DDMA-MIMO Observations With the MU Radar: Validation by Measuring a Beam Broadening Effect

The phased-array radar, basically developed with the defense systems, has mainly utilized for the atmospheric radar and the wind profiling radar in the field of meteorological remote sensing, and recently it has also applied to the weather radar for research purposes. As a further development of phased-array technology, the “multiple-input–multiple-output (MIMO) technique,” which has been developed in the field of communication systems, has been applied to radars. With the MIMO radar, it is possible to create a virtual antenna aperture plane beyond the actual antenna and to reduce the actual antenna size compared to that of the conventional antenna while maintaining the angular resolution. This effect is expected to reduce costs, which is one of the major hurdles in expanding phased array radars instead of parabolic antenna systems. In order to confirm the effect, an experimental observation was performed using the MU radar which is a VHF-band phased array atmospheric radar with multichannel receivers. The MIMO technique requires orthogonal waveforms on each transmitter to identify the transmitted signals with multiple receivers, and various methods are known to realize orthogonality. In this article, we focus on the “Doppler division multiple access (DDMA)” MIMO technique, with which slightly different frequencies are selected as transmit waveforms to separate in each receiver in the Doppler frequency domain. The observation results by measuring a beam broadening effect with the MU radar indicate that it will be a key technique for the atmospheric radar in the near future.


I. INTRODUCTION
T HE phased-array radar, developed by defensive radars, has been mainly utilized for the atmospheric radar and the wind profiling radar and recently, it has also been applied to the weather radar for research purposes. As a further development of the phased-array technology, "multiple-input-multiple-output (MIMO) technique," which was developed in the field of communication systems, has been applied to radars.
In recent years, various phased-array applications have been developed which performs digital beam forming (DBF) with multiple receivers. By using this technique, we can obtain multiple receive beams in a single scan, which reduces the time interval dramatically; however, in general, conventional phased array radars have no function that each transmit signal can be distinguished at the receivers, therefore, they are categorized as a "single-input-multiple-output (SIMO)" radar. The MU radar, e.g., [1], [2], which has been operated for more than 30 years with 465 multiple transmitters, phase shifters, and corresponding antennas to orient the direction of electrically, is one of the advanced atmospheric radars and has been operated as an SIMO radar, whereas it can be also operated as an MIMO radar with additional settings explained later.
The most definitive difference between SIMO and MIMO radars is the orthogonality of the transmitter signal. A phasedarray radar that separates the transmit waves can be called "multiple-input" transmitters and it is important to understand how we can separate transmit waves from the receive signals at the targets. It should be noted that adopting orthogonal waveforms in each transmitter can separate transmit waves. To realize their orthogonalities, some methods have been introduced by Sun et al. [3] in which each method has its own limitation and significance to choose the correct method for various applications.
In addition, it is necessary to consider not only the phase differences between receive signals but also those between transmit signals to create a virtual antenna aperture. For this reason, the coherent MIMO radar, which provides more capable antennas, will be discussed in this article.
The MIMO radar can be expected to improve angular resolution as described before, whereas, 3-D-high-resolution observations has been already achieved with multiple frequencies and multiple receivers. This technique is based on the combination of range imaging (RIM) toward the range direction with frequency offsets and Fourier or Capon beamformer toward spatial distribution with multiple antenna arrays [4]. The MIMO technique is similar to these methods, however, quite different from the point that it is realized by using orthogonality of transmit signals which are radiated simultaneously. This fact implies MIMO radar can be combined to the 3-D radar imaging technique which has been researched before, toward horizontal space.
An MIMO approach for the atmospheric radars has been applied by Urco et al. [5], [6] for atmospheric and ionospheric synthesis radar imaging observation, trying to use "time division multiple access (TDMA)," "code division multiple access (CDMA)" and a polarization diversity method. In this article, the DDMA method, which is useful for lower frequency radar and is superior to TDMA in terms of information density [7], is adopted and the first observation result is introduced after discussing the signal model of the MIMO radar and the revised system of the This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ MU radar. However, it causes gain loss, because of the transmit resources are used for increment of capable receive antennas. The gain depends on the MIMO array layout and evaluated by Liao et al. [8].
The basic principle, which covers the effect of target Doppler velocities, amplitude, and phase differences between transmitters, is reviewed in Section II. The theory of the DDMA-MIMO signal model, which utilizes slow-time domain, is presented in Section III. The system configuration of the MU radar as a MIMO radar, observation results, and the future work are described in Section IV. Finally, the effectiveness of the MIMO radar is summarized in Section V.

