Principles of Modern Radar - Volume 2 1891121537

10.2 Space-Time Signal Representation 459MVMVDRPCIPDIPSDPRFPRIRD-STAPRFRR-STAPRxSNRSINRSLASTAPULAUDSFminimum varianceminimum variance distortionless responseprincipal components inversepost-detection integrationpower spectral densitypulse repetition frequencypulse repetition intervalreduced-dimension STAPradio frequencyreduced-rank STAPreceiversignal-to-noise ratiosignal-to-interference-plus-noise ratioside-looking arrayspace-time adaptive processinguniform linear arrayusable Doppler space fraction10.2 SPACE-TIME SIGNAL REPRESENTATIONIn this section we describe spatial, temporal, and space-time signal representation. SinceMTI radar is our primary concern, we consider temporal signal properties in the contextof slow-time (pulse-to-pulse) sampling. We begin by describing spatial sampling usinga multichannel array, along with the corresponding spatial filtering operation known asbeamforming. Thereafter, we discuss temporal sampling and Doppler processing as mathematicalextensions of the spatial case. We then extend the separate spatial and temporaldevelopments to describe simultaneous space-time signals.10.2.1 Spatial Sampling and BeamformingUsing an array of spatial sensors, the radar determines the direction of a propagating electromagneticwave by effectively measuring time difference of arrival as a phase variationacross the aperture [2,9]. Specifically, the array digitizes the voltage outputs of M channels,thereby gathering spatial information embodied in the relative phase of each of the measuredsignals. A collection of antenna elements, known as a subarray, feeds each receiverchannel. In our discussion we assume the following conditions: the far-field approximationapplies, indicating a planar, propagating wave; the signals are narrowband, suggesting thesignal bandwidth is a very small fraction of the signal carrier frequency; M equal sizesubarrays comprise the overall antenna array; and the array configuration exhibits uniformspacing with horizontal, linear orientation. Figure 10-1 depicts the corresponding scenarioof a propagating plane wave impinging on a uniform linear array (ULA).The signal direction of arrival (DOA), frequency, and physical location of a givensubarray influence the varying phase shift the propagating wave induces among each ofthe channels. Letting τ s,m represent the delay between the time it takes the wavefront toarrive at a suitably designated reference point and the m-th subarray and defining ω c asthe center frequency of the propagating signal in radians, the corresponding phase shift isγ s,m = τ s,m ω c (10.7)