(t_1\right)S_l^{*}\left(t_1\right)\right\}}^{*}\left\{S_r\left(t_2\right)S_l^{*}\left(t_2\right)\right\}-\left\{S_r\left(t_1\right)S_l^{*}\left(t_1\right)\right\}{\left\{S_r\left(t_2\right)S_l^{*}\left(t_2\right)\right\}}^{*}\right]}{\left\{S_r\left(t_1\right)S_l^{*}\left(t_1\right)\right\}{\left\{S_r\left(t_2\right)S_l^{*}\left(t_2\right)\right\}}^{*}+{\left\{S_r\left(t_1\right)S_l^{*}\left(t_1\right)\right\}}^{*}\left\{S_r\left(t_2\right)S_l^{*}\left(t_2\right)\right\}}\right]$ wherein is the frequency change rate, wherein t.sub.1−t.sub.2 is the time difference between the first time point t.sub.1 and the second time point t.sub.2, wherein tan.sup.−1 is an inverse tangent (arctangent) function, wherein j.sup.2=−1, wherein{S.sub.r(t.sub.1)S.sub.1*(t.sub.1)} is the first correlations at the first time point t.sub.1, where S.sub.r(t.sub.1) and S.sub.l(t.sub.1) are the reception signals of the first and second sub-array beams at the first time point t.sub.1, wherein{S.sub.l(t.sub.2)S.sub.1*(t.sub.2)} is the second correlations at the second time point t.sub.2, wherein S.sub.r(t.sub.2) and S.sub.l(t.sub.2) are the reception signals of the first and second sub-array beams at the second time point t.sub.2, and wherein * is a complex conjugate operator.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 is a diagram illustrating an arrangement of a first example embodiment of the present invention.

(2) FIG. 2 is a diagram illustrating an arrangement of a second example embodiment of the present invention.

(3) FIG. 3 is diagram illustrating an arrangement of a third example embodiment of the present invention.

(4) FIG. 4 is a diagram illustrating an arrangement of a fourth example embodiment of the present invention.

(5) FIG. 5 is a diagram illustrating a related art.

(6) FIG. 6A is a diagram illustrating the related art.

(7) FIG. 6B is a diagram illustrating the related art.

(8) FIG. 7 is a diagram illustrating an arrangement of a fifth example embodiment.

DETAILED DESCRIPTION

(9) First, example embodiments of the present invention will be described with reference to the drawings. FIG. 1 is a diagram illustrating an arrangement of a first example embodiment of the present invention. Referring to FIG. 1, a direction of arrival (DOA) estimation apparatus (system) 100 includes a transmission waveform setting part 101 whereby a user sets a transmission waveform in advance, a transmission part 102 configured to transmit a wave(s) of the transmission waveform that has been specified, and a plurality of reception parts (echo sounder receivers) 103 configured to receive a waves including a reflected wave from a target. The plurality of reception parts 103 constitute an array (echo sounder receiver array) 105 which includes a plurality of sub-arrays (sub-arrays) 104. Each sub-array 104 is constituted from a predefined number (a plurality) of the reception parts 103 that are close to one another in distance. In the example in FIG. 1, the array 105 includes two sub-arrays 104.

(10) The direction of arrival estimation apparatus (system) 100 includes, for each sub-array 104, a phasing part 106 configured to perform a process including aligning phases of reception signals received at the reception parts 103 of the sub-array 104 to generate a sub-array beam. The direction of arrival estimation apparatus (system) 100 further includes an arrival time difference calculation part 107 configured to calculate correlations between the respective reception signals of the sub-array beams of the two sub-arrays 104 at a plurality of times, and calculate a difference between times of the reflected waves arriving at the two sub-arrays 104 from the target, based on an operation on the correlations at the plurality of times, and a direction of arrival calculation part 108 configured to calculate (estimate) a direction of arrival of the target based on the difference between the arrival times of the reflected waves from the target arriving at the two sub-arrays 104. It is noted that the arrival time difference calculation part 107 calculates a phase difference of the reflected waves from the target between the sub-arrays 104. These respective parts may be configured as a unit apparatus (direction of arrival estimation apparatus) constituted from a single unit, may be configured as two apparatuses (units) of a transmission apparatus (including the transmission waveform setting part 101 and the transmission part 102, for example) and a reception apparatus (including sub-arrays 104.sub.a and 104.sub.b, phasing parts 106.sub.a and 106.sub.b, the arrival time difference calculation part 107, and the direction of arrival calculation part 108, for example), or may be configured as a system (direction of arrival estimation system) constituted from three or more units. The following describes an outline of a process of each part.

