The Impact of Antenna Array Calibration Errors on MIMO and Multi-Channel Synthetic Aperture Radar Imaging

In radar imaging, any lateral resolution is either attained using antenna arrays or by synthetic aperture radar (SAR) principles. Recently, the combination of both approaches to a multi-channel SAR (MCSAR) has gained increased attention. However, antenna arrays are impaired by unknown calibration errors, particularly phase errors, amplification deviations, mutual coupling, and phase center position deviation of the antenna element. Since, first, established calibration procedures are often non-ideal and, second, the calibration parameters usually vary over time due to aging and changes in temperature, calibration errors distort the measurements during the radar operation. Therefore, current research focuses on finding new practical calibration procedures. However, hardly any research has been conducted to analyze the basic effects which erroneous calibration parameters have on the reconstructed image in MCSAR or MIMO radar. In this work, we analyze the effects of multiple calibration error types on MCSAR and MIMO radar imaging. Both measurement principles lead to periodic error patterns, resulting in deterministic distortions. The derived relations can be used to precisely predict the position and number of ghost targets caused by calibration errors for known measurement setups and targets. For this, no information about the actual calibration errors is required. Furthermore, worst-case distortion levels are provided for different bounded errors allowing rough estimates of the impact of different error types on the image's distortion level. The findings are validated via measurements using a 76$\,$GHz FMCW MIMO radar.


I. INTRODUCTION
Today, imaging radar sensors are used in many applications ranging from remote sensing [1], [2], [3] over nondestructive testing [4] and security scanning [5] to the broad area of automotive radar [6], [7], [8]. Generally, imaging radar collects cross-range information by utilizing either antenna arrays in a single-input multiple-output (SIMO) or multiple-input multiple-output (MIMO) manner [9], [10], or by moving a single antenna to create a synthetic aperture radar (SAR) [2], [11]. The combination of both approaches, i.e., moving complete antenna arrays, leads to the principle of multi-channel SAR (MCSAR), either implemented as SIMO-SAR or MIMO-SAR. MCSAR has been investigated primarily in remote sensing to further increase the imaging resolution and relax the demand on the pulse repetition frequency (PRF) [3], [12], [13], [14]. In the automotive radar field, the MIMO concept is established to create large apertures with limited sensor size. In contrast, SAR imaging is a topic that has only emerged in the past years but has gained increasing interest because of its high-resolution possibilities [7], [8], [15], [16], [17], [18]. Especially for radars with a side-looking arrangement, the car's movement can be exploited to directly apply the SAR principle. While most of this research focuses on single antenna SAR for now, the extension to MCSAR is a logical next step, which has already been investigated in e.g. [19], [20], because most automotive radars are already equipped with MIMO antenna arrays. When using antenna arrays, the imaging performance is generally deteriorated due to mismatches between the channels caused by gain and phase errors, shifted antenna phase centers, and mutual coupling between the antennas [21]. Therefore, antenna arrays should be calibrated to eliminate or reduce these effects. Calibration procedures usually require measurements to targets at known positions in a multipathfree far-field environment, such as an anechoic chamber [22], [23], [24]. Using elaborated approaches, as in [25], [26], allows to reduce the high requirements on the calibration setup. Nevertheless, radar sensors, especially in the automotive sector, typically suffer from some residual calibration error because no sufficiently frequent recalibration is performed to maintain high imaging quality [27]. As reported in e.g. [28], [29], [30], such remaining mismatches between the antennas in a MCSAR setting and even in single MIMO measurements lead to replicas in the radar image, generally referred to as ghost targets, because channel specific errors are repeated over the full aperture. To provide a visual intuition on the examined research content of this paper, this creation of ghost targets is illustrated in Fig. 1. Here, a SIMO-SAR measurement using a 76 GHz frequency-modulated continuous-wave (FMCW) MIMO-radar with 8 receive channels was performed to an automotive target scene shown in Fig. 1(b). The MIMO array is shown in Fig. 1(a). For the SIMO-SAR measurement, only the leftmost TX antenna was used. The measurement was performed such that a synthetic aperture of the length 12 cm was sampled uniformly. Then, radar images were reconstructed via backprojection [31]. First, the calibration data found via the offline calibration proposed in [25] was used and second, additional phase errors of ±20 • were added to the individual channels during the reconstruction. The resulting images are shown in Figure 1(c) and 1(d), respectively. While the scene in Fig. 1(b) is well reconstructed in Fig. 1(c), the radar image in Fig. 1(d) is strongly degraded via numerous ghost targets. These ghost targets occur as replicas of the true targets within regular distances, thereby, forming distinct patterns. Such artifacts within the radar image, which can also be caused by multipath or interference, are a relevant safety issue in autonomous driving. Therefore, on the one hand, the identification and reduction of these ghost targets is subject of current research [32], [33]. On the other hand, there is a need to detect occurring errors in the calibration, which can be caused by aging, temperature change, and deformation, to initiate re-calibration of the sensor [34]. Nonetheless, only little research has been conducted to analytically investigate the effects of calibration errors on MCSAR radar imaging and the corresponding creation of ghost targets. In [35], the authors discuss the impact of phase and amplitude errors on the radar image's ambiguity level in the radar image without discussing the ghost target positions. [30] exemplarily shows this relation for a two antenna array but not forarbitrary arrays arrays. Furthermore, neither [35] nor [30] discuss the influence of mutual coupling, which is relevant in most automotive radar sensors using closely spaced antennas either in the receiver (RX) or the transmitter (TX) array.
In MIMO radars, a virtual aperture is created by the convolution of the RX and TX subarrays [36]. They are often realized as uniform linear arrays (ULAs), leading to a periodic evaluation of the repeated subarray during the formation of the virtual aperture. In most scenarios of SAR measurements, including automotive applications, a constant position shift of the radar between consecutive measurements is assumed due to a fixed radar velocity and pulse repetition frequency [17], [18]. Then, both MIMO and MCSAR can be regarded as a spatial sampling process with periodically occurring errors due to the periodic evaluation of the channels. An almost equivalent sampling process is conducted by time-interleaved analog-to-digital converters (ADCs), where M ADCs with identical sampling rates are combined in a time-interleaving architecture to one ADC with M times the sampling rate [37], based on the theory of the generalized sampling expansion [3], [38]. Here, amplitude, timing, and further mismatches of the sub-ADCs lead to deterministic spurious signals, which are equivalent to the discussed ghost targets in radar images. For time-interleaved ADCs, these effects are very well analyzed and documented, e.g. in [39], [40]. The applicable findings will be applied and extended in this paper to analyze the impact of different calibration errors on MCSAR imaging. Thereby, the creation of calibration error caused ghost targets as in Fig. 1(d) will be explained thoroughly. A closed form solution will be given to predict the number and the position of occurring ghost targets in the case of calibration errors, without requiring any knowledge about the type of errors. Furthermore, for known errors, a relation for the height of the ghost targets will be derived. Based on this, we will provide rules for the worst-case distortion level in the case of unknown but bounded errors. To validate these theoretic findings in a realistic scenario, measurements using a 76 GHz FMCW MIMO radar are shown, illustrating the predicted effects and, thereby, demonstrating the good applicability of the derived relations.
The remainder of this paper is organized as follows: The considered error types of antenna arrays are modeled in Section II. The derivation of the erroneous SIMO-SAR signal model including extensions for MIMO-SAR and MIMO radar is shown in Section III. Section IV discusses the impact of a resampling step for non-uniform sampling on the error propagation. Based on the previous findings, boundaries for the worst-case signal-to-distortion ratios assuming different error types are derived in Section V. Section VI comprises a summary and discusses implications for future research. Finally, measurements are shown to validate the findings in Section VII, followed by a conclusion.
Notation: Continuous signals are written as f (t ), whereas discrete signals are written as f [n] = f (nT ). Vectors and matrices are written in bold lower and upper case letters, respectively. The nth entry of a vector x is given by x[n]. The transpose of a vector or matrix is given by (·) T and the inverse of a quadratic matrix by (·) −1 . a b denotes the modulo operation a modulo b, and ⊗ denotes the operator of the Kronecker product. δ(x) denotes the Dirac delta function. The mean-value calculation is expressed by the operator · . ∠(·) calculates the angle of a complex number. Finally, · and · denote the ceiling and floor function, respectively.