A. Basic Principle
Let the M transmit signals, mth transmit signal be x m (t, θ 0 ) = a m (θ 0 )φ m (t) and the nth receive signal y n (t, θ 0 ) is defined as where v n (t) is receive noise, a m (θ 0 ) and b n (θ 0 ) are transmit and receive phase shifts corresponding to the target angle θ 0 , φ(t) ∈ C M is a normalized transmit waveform column vector composed of M transmitters, a(θ 0 ) ∈ C M is a transmit steering column vector corresponding to the transmission angle θ 0 , and α is a (complex-valued) backscatter coefficient. To expand y n (t, θ 0 ) to the N receivers, a receive signal column vector y(t, θ 0 ) ∈ C N is defined as where b(θ 0 ) ∈ C N is a receive steering column vector corresponding to receive angle θ 0 (in this case, transmit and receive angles are defined to be the same), and v(t) ∈ C N is a receive noise-column vector.
After range processing with time lag τ and matched filters φ(t − τ ) H (suffix H means Hermitian transpose) to separate the transmit waveforms, the receive signal matrix Z(θ 0 ) ∈ C (N,M ) is written as ( [9], [10]) where is the M × M MIMO signal correlation matrix that describes the correlation among the transmit waveforms, and is the filtered receive noise matrix. The N × M data matrix of (3) may be vectorized by stacking its columns of Z(θ 0 , τ) and we define the receive MIMO signal by Equation (6) is rewritten with the well-known relationships of the vectorization operator shown as [11] vec{XYZ} = {Z T ⊗ X}vec{Y} where X is an arbitrary N × K matrix, Y is an arbitrary K × L matrix, Z is an arbitrary L × N matrix, I N is a unit matrix of N × N , and "⊗" indicates the Kroneker product where the MIMO steering vector s(θ 0 , τ) for the beam angle θ 0 and the filtered noise columns vector e(τ ) are defined as If we assume that the transmit signals of φ(t) are orthogonal with each other and that each matched filtered range response is identical defined as R φ (τ ), then each element of MIMO signal correlation matrix can be expressed as and (8) is rewritten as According to (12), the radar can utilize its transmit freedom called the MIMO steering matrix, which is expressed as a(θ 0 ) ⊗ b(θ 0 ) ∈ C (NM,1) , by transmitting orthogonal waveforms. Fig. 1 shows the principal image of the MIMO radar with transmit orthogonality in the case of M = N = 3.

B. Principle With Doppler Effect, Transmit Amplitude, and Phase Difference
The ideal transmit signals are expressed as x(t, θ 0 ) = a(θ 0 ) φ(t) using the Hadamard product . However, in an actual radar system, we must consider where and how we execute the matched filter process.
In general, the matched filter φ(t) H is not applied directly because the transmit power is too high to measure the waveforms. To overcome this limitation, a low-power signal was used as the reference signal.
We may decide to redefine the reference signal as the point before high-power amplifiers, and the transmit waveform is replaced as where p ∈ C M is the (time constant) amplitude and phase offset function usually caused by hardware difference between transmitters.
The transmitter amplitude and phase offset function p can be expressed as where A m is the amplitude offset and δ m is the phase offset in the mth transmitter. Equation (2) is rewritten by considering the Doppler shift from the target as follows: where αe j2πνt is the complex backscatter coefficient with Doppler shift frequency of the target ν.