(11) When a user sets a transmission waveform using a keyboard (not illustrated) or the like, the transmission waveform setting part 101 stores the transmission waveform that has been set. It may be so configured that the transmission wave setting part 101 specifies modulation such as frequency modulation or phase modulation.

(12) The transmission part 102 includes a transmitter and electric circuits (electronic circuits) to convert a digital signal to an analog signal. In the case of a sonar, for example, the transmission part 102 includes a transducer constituted from a piezoelectric element or the like which converts an electronic signal (electrical energy) to an acoustic signal (acoustic energy). After the electric circuit has converted the transmission waveform stored in the transmission waveform setting part 101 to an analog signal (electronic signal), power of the analog signal is amplified, and a sound wave is transmitted into sea from the transducer. In the case of a radar, for example, the transmission part 102 includes an antenna. The transmission part 102 converts the transmission waveform stored in the transmission waveform setting part 101 in the form of digital codes to an analog signal (electronic signal) using an analog-to-digital converter or the like, mixes the analog signal with a local oscillation signal using a mixer, thus performing frequency-conversion to an RF signal. Power of the RF signal is amplified, so that a radio wave is emitted into air from the antenna.

(13) Each reception part 103 includes a receiver (echo sounder receiver) and an electric circuit (electronic circuit) such as an analog-to-digital converter to convert an analog signal output from the receiver to a digital signal. In the case of the sonar, for example, the reception part 103 includes a transducer (transducer) constituted from a piezoelectric element or the like to convert a sound wave from under water to an electronic signal (analog signal), converts, into the digital signal, the electronic signal (analog signal) obtained by the conversion of a sound wave under water including a reflection wave from a target which reflects a wave transmitted from the transmission part 102 and then received by the transducer, and outputs a digital signal. In the case of the radar, for example, the reception part 103 includes an antenna. The antenna receives a reflected signal. An LNA (Low Noise Amplifier: low noise amplifier) performs low noise amplification on the reception signal from the antenna. The resulting signal is mixed with a local oscillation signal by a mixer to generate a beat frequency (intermediate frequency) output, which is converted by an analog-to-digital converter to a digital signal for output.

(14) Each of two sub-arrays 104.sub.a and 104.sub.b is constituted from the predefined number (a plurality) of the reception parts 103 that are close to one another in distance. When no particular distinction does not need to be made between the sub-arrays 104.sub.a and 104.sub.b for description, the sub-arrays 104.sub.a and 104.sub.b are referred to as the sub-arrays 104.

(15) When all the reception parts 103 are linearly arranged as illustrated in FIG. 5, for example, the array 105 is constituted from all the reception parts 103. Generally, the array 105 is referred to as a “line array”, a “linear array”, or the like. This array 105 in the form of a straight line is divided into two parts at a center thereof. A set of the reception parts 103 on the right and left sides equally divided into two parts can be set to the sub-arrays 104. This array 105 in the form of the straight line is horizontally disposed and the sub-array 104.sub.a on the right side with respect to the target is referred to as a right sub-array, and the sub-array 104.sub.b on the left side with respect to the target is referred to as a left sub-array, for example.

(16) The phasing parts 106.sub.a and 106.sub.b are respectively provided in association with the sub-arrays 104.sub.a and 104.sub.b, and each of the phasing parts 106.sub.a and 106.sub.b aligns phases of signals received at the reception parts 103 for each sub-array 104 to generate a sub-array beam. When no particular distinction does not need to be made between the phasing parts 106.sub.a and 106.sub.b for description, the phasing parts 106.sub.a and 106.sub.b are referred to as phasing parts 106.