II. CALIBRATION ERROR MODELS
In this section, different calibration errors will be described and their respective transfer function will be formulated. In this work, we limit the discussion to the linear phase and amplitude errors, phase center displacement and mutual coupling. To establish the signal model, a SIMO radar, comprising N r RX antennas is assumed with its RX antennas located along the x-axis at the positions x n r , n r ∈ [1, N r ] and the TX at x = 0. This basic measurement principle is illustrated in Fig. 2. For the sake of simplicity, all derivations will be made for a continuous wave (CW) radar emitting a transmit signal with the carrier frequency f c . Nevertheless, the general findings within this paper can be easily applied to common radar types, such as the FMCW radar. This will be demonstrated in Section VII, where the findings within this work are validated using a 76 GHz FMCW MIMO radar. The transmit signal s tx (t ) is reflected at a target and received by the n r th RX antenna with a delay τ r , which depends on the target and antenna position. Generally, the received signal will be evaluated at the carrier frequency f c . Thus, the ideally received frequency domain signal of the n r th antenna can be written as where ϕ 0 introduces an additional phase shift caused by the target reflection and A 0 includes the path attenuation and the attenuation caused by the reflection at the target. Next, τ r (x n r ) is expressed as a continuous function τ r (x) along the x-axis, which is evaluated at the location of the active antenna to obtain Then, the phase change over the aperture can be used to define a spatial angular frequency In the following, the standard far-field assumption will be used. Thereby, the impinging signal is described by an incident angle θ 0 and has a constant spatial frequency k(x) = −k 0 . Here, k 0 is defined as with the carrier wavelength λ c = c 0 f c , where c 0 denotes the speed of light. k(x) = −k 0 was chosen to establish a positive spatial frequency for the positive angle θ 0 in Fig. 2. A received signal caused by multiple targets in the far-field is created by superposition.
In the presence of errors, the received signal can be stated as where the channel-specific error transfer function H error,n r may depend partly on the spatial angular frequency and will be split up further in the following to represent the different error sources. First, phase errors are considered, which represent the dominating error source of antenna arrays. This is caused by the use of small wavelengths, which are necessary to enable a high measurement sensitivity of the overall system but result in large errors even for small differences in the length of the waveguides. Generally, the phase error depends on the frequency but it reduces to a constant phase ϕ n r per channel for the fixed frequency f = f c , such that it can be modeled as Since, in most cases, relative phases are evaluated in radar imaging, only phase mismatches influence the imaging process, whereas absolute phase errors can be ignored. Hence, we can further assume without loss of generality that Second, amplitude errors are caused by differences of the antenna elements' gain and the attenuation within the channel processing chain. The channel gains at f c will be modeled with a fixed amplitude A n r per channel as Again, only amplitude mismatches are relevant for the reconstruction.
As a third error type, mutual coupling will be modeled. Mutual coupling at RX arrays occurs when an impinging signal is reflected at one antenna n r and then received at another antenna n r . Further coupling is caused by the leakage between the transmission lines within the radar electronics. To describe the impact of the n r th antenna's signal onto the n r th antenna's, a transfer function h n r ,n r is introduced, which models this specific coupling path as h n r ,n r (t ) = c n r ,n r δ(t − τ n r ,n r ), with a delay τ n r ,n r and an attenuation factor c n r ,n r . In (10), it is assumed that the first reflection between the antennas dominates coupling since multiple reflections will lead to negligible signal amplitudes. This model for mutual coupling is typically used in band-limited systems for a simplified discussion [24]. In the frequency domain at f c , the coupling transfer function is then given as H n r ,n r ( f c ) = c n r ,n r e −j2π f c τ nr ,n r = c n r ,n r , where the constant phase shift introduced by the delay at a fixed frequency is included in the adapted, now complex, coupling factor c n r ,n r , where c n r ,n r = 1 holds. Then, applying (11) to write the mutual coupling impaired receive signal (6) as c n r ,n r S ideal,n r ( f c ), (12) allows to express mutual coupling using the familiar matrix notation with the coupling matrix C, whose entries are given by c n r ,n r , the true receive signal vector S( f c ) = (S 1 ( f c ), . . . , S N r ( f c )) T and the equivalently formed ideal receive signal vector S ideal ( f c ). Note that here exclusively, the vectors S( f c ) and S ideal ( f c ) are written as upper case bold letters, not following the notation introduced in Section I, because they denote signals in the frequency domain. Finally, to model the influence of displaced phase centers, we define an ideal 2D antenna position as p n r = (x n r , 0) and a displaced true phase center position p n r = (x n r + e x nr , e y nr ). The transfer function then depends on the incident angle and follows as with the distance difference caused by the displacement d n r (θ 0 ) = e 2 x nr sin 2 θ 0 + e 2 y nr cos 2 θ 0 .
For the combination of the discussed error types, they are assumed to occur in the order phase center displacement, mutual coupling, phase, and amplitude error. Hence, for an impinging wave with k 0 , (6) can be stated with the individual error transfer functions as with the distance x n r = x n r − x n r between the n r th and the n r th antenna. (16) shows that for a fixed spatial frequency k 0 the influence of all error types can be expressed by a single constant complex error factor α n r (k 0 ) per antenna. Furthermore, note that phase and amplitude errors form a constant error factor independent of the spatial frequencies.
We will limit our discussion to amplitude, phase, and coupling errors for the remainder of this paper. However, the impact of phase center displacement is closely related to mutual coupling because both cause a spatial frequency dependent phase shift and can be incorporated into α n r (k 0 ), as shown above.

