In most pulsed radar applications, a Linear Frequency Modulated (LFM) signal is used, since it provides an optimal compromise between requirements such as range resolution or unique range-Doppler coupling (cf. Sect. 5). This waveform, also known as a chirp, is given in the base-band time domain by: 1sBB,chirp(t)=exp⁡jπKrt2⋅recttτp, where t is the time variable, the rect-function represents the pulse envelope of pulse duration τp and Kr is the modulation rate defined as 2Kr=±Bτp with the signal bandwidth B. A plot of a typical up-chirp with B=20 MHz and τp=10 µs (values have been chosen for ideal illustration purposes) in time-domain is given in Fig. a.

It is known that the Fourier transform of the chirp-signal-envelope is not rectangular due to ringing artifacts at the edges of the spectrum. These result from the Fourier series expansion and are known as the Gibbs phenomenon . In radar, they result in higher sidelobes of the impulse response after range compression. However, in a single-channel system these artifacts can be easily compensated by windowing during the pulse compression process. For multichannel radars, this is not possible because the discontinuities are also located within the waveform spectrum, not just at either end. Therefore, a more detailed analysis is required, which is given in Sect. 5. The Fourier transform of a chirp signal can be expressed as 3SBB,chirp(f)=FsBB,chirp(t)=12π∫sBB,chirp(t)⋅exp⁡-j2πftdt which is plotted in Fig. b for B=20 MHz and τp=10 µs.

All of the following suggested MIMO waveforms have this chirp-like structure.

The requirements for MIMO radar and MIMO SAR are quite different. In conventional MIMO radar applications, a stringent separation of the waveforms is often not necessary because they are typically used to detect a few point-like targets . When using signal codes, e.g. Barker codes, a part of the energy is smeared over the scene after pulse compression, but this energy does not disappear. In most radar applications to detect specific targets with a sufficiently low false alarm rate, these codes are adequate . This is because the interference is very small compared to the optimal impulse response, and the interfering energy can be shifted to spatial locations where no targets are present . This concept can also be adopted for more advanced coding and multiplexing that achieve orthogonality in the discrete frequency domain . For SAR, as a typical example of an imaging radar, orthogonality must be achieved with previously known waveforms for each delay and for the imaged scenario consisting of a multitude of point and distributed targets (e.g. rain forests or crop fields). This is because the unwanted, correlating energy from the other waveforms will add up for every single point target in the observed area after coherent combination. Even when using existing orthogonal codes, the interfering energy smears to the positions of other targets and cannot be resolved. The echoes of the different MIMO transmit signals will completely overlap in the channel and it will be impossible to separate them later in the receiver. Therefore, it is desirable to achieve full orthogonality for arbitrary shifts τ between the signals sk(t) and si(t) : 4∫sk(t)⋅si*(t-τ)dt=0∀τ∈R,k,i∈N0+ and k≠i, where sk(t) and si(t) denote two arbitrary waveforms. In any case, this condition simply cannot be achieved with waveforms occupying the same frequency band. This is justified by the fact that in direct analogy to the convolution theorem, there is always a correlation between two signals if they share the same spectrum. The only possible solution is to merge several filtering methods, as proposed in , or . They claim and prove that filtering not only in the time-frequency domain but also in the spatial/angular domain via beam-forming allows perfect orthogonal conditions.

In the past, the use of alternating or even simultaneous up- and down-chirp signals as a kind of orthogonal waveform has been extensively discussed. This modulation concept was examined in more detail in . Limited to a maximum number of only NTx=2 transmit channels, this type of waveform does not lead to the desired results even for small observation areas. The reason for this can be seen in the plot of the frequency ramps (Fig. ). s0(t) and s1(t) already intersect in the time-frequency plane for a single point target – at a certain point of time, which inevitably leads to cross-correlation interference. Each additional time shift caused by additional point targets results in even more interference. For a SAR with homogeneous extended targets (e.g. rain forests or deserts), the SNR will tend to zero, as proven in .

With the recently described knowledge, we will omit this waveform type from further analysis and acknowledge it for completeness.

OFDM is generally a multicarrier modulation technique. The individual transmit signals occupy the same frequency spectrum, while orthogonality is achieved by periodically interleaving sub-carriers. The OFDM chirp signals were introduced in . In his subsequent publications, this multiplexing technique was further investigated . Originally defined for two transmit channels only, it is straight forward to extend the scheme to more channels, as will be shown next.