(17) In the above example described with reference to FIG. 5, for example, the beam generated for a right sub-array 104.sub.r (the sub-array 104.sub.a in FIG. 1) of the array in the form of a straight line, which is horizontally disposed, will be referred to as a “right beam”, while the beam generated for a left sub-array 104.sub.l (the sub-array 104.sub.b in FIG. 1) will be referred to as a “left beam”. Any known method may be employed for beam generation.

(18) In this example embodiment, as a transmission signal, an LFM (Linear Frequency Modulation: linear frequency modulation) transmission waveform (pulse compression waveform obtained by linear chirping) is used which is expressed as follows:

(19) $\begin{array}{cc}{S}_t\left(t\right)=\exp \left\{2\pi j\left(f_{0}t+\frac{\xi t^2}{2}\right)\right\}& \left(20a\right)\end{array}$
where an amplitude is set to 1, and a length (pulse length) of the transmission signal is set to T.

(20) By performing time-differentiation on a phase ϕ(t):

(21) $\begin{array}{cc}\varphi \left(t\right)=2\pi \left(f_{0}t+\frac{\xi t^2}{2}\right)& \left(20b\right)\end{array}$
we have an instantaneous frequency f(t) given by the equation (20c):

(22) $\begin{array}{cc}f\left(t\right)=\left(\frac{1}{2\pi }\right)\frac{d}{dt}\varphi \left(t\right)=f_0+\xi t& \left(20c\right)\end{array}$

(23) When a frequency change rate (chirp rate) ξ is positive, the instantaneous frequency f(t) linearly increases from a sweep start frequency f.sub.0 at time t=0 until time T (where a sweep end frequency=f.sub.0+ξT). When the frequency change rate (chirp rate) is negative, the instantaneous frequency f(t) linearly decreases from the sweep start frequency f.sub.0 at time t=0 until time T. It is noted that in the equations (20a) and (20b), a phase ϕ(0) at time t=0, which is an initial phase, is set to 0.

(24) Here, it is assumed that the transmission signal of the equation (20a) was transmitted from the transmission part 102 and an echo from a target has arrived at the right beam at time t=t.sub.r0. In this case, a reception signal S.sub.r(t) of the right beam is expressed by the following equation (21a), where A.sub.r(t) is an amplitude (real value):

(25) 0$\begin{array}{cc}{S}_r\left(t\right)=A_r\left(t\right)\exp \left[2\pi j\left\{f_0\left(t-t_{r0}\right)+\frac{{\xi \left(t-t_{r0}\right)}^2}{2}\right\}\right]& \left(21a\right)\end{array}$

(26) When it is assumed that an echo from the target has arrived at the left beam at time t=t.sub.l0, a reception signal S.sub.l(t) of the left beam is expressed by the following equation (21b), where A.sub.l(t) denotes an amplitude (real value):

(27) $\begin{array}{cc}{S}_l\left(t\right)=A_l\left(t\right)\exp \left[2\pi j\left\{f_0\left(t-t_{l0}\right)+\frac{{\xi \left(t-t_{l0}\right)}^2}{2}\right\}\right]& \left(21b\right)\end{array}$

(28) The arrival time difference calculation part 107 calculates multiplication (product) of the reception signal S.sub.r(t) of the right beam from the phasing part 106.sub.a and a complex conjugate of the reception signal S.sub.l(t) of the left beam from the phasing part 106.sub.b.

(29) The result of complex conjugate multiplication is given by the following equation (22):

(30) $\begin{array}{cc}{S}_r\left(t\right)S_l^{*}\left(t\right)=A_r\left(t\right)A_l\left(t\right)\exp \left[2\pi j\left\{f_0\left(t_{l0}-t_{r0}\right)-\xi \left(t_{r0}-t_{l0}\right)t+\frac{\xi \left(t_{r0}^2-t_{l0}^2\right)}{2}\right\}\right]& \left(22\right)\end{array}$

(31) This result of the multiplication is referred to as a “sub-array correlation”. The arrival time difference calculation part 107 sequentially calculates and stores the “sub-array correlation in a storage part (not illustrated).