III. SIGNAL MODEL FOR ERRONEOUS MCSAR
This section will first derive the signal model for an erroneous SIMO-SAR. The extension to MIMO and MIMO-SAR will be presented later in the section. The basic MCSAR measurement principle is depicted in Fig. 3 for a SIMO-SAR. To derive the signal model of SIMO-SAR with calibration errors, the following assumptions are made: First, the radar is moved along the x-axis with constant speed to create the synthetic aperture. We will assume that the synthetic aperture is extended infinitely for mathematical simplicity. Measurements are taken with a distance of L sar , hence, at the mth measurement point, the TX antenna is located at x t = (m − 1)L sar and the n r th RX antenna is located at x n r = x t + l n r , where l n r denotes the constant distance between the n r th RX and the TX antenna, see Fig. 3. Second, as before, the target is assumed to be located in the far-field with an incident angle θ 0 , so (5) holds, and all derivations will be made using k 0 . Note that an infinitely long synthetic aperture contradicts the far-field assumption but simplifies the subsequent derivation. All relations found in this work are also valid if only a few measurements are performed or the far-field condition is not fully met, as will be demonstrated by the simulations within this section and the measurements in Section VII. Third, the transmit signal S t ( f ) is assumed to have a limited bandwidth such that (13) and, thus, (16) holds [25], which should be valid for most radar systems.

A. DERIVATION OF THE SIGNAL MODEL
The received signal of the n r th antenna with calibration errors expressed by the error factor α n r (k 0 ) is formulated generally as a continuous function of the TX position along the aperture as Here, the RX and TX transfer functions H n r (x t ) and H t (x t ) are given by and describe the impact of the time delay caused by the path between the target and the RX and TX antenna, respectively. Additional constant phase and attenuation factors were discarded for simplicity. Because the evaluation of the received signal along the aperture is usually performed at fixed evaluation frequencies, the form of the band-limited transmit signal S t ( f ) does not influence further discussions and will be discarded in the following. Hence, all derivations will be made solely using the transfer functions, which fully contain the measured information of interest. The SAR measurement can then be modeled as a spatial sampling process with the spatial angular sampling frequency k s = 2π L sar , yielding the sampled continuous signal as In contrast to the single SIMO or MIMO case, where the TX antennas are static, both the TX and the RX antenna positions are moved during a SAR measurement, yielding the actual measured signal frequency k MCSAR m = 2k 0 . Hence, in SIMO-SAR, the incident angle θ 0 can be determined from the measured signal frequency via which shows that an effective sampling spatial frequency of 4 λ c is required to unambiguously reconstruct the angle range . Therefore, it is suitable to define equivalent antenna positions l nr 2 , which allow considering the bi-static RX-TX measurements as equivalent mono-static measurements, where the sampling positions during the mth measurement position are given by mL sar + l nr 2 [3], [41]. Then, applying the Fourier transform (FT) along the spatial dimension x t to (20) yields with spectral replicas of the continuous spectrum with a spacing of the spatial sampling frequency 2π L sar . Following [39], [40], all N r receive signals are combined to get the full measurement, which according to the generalized sampling expansion will increase the effective sampling frequency to N r 2π L sar [38]. Hence, considering (21), an unambiguous reconstruction of all frequencies resulting from θ 0 ∈[− π 2 , π 2 ] should be possible, as long as the SAR shift between two subsequent measurements is limited to To combine the signals, the discrete sampling sequences are concluded from (20) as upsampled by the factor N r , such that [39] and combined in an interleaved manner to achieve a single sequence with sampling rate N r 2π L sar , yielding The FT of the combined sampled signal can then be stated using (22) as Due to arbitrary antenna positions l n r and SAR shifts L sar , the combined sampling positions are non-uniform, see Fig. 3. Therefore, analogous to the signal model of time-interleaved ADCs [39], the equivalent mono-static antenna positions l nr 2 , introduced before, will now be expressed by the positions (n r − 1) L sar N r , which would yield uniform sampling in a SIMO-SAR scenario, and a deviation l n r from this position, such Then, the current sampling position l s (m, n r ) of the n r th antenna at the mth radar measurement position is given by Inserting (28) into (27) then leads to where at a) the sampling property of the Delta dirac function was used [42]. At b), the sum over n r was interpreted as a discrete Fourier transform (DFT) [39], [43], showing that the spectral replicas' weighting factors β p are given by the Fourier coefficients of the channel properties, namely α n r (k 0 )e −j2k 0 l nr . As a consequence of the DFT, these weighting factors are periodic so that we can state

B. INTERPRETATION
The established relation (30) describes the combined receive spectrum of all antennas. For uniform sampling and ideal calibration, i.e. when l n r = 0 and α n r = 1 holds for all n r , this combined spectrum is equivalent to one caused by ideal sampling with a N r times increased sampling frequency. Otherwise, spectral replicas of the ideal continuous receive spectrum emerge. Two main effects cause these spectral repetitions. These are the non-uniform sampling via e −j2k 0 l nr in (31), which is a well-known effect and discussed, e.g. in [3], and the calibration errors per channel via α n r (k 0 ) in (31). Both of these effects cause peaks at the same frequencies which are only defined by the chosen SAR shift L sar and the spatial frequency k 0 caused by the true target. These spatial frequencies k p can directly be interpreted as incident angles θ using (21). Here, the spectral repetitions caused by non-uniform sampling generate ghost targets, when simple matched-filter-based reconstruction methods like fast FT or backprojection (BP) [31] are used. Also, inversion-based reconstruction methods, such as MUSIC and compressed sensing, are susceptible to modeling errors. They often fail due to instability or yield wrong solutions if such spectral repetitions occur. However, the effects of non-uniform sampling are generally invertible, e.g. via resampling, since the antenna geometry is usually known [3]. The effects of non-uniform sampling and this resampling step on the calibration error propagation will be discussed in the following subsection in detail.
In comparison to non-uniform sampling, the calibration errors are unknown. Hence, reconstructions cannot compensate the effects of the unknown calibration errors and, therefore, will lead to ghost targets in the final image. Naturally, the mismatches will cause N r ghost targets within the spatial frequency band k ∈ (− N r 2L sar , N r 2L sar ]. However, in radar imaging, the frequencies of interest result from the hypothetical angles θ ∈ (− π 2 , π 2 ]. In practice, these two intervals do not necessarily coincide, especially in automotive radar, where the driving speed influences the SAR step size. To determine the number of ghost targets in this frequency band, (32) and (21) allow us to formulate the condition for a spectral replica with index p rep to be located inside this interval. Hence, the relevant p rep are given by with Z * denoting integer numbers without 0, because p = 0 in (30) belongs to the true target and is not considered a ghost target. The lower and upper bounds are Consequently, this leads to a total of N rep = p rep,u − p rep,l ghost targets, which are located at the incident angles (36) and their amplitudes are given by β p rep (k 0 ) according to (31). Note that the desired target amplitude at p = 0 is also weighted by β 0 (k 0 ). To illustrate the effects, simulations were performed assuming a SIMO-radar with N r = 8 RX antennas arranged as a ULA with λ c 2 spacing, i.e. the antennas are located at l n r = (n r − 1) λ c 2 . The same RX array will be used in the other simulations for the remainder of this work for an easy comparison. Here, the radar was moved to 64 measurement positions for a SIMO-SAR measurement with two different SAR step sizes. The resulting spectra are shown in Fig. 4. The frequency axis is exchanged by the corresponding θ -axis. For the calibration error, only phase errors were assumed, with ϕ n r evenly distributed in (− π 10 , π 10 ). In Fig. 4(a), the SAR step size is L sar = N r λ c 4 , leading to ideal uniform sampling. Here, the spectral replicas are solely caused by calibration errors. In Fig. 4(b), L sar = 0.55N r λ c 4 was chosen, which leads to oversampling of the relevant frequency interval. As a consequence, there are fewer spectral replicas visible because some spectral replicas occur at frequencies outside the reconstructed interval θ ∈ (− π 2 , π 2 ]. For an even stronger oversampling with L sar ≤ λ c 4 all spectral replicas will lie outside the reconstruction interval θ ∈ (− π 2 , π 2 ]. This corresponds to the case where a single antenna would be sufficient to satisfy the Nyquist criterion within the SAR measurement. Furthermore, the SAR shift L sar = 0.55N r λ c 4 leads to non-uniform sampling, and the spectral replicas are, therefore, caused by both calibration errors and sampling errors.

