WAVEFORMS FOR MIMO RADAR

4.5 WAVEFORMS FOR MIMO RADAR

We have seen how the choice of transmitted waveforms determines the characteristics of a MIMO radar by considering the correlation matrix, R_φ. The challenge of MIMO radar is to design families of waveforms that possess a desired correlation matrix. Also, note that the previous analysis considered only the zero-lag correlation matrix. By extending this analysis to consider the correlation matrix as a function of delay, we can study the behavior of the range sidelobes. An open research topic is to identify waveforms that enable the benefits of MIMO radar without unacceptably compromising other aspects of radar performance.

4.5.1 Classes of Waveforms for MIMO Radar

Three techniques exist to minimize the cross-correlation among a suite of waveforms, which correspond to exploiting orthogonality in time, frequency, and/or code.

Time Division Multiplexing (TDM) An obvious method to decorrelate waveforms is to simply transmit them at different times. This is possible in some applications, but in others it may not be compatible with requirements of the radar timeline. This will also increase data handling requirements since each transmitted waveform must be digitized separately.

Frequency Division Multiplexing (FDM) Another natural way to limit correlation be-tween waveforms is to offset them in frequency. The drawback of this is apparent in systems that rely on coherent processing across a number of frequencies. In such cases, the target reflectivities may vary if the chosen frequency is too large, limiting coherent gain. Also, coherent imaging techniques like SAR rely on each spatial sample having the same frequency support; otherwise, artifacts will be present in the image.

Code Division Multiplexing (CDM) In wireless communications, it is frequently nec-essary for a number of users to access a particular frequency band at the same time.

A common solution is to apply code division multiple access (CDMA) techniques.

Even though each user transmits at the same time and at the same frequency, each signal is modulated by a unique phase code.

Another approach for CDM is to employ noise like waveforms. These have the drawback that they may have a high peak-to-average power-ratio, which requires transmit amplifiers that are more linear and thus have lower gain.

The synthesis of waveforms that are reasonably orthogonal that possess desired side-lobe responses is a key challenge in the realization of a MIMO radar system. It is important to realize that different radar applications have different sidelobe requirements. For exam-ple, in moving target indication (MTI) radar, the peak sidelobe ratio is paramount. This is contrasted with applications where a continuum of scatterers is expected, as in SAR and weather radar, where the integrated sidelobe ratio tends to drive performance.

Some may argue that TDM and FDM are not properly MIMO waveforms. They are included here to emphasize that MIMO radar is a generalization of traditional radar systems. For example, a number of radars operating cooperatively using difference fre-quencies may be considered as a single MIMO radar system. The utility of developing a theory of MIMO radar is to discuss this and many other configurations using a common framework. Including these waveforms greatly expands the class of MIMO radars.

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4.5.2 MIMO Range Response

In the previous section, we described the angular point spread function of a MIMO radar, which described the sidelobes of the effective antenna pattern. Similar sidelobe structures will exist in the range domain, just as they do in traditional radar systems.

We extend the definition of the MIMO signal correlation matrix of (4.11) to R_φ(τ)=

 _∞

−∞(t)(t − τ)^H dt (4.33)

Note that our analysis of the spatial characteristics of a MIMO radar focused on the case whereτ = 0, the range bin of interest. This dependence on lag, τ, will allow us to characterize the influence a target at a particular range will have in bins up- and down-range.

The range response of a set of MIMO waveforms with correlation matrix R_φ(τ) is given by

h(t, θ; θ0) = a(θ0)^HR^∗_φ(t) a (θ)



a(θ0)^HR^∗_φ(0) a (θ0) (4.34) This result is extended to consider a Doppler offset in addition to a range offset in [15].

4.5.3 Example: Up- and Down-Chirp

Since many radar systems use linear frequency modulated (LFM) waveforms, also called an LFM chirp, a natural method for generating two approximately orthogonal waveforms is to use one chirp with increasing frequency and another with decreasing frequency.

Time-frequency representations describing a notional pair of such signals is shown in Figure 4-8. Note that these signals possess the same frequency support.

FIGURE 4-8

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−5 0 5

−100

−80

−60

−40

−20 0

Time (µs)

Power (dB)

−5 0 5

−100

−80

−60

−40

−20 0

Time (µs)

Power (dB)

Range Response Auto−Correlation Cross−Correlation

FIGURE 4-9 The correlation

properties and range response of the up-and down-chirp waveforms.

As shown in Figure 4-9, each waveform has a desirable auto-correlation, and the peak of the correlation is well below that of the auto-correlation. However, the cross-correlation does not decay as the delay offset increases. This is apparent in the range response, which for a broadside target is the sum of the auto- and cross-correlations, as stated in (4.34).

The resulting range response when using an up- and a down-chirp as MIMO wave-forms has the same range sidelobe structure near the peak as a single LFM, where the auto-correlation function dominates the MIMO range response. Instead of decaying to very low levels the contribution of the cross-correlation of the waveforms is apparent, even for relatively large delays.

From this analysis, we see that the simple case of an up- and down-chirp may provide acceptable peak sidelobe performance. Another figure of merit in waveform design is the integrated sidelobe level, which characterizes not simply the largest sidelobe but also includes the effect of all of the sidelobe energy. Clearly, the integrated sidelobe level is compromised by transmitting the second quasi-orthogonal LFM waveform.

This example of two waveforms was presented not as a recommendation for use in a MIMO radar but instead to present an example of the analysis that is required in choosing waveforms. We have seen previously that the zero-lag of the correlation matrix in (4.33),

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namely, R_φ(0), characterizes the antenna performance of a MIMO radar. Also, the struc-ture of the matrix R_φ(τ) for τ = 0 characterizes the range sidelobe performance, which is also critical to the operational utility of a radar system. The MIMO signal correlation matrix once again describes the capability of a set of waveforms to realize an effective MIMO radar.