For MIMO-SAR, the OFDM technique is applied to a standard chirp signal, where the total signal duration is τp. In contrast to OFDM in communications, where the entire spectrum is occupied at each instance of time, in this context the individual sub-carriers are swept in time due to the chirp-like waveform characteristic. Another difference from classical OFDM is the constant signal envelope with a rectangular shape in the time domain due to the pulse-like character. While a standard chirp occupies the entire frequency spectrum B, the OFDM chirps share it with an interleaved behavior. An example for two transmit signals is shown in Fig. . The left side shows the orthogonal sub-carrier scheme for NTx=2 and the right side shows the corresponding time-frequency representation. It can be seen that there is a frequency offset of Δf. This so-called sub-carrier offset results in the desired spectral shift of the second waveform to achieve the interleaved carrier structure. Due to the DFT-typical circular time shift property, the suppression of each second discrete frequency sample results in a repetition of the waveform in the time-domain.

With reference to Eq. () an extension of the original definition of the OFDM chirp waveforms from to an arbitrary number of NTx leads to: 5sBB,k(t)=A⋅∑m=0NTx-1exp⁡jπKrNTxt+τp2-τp(1+2m)2NTx2⋅rectNTx(2t+τp)-τp(1+2m)2τp⋅exp⁡j2πΔfkt. A is the amplitude, which is the same for each transmit waveform and m is the index of the repeated sub-chirp. The last exponential function in Eq. () results in the sub-carrier offset of the kth transmit signal. All other terms are equal to the standard chirp except for a shift along the time axis for the signal repetition. The sub-carrier offset Δf depends on the selected number of sub-carriers Nc and the pulse duration τp: 6Δf=1Ncτp. Nc is the product of the sampling frequency fS, which must satisfy the sampling theorem and the signal duration

Oversampling is not considered in favor of understanding.

The Multiple Sub-Pulse (MSP) mode described in , , is a mode where multiple chirp-signals of the same length occupying the same frequency band are transmitted (Fig. ). The individual sub-pulses are transmitted within the same pulse repetition interval and form a single unit. Due to the high time-spread of the propagation channel, the sub-pulses overlap during propagation and must be separated using MIMO-techniques.

The main changes of a sub-pulse in direct comparison to the single linearly frequency modulated chirp waveform of Eq. () are the value of the modulation rate, the reduction of the total pulse duration to τsub and a shift on the time-axis to the individual sub-pulse center tc,k. The common chirp-like characteristic is still preserved. An illustrative MSP signal with the same parameters as for the OFDM chirp signals (B=10 MHz, τp=10 µs, NTx=2, f0=B/2) and a gap of τs=2 µs is shown in the time-domain in Fig. , while Fig.  shows the frequency ramps with continuous and equal sweep rate for each sub-pulse.

To have a common basis for later waveform comparison, τp is set equal for the total signal duration of all signals introduced in this paper. As mentioned before, in the MSP mode, the duration of each sub-pulse is reduced to τsub, whith positive or negative gaps of value τs between them. The sub-pulse duration, which is the NTxth fraction of τp minus the sum of all gaps is 8τsub=τp-(NTx-1)τsNTx. Since all sub-pulses have the full signal bandwidth to preserve the range resolution, the chirp must sweep within the shorter time-interval of τsub and the modulation rate of Eq. () must be redefined to a higher chirp-rate: 9Kr,sub=±Bτsub=±B⋅NTxτp-(NTx-1)τs. For completeness, the shift of the sub-pulses along the time-axis and thus the center of the kth sub-pulse can be expressed by 10tc,k=-τp2+τsub2+kτsub+kτs. The final description of the kth sub-pulse reads as follows: 11sBB,k(t)=A⋅exp⁡jπKr,sub(t-tc,k)2⋅rectt-tc,kτsub.

Another promising waveform for MIMO-SAR was proposed in , later refined in and first verified by an experiment in . In the following we will call it the Segmentally-Shifted Chirp (SSC) waveform. Basically, it consists of shifted chirp waveforms with the same modulation rate but different sub-pulse durations and bandwidths. Each two sub-pulses are combined into a single transmit signal of full bandwidth B and full pulse duration τp. To give an example in Fig. , the SSC signal is plotted in the time-domain for B=10 MHz, τp=10 µs, NTx=2 and f0=B/2. The upper plot shows the first waveform, which consists of segments of the first waveform within τp that have been repositioned. In the absence of the first transmit signal, each waveform consists of two sub-pulses, using the full bandwidth and full pulse duration.