Equation (3) added by Doppler term is also written as
where diag(p) denotes the diagonal matrix of vector p, and X φ (τ, ν) is the ambiguity function matrix [12] defined as The exponential term of (17) implies that the off-diagonal components of the correlation matrix does not always become zero, and therefore cannot maintain orthogonality between the transmitters due to the target Doppler shift. However, when dealing with pulsed Doppler radars having a pulse repetition time T IPP , the integration time is within the pulsewidth time t w , and the target velocities are relatively small, the Doppler term can be taken as a constant value around t = τ . For example, if ν and t w are taken to keep |2πνt w | < 0.01 radian (0.57 • ) phase rotation in the pulse length, the exponential term can be taken as the same value while pulsewidth duration. This is good approximation because it considers the exponential term as a constant during integration to divide this term into the correlation part and the following equations are derived [13]: where τ denotes the correlation time of T IPP duration. Clearly, from (18), we can separate the target Doppler component from the MIMO signal correlation matrix. This separation makes the radar hardware configuration easier, which means that the fast time signal processing (corresponding to the integration time) and the slow time signal processing (corresponding to the discrete time) can be independently constructed. Note that the term of e j2πντ , phase rotation by time lag, is multiplied by all the components commonly, therefore it will be omitted to simplify the equations.
The products of the target Doppler velocities and the pulse length should also be considered to keep the value small. Table I shows an example of phase rotation in the fast-time signal processing substituted by the typical transmit frequency, pulse length, and maximum Doppler velocity of the target. From this table, it can be seen that the Doppler effect is negligible for the lower frequency radar; however, it may influence the assumption when using L-band or C-band radio frequency.
As described before, two conditions of lower transmit frequency and small target Doppler speed are enabled to establish the approximation in (18), which Doppler ambiguity term has been deleted. The MU radar, radiates VHF transmit frequency (46.5 MHz) and normally < 20 ms −1 target, may have the decision to apply this approximation; however, there would be a case that Doppler ambiguity cannot be ignored, (17) needs be strictly applied with the consideration of Doppler ambiguity. In general, the choice of DDMA is recommended for lower transmit frequency.

III. DDMA-MIMO OBSERVATION
DDMA-MIMO method is commonly used and has proved the effectiveness of its orthogonality (e.g., [14], [15]), whereas Sun et al. [3] has pointed out that it requires wide Doppler unambiguity so that lower frequency radar is suitable in general.
In this section, L pulse trains transmit waveforms are introduced to discuss more specifically to discuss DDMA-MIMO principle, which leads to clarify the important points when applying DDMA-MIMO method to the MU radar. 1) , and summation of noise columns vector e(τ ) ∈ C (MN,1) using the lth receive noise column vector {v(t)} l ∈ C N as [16]

A. Expansion to L Pulse Trains
The ambiguity function of periodic pulse radar signals X(τ, ν) can be expressed by [13] on condition that product of the target Doppler shift frequency times pulsewidth νt w ≈ 0 and the influence of second trip echoes is not affected in the signal processing described before. The receive MIMO signal added to the target Doppler shift is rewritten from (16) where the MIMO steering vector for the beam angle θ 0 , lag time τ and the target Doppler frequency ν is defined as and lth pulse train component s(θ 0 , τ, ν, l) is also expressed as