C. IMPLICATIONS FOR MIMO RADAR AND MIMO-SAR
In MIMO radar, the virtual antenna array is created by the convolution of the TX array with the RX array [36]. Usually, the arrays are arranged such that one has small spacing ensuring unambiguous measurements, whereas the other is widely spaced, creating a large aperture. Without loss of generality, we assume the RX array to be closely spaced, as illustrated in Fig. 5(a). Then, a single MIMO measurement can be interpreted as an equivalent SIMO measurement with a single TX at the first TX's position and a N t N r sized virtual RX array, which consists of N t repetitions of the actual RX array at the TX antenna positions. A single MIMO measurement does not lead to an interpretable image if both the TX and RX array suffer from significant calibration errors. However, for TX arrays with uniform structure, channel mismatches within the RX array yield a similar error pattern structure as the discussed SIMO-SAR case whenever calibration errors within the TX array are negligible. This effect is illustrated in Fig. 5(a), which shows the error pattern within the virtual array resulting from a MIMO array with a calibrated TX array. Note that the widely spaced TX array is much easier to calibrate than the closely spaced RX array because coupling effects can be neglected; hence, the assumption of a well-calibrated TX array often is justified in practice. Now, the same derivations as for SIMO-SAR can be made to find identical distortion effects occurring in single MIMO radar measurements. The derivation is very similar, with the only difference that for a single MIMO measurement the measured frequency is k MIMO m = k 0 instead of k MCSAR m = 2k 0 for SIMO-SAR as in (21). This is caused by the fact that the MIMO array's virtual TX antenna is stationary compared to the moved TX antenna of a SAR. However, the unambiguous spatial frequency band is also doubled and the angle can be reconstructed from the measured frequency k MIMO Therefore, the MIMO ghost targets appear at Often, a MIMO radar is designed to yield a uniformly distributed equivalent SIMO array with λ c 2 spacing. To do so, the RX distance is λ c 2 , whereas the TX distance is chosen as L tx = N r λ c 2 to create uniform sampling at the Nyquist border.  Fig. 4, which was extended by a four antenna TX ULA, such that a uniform virtual array is achieved. In both cases, the RX array is impaired by the phase calibration errors used in Fig. 4.
Thus, a MIMO measurement is equivalent to a SIMO-SAR measurement whenever L tx = 2L sar holds, see (23).
Using the equivalent SIMO assumption for a MIMO radar, the extension to a MIMO-SAR measurement is also straightforward. For the general case of calibration errors on both arrays, a virtual SIMO array with N r N t elements can be assumed, such that all previous findings for the SIMO-SAR case can be applied without any adaptions. This is verified by a simulation shown in Fig. 6. Here, the same RX ULA including the same error pattern was used as in Fig. 4. However, the RX array is now extended to a MIMO radar using a N t = 4 element TX ULA with L tx = N r λ c 2 spacing. Here, the black spectrum shows the N r N t artifacts caused by additional phase errors on the TX array, which are evenly distributed in (− π 10 , π 10 ), yielding a total of N t N r = 32 ghost targets. However, a special case occurs if again the TX is calibrated to the degree that their remaining error factors are negligible. The resulting error structure is illustrated in Fig. 5(b). For a calibrated TX array, the repetition frequency of the error factors along the synthetic aperture is increased by the factor N t . Hence, only every N t th artifact will be non-zero due to the DFT, which determines the artifact heights. Thereby, the error pattern of the MIMO-SAR measurement is identical to the SIMO-SAR case, as it is visible by the resulting spectrum in Fig. 6.