In order to satisfy the orthogonality condition, the sub-chirps follow a certain sequence, which can be clearly seen in the time-frequency plots in Fig. a and b for two and in Fig. c for four activated transmit channels. While the modulation rate is the same for each waveform as defined in Eq. (), we need two sub-pulse centers tc1,k and tc2,k for each split sub-pulse. They are defined as 12tc1,k=kτp2NTx and 13tc2,k=-(NTx-k)τp2NTx. The duration of the sub-chirps are 14τsub1,k=(NTx-k)τpNTx and 15τsub2,k=kτpNTx. It becomes immediately obvious that τsub1,k+τsub2,k=τp. Using Eqs. ()–() for the standard chirp signal, the SSC waveforms can be written in a closed expression as 16sBB,k(t)=A⋅exp⁡jπKrt+kτpNTx2⋅exp⁡-jπBkNTxt+kτpNTx⋅rectNTxt+kτp(NTx-k)τp+A⋅exp⁡jπKrt-(NTx-k)τpNTx2⋅exp⁡jπB(NTx-k)NTxt-(NTx-k)τpNTx⋅rectNTxt-(NTx-k)τpkτp for k>0. For k=0 Eq. () must be taken, since the first transmit waveform is the standard chirp waveform, with full bandwidth and full pulse duration.

The chirp diverse waveforms were proposed by Wang and published in , , . In this OFDM-based technique, multiple short sub-pulses are generated and modulated on individual sub-carriers. The number of carriers is smaller than in classical OFDM signals, but usually larger than the number of transmit channels. Furthermore, the number of simultaneously active sub-carriers is independent of the number of sub-pulses, and up- and down-chirp modulation can be used in parallel. As suggested in , a random matrix modulation concept helps to optimize the auto- and cross-correlation interference between the waveforms. A set of many short sub-chirp signals shifted in the time-frequency plane is generated. An example with NTx=2 is shown in the time-frequency plane in Fig. .

Simulation results with narrow swath widths show in that this waveform type is promising for MIMO-SAR. However, it was shown in , that this multiplex concept does not work for distributed scenarios with millions of scatterers. As it can be seen from the time-frequency plots in Fig. , even small time shifts of s0 and s1 lead to overlapping of sub-chirps and thus to cross-correlation interference.

First, we focus on a single waveform with optimal characteristics for radar. The basic auto-correlation of a single waveform can then be represented graphically using Woodward's AF . The ambiguity function of the kth waveform is defined as the squared magnitude of 17χk(τ,fν)=∫sk(t)⋅sk*(t-τ)ej2πfνtdt. It basically reflects the normalized matched filter at different combinations of range delays τ and Doppler shifts fν. In the typical 3D surface plot representation of the AF, these two parameters correspond to the axes, while the magnitude is given by χk2. This gives an idea of how the radar system responds to targets at different ranges and velocities. For a standard radar chirp signal with B = 300 MHz and a pulse duration of τp = 120 µs, the ridge in the ambiguity function of Fig.  represents the area of maximum response. The width and shape of the ridge provide insight into the resolution of the radar in both the range and velocity domains, with a narrower ridge indicating better resolution. More specifically, the AF plot in the center at fν=0 Hz along the τ axis shows the impulse response of the waveform with its 3 dB width as range resolution.

In recent years, a number of MIMO radar Ambiguity Functions have been derived. They focus on different properties of the radar and are therefore not directly comparable. In 2007, developed a very good evaluation of these functions and introduced a generalized MIMO AF, which was later further analyzed by Chen . San Antonio showed that the MIMO Ambiguity Function depends not only on the signal itself, but also on the radar system, the beam-forming algorithm used, and the target. Basically, the MIMO AF shows that the unwanted correlated signal energy can also be suppressed by proper beam-forming. This equation takes into account all the parameters of the radar system, including the element spacing, the antenna beam patterns, the channel characteristics and the backscattering characteristics of the targets. The function presented in depends on 16 independent parameters.