B. DDMA-MIMO Principle
The DDMA method utilizes the pulse-to-pulse phase difference, which is derived from the Doppler effect, and slightly different frequencies are selected as transmit waveforms to be separated in the Doppler frequency domain.
The DDMA can divide their transmit signals with Doppler division, which means that we can use time slots per interpulse period (or pulse-repetition-frequency), called the "slow-time" direction. Therefore, DDMA is executed after the ranging process [17]. In other words, the range response process and orthogonal transmit signal separation process are divided when using the slow-time direction.
DDMA requires a transmit phase difference of 2πf md between T IPP to obtain the orthogonality of the transmitter, and transmit signals u(t) ∈ C (M,L) is defined as [16] Now we focus on the second exponential component of (33). In general, the discrete Fourier transformation toward t l = lT IPP (l = 0, 1, 2 · · · L − 1) (time series of every T IPP ) is executed instead of the summation according (21), and the frequency distribution toward f l = 1/(lT IPP ) is defined by the expression with Fourier transform expression "F" as follows: where is the Doppler frequency distribution of the target which depends on its characteristics. Now, we can assume that atmospheric and/or precipitation motions which are our interests in meteorology have limited frequency distribution between minimum Doppler frequency f td−min and maximum one f td−max . To divide the transmitted waveforms in the receiver, DDMA uses a Doppler frequency bandpass (rectangular) filter toward the slow-time direction as and the ambiguity function multiplied Doppler frequency bandpass filter becomes where f md should be selected to keep the condition that −f md /2 < f td−min and f td−max < f md /2. Again, the DDMA-MIMO steering vector of lth pulse train for beam angle θ 0 and the target Doppler frequency ν after filtering process is defined as where h[lT IPP ] ≡ sinc[f md lT IPP ] is the time-domain expression of the bandpass filter and " * " denotes convolution integration.
Equation (38) indicates that the transmitters with DDMA have a complete orthogonality with each other if the target Doppler distribution is within the Doppler frequency bandpass filter. Although the influence of the Doppler filter has been shown to be a sinc function in (38), this effect may be minimized to apply other window functions instead of a rectangular window.
On the other hand, these ideal characteristics may require very wide Doppler unambiguity range, which restricts the maximum target range. This may cause limitations in using the radar for higher transmit frequencies.

C. DDMA-MIMO With Frequency Offset Principle
DDMA-MIMO is also realized with a transmit frequency offset slightly. We focus on the frequency offset f md instead of the phase offset between transmitters the phase and frequency offset {w(t)} m,l and {f (t)} m,l are expressed as [16] {w(t)} m,l = 1 Equation (41) can be written as follows [15]: The first term of the right-hand side stepped by T IPP in (42) is the same as (32) and the second term means transmit signals have linear phase increment of 2π(m − 1)f md t toward fast-time but reset to zero by T IPP . Moreover, in case that 2π(m − 1)f md t w (m takes 1 to M ) is relatively small, the same discussion in Section II-B can be applied that the exponential term exp[j2π(m − 1)f md (t − lT IPP )] is regarded as the constant value at τ while integral duration toward the first time in each transmit signal. However, this term has a unique value for the transmit position so that phase difference between transmitters need to be calibrated whereas target Doppler phase rotation is able to omitted. This range-direction phase offset defined q(τ ) ∈ C M is expressed as q(τ )= e −j2π0f md τ ,· · ·, e −j2π(m−1)f md τ ,· · ·, e −j2π(M −1)f md τ T (43) and DDMA-MIMO steering vector when using frequency offset is expressed by multiplying (43) into (38) For the radar system, the frequency offset method can be implemented easier than the phase offset method because the frequency accuracy is very stable and it is very easy to set up a system that only requires the addition of frequency resources for the MU radar when applying as the MIMO radar. Besides, the low frequency of the MU radar also has an advantage that the maximum offset frequency 2π(M − 1)f md w can be taken to almost zero to apply for DDMA.
For example, the maximum offset frequency is selected to 2(M − 1)f md = 30 Hz and pulsewidth w = 50 μs, pulse rotation becomes 0.009 rad shown as Table I that is sufficient to meet the condition. On the other hand, in the case that higher frequency for phase offset is chosen, strict evaluations with the ambiguity function may be required which has the same behavior as faster Doppler targets. In summary, the phase offset between transmitters expressed by (14) and the range-direction phase offset by (43) needs to be implemented to the signal processor from the result of (44) when constructing DDMA-MIMO radar system with frequency offset. It should be noted that DDMA with the frequency offset method uses to generate pulse-to-pulse phase difference, therefore it is not categorized as FDMA.
In general, it is known that the DDMA-MIMO channel matrix is expressed simply as a(θ 0 ) ⊗ a D (νT IPP ) ⊗ b(θ 0 ) with the Doppler term expression a D (νT IPP ) shown by and [7] and [13], for example. The result of (44) is consistent to this formula if it is expressed as summation of lth pulse trains and eliminated the term of [diag(p) · diag(q(τ )) ⊗ I N ], which means it may be used directly to the MU radar. However, the DDMA with frequency offset method and two offset corrections (amplitude/phase difference between transmitters and range phase offset on receivers) need to be explained to apply for the MU radar, therefore, this article does not use this simple formula and describes about these effects from the principles shown by [9].