IV. COMPENSATION OF NON-UNIFORM SAMPLING
As discussed before, in radar imaging, typically reconstruction methods are used, which require uniformly spaced sampling points [3], [44], [45]. Therefore, first, a resampling step must be performed. Several resampling approaches were proposed in literature as in [3], [45], [46], whereas most are based on a reinterpretation of the generalized sampling expansion given in [47]. We will now discuss the resampling in a very general manner and analyze the consequences of the impact of calibration errors on the reconstructed spectrum.
For the resampling process, we reformulate the SIMO-SAR measurement process in matrix-vector notation, assuming N sar SAR measurement positions as where x Y c ∈ C N sar N r denotes the discrete representation of the impinging receive signal's continuous spectrum and H ∈ C N sar N r ×N sar N r denotes the ideal measurement matrix describing the sampling process as given by the RX and TX transfer functions (18) and (19), as well as the inverse discrete Fourier transform (iDFT). The diagonal block matrix C K = I N sar ⊗ C ∈ C N sar N r ×N sar N r , with the identity matrix I N sar of the dimension N sar , applies the coupling matrix to all sampling points. Note that in this representation, phase and amplitude errors are assumed to be included in C. Finally, y nu ∈ C N sar N r contains all non-uniform sampling values of all channels as given by (26). We now define a corresponding vector of the non-uniform sampling positions l nu for this measurement vector. In (29), the sampling positions were described by the index m of the radar position and the index n r of the receive antenna. Now, the sampling positions are expressed by a single index p, such that the pth entry of l nu is given by where l u = L sar N r is the sampling point spacing, which would lead to uniform sampling with the same sampling rate and l p Nr are the respective periodic deviations. Furthermore, we define a vector of uniform sampling positions l u , assuming the same sampling rate as in the non-uniform case, where the pth sampling position is given by The goal is now to find an alternative receive signal y uni , which contains the sampling values corresponding to the uniform sampling positions l u . To do so, the SIMO-SAR measurement equation in (39) is reformulated as where at a), the measurement matrix is divided into a uniform sampling matrix H uni , which describes the sampling process at the uniform sampling positions l u and is essentially an iDFT matrix, thereby causing no information loss, and a resampling matrix H res . Consequently, in b) we can define an ideal uniform receive signal y uni,ideal without the influence of calibration errors. This demonstrates that H res represents the required linear combination of uniform sampling values to form the corresponding non-uniform sampling values. During the resampling process, this transformation is reversed by taking the inverse 1 H −1 res to get the reconstructed uniform receive signal y uni,rec via y uni,rec = H −1 res y nu = H −1 res C K H res y uni,ideal = C res y uni,ideal .
For an error-free measurement, C res becomes an identity matrix and, hence, the uniform signal can be reconstructed perfectly. Otherwise, a comparison of (42) and (43) shows that both non-uniform sampling and calibration errors cause the measured signal to be a linear combination of ideal samples caused by uniform sampling, demonstrating an analogy of both error sources. The linear combination then causes correlations within the spectrum and, therefore, spectral replica.
To further analyze the impact of calibration errors after resampling, the matrices' structures are exploited. Since y uni,ideal is the result of an iDFT, it can be expanded in a cyclic manner. Recalling the periodic structure of the non-uniform sampling positions in (40) shows that H res is a block-circulant matrix as in [48], where each block is of the size N r × N r . Since this structure is preserved under inversion [48], the same holds for H −1 res . Then, C res in (43) is the product of a cyclic block matrix and a diagonal block matrix and, hence, also a cyclic block matrix [49]. Therefore, the measurement error caused by calibration errors has a periodic structure, even if a resampling step is performed. The pth resampled measurement value of y uni,rec can then be written as where c res p,p denotes the (p, p )th entry of C res . Here, an error factor per sampling value is defined equivalently to (16). Using the periodicity of the mismatches, which were shown before, only the first N r factors must be calculated because α n r (k 0 ) = α p Nr (k 0 ) (45) holds. Hence, the error factors after the resampling step again only depend on the active antenna and can therefore be used identically to α n r (k 0 ) in the previous section. This is further illustrated by the simulation shown in Fig. 7. Here, for an have full rank, i.e. the inverse exists, if the generalized sampling expansion is valid. This is true as long as no sampling points coincide, which is assumed to be the case. However, if sampling points coincide, the sampling rate is effectively reduced slightly, and the inverse would have to be replaced by the pseudoinverse. Finally, note that the reconstruction quality strongly depends on the eigenvalue distribution of H res . Receive spectrum of the non-uniformly sampled SIMO-SAR with phase errors, which was used in Fig. 4(b) before and after performing a resampling step to compensate the non-uniform sampling.
easy comparison, the black spectrum shows the SIMO-SAR spectrum from Fig. 4(b), which occurred for an erroneous RX array and a SAR shift of 0.55N r λ c 4 , thereby causing non-uniform sampling. The orange spectrum shows the same signal after applying a resampling step onto uniform sampling positions. While the height of the ghost targets is reduced due to the corrected sampling errors, the remaining calibration errors still cause ghost targets to appear at the predicted positions because the periodic pattern is preserved during the resampling step.
Since we have shown that non-uniformly sampled signals yield the same error structure as uniform sampling, for the sake of simplicity, in the remainder of this work, we limit all further derivations to the case of uniformly sampled signals. Consequently, l n r = 0, ∀n r , is assumed and (31) can be simplified, yielding α n r (k 0 )e −jp(n r −1) 2π Nr . (46)

V. IMPACT ON SIGNAL-TO-DISTORTION RATIO
In this section, the level of distortion caused by calibration errors is analyzed by determining worst-cases for different error types. To do so, first, a signal-to-distortion ratio (SDR) is defined as SDR(k 0 ) = 10 log 10 P S P D = 10 log 10 |β 0 (k 0 )| 2 − 10 log 10 with the signal power P S and the distortion error power P D . These can be directly calculated using (46) by recalling that β 0 contains the desired signal, whereas all β n r with p N r = 0 contain the unwanted distortion error power. The signal and distortion power are now reformulated to find an alternative representation. First, the signal power can be expressed using α n r (k 0 ) as [40] P S (k 0 ) = |β 0 (k 0 )| 2 where at a) the definition of the absolute of a complex number was used with |x| = Re 2 x + Im 2 x, and the real and imaginary parts' mean are denoted γ 1 and γ 2 , respectively. They can also be expressed as |α n r (k 0 )| cos(∠α n r (k 0 )) (49) |α n r (k 0 )| sin(∠α n r (k 0 )).
Second, the distortion power is given by the total power minus the power of the desired signal as where at a) Parseval's theorem was used [40], and at b), the previously found alternative expression for the signal power was inserted. Hence, the SDR follows as SDR = 10 log 10 Next, the worst-case scenarios for different error types, namely isolated amplitude and phase errors, combined phase and amplitude errors, as well as coupling errors will be discussed. Worst-case SDRs will be derived, which allow estimating the level of distortion occurring in the radar image for a maximum expected error of the different error types. For example, large temperature differences, which lead to the circuit board's contraction or expansion, will change of the line lengths of the different channels and, thereby, cause phase errors. With the worst-case SDRs derived in the following, the expected distortion level can be calculated for a maximum expansion and, thereby, a maximum phase error. The worstcase analysis for isolated amplitude and phase errors is closely related to the one made for gain and timing mismatches of time-interleaved ADCs in [40]. Therefore, the derivation of these two cases will be rather compact. Furthermore, note that when only amplitude and phase errors are considered, while coupling is neglected, the dependency on k 0 can be discarded.

A. AMPLITUDE ERRORS
First, only amplitude errors will be discussed. For this purpose, they are defined as A n r = A n r − A n r with | A n r | < A n r . Then, the error factors are given by α n r = A n r + A n r , where A n r = 0 holds. By inserting this relation in (48) and (51), the related SDR A follows as To find a worst-case SDR, A n r is assumed to be bounded by a maximum value A max with | A max | < A n r . Then, [40] showed, that the worst-case SDR, indicated by the superscript wc, occurs, when all A n r equal ± A max . The worst-case SDR can then be simplified to

B. PHASE ERRORS
The derivation of the worst-case SDR for phase errors is similar and for an extensive derivation the reader is referred to [40]. Again, phase errors are defined as ϕ n r = ϕ n r − ϕ n r , whereas we recall that ϕ n r = 0 holds. Then, the error factors follow as α n r = e j ϕ nr and the SDR for only phase errors SDR P can be found by inserting this in (49), (50), and (52), leading to SDR P = 10 log 10 γ 1 2 + γ 2 2 1 − γ 1 2 + γ 2 2 = −10 log 10 1 γ 1 2 + γ 2 2 − 1 = −10 log 10 1 cos ϕ n r 2 + sin ϕ n r 2 − 1 . (55) Again, for the worst-case scenario ϕ n r is assumed to be bounded by a maximum error ϕ max . Then, the minimum SDR can be found, by minimizing both cos ϕ n r 2 and sin ϕ n r 2 . cos ϕ n r 2 can be minimized if for all errors | ϕ n r | = ϕ max holds. sin ϕ n r 2 is minimized to its minimal value zero, if either all ϕ n r = 0 or the same number of errors equal plus and minus ϕ max , respectively, causing pairs of sin ϕ max and sin(− ϕ max ) = − sin ϕ max , which add up to zero [40]. Hence, the latter yields the worst-case SDR for only phase errors, which is then given by SDR wc P = −10 log 10 1 cos 2 ϕ max − 1 . (56)