(32) The arrival time difference calculation part 107 calculates a product of a sub-array correlation {S.sub.r(t.sub.1)S.sub.l*(t.sub.1)} at time t.sub.1 and a complex conjugate of a sub-array correlation {S.sub.r(t.sub.2)S.sub.l*(t.sub.2)} at time t.sub.2 and calculates a product of a complex conjugate of the sub-array correlation {S.sub.r(t.sub.1)S.sub.l*(t.sub.1)} at time t.sub.1 and the sub-array correlation {S.sub.r(t.sub.2)S.sub.l*(t.sub.2)} at time t.sub.2.
{S.sub.r(t.sub.1)S.sub.l*(t.sub.1)}{S.sub.r(t.sub.2)S.sub.l*(t.sub.2)}*=A.sub.r(t.sub.1)A.sub.l(t.sub.1)A.sub.r(t.sub.2)A.sub.r(t.sub.2) exp{2πjξ(t.sub.r0−t.sub.l0)(t.sub.2−t.sub.1)}  (23a)
{S.sub.r(t.sub.1)S.sub.l*(t.sub.1)}*{S.sub.r(t.sub.2)S.sub.l*(t.sub.2)}=A.sub.r(t.sub.1)A.sub.l(t.sub.1)A.sub.r(t.sub.2)A.sub.r(t.sub.2) exp{−2πjξ(t.sub.r0−t.sub.l0)(t.sub.2−t.sub.1)}  (23b)

(33) Accordingly, from the equation (5), the following equation holds:

(34) $\begin{array}{cc}\frac{j\left[\begin{array}{c}{\left\{S_r\left(t_1\right)S_l^{*}\left(t_1\right)\right\}}^{*}\left\{S_r\left(t_2\right)S_l^{*}\left(t_2\right)\right\}-\\ \left\{S_r\left(t_1\right)S_l^{*}\left(t_1\right)\right\}{\left\{S_r\left(t_2\right)S_l^{*}\left(t_2\right)\right\}}^{*}\end{array}\right]}{\begin{array}{c}\left\{S_r\left(t_1\right)S_l^{*}\left(t_1\right)\right\}{\left\{S_r\left(t_2\right)S_l^{*}\left(t_2\right)\right\}}^{*}+\\ {\left\{S_r\left(t_1\right)S_l^{*}\left(t_1\right)\right\}}^{*}\left\{S_r\left(t_2\right)S_l^{*}\left(t_2\right)\right\}\end{array}}=\tan \left\{2\mathrm{\pi \xi }\left(t_{r0}-t_{l0}\right)\left(t_2-t_1\right)\right\}& \left(24\right)\end{array}$

(35) A difference t.sub.l0−t.sub.r0 between times when the echo from the target arrives at the right beam of the sub-array 104.sub.a and when the echo from the target arrives at the left beam of the sub-array 104.sub.b is given by the following equation (25), based on results of the calculations of the sub-array correlations at time t.sub.1 and time t.sub.2:

(36) $\begin{array}{cc}{t}_{l0}-t_{r0}=\frac{1}{2\mathrm{\pi \xi }\left(t_1-t_2\right)}{\tan }^{-1}\left[\frac{j\left[\begin{array}{c}{\left\{S_r\left(t_1\right)S_l^{*}\left(t_1\right)\right\}}^{*}\left\{S_r\left(t_2\right)S_l^{*}\left(t_2\right)\right\}-\\ \left\{S_r\left(t_1\right)S_l^{*}\left(t_1\right)\right\}{\left\{S_r\left(t_2\right)S_l^{*}\left(t_2\right)\right\}}^{*}\end{array}\right]}{\left\{S_r\left(t_1\right)S_l^{*}\left(t_1\right)\right\}{\left\{S_r\left(t_2\right)S_l^{*}\left(t_2\right)\right\}}^{*}+{\left\{S_r\left(t_1\right)S_l^{*}\left(t_1\right)\right\}}^{*}\left\{S_r\left(t_2\right)S_l^{*}\left(t_2\right)\right\}}\right]& \left(25\right)\end{array}$