A. System Configuration
The MU radar [1], located in Shigaraki, Shiga, Japan, is a state-of-the-art atmospheric radar that has been under operation since 1984. It has one modulator, 475 elements of VHF-band (46.5 MHz) cross-Yagi phased array antennas, and 25 plus additional four receivers as shown in [18], therefore, it is categorized as an SIMO radar. However, this radar can also play the role of MIMO radar quite easily because of its flexibility. Now, we can classify transmit antennas into six parts and apply orthogonal waveforms with synchronized signal generators (SGs) that have slightly different frequencies. Each receiver is divided into six orthogonal transmitter signals with Doppler matched filters to realize MIMO virtual receive array. Fig. 2 shows the actual antenna layout on transmit and the MIMO virtual array on receive image.

1) Transmitter Configuration:
The transmitter configuration is shown in Fig. 3. The MU radar radiates 46.5 MHz RF transmit signal by mixing a 5 MHz modulated IF-signal and a 41.5 MHz CW-local signal, those of which are divided into six part toward to Booth A to F. Orthogonalities on transmission are realized by replacing the original local signal with a separate local signal  generated with each SG. The interesting part of this configuration is that the compensation algorithm of the amplitude and phase differences between the transmit signals, which is expressed by the diag part of (38), is added. As explained before, φ(t) is not used as a matched filter directly, therefore, we need to know their initial amplitude and phase in the process of beam forming. To acquire amplitude and phase information of these transmit signals, 20 dB couplers are added to each transmitter to extract local signals.
2) Receiver Configuration: The receiver configuration is shown in Fig. 4. The receive echoes corresponding to each transmit signal are mixed at the signal processing stage (until the pulse compression stage). To divide into orthogonal signals, the Doppler filter toward to slow-time direction, which is described by (38), is applied. It should be noted that matched filter processes corresponding to each transmit frequency should be selected; however, the signal processing for the down conversion of the MU radar cannot be changed because the same analog local signals are used in each receiver and the matched filter is not adopted directly. For the above reason, we must include a digital down-conversion part to realize the DDMA-MIMO matched filter after the analog down conversion part and pulse compression part.
To digitize and synchronize the divided transmit amplitude and phase signals, an additional receiver is utilized. Their frequencies are around 41.5 MHz, so another synchronized SG (CW = 36.5 MHz) is used for the down converter to adjust the receiver's specifications (IF = 5 MHz input).
Ideally, the same number of receivers corresponding to each reference signal should be prepared; however, this could result in a shortage of receivers (in this case, six receivers are requested against four additional receivers). To overcome this limitation, we have adopted a method in which the transmit waveforms are combined in the analog region and subsequently detected by one receiver to separate them using Doppler filters during digital signal processing. It should be noted that the orthogonal transmit signal interval is set to 3.125 Hz which is corresponded to 10.078 m/s Doppler speed, therefore aliased echoes caused by adjacent transmit signal are seen in Fig. 5. The reason is that atmospheric echoes has around 2-5 m/s Doppler shift, which becomes visible appearing beyond (fixed) Doppler filters. It has a possibility to degrade the orthogonality in general, however, in this case it can keep orthogonalities, because that adjacent echoes are not overlapped with each other obviously from the Fig. 5.