C. COMBINED AMPLITUDE AND PHASE ERRORS
Combined amplitude and phase errors are modeled as α n r = (A n r + A n r )e j ϕ nr . The SDR is then given by SDR AP = −10 log 10 Finding the amplitude and phase errors, which cause a worst-case SDR in this case corresponds to finding the solution of argmin ϕ nr , A nr (A n r + A n r ) cos ϕ n r 2 (A n r + A n r ) 2 For now, it is assumed that, for the solution of (58), the second summand is minimized to its minimal value being zero. Then, the optimization task is reduced to argmin ϕ nr , A nr (A n r + A n r ) cos ϕ n r 2 (A n r + A n r ) 2 .
(59) Independent of the amplitude errors A n r , (59) is minimized by choosing | ϕ n r | = ϕ max , such that cos ϕ n r is minimal. Then, cos ϕ n r is constant and, consequently, (A n r + A n r ) cos ϕ n r in the numerator of (59) will also be constant since A n r = 0 holds. This allows to further reduce (59) to which is solved by choosing pairwise A n r = ± A max . The found relations A n r = ± A max as well as | ϕ n r | = ϕ max solve (58) as long as the second part of (58) equals zero. This, in turn, can be assured by further choosing conjugate phase errors ± ϕ max for channel pairs with the same amplitude errors. Hence, by inserting α n r = (A n r ± A max )e j± ϕ max and α n r = (A n r ± A max )e j∓ ϕ max in (57), the worst-case SDR for combined phase and amplitude errors is calculated as SDR wc AP = −10 log 10 (A n r + A max ) 2 + (A n r − A max ) 2 A n r 2 cos 2 ϕ max − 1 = −10 log 10 A n r 2 + A 2 max A n r 2 cos 2 ϕ max − 1 . In order to illustrate the impact of calibration errors, simulations were conducted assuming a uniform antenna array with N r = 8 receive channels with λ c 2 spacing, which, if e.g. operated at f c = 76 GHz, would correspond to a typical automotive radar. L sar = N r λ c 4 was chosen to achieve uniform sampling. Fig. 8 shows the worst-case SDRs for certain maximum values for phase and amplitude errors along the channel, demonstrating that even small errors can significantly degrade performance. For example, if only phase errors occur, the SDR is reduced by up to 20 dB for a maximum phase error of 5.7 • , whereas a sole maximum relative amplitude of 10% leads to the same performance degradation. Furthermore, to verify the applicability of the worst-case SDRs as boundaries, 5000 combinations of phase and amplitude errors per channel were randomly generated within the limits defined by ϕ max = 8 • and A max A nr = 10%, respectively, for an 8 antenna RX array. This was repeated for only phase, only amplitude, and combined errors. Fig. 9 shows histograms of the according SDRs and the worst-case SDRs as dashed lines. For all simulated cases the worst-case SDRs clearly limit the occurring histograms, and the mean SDRs are approximately 5.7 dB higher in all cases. Furthermore, for combined errors, the SDR is lowered and the histogram becomes narrower because the combination of multiple random variables has an averaging effect.

D. COUPLING ERRORS
In the following, the impact of coupling errors will be discussed. For this purpose, a coupling matrix is assumed whose main diagonal equals one. All other entries, representing the actual coupling, have amplitude values that are fixed, known, and symmetric, meaning |c n r ,n r | = |c n r ,n r | holds. Thereby, we account for the fact, that the coupling amplitude is mainly given by the antenna distance, and, therefore, can be assumed to be approximately known and symmetric [24]. It was discussed in [25] that the assumption of constant coupling is valid in typical radar systems with limited bandwidth such as FMCW radar. In special cases, where the above assumptions are not justified, the following derivation of a worst-case SDR may still be useful to provide a rough estimate of the expected image distortion caused by mutual coupling. The phase values of the coupling entries c n r ,n r can take any value in [−π, π ), i.e. they are completely random. Furthermore, a fixed incident angle, i.e. a fixed spatial frequency k 0 is assumed. Then, the error factors α n r (k 0 ) can be calculated as α n r (k 0 ) = 1 + N r n r =1,n r =n r c n r ,n r e jk 0 x n r (62) This shows, that, unlike amplitude and phase errors, calibration errors cause an additive error on α n r (k 0 ). For any worst-case scenario, all error factors will have the greatest possible amplitude. This will occur, when the coupling factors c wc n r ,n r are added with an identical phase during the calculation of the worst-case error factors α wc n r (k 0 ), such that α wc n r (k 0 ) = 1 + with a, yet to determine, worst-case phase per channel ψ wc n r and a fixed total coupling amplitude |c n r |. The worst-case SDR caused by coupling SDR wc C is then given by (64), shown at the bottom of the next page. Note that SDR wc C is not a function of the spatial frequency, and, therefore, the following worst-case analysis is generally valid.
Due to the different coupling amplitudes per channel in (64), the values for ψ wc n r are difficult to find. Therefore, we will instead define an approximate SDR wc C,a by assuming |c n r | ≈ |c|, with a fixed value |c|. This approximation is mostly justified for the common case of ULAs. Then, the assumption is valid for the array's inner antennas, because coupling occurs mainly between adjacent antennas. For other antenna arrangements, this may still serve as an approximate value, which will be used here for simplification of the subsequent derivation of the worst-case SDR. Hence, |c n r | = |c| allows to rearrange (64) to yield SDR wc C,a = min As shown in the Appendix, the worst-case SDR can again be found by choosing all ψ wc n r to match plus/minus a, for now unknown, value ψ wc and ψ wc n r = 0, such that (65) can be reformulated to yield SDR wc C = min ψ wc 10 log 10 1 |c| + cos ψ wc 2 1 − cos 2 ψ wc , whose derivative can be used to find analytically ψ wc = arccos(−|c|).
Then, the worst-case SDR for coupling errors is given by SDR wc C = 10 log 10 1 |c| + |c| which is shown in Fig. 10 for different values of |c|. Next, a closer look is taken at the specific coupling structure of ULAs, which represent an important and often used special case of antenna arrays. Because of the uniform antenna arrangement, the coupling matrix is usually assumed to have a Toeplitz structure; hence, it is fully described by N r − 1 complex factors c t,|n r −n r | with n r = n r and C = toeplitz (1, c t,1 , . . . , c t,N r −1 ). Note that in this work, a Toeplitz matrix is defined as a symmetric matrix, not a Hermitian and it can be seen that the above-discussed worst-cases for coupling can not occur in ULAs. In fact, for large arrays and c t,|n r −n r | → 0 for increasing |n r − n r |, all inner α n r (k 0 ) will be identical. In this case, the distortion power caused by coupling is almost completely canceled out by the Toeplitz structure, demonstrating a favorable property of ULAs. For smaller arrays, where most c t,|n r −n r | are greater than zero, note that within the sum in (69) the c t,|n r −n r | has a symmetric influence along the array, whereas e jk 0 x n r is conjugate symmetric. Recalling that the phase-only worst-case discussed above occurs when ∠α n r is pairwise ± ϕ max and that ∠α n r = 0 holds, shows that the worst-case SDR for Toeplitz structured coupling matrices occurs when for all coupling factors ∠c t,|n r −n r | = 0 holds and the phase error is solely introduced SDR wc C = min ψ wc nr 10 log 10 ⎛ ⎝ 1 + |c n r | cos ψ wc n r 2 + |c n r | sin ψ wc n r 2 1 + 2|c n r | cos ψ wc n r + |c n r | 2 − (1 + |c n r | cos ψ wc n r ) 2 + |c n r | sin ψ wc by the conjugate symmetric e jk 0 x n r . The corresponding SDR can then be calculated using (69) and (52).
To conclude this section, simulations were performed to illustrate the impact of coupling errors. For this purpose, the same exemplary ULA as for previous simulations was assumed. To estimate the amplitudes of the coupling matrix for this exemplary array, a field simulation of an array of 8 rectangular linear polarized patch antennas at 76 GHz with λ c /2 distance was performed. Based on this simulation, the coupling matrix was chosen as |C| = toeplitz(1, 0.12, 0.06, 0.04, 0.03, 0.02, 0.01, 0.01). Furthermore, a second ULA comprising N r = 16 antennas was considered, where |C| was expanded by zeros, i.e. by vanishing coupling for large antenna distances. To analyze the impact of fully Toeplitz-structured coupling matrices compared to arbitrary coupling, two different simulations were performed for both array sizes. For an arbitrary coupling matrix the phases of all non-diagonal entries ∠c n r ,n r and for a Toeplitz matrix N r − 1 phases ∠c t,|n r −n r | were drawn from a uniform distribution U (−π, π]. This was repeated 5000 times for both array sizes and both realizations forms to simulate randomly generated coupling errors. Each time the resulting SDRs were calculated. The distributions of these SDRs are shown in Fig. 11 as histograms for the 8-antenna array at the top and the 16-antenna array at the bottom. Furthermore, the worst-case SDRs are shown as dashed and dotted lines for the arbitrary and Toeplitz-shaped coupling matrices, respectively. Here, |c| = N r n r =1 |c n r ,n r | − 1 was chosen as the mean maximum coupling amplitude per channel to calculate the worst-case SDR in (68). For both the arbitrary and the Toeplitz arrays, the distance between the mean SDRs and the worst-case SDRs is approximately the same. The histograms of the arbitrary arrays are narrower because more random variables are combined, which has an averaging effect. Generally, as expected, the SDR is much higher when the Toeplitz assumption is met. Comparing the 8-and 16-antenna array shows a further increase of the Toeplitz SDR because more coupling distortion power is canceled out. For the arbitrary array, both the mean and worst-case SDR decrease when the larger array is used. This is caused by the increase in the mean coupling amplitude |c| since it comprises more inner antennas, which receive high coupling power from both directions of neighboring antennas, thereby increasing the distortion power.

