Misspecification Under the Narrowband Assumption: A Cramér-Rao Bound Perspective

To efficiently extract estimates about the propagation behavior of electromagnetic waves in a radio environment it is common to invoke the narrowband-assumption. It essentially states that the relative bandwidth of the measurement system is so low that the frequency response of a single propagation path only depends on it Time-of-Flight and the response of the measurement device can be calibrated independently of the measured channel. Recent advances into higher relative bandwidths and antenna arrays with larger spatial aperture render this assumption less likely to be satisfied, which leads to a model mismatch during estimation. In this case estimates are inherently biased and have a special statistical behavior. This behavior can be captured by the so-called Misspecified Cramér-Rao Bound, which formulates a lower bound for the variance of estimates that are biased due to model mismatch. We analyze this bound in contrast to the traditional Cramér-Rao Bound and show the shortcomings in the setting of joint ToF-DoA estimation in the mmWave spectrum. The conducted numerical studies also show that planar array geometries inherently suffer from violation of the narrowband assumption irrespective of the individual elements' frequency response, whereas circular structures show it to a lesser degree.