Because several factors are involved, an analytical solution of this function is not readily possible. For this reason, Chen defined in a simplified form, also known as the Cross Ambiguity Function, which focuses on the fundamental correlation between multiple transmit waveforms. For the derivation, Chen introduced constraints that are valid for most MIMO radar systems:

No analog beam-forming/no analog spatial filtering

Figure  shows the plots of the Cross Ambiguity Functions for the OFDM chirp signals, the MSP mode, the SSC waveforms and for the Chirp Diverse waveforms for NTx=2. Each set of waveforms has a total bandwidth of 300 MHz and a total pulse duration of 120 µs. An important extraction from the three-dimensional AF plots are the zero-Doppler cuts, which express the impulse response for a point target in a static scene (fν=0 Hz). They can be found on the right side of Fig. . It can be clearly seen that the desired (main-lobe) in the center has exactly the same shape as for a standard up-chirp, except for the Chirp Diverse waveforms. The ambiguities resulting from cross-correlation with the second transmit waveform are concentrated on dedicated, localized ridges for the three waveform types. This allows removal of this unwanted energy by time-varying beam-forming in post-processing. The AF of the Chirp Diverse waveform shows pronounced interference that is randomly distributed throughout the entire time-frequency plane.

A noticeable artifact in the OFDM AF plot is that the ridge in the center is not completely filled due to the interleaving behavior of the sub-carriers. The repetitive structure of each OFDM chirp also introduces auto-correlation interference, which are superimposed on the cross-correlation interference. However, due to the sub-carrier spacing, the waveforms can be separated. Since only half of the full waveform is correlated at each each ambiguous ridge, only half of the signal energy is focused on such an ambiguity. In addition, each of the repeated sub-pulses implies the full signal bandwidth. The width of the ridge, and therefore the resolution, is also the same.

Basically, the MSP mode is just a sequence of repeated chirp waveforms. As a result, the signal ambiguities have the same energy and resolution as the main impulse response. The result is an exact replication of the main impulse response.

With the exception of the first waveform, all other waveforms in the SSC waveform mode consist of two shifted sub-pulses. This results in two ambiguities for each subsequent transmitted signal. Each ambiguity shares half of the signal energy and half of the total bandwidth. This means that the signal ambiguities to be suppressed have half the amplitude and half the resolution of the auto-correlation. It is worth noting that in the later application, due to the projection of the signal onto the spherical Earth ground, the positions of the signal ambiguities are not symmetrical, leading to a more complicated pattern synthesis for DBF suppression.

Because the Chirp Diverse waveform typically consists of several short sub-chirps with overlapping spectra, certain sub-pulses will correlate at certain time or frequency shifts. This is also the case for a single transmit waveform. Accordingly, the energy of both waveforms is smeared in the AF plot. This fact alone makes this waveform type inapplicable for MIMO-SAR.

An advantage of the OFDM, MSP and SSC waveforms is that they all have the same requirements for additional filtering because the separation between the signal ambiguities and the main response is the same. As already mentioned by and , the energy in the auto- and cross-correlation interferences can be suppressed by DBF. Using a priori knowledge of the scene geometry, antennas and waveforms, appropriate antenna patterns can be synthesized during post-processing to enable MIMO-SAR without interference from the radar's own waveforms.

By definition, resolution is the ability to detect two identical objects separated by a minimum distance δrg and to resolve them from each other . The range resolution is along the radial direction from the radar, while the azimuth and elevation angles are constant. To separate the backscattered signals from these two targets, it is mandatory to separate at least the two leading edges of the pulses from each other, since the radar signals are usually modulated (also called intrapulse modulation). The maximum achievable range resolution of a radar is defined as 21δrg=wwin⋅c2B, where wwin is a scalar, characteristic value for the window function used during the range compression process. Common values are wwin=0.886 for a rectangular window and wwin=1.3 for a Hann window . This maximum resolution can only be achieved if the transmit signal envelope is perfectly rectangular, which is the case for all four proposed waveform types (OFDM, MSP, SSC, and Chirp Diverse waveforms). In all other cases, any variations in the pulse shape will result in reduced resolution.

Instead of using Eq. (), it is also possible to determine the resolution from the 3 dB-width of the impulse response, which corresponds to the centered sinc-function in the zero-Doppler cuts of the AF (Fig. ). Both methods give the same result. Simulations for different numbers of Tx waveforms NTx have shown that each of the previously defined waveforms shares the full bandwidth and therefore has the same resolution characteristics.

In this subsection, we analyze the effects on the Integrated Sidelobe Ratio (ISLR). By definition, the ISLR is an indicator of the amount of energy in the sidelobes of the impulse response relative to the amount of energy in the main-lobe of the impulse response . Thus, a low ISLR is preferred. In Fig.  both amounts of energy are marked in green and blue, respectively. Typically, the energy in the side-lobes is not fully accounted for. While the impulse response function has a total width of τp after pulse compression, the ISLR uses only the time segment ΔT. Following the definition of the ISLR , for a given observation time ΔT we get: 22ISLR(ΔT)=∫-ΔT/2-TNBW/2χk′(τ,0)dτ+∫TNBW/2ΔT/2χk′(τ,0)dτ∫-TNBW/2TNBW/2χk′(τ,0)dτ∀TNBW≤ΔT≤τp, where TNBW is the null-to-null-beamwidth and approximately twice the 3dB-width of the impulse response.