VI. SUMMARY OF THE FINDINGS AND IMPLICATIONS FOR FUTURE RESEARCH
In the previous sections, we analyzed the impact of calibration errors on MCSAR and MIMO systems. In these imaging systems, the signal of the receive and/or transmit array is evaluated periodically during the spatial sampling process of the incoming signal. In the case of erroneously calibrated arrays, this leads to a periodic error function on the measured signal, which then leads to repetitions of the true spectrum within the measured spectrum and causes the error power in the final image to accumulate in a few distinct ghost targets. The positions of these ghost targets are independent of the underlying calibration error type. They can be predicted only based on the true target position and knowledge about the radar movement. Furthermore, the artifacts' heights can be calculated for known errors as the periodic error function's Fourier coefficients. The respective equations are summarized in Table 1. We also proposed a simple measure for the distortion power caused by calibration errors in the form of an SDR and provided worst-case SDRs for bounded errors of different types. Again, these are summarized in Table 1.
The described effects will arise in naturally in many radar imaging applications of both MIMO and MCSAR. For example, in automotive radar, MIMO arrays are already used and SIMO-SAR raises increased interest for e.g. the search for parking spaces [20]. As another example, radar-based personnel scanners at airports use large MIMO arrays [5], where errors in individual arrays will cause the established ghost target patterns. In these safety-critical imaging applications, targets with a high dynamic range should be detectable. Often, classifying algorithms are used to find and identify target clusters as e.g. in [50], [51], [52]. However, the deterministic ghost target positions will lead to replicas of these full target clusters within the image. Hence, these repetitions will have the same characteristic features as the original target clusters and, thus, will be classified accordingly by respective algorithms. This impedes any feature-based identification of ghost targets as in [53].
This work's findings about ghost target formation may be used for automatic classifications of detections as actual targets or ghost targets, based on e.g. finding the characteristic artifact pattern within the image. Furthermore, it can be used to find novel metrics for the calibration quality based on the artifacts' amplitudes, thereby allowing an evaluation of the radar performance at runtime and notifying the user whether there is a need for re-calibration. Additionally, the accumulation of error power in small and known parts of the radar image may be used to establish novel calibration procedures. In conventional online calibration concepts, the unknown scattering scene usually does not allow a clear distinction of the correct and the erroneous signal, since the error power is assumed to be distributed over the full image. Using this paper's findings, artifact positions containing mainly error power can be identified. Therefore, a calibration cost function, e.g. some variation of the established SDR, can be created only based on the artifact's positions while discarding the remaining unknown image. This simplifies online-calibration in unknown environments drastically. Offline calibration approaches, as in [25], [26], already exploit the sparsity of an unknown target scene to perform simultaneous calibration and imaging. By exploiting the formation of ghost targets in a proper measurement setup, the known separation of signal and error power may significantly lower the requirements on the imaging part of such a calibration.

VII. EXPERIMENTAL VERIFICATION
In order to validate the theoretical findings within this work, measurements were performed in a realistic automotive imaging scenario. The measurement scenario is shown in Fig. 12. Here, a commercial MIMO radar using FMCW measurements with a carrier frequency f c = 76 GHz and a bandwidth of 1 GHz was mounted on a linear unit, which provided the motion for SAR measurements. The MIMO array consisted of N r = 8 element RX ULA with an antenna distance of λ c 2 and a N t = 4 element TX ULA with an antenna distance of 7 λ c 2 . Since, in the case of SIMO-SAR and MIMO-SAR, the targets are partly located in the near-field of the synthetic aperture, a backprojection algorithm was used for reconstruction. The radar was calibrated using the concept proposed in [26], which provides separate coupling matrices for the TX and RX array. To validate the findings of this work, this calibration was downgraded during the reconstruction to produce calibration errors.   Fig. 13(b) for the cases that the radar is calibrated, phase error, amplitude error, and coupling error impaired. The expected ghost target positions are shown as red lines.