(37) Assuming that the frequency f.sub.c is set to a center frequency (=(sweep start frequency+sweep end frequency)/2) of the transmission waveform, for example, the arrival time difference calculation part 107 may calculate a phase difference between reflected wave signals from the target at the sub-arrays 104.sub.a and 104.sub.b in FIG. 1, using:
2πf.sub.c(t.sub.l0−t.sub.r0)  (26)

(38) The above equation (9) regarding the direction of arrival θ is here listed again:

(39) $\begin{array}{cc}\sin \theta =\frac{c\left(t_{l0}-t_{r0}\right)}{d}& \left(27\right)\end{array}$
where c is s sound velocity in the case of the sonar and is a light velocity in the case of the radar, and d is a distance between sub-array beams (distance between sub-arrays).

(40) When the right side of the equation (25) is substituted into the t.sub.l0−t.sub.r0 in the equation (27), the following equation (28) holds:

(41) $\begin{array}{cc}\sin \theta =\frac{c\left(t_{l0}-t_{r0}\right)}{d}=\frac{c}{2\mathrm{\pi \xi }\left(t_1-t_2\right)d}{\tan }^{-1}\left[\frac{j\left[\begin{array}{c}{\left\{S_r\left(t_1\right)S_l^{*}\left(t_1\right)\right\}}^{*}\left\{S_r\left(t_2\right)S_l^{*}\left(t_2\right)\right\}-\\ \left\{S_r\left(t_1\right)S_l^{*}\left(t_1\right)\right\}{\left\{S_r\left(t_2\right)S_l^{*}\left(t_2\right)\right\}}^{*}\end{array}\right]}{\begin{array}{c}\left\{S_r\left(t_1\right)S_l^{*}\left(t_1\right)\right\}{\left\{S_r\left(t_2\right)S_l^{*}\left(t_2\right)\right\}}^{*}+\\ {\left\{S_r\left(t_1\right)S_l^{*}\left(t_1\right)\right\}}^{*}\left\{S_r\left(t_2\right)S_l^{*}\left(t_2\right)\right\}\end{array}}\right]& \left(28\right)\end{array}$

(42) The direction of arrival calculation part 108 can obtain a direction of arrival θ of a target by substituting, into an inverse sine function sin.sup.−1, a value obtained by dividing the result of multiplication of the arrival time difference t.sub.l0−t.sub.r0 of the echoes from the target and the velocity c of the reflected wave by the distance d between the right and left sub-array beams. That is, the direction of arrival calculation part 108 obtains a value of the right side in the second row of the equation (28) which involves:

(43) a value of the inverse tangent function of a result of an operation (operation in square brackets [ ] of tan.sup.−1 in the equation (28)) in which a first product of the sub-array correlation {S.sub.r(t.sub.1)S.sub.l*(t.sub.1)} at time t.sub.1 and a complex conjugate of the sub-array correlation {S.sub.r(t.sub.2)S.sub.l*(t.sub.2)} at time t.sub.2, which is {S.sub.r(t.sub.1)S.sub.l*(t.sub.1)} {S.sub.r(t.sub.2)S.sub.l*(t.sub.2)}*, and a second product of a complex conjugate of the sub-array correlation {S.sub.r(t.sub.1)S.sub.l*(t.sub.1)} at time t.sub.1 and the sub-array correlation {S.sub.r(t.sub.2)S.sub.l*(t.sub.2)} at time t.sub.2, which is {S.sub.r(t.sub.1)S.sub.l*(t.sub.1)}*{S.sub.r(t.sub.2)S.sub.l*(t.sub.2)}, are added in a denominator, while the first product is subtracted from the second product in a numerator;

(44) the information on the transmission wave (frequency change rate ξ in the equation (28));

(45) the distance d between the right and left sub-array beams;

(46) a time difference (=t.sub.1−t.sub.2); and

(47) the sound or light velocity c.

(48) The direction of arrival calculation part 108 calculates the value of the inverse sin.sup.−1 of the value of the right side of the equation (28) to calculate the direction θ of the target. Herein, the sound velocity or the light velocity c and the distance d between the sub-arrays are given in advance. Times t.sub.1 and t.sub.2 can be arbitrarily specified by a user.

