Ambiguity Function Analysis of the Frequency-Hopped Code Selection Scheme

To alleviate contention in the frequency spectrum, dual-function radar communications (DFRC) strategies modulate the transmitted radar waveform to embed communication symbols. The frequency-hopping code selection (FHCS) scheme has been proposed as an effective DFRC approach for frequency-hopping multiple-input multiple-output (FH-MIMO) radar. By encoding the symbols via the selection from a set of permissible FH codes, the FHCS scheme embeds information in the radar fast-time, thus increasing the achievable bit rate. The resulting impact on the radar performance, however, demands analysis of the average ambiguity function (AF) resulting from the complete FHCS codebook, which quickly becomes challenging as the number of hops increases. In this paper, we formulate a simplified expression for the average AF that avoids the need to enumerate the symbol dictionary, and can evaluate the zero-Doppler AF profile for any scenario. To this end, we define a new measure in the form of a similarity function which enables the calculation of the average AF across the FHCS codebook. Through numerical examples, we show that the expression for the simplified AF successfully captures the zero-Doppler AF profile and gives much needed insight to how the FHCS codebook impacts the radar performance.