B. Observation Result
It is also noted that it is important for DDMA not to become invisible adjacent transmit signals by filter, but to be selected offset frequency to avoid overlapping with adjacent target echoes in the Doppler region. Fig. 7 shows a DDMA-MIMO observation result at the height of 5.5 km, which displays the virtual receive aperture (fullsynthesized) and the real aperture (used receive echoes with from A-booth transmitters only) using the same data of Fig. 5. This comparison is also suggested that narrower beam width generated by virtual antenna helps sharper Doppler spectrum width.
Generally, when the radar beam is relatively narrow (beam width is a few degrees) for the volume target, the relationship between the radar beam width and the beam broadening is known to follow an approximation of where σ 1 2 B is the half spectrum width of the Doppler velocity, θ 1 2 is the half-power half-width of the effective radar beam, and v h is the total background horizontal wind [19].
As shown by [20], the observed spectrum width σ 1/2 is expressed by the convolution of the turbulence σ 1/2turb and the beam broadening as follows:  If we regard that σ 1/2B is dominant compared to σ 1/2turb because of large horizontal velocities, σ 1/2B is proportional to θ 1/2 , and Fig. 7 implies that this assumption is substantiated. Fig. 8 displays height profiles of Doppler shift, square of the half spectrum width toward the eastward direction of zenith angle 10 • observed at 1620-1640 JST November 22, 2019 as MIMO observation. Compared with the Doppler shift between the MIMO virtual array and real aperture (SIMO observation), these profiles are consistent. This fact indicates the Doppler shift estimation process works correctly. On the other hand, square of the half spectrum width calculated with the same process has different profiles.
As [20], observed spectrum width is greatly influenced with the antenna beam width, therefore the comparison between the observed spectrum width and the beam width could be a criterion. Fig. 9 displays the array factor of the MU radar. The beam width is approximately θ 1/2SIMO ≈ 1.8 • and θ 1/2MIMO ≈ 1.4 • , respectively, and the ratio θ 2 1/2SIMO /θ 2 1/2MIMO ≈ 1.65. The height profile of squared spectrum width observed with the MIMO virtual and the real aperture is also shown on the right side of Fig. 8, in which the ratio of the beam width is displayed. From this figure, it can be seen that these ratios are comparable. Differences between these height profiles are estimated that turbulence component is dominant or weaker horizontal wind is affected.
As discussed earlier, these results indicate that the MIMO observation can generate narrower beam patterns. Further, we need to investigate it more quantitatively in another way.
Moreover, it may be unfair to compare MIMO and the fullarray SIMO method with the identical radar as described by [8], because the MIMO radar compensates the transmit aperture and the transmit power resources with the virtual receiver aperture, which causes weak target detection ability than the identical fullarray antenna system. In other words, narrower receive beam width with a virtual antenna and freedom of M × N receivers for clutter suppression are obtained in exchange for the maximum target detection range to build identical resources for the system design.

V. CONCLUSION
In this article, the basic principles of the MIMO radar were described, especially the DDMA-MIMO method, which utilizes pulse-to-pulse phase difference between the transmitters to achieve orthogonality. Furthermore, it was proved that a transmitter's frequency offsets could be also adopted to realize DDMA-MIMO radar.
Next, to apply the DDMA-MIMO method using the MU radar, the received MIMO signal model was expanded to L pulse trains, and it was explained how the orthogonality characteristics were obtained using Doppler filters. In addition, the compensation method for the amplitude and phase difference of each transmitter was introduced along with the configuration of the MU radar.
Finally, the observations with the MIMO-MU radar were compared with that of the SIMO-MU radar with measuring a beam broadening effect, and the results showed that a narrower beam width is achieved using the MIMO method, which helps in creating a virtual antenna aperture plane beyond the actual antenna. In 1999, he joined Mitsubishi Electric Corporation, Tokyo, Japan, where he has been engaged in the development of active phased array radar system. His research interests include the development of phased array radar systems for atmospheric and weather radars.
Dr. Matsuda was the recipient of the 2022 Gambo-Tatehira Award of the Meteorological Society of Japan on behalf of Mitsubishi Electric Corporation.