(49) It may be so arranged that the arrival time difference calculation part 107 calculates the sub-array correlation {S.sub.r(t.sub.1)S.sub.l*(t.sub.1)} at time t.sub.1 and the sub-array correlation {S.sub.r(t.sub.2)S.sub.l*(t.sub.2)} at time t.sub.2 and stores the sub-array correlations in the storage part (not illustrated), the direction of arrival calculation part 108 receives the sub-array correlation {S.sub.r(t.sub.1)S.sub.l*(t.sub.1)} at time t.sub.1 and the sub-array correlation {S.sub.r(t.sub.2)S.sub.l*(t.sub.2)} at time t.sub.2 that have been stored in the storage part (not illustrated) of the arrival time difference calculation part 107 and then calculates the above equations (23a), (23b), and (25) to find the direction of arrival of the target, based on the above equation (28). It is noted that as an alternative arrangement, the arrival time difference calculation part 107 and the direction of arrival calculation part 108 may be configured as one unit (processing unit).

(50) When the inequality expression (16a) regarding the range of an inverse tangent function tan.sup.−1 is applied to the inverse tangent function tan.sup.−1 in the equation (28), the following holds:

(51) $\begin{array}{cc}-\frac{c}{4\xi \left(t_1-t_2\right)d}<\sin \theta <\frac{c}{4\xi \left(t_1-t_2\right)d}& \left(29\right)\end{array}$

(52) The difference (t.sub.1−t.sub.2) between time t.sub.1 and time t.sub.2 in the above inequality expression (29) can be arbitrarily set. Therefore, a degree of freedom is high with respect to direction of arrival estimation. That is, a restriction with respect to a direction estimable range which depends on a distance between sub-array beams can be at least eliminated and accuracy of direction of arrival estimation can be ensured.

(53) When the pulse length T of the transmission waveform is 1 second and a frequency amplitude (difference between the sweep start frequency and the sweep end frequency) is 1 kHz, for example, ξ=1 kHz/second. It is assumed that the sound or light velocity c=1500 m/second and the distance d between the right and left beams=1 m, and
(t.sub.1−t.sub.2)=0.25 seconds  (30)

(54) In this case, the restriction to the direction of arrival, based on the above inequality expression (29) becomes as follows:
−1.5<sin θ<1.5  (31)
An arbitrary direction of arrival can be estimated without unambiguity.

(55) According to this example embodiment which uses the inverse trigonometric functions, by holding a numeral table or the like in advance in a storage part, a processing load can be kept low. As a result, the direction of arrival estimation apparatus with low processing load can be mounted on a drone or the like which is not equipped with a large computational capability.

(56) In the present example embodiment, the pulse waveform (LFM pulse compression waveform, but not limited to LFM) is used as the transmission waveform. There is no limitation on the pulse length. Therefore, the example embodiment described above can also be applied to a sonar or a radar that uses a continuous wave (such as FMCW (Frequency Modulation Continuous Wave)). The example embodiment described above can also be applied to a sensor (whose reception part is constituted from sub-arrays) configured to sense infrared light, a light wave or the like.

(57) FIG. 2 is a diagram illustrating an arrangement of a second example embodiment. Referring to FIG. 2, the second example embodiment includes a direction averaging part 109, in addition to the configuration of the first example embodiment.

(58) By storing, in a storage part not illustrated, directions obtained at three or more different times with respect to sub-arrays 104 and averaging the directions by the direction averaging part 109, direction estimation accuracy of a target can be improved.

