- Research Article
- Open Access
Low-Complexity Design of Frequency-Hopping Codes for MIMO Radar for Arbitrary Doppler
© Badrinath S. et al. 2010
- Received: 8 February 2010
- Accepted: 30 September 2010
- Published: 7 October 2010
There has been a recent interest in the application of Multiple-Input Multiple-Output (MIMO) communication concepts to radars. Recent literature discusses optimization of orthogonal frequency-hopping waveforms for MIMO radars, based on a newly formulated MIMO ambiguity function. Existing literature however makes the assumption of small target Doppler. We first extend the scope of this ambiguity function to large values of target Doppler. We introduce the concept of hit-matrix in the MIMO context, which is based on the hit-array, which has been used extensively in the context of frequency-hopping waveforms for phased-array radars. We then propose new methods to obtain near optimal waveforms in both the large and small Doppler scenarios. Under the large Doppler scenario, we propose the use of a cost function based on the hit-matrix which offers a significantly lower computational complexity as compared to an ambiguity based cost function, with no loss in code performance. In the small Doppler scenario, we present an algorithm for directly designing the waveform from certain properties of the ambiguity function in conjunction with the hit-matrix. Finally, we introduce "weighted optimization" wherein we mask the cost function used in the heuristic search algorithm to reflect the properties of the required ambiguity function.
- Cost Function
- Simulated Annealing
- Ambiguity Function
- MIMO Radar
MIMO radar is a recent evolution of radar that utilizes multiple transmitters and receivers [1, 2]. MIMO radar waveforms can have any degree of coherence with each other, ranging from complete coherence (in which case it is equivalent to a phased-array radar) to complete incoherence (orthogonality). The choice of radar waveforms  plays an important role in the resolution characteristics of the radar. The optimization of radar waveforms for the phased-array radar, which is also viewed as single-input multiple-output (SIMO) radar, focuses on obtaining a desirable ambiguity function in terms of range and Doppler resolutions. On the other hand, MIMO radars provide spatial resolution and spatial diversity in addition to range and Doppler resolution.
Frequency-hopping codes have been used in pulse compression radars  because of their highly desirable ambiguity properties. The design of frequency-hopping codes for SIMO radars to obtain desired ambiguity functions  has been well studied. A method to design discrete frequency-coding waveforms for the netted radar has been proposed in  and a modified Genetic algorithm to design orthogonal discrete frequency-coding waveforms for MIMO radar has been presented in . The hit-array [8, 9] has also been extensively used for waveform design in the SIMO context.
Recently, Chen and Vaidyanathan [10, 11] have dealt with the design of frequency-hopping codes for MIMO case based on the optimization of a newly formulated MIMO ambiguity function . Their approach to designing radar waveforms is to first parameterize these waveforms and then apply simulated annealing to find a near-optimal set of parameters using a "cost function" that allows comparison of different parameter sets. The formulation assumes small target Doppler resulting in a simpler cost function. This formulation, however, is inapplicable in the presence of large target Doppler.
In this paper, we first extend the scope of this ambiguity function to large values of target Doppler, and then introduce the concept of hit-matrix in the MIMO context. Next, we propose new methods to obtain optimal waveforms in both the large Doppler and small Doppler scenarios. In the case of large Doppler, we propose a cost function based on the hit-matrix which offers a significantly lower computational complexity as compared to an ambiguity-based cost function, with no loss in code performance. In the small Doppler case, we present an algorithm for directly designing the waveform from certain properties which can be obtained from the ambiguity function in conjunction with the hit-matrix. Finally, we introduce a "weighted optimization" wherein we weight the cost function, used in the heuristic search algorithm, to reflect the properties of the required ambiguity function.
In Section 2, we present our model for MIMO radar and frequency-hopping waveforms. In Section 3, we reformulate Chen and Vaidyanathan's MIMO ambiguity function such that it is applicable to any general value of Doppler. In Section 4, we describe the use of hit-matrix as an optimization tool. The hit-matrix corresponds to a digitized version of the ambiguity function which is relatively simple to compute. In Section 5, using a hit-matrix-based cost function, we use simulated annealing to search for the best frequency-hopping codes under the large Doppler scenario. In Section 6, we present an algorithm for directly obtaining a waveform corresponding to a good ambiguity function in the small Doppler case. In Section 7, we propose a method to obtain waveforms satisfying certain conditions on the ambiguity function. A heuristic search performed using a weighted cost function, with the weights representing conditions on ambiguity function, illustrates our proposed method. Section 8 concludes this paper.
2.1. MIMO Radar Model
2.2. Frequency-Hopping Waveforms
The pulse spacing vector plays a role in shaping the Doppler resolution of the waveforms. In this paper, however, we will not be dealing with optimization of this vector.
In the subsequent sections, we do not assume small Doppler and work with (17). We next describe the hit-matrix formalism in Section 4 and waveform optimization for the large Doppler case in Section 5.
Ambiguity Function and the Hit-Matrix
It can be shown that the height of the peak in at is constant, and hence this cost function favors codes that have their sidelobes flattened out over the delay, Doppler and spatial frequency dimensions. Increasing the value of increases the penalty on higher sidelobes. can be evaluated for each code using Riemann sums with a sufficient number of bins instead of integrations. However, this results in a high computational complexity which increases as , where is the time-bandwidth product of the code.
which favors the codes that have their sidelobes flattened out over the delay, Doppler and spatial frequency dimensions. Evaluation of is far less computationally intensive than , and increases only as . This allows the heuristic search algorithms using this cost function to rapidly traverse the code space, thereby allowing good codes to be found faster.