A. SIMO-SAR MEASUREMENT
First, a SIMO-SAR measurement was performed by moving the radar to 30 positions separated by L sar = N r λ c 4 , thereby creating a uniformly sampled synthetic aperture of 23 cm. For the SIMO radar measurements, only the first TX was used. The reconstructed, fully calibrated 2D-SIMO-SAR image is shown in Fig. 13(a), clearly showing the reconstructed poles and the target cluster representing the car. To downgrade the calibration, first, a 20 • phase error was added to the main-diagonal entries such that for half of the antennas ϕ n r = 20 • and for the other half of the antennas ϕ n r = −20 • holds. Thereby, the worst-case condition, as discussed in Section V-B is fulfilled. The reconstructed image with additional phase errors is shown in Fig. 13(b). Here, additionally, the predicted angles based on (36) are plotted for each marked target in Fig. 12 as dotted lines as well as the arc of equal range as a dashed line. The comparison of the calibrated with the erroneous reconstruction demonstrates the deterministic formation of ghost targets due to the calibration errors, which occur at the predicted angles at the same range as the actual target. This verifies the theoretical findings in Section III. In the case of the car, Fig. 13(b) clearly illustrates how entire target clusters are repeated within the image leading to replicas of the entire car target pattern.
To further analyze the impact of different calibration errors on actual measurements, the azimuth profiles of both the calibrated and phase error impaired reconstruction along the dashed green through pole 1 are shown in Fig. 14 together with their corresponding SDRs and the expected artifact angles as red lines. An additional azimuth profile was created to demonstrate more error scenarios, where coupling errors are manually introduced by applying only the main diagonal of the RX calibration matrix during reconstruction. Furthermore, Fig. 14 also shows a reconstruction, in which a 15% amplitude error was added similarly to the phase errors before, such that for half of the array Finally, the SDRs are analyzed. Both amplitude and phase errors were added such that they fulfill the respective worstcase SDR conditions. In case of phase errors, the resulting SDR is 10.9 dB whereas the worst-case SDR was predicted via (56) to be SDR wc P = 8.8 dB. In case of amplitude errors the measured SDR is 17.1 dB compared to a predicted worst-case SDR via (54) of SDR wc A = 16.5 dB. Hence, both measured SDRs match the expected worst-case well and only show a slightly smaller distortion. However, as discussed above, some difference is also expected since the artifacts at high angles are slightly decreased due to FMCW range resolution. Hence, this demonstrates a good agreement of the true distortion power with the predictions. Furthermore, the SDR caused by non-compensated coupling in the used ULA is 21.6 dB, which closely matches the mean value 22.3 dB for a 8-element ULA patch antenna array with a Toeplitz structured coupling matrix, which was simulated in Fig. 11 and is very similar to the radar used within the measurements. The SDR is in the expected order, thereby demonstrating both the predictions' applicability and the justification of the Toeplitz assumption.

B. MIMO-SAR AND MIMO MEASUREMENT
Second, a MIMO-SAR measurement was performed to the same scene shown in Fig. 12. Here, only N r = 7 RX antennas were used to ensure uniform sampling during MIMO and MIMO-SAR evaluation. The radar was moved to 10 positions separated by L MIMO sar = N r N t λ c 4 , thereby creating a uniformly sampled synthetic aperture of 27 cm. To simulate calibration errors, again, ±20 • phase errors were added to both the TX and RX array during the reconstruction. The resulting reconstructed image is shown in Fig. 13(c). Even though the amplitude per artifact is smaller than in the SIMO-SAR measurement, it can be seen clearly that the artifact positions  match the predicted positions indicated by the dotted lines. In Section III-C, we predicted that sole errors on the RX array only lead to specific artifacts. For further analysis, azimuth profiles through pole 1 were reconstructed, where the phase errors were applied to only the RX and to both the TX and RX array, respectively. The results are shown in Fig. 15, where the solid red lines show the expected artifact positions, which are mainly influenced by the RX calibration error, and the dashed red lines show the remaining artifact positions. Again, the measurements closely match the predicted behavior, further validating this work's theoretical findings.
Finally, a single MIMO measurement was performed with phase errors added to the RX array while the TX array remained calibrated. Fig. 16 shows the resulting azimuth profiles. Again, most of the distortion power caused by the calibration error is accumulated at the predicted positions, demonstrating that the discussed effects also occur within single MIMO measurements with only 4 TX antennas and, hence, 4 repetitions of the RX array.
In summary, the measurements validated the predicted formation of artifacts at the expected positions in all discussed imaging setups SIMO-SAR, MIMO, and MIMO-SAR. Also, the good applicability of the proposed worst-case SDR estimations was demonstrated. Furthermore, for MIMO-SAR, the impact of a partial calibration on the TX side on different artifacts was demonstrated. The reconstructed images of a realistic scene in Fig. 13 showed the applicability of the theoretic findings also on complex scattering clusters, which may e.g. lead to the multiple detections of the same objects or road users even within elaborated detection algorithms in automotive applications.

VIII. CONCLUSION
In this paper, it was shown that calibration errors form periodic repetitions of the spatial spectrum in all imaging systems, which repeatedly evaluate the identical array. In the radar image, this causes the error power to be accumulated at few known positions, thereby creating artifacts whose amplitudes depend on the calibration errors. Hence, it is possible to predict and recognize calibration error-induced ghost targets during the radar operation without knowing the actual calibration errors. This work derived the necessary equation tool-set to describe these effects fully and provided means to estimate the worst-case distortion level for bounded errors of different error types. Thereby, it provides the groundwork for future research enabled by the findings in this work, which should mainly focus on ghost target identification, calibration quality monitoring, and offline and online calibration.

APPENDIX
In this appendix, it is shown, that the worst-case SDR in the case of coupling errors can be found by choosing all ψ wc n r to be ±ψ wc , while ψ wc n r = 0. Finding the worst-case corresponds to the finding the solution to the minimization problem from (65) where the identity cos 2 x + sin 2 x = 1 was used at a), while taking into account the possibly positive and negative square root. Next, we substitute x n r := cos ψ wc n r , while limiting x n r to be within [−1, 1]. This yields the equivalent problem Next, assuming that x n r = k holds for the solution of (71), it can be written as x n r = k.
Discarding all constant terms and negating the remaining term again leads to min x nr ± 1 − x 2 n r 2 subject to x n r ∈ [−1, 1] x n r = k.
Because of its quadratic form, the minimal possible value of the objective function is zero. Now, we assume that for all n r x n r = k holds. Then, (74) becomes zero by choosing pairwise the negative and positive root value, thereby providing a solution to the optimization, which does not depend on the value of k. Hence, re-substituting x n r = cos ψ wc n r shows that a worst-case solution occurs for any given k, when both ψ wc n r = ± arccos k and ψ wc n r = 0 hold. Finally, for a concise representation, we substitute ψ wc = arccos k, which allows to reduce the original optimization to min ψ wc 1 |c| + cos ψ wc 2 1 − cos 2 ψ wc . (75)