(59) To take an example, assuming that a direction of arrival of the target calculated using the equation (28) based on results of complex conjugate multiplications of {S.sub.r(t.sub.1)S.sub.l*(t.sub.1)}*{S.sub.r(t.sub.2)S.sub.l*(t.sub.2)} and {S.sub.r(t.sub.1)S.sub.l*(t.sub.1)} {S.sub.l(t.sub.2)S.sub.l*(t.sub.2)}* which are calculated using sub-array correlations {S.sub.r(t.sub.1)S.sub.l*(t.sub.1)} and {S.sub.r(t.sub.2)S.sub.l*(t.sub.2)} of reception signals of right and left beams obtained at time t.sub.1 and time t.sub.2 is θ.sub.A, and a direction of arrival of the target calculated from the 0 (28) using results of complex conjugate multiplications of {S.sub.r(t.sub.3)S.sub.l*(t.sub.3)}*{S.sub.r(t.sub.4)S.sub.l*(t.sub.4)} and {S.sub.r(t.sub.3)S.sub.l*(t.sub.3)} {S.sub.r(t.sub.4)S.sub.l*(t.sub.4)}* calculated using sub-array correlations {S.sub.r(t.sub.3)S.sub.l*(t.sub.3)} and {S.sub.r(t.sub.4)S.sub.l*(t.sub.4)} of reception signals of right and left beams obtained at time t.sub.3 and time t.sub.4 is θ.sub.B, the direction averaging part 109 outputs, as the direction of arrival of the target, the following average between the θ.sub.A and the θ.sub.B:

(60) $\begin{array}{cc}\frac{{\theta }_A+{\theta }_B}{2}& \left(32\right)\end{array}$

(61) Alternatively, as an example of the directions obtained at the three or more times, the sub-array correlations of the reception signals of the right and left beams obtained at time t.sub.1, t.sub.2, and t.sub.3 are used. When the direction of arrival of the target estimated from the equation (28) based on the results of the complex conjugate multiplications of {S.sub.r(t.sub.1)S.sub.l*(t.sub.1)}*{S.sub.r(t.sub.2)S.sub.l*(t.sub.2)} and {S.sub.r(t.sub.1)S.sub.l*(t.sub.1)} {S.sub.r(t.sub.2)S.sub.l*(t.sub.2)}* calculated using the sub-array correlations {S.sub.r(t.sub.1)S.sub.l*(t.sub.1)} and {S.sub.r(t.sub.2)S.sub.l*(t.sub.2)} of the reception signals of the right and left beams obtained at time t.sub.1 and t.sub.2 is set to θ.sub.A, and when a direction of arrival of the target estimated from the equation (28) based on results of complex conjugate multiplications of {S.sub.r(t.sub.2)S.sub.l*(t.sub.2)}*{S.sub.r(t.sub.3)S.sub.l*(t.sub.3)} and {S.sub.r(t.sub.2)S.sub.l*(t.sub.2)} {S.sub.r(t.sub.3)S.sub.l*(t.sub.3)}* calculated using the sub-array correlations {S.sub.r(t.sub.2)S.sub.l*(t.sub.2)} and {S.sub.r(t.sub.3)S.sub.l*(t.sub.3)} of the reception signals of the right and left beams obtained at time t.sub.2 and time t.sub.3 is set to θ.sub.B, the direction averaging part 109 outputs, as the direction of arrival of the target, an average calculated from the equation (32) indicating the average of the θ.sub.A and the θ.sub.B. Alternatively, the direction averaging part 109 may calculates, as the direction of arrival of the target, an average between the direction of arrival of the target estimated from the equation (28) based on the results of the complex conjugate multiplications of the sub-array correlations of the reception signals of the right and left beams obtained at time t.sub.1 and time t.sub.2 and a direction of arrival of the target estimated from the equation (28) based on results of complex conjugate multiplications of the sub-array correlations of the reception signals of the right and left beams obtained at time t.sub.1 and time t.sub.3.

(62) FIG. 3 is a diagram illustrating an arrangement of a third example embodiment of the present invention. Referring to FIG. 3, a direction of arrival estimation apparatus (system) 100 in the third example embodiment includes a fitting part 110, in addition to the configuration of the first example embodiment.

(63) The fitting part 110 stores directions θi (i=1, . . . m: m>=3) obtained at different time points that are three or more, and performs fitting of a straight line or a curve (such as fitting the straight line by linear regression or the like or polynomial curve fitting) to these directions θi (i=1, . . . m: m>3), which contributes to improvement in direction estimation accuracy of a target. Further, The fitting part 110 can track a time variant direction of arrival of the target.