Randomly draw a code matrix from such that the code is orthogonal, that is, for .
Randomly draw from .
- (3)Set , and repeat steps 3(a) to 3(c) times.
Randomly draw from and from .
Select from with .
Set . (At this point, we will have the original code , and obtained from the above steps.)
Randomly draw from .
If , then set .
Set and .
If a sufficiently small value of has been obtained, terminate the algorithm. Otherwise, return to step 2.
Consider as an example the optimization of frequency-hopping codes for . The total number of possible codes of this size is . We found the number of Riemann sum bins, required for reasonable accuracy in the evaluation of for this code size, to be 100 each along the delay and Doppler dimensions, and 20 each along the two spatial frequency dimensions. Hence, while requires the computation and summation of terms per iteration, only needs 576 computations per iteration, resulting in a significant decrease in complexity.
5.1. Simulation Results
We present a number of examples to demonstrate the effectiveness of the hit-matrix-based cost function in designing good frequency-hopping codes. We have generated codes of various sizes by simulated annealing using both and at . The code parameters used were , and for each value of . Other parameters used were , and . For simulated annealing, we set the parameters , , and .
The above expression is independent of the code matrix, and thus the ambiguity function in this form cannot be optimized along the spatial frequency dimensions at .
. Noting that each peak, given by , corresponds to a hit which is reflected in (34), and the ambiguity function consists of a weighted sum of hits with the weights being of the form , optimizing the hit-matrix is analogous to optimizing the upper bound on the ambiguity function along the spatial frequency dimensions.
Note from (38) that the terms (corresponding to the waveform pulses), which are contained in , do not affect the Doppler resolution. The objective behind waveform design is to obtain a set of waveforms with a desirable MIMO ambiguity function. In  a heuristic search (simulated annealing) is performed over the space of all code-matrices, to acquire a code C which minimizes the cost function.
In the next subsection, we explain our proposed algorithm.
The Hit-Matrix under the Small Doppler Assumption
and refers to the Kronecker delta function.
- (1)Condition A. Within the constraints defining the code matrix ( ), the sidelobe levels must be reduced. This means that the total number of hits, given by
should be made as small as possible.
- (2)Condition B. A key aspect of waveform design is to achieve high mainlobe to peak sidelobe ratio in its ambiguity function. This is reflected in . We know from the previous condition that the sum total of all the elements of over all values of equals . Also, is constant and is equal to . Our objective, therefore, is to spread out the remaining elements corresponding to peaks over the summation terms of (excluding ) thereby minimizing the peak sidelobe. This can be expressed in compact form as follows
6.1. Waveform Design
In this section, we propose a direct waveform design algorithm that yields codes that satisfy both Conditions A and B described above. This algorithm requires the number of frequencies to be constrained to (the consequences of such a restriction are discussed in the next subsection).
Since and , we can write . Hence, minimizing for a given value of is equivalent to minimizing . Now, from (43) we note that if the pair of entries and in the code matrix are equal, they will contribute a value of 1 (or one "hit") to at . Thus we can say that will equal the number of pair of entries with identical values that can be found in the code matrix.
Using this alternate interpretation of , we proceed to show that under the constraint , will be minimized only when each of the usable frequency values occur exactly twice in the code matrix. To see this, consider an example with , which means that the code matrix has 6 entries and 3 frequency choices are available. Let , and correspond to the 3 frequencies. If we choose each of these three frequencies twice, that is, we make the selection , the possible pairs of entries with identical frequencies will be , and , thus giving . Suppose we pick , the number of possible same-frequency pairs will be four: one pair and three pairs giving , one higher than the case where each frequency was used twice. Other combinations can also be shown to yield a higher value of . A similar argument can be extended to other values of and as well.
which implies that all the values of for must equal either or .
6.2. Related Discussion
Values of for which codes can be designed with the proposed algorithm ( and ).
6.3. Simulation Results
where is the weight applied to th element of (see (20)). We now discuss possible applications of such a weighted cost function.
In this paper, we have shown how the MIMO radar ambiguity function for orthogonal frequency-hopping waveforms can be extended to general values of Doppler. We have also presented the hit-matrix as an analysis tool for these waveforms. To enable the optimization of these waveforms under the large Doppler scenario using simulated annealing, cost functions have been presented based on the ambiguity function as well as the hit-matrix. The codes obtained using both cost functions are shown to have similar performance based on their ECDF curves. The hit-matrix-based cost function has a significantly lower computational complexity, and can be useful when searching for codes with high time-bandwidth products, where using a ambiguity-based cost function is infeasible.
Under the small Doppler scenario, an algorithm has been proposed which directly computes the code matrix of a given size. It has been shown to perform as well as the heuristic search proposed by Chen and Vaidyanathan . The use of weighted cost functions to optimize the ambiguity function within a subregion of the delay-Doppler space has also been explored. This method of "Weighted Optimization" also addresses the problem of waveform design for intermediate Doppler.
The authors gratefully acknowledge the useful comments and suggestions of the reviewers which improved the clarity of the paper. One of the authors, V. U. Reddy, wishes to acknowledge the partial support he received from CR Rao Advanced Institute of Mathematics, Statistics and Computer Science for this work.
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