IEEE - Aerospace and Electronic Systems - January 2020 - 39

Levanon

Figure 10.
(Top) True spectrum, after LPF, of an 8.2-Hz single tone; (Bottom) FFT outputs of 16 samples. Noise free case.

Figure 11.

(Top) Spectrum after LPF, of a noisy 8.2-Hz single tone; (Bottom) FFT outputs of the 16 samples. SNR ¼ -2 dB per sample.

estimation with better accuracy than the resolution
Dn ¼ Df ¼ 1=ðMT r Þ ¼ 1=CPI. In a high-SNR scene,
zero-padding the FFT can be used. Furthermore, if the
exact Doppler frequency is located between two nominal
frequencies, corresponding to two neighboring FFT outputs, the estimated frequency need not be the nominal frequency of the FFT cell whose output is the highest.
Interpolation can be used to estimate a more exact location
of the spectrum peak. A qualitative example is demonstrated in Figures 10 (noise-free) and 11 (SNR ¼ -2 dB,
per sample).
In the noise-free case (see Figure 10), interpolating the
lower subplot would have correctly placed the spectral
peak to the left of the 9.375 Hz cell. In the low-SNR case
(see Figure 11), interpolation would have erroneously
placed the peak at about half way between 9.375 and the
12.5 Hz cells.

PROCESSING HIGH-SNR SCENES
In rare radar scenes, where the signal-to-clutter plus
noise ratios are high, the concept of Doppler measurement accuracy (or precision) becomes relevant, including its dependence on SNR. By accuracy, we refer to
JANUARY 2020

the noise-induced error standard deviation. Mean
error, usually caused by other error sources, will be
ignored [7].
Returning to the Doppler shifted, noise-free, phase
evolution in Figure 5, our ad-hoc Doppler measuring concept will be based on measuring the phase of each pulse
(after compression). Assuming no change of the target's
range rate during the CPI, the Doppler frequency will be
derived from the slope of the unwrapped line connecting
the measured phases.

PHASE MEASUREMENT
The phase of a single compressed pulse will be determined
by tanÀ1 ðQp =Ip Þ, where Qp ; IP are the quadrature and inphase values of the sample corresponding to the peak of
the matched filter response (i.e., the correlation of the
received coded pulse and the noise-free coded reference).
In high-SNR scenes finding the closest sample to the peak
is rather simple if there are enough samples per code
elements.
Figure 12 (top) shows the complex envelope of the
Barker 13 signal used in the phase measurement example.
xs is the pre LPF waveform and ys is the post LPF waveform. Note that the complex envelope is real, having no

IEEE A&E SYSTEMS MAGAZINE

39



IEEE - Aerospace and Electronic Systems - January 2020

Table of Contents for the Digital Edition of IEEE - Aerospace and Electronic Systems - January 2020

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