(64) FIG. 4 is a diagram illustrating an arrangement of a fourth example embodiment of the present invention. Referring to FIG. 4, a direction of arrival estimation apparatus (system) 100 in the fourth example embodiment includes a plurality of sub-arrays that are three or more sub-arrays, and includes a sub-array averaging part 111 configured to calculate and store a direction of arrival of a target for each combination of two of the sub-arrays and to average directions that have been stored. Since the sub-array averaging part 111 performs estimation of direction of arrival of a target using the three or more sub-arrays, direction estimation accuracy can be improved more than in a case where there are only two sub-arrays. Referring to FIG. 4, reception signals from a first sub-array 104.sub.a and a second sub-array 104.sub.b are respectively subjected to phasing addition by phasing parts 106.sub.a and 106.sub.b. Then, an arrival time difference calculation part 107.sub.a and direction of arrival calculation part 108.sub.a calculate a first direction θ.sub.1. Reception signals from a third sub-array 104.sub.c and a fourth sub-array 104.sub.d are respectively subjected to phasing addition by phasing parts 106.sub.c and 106.sub.d. Then, an arrival time difference calculation part 107.sub.b and direction of arrival calculation part 108.sub.b calculate a second direction θ.sub.2. The sub-array averaging part 111 calculates, as the direction of arrival of the target, a value as follows obtained by calculating an average between the first direction θ.sub.1. calculated by the direction of arrival calculation part 108.sub.a and the second direction θ.sub.2. calculated by the direction of arrival calculation part 108.sub.b:

(65) $\begin{array}{cc}\frac{{\theta }_1+{\theta }_2}{2}& \left(33\right)\end{array}$

(66) It may be so configured, for example, that three sub-arrays (such as the sub-arrays 104.sub.a, 104.sub.b and 104.sub.c) are provided in FIG. 4, the direction of arrival calculation part 108.sub.a calculates the first direction based on reception signals at two sub-arrays 104.sub.a and 104.sub.b, the direction of arrival calculation part 108.sub.b calculates a second direction based on reception signals at two sub-arrays 104.sub.b and 104.sub.c, and the sub-array averaging part 111 calculates, as the direction of arrival of the target, a value obtained by calculating an average between the first direction calculated by the direction of arrival calculation part 108.sub.a and the second direction calculated by the direction of arrival calculation part 108.sub.b.

(67) FIG. 7 is a diagram illustrating a fifth example embodiment of the present invention, and is a diagram illustrating a configuration when the direction of arrival estimation apparatus is implemented in a computer apparatus 200. Referring to FIG. 7, the computer apparatus 200 includes a processor 201, a memory 202 such as a semiconductor memory (or that may be an HDD (Hard Disk Drive) or the like) that is a RAM (Random Access Memory), a ROM (Read Only Memory), or an EEPROM (Electrically Erasable Programmable Read-Only Memory), a display device 203, and an interface 204 (bus interface) that is connected to the transmission part 102 and the reception part 103 in FIG. 1. The processor 201 may be a DSP (Digital Signal Processor). By executing a program stored in the memory 202, the processor 201 executes processes of the phasing part 106, the arrival time difference calculation part 107, and the direction of arrival calculation part 108 in FIG. 1. Alternatively, the processor 201 may execute processes of the direction averaging part 109 and the fitting part 110. The memory 202 may be used as the storage part (not illustrated in each of FIGS. 1 to 4) for the arrival time difference calculation part 107, the direction of arrival calculation part 108, and the transmission waveform setting part 101, and so on.

(68) Each disclosure of the above-listed Patent Literatures 1 to 5 is incorporated herein by reference. Modification and adjustment of each example embodiment and each example are possible within the scope of the overall disclosure (including claims) of the present invention and based on the technical concept of the present invention. Various combinations and selections of various disclosed elements (including each element in each claim, each element in each example, each element in each drawing, and the like) are possible within the scope of the claims of the present invention. That is, the present invention naturally includes various variations and modifications that could be made by those skilled in the art according to the overall disclosure including the claims and the technical concept.