A typical value for the impulse response of a chirp signal is ISLR = -9.8 dB. The RCS of different targets within a standard SAR scene will usually exceed this value, and sidelobes from a strong scatterer can blur the rest of the image. Therefore, it is recommended to apply windowing to overcome the steep slopes at the edges of the chirp spectrum. For example, Hann windowing can significantly reduce the ISLR to about -21 dB, while degrading the resolution by a factor of wwin (see Eq. ). Windowing affects the ISLR mainly for small values of ΔT. Especially for MIMO-waveforms, where ambiguities naturally occur, for larger ΔT (ΔT>τp/NTx) the cross- and auto-correlation interferences lead to significant contributions to the ISLR. Since they are typically suppressed by DBF and null-steering techniques during post-processing, analyzing the full range of ΔT is mainly useful to study the waveforms, but not for a fair comparison.

For a meaningful analysis, the ISLR was calculated for different values of ΔT, NTx and window types. To have a common base, the following simulation parameters were used: B=300 MHz, τp=120 µs and NTx=4. The results can be found in Fig.  and Table  and general observations can be made. Across all plots in Fig. , the ISLR increases in a stepwise manner. As indicated by the zero-Doppler cuts of the AF, observation times outside the interval -τpNTx,τpNTx produce strong ambiguities, which cause the ISLR value jumps shown in Fig. . For sure, after the first step at τp/NTx the ISLR must always be higher than 0 dB, because not only the full energy of the second transmit waveform applies, but also the energy of the side-lobes of all remaining signals.

The maximum available energy is mainly limited by the instrument resources, the duty cycle, and the gain and efficiency of the amplifier used in the transmitter. The energy of a signal sk(t) is defined as: 23Ek=∫-∞∞|sk(t)|2dt, which in turn is the auto-correlation (zero lag) of the transmit signal function. In particular, for the signal types mentioned above, the energy is independent of the waveform index k, because in a set of waveforms s(t), each chirp has the same pulse duration. To have a common basis for later illustrative case studies, we define that the total transmit signal energy and the pulse duration τp are the same for each waveform type. Consequently, the signal amplitude must be changed for each case.

Beyond the waveform perspective, system-level aspects must also be considered. In general, the available antenna aperture in MIMO-SAR configurations, especially those employing spatial multiplexing, is typically divided among the transmit sub-apertures (NTx channels). This reduces the effective radiated energy per channel compared to an equivalent SISO or SIMO system. This effect must be considered when comparing the overall system performance. However, the actual energy distribution depends fundamentally on the specific hardware implementation. For example, hardware architectures with separate or shared Transmit-Receive (T/R) modules for each waveform channel. Considering fully polarimetric MIMO-SAR configurations, the two orthogonal waveforms must pass through individual T/R modules. They can then be fed to two orthogonal ports of a single dual-polarized antenna or to two distinct antennas.

In SAR and Ground Moving Target Indication (GMTI), it is necessary to have a unique relationship between time and Doppler frequency. Otherwise, after range and azimuth compression, targets would appear multiple times as ambiguities in the SAR image. An approximation for the Doppler bandwidth of a SAR sensor is : 27ΔfDop≈vEδsa. A low Earth orbit satellite like TerraSAR-X has a ground velocity of about vE=7000 m/s. Assuming an azimuth resolution of 1.0 m, the Doppler bandwidth will be ΔfDop≈7.0 kHz. Typical moving targets (cars, ships, …) obviously cause much smaller Doppler frequency shifts. This means that the waveforms must be orthogonal at least for ΔfDop.

Nowadays, generating arbitrary waveforms with digital hardware is not a problem, but post-processing on-board the satellite is a demanding issue. Due to the limited data rate from the satellite to the ground station, it is necessary to pre-process the received multi-channel raw-data on-board. While the MSP and SSC waveforms can be processed in real-time on-board the satellite using the algorithm proposed in , the OFDM chirp signals require a complex signal separation processing . The processing strategy proposed by Wang for the Chirp Diverse waveforms is a simple range compression with the corresponding transmit signal as in typical SAR. It turns out that this is a significant advantage of this waveform type, although it has many other drawbacks.