Page 389 - Fundamentals of Radar Signal Processing

Many other interpolation-based estimators have been proposed for the
DFT; several are described in Jacobsen and Kootsookos (2007) and MacLeod
(1998). They include versions of Eqs. (5.96) and (5.97) that use the complex

DFT data Y[k] instead of its magnitude and achieve significantly better
frequency estimation accuracy in noise-free data, as well as adjustments to the
weighting coefficients in either the complex or magnitude version which
improve accuracy when a window is used on the data. Another family of
interpolators uses the apparent peak and the larger of its two neighbors to

compute Δk as

(5.98)

where Y[k ] is the larger of Y[k – 1] and Y[k + 1]. While this estimator uses
0
0
±
only two DFT values explicitly, it implicitly uses three since two must be
examined to determine which is Y[k ]. Complex and magnitude-only versions of
±
this estimator also exist. Its frequency estimation performance is surprisingly
good in the absence of noise.
These interpolation techniques are not limited to Doppler frequency
estimation only. The same issues of sampling density and straddle loss arise at
the output of the matched filter in fast time, for instance, and the same

interpolation techniques can be applied to improve the range estimates. The
application to time delay (range) estimation, along with simulation results for
this and alternative techniques, is discussed in Chap. 7.
The above results are all for noise-free data, an unrealistic assumption.
Estimation accuracy and the effect of interpolation algorithms will be revisited
more formally in Chap. 7. It will be seen there the two-point interpolator is of
little value in the presence of noise, but that the quadratic interpolator is still
effective at high SNR. Other effects unrelated to interpolation dominate the

estimation precision at mid to low SNRs.

5.3.7 Modern Spectral Estimation in Pulse Doppler Processing
So far, the DFT has been used exclusively to compute the spectral estimates
needed for pulse Doppler processing. Other spectral estimators can be used.
One that has been applied to radar is the autoregressive (AR) model, which

models the actual spectrum Y(ω) of the slow-time signal with a spectrum of the
form