Doppler Analysis

NLFM waveforms are less Doppler tolerant [24–26] than their LFM counterparts. NLFM waveforms exhibit high near-in sidelobes in the presence of small, uncompensated, frac-tional Doppler shifts (i.e., f_d/β). Doppler shifts on the order of a 1/4 cycle across the uncompressed pulse width are sufficient to significantly elevate the near-in sidelobes. Fig-ure 2-21 and 22 contain plots of the compressed response with a 1/4 and full cycle of Doppler, respectively. The Doppler shifts used in this example (0.25 MHz and 1 MHz, respectively) are not commonly realized, but several cycles of Doppler may be observed when employing longer pulses (e.g., 1 msec) operating against fast moving targets. In this example, the peak sidelobes have increased from−40 dB to approximately −28 dB and –15 dB, respectively. As with the LFM waveform, a loss in the peak amplitude and resolution are experienced with increasing Doppler shift. Johnston [24, 25] provides an excellent discussion on Doppler sensitivity.

2.4.5 Summary

NFLM waveforms achieve low range sidelobes without the SNR loss associated with an amplitude taper. Frequency modulation is used to shape the waveform’s spectrum, and the square of the spectrum magnitude determines the shape of the time-domain response.

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−70

−60

−50

−40

−30

−20

−10 0 10

time delay (x10⁻⁸ sec)

dB

FIGURE 2-21 A quarter cycle of Doppler shift across the uncompressed pulse starts to elevate the near-in sidelobes of the NLFM waveform.

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2.5 Stepped Frequency Waveforms 61

The first term to the right of the equal sign is a constant and is associated with the round-trip delay and the initial transmit frequency. The second term is a complex phasor rotating at a rate defined by the frequency step size and the range to the target. The phase rotation is similar to that observed when employing a pulsed Doppler waveform. To simply the expressions, the amplitude of the return is set to unity.

Fourier analysis is applied to the complex samples in equation (2.112) to extract the location of the return in range. Consider the DTFT defined by

X(ω) =

N−1

n=0

x(n) exp (− jωn) (2.113)

where x(n) are the measured returns collected from N pulses, and ω is the digital frequency with units of radians/sample. The DTFT represents a filter bank tuned over a continuum of frequencies (or rotation rates), which in this case correspond to different ranges.

The samples are often viewed as being collected in the frequency domain, and the returns are then transformed to the time (or range) domain. The frequency-domain in-terpretation is based on the assertion that each pulse is measuring the target’s response (amplitude and phase) at a different frequency. From this perspective, an inverse DTFT would naturally be applied; however, it is the rotating phase induced by the change in frequency that is important. Either a forward or inverse DTFT may be applied as long as the location of the scatterers (either up- or down-range) is correctly interpreted within the profile, and the return is scaled to account for the DTFT integration gain.

For a point target located at range R0, the output of the DTFT is a digital sinc defined by

c f , and R is a continuous variable representing range. Equation (2.114) represents the range compressed response. The termωR0centers the response at a particular range or frequency. The shape of the compressed response is examined by setting R₀= 0 or

A plot of the compressed response is shown in Figure 2-25. The response consists of a main lobe and sidelobe structure with peak sidelobes 13.2 dB below the peak of the main lobe.

The DTFT is periodic and repeats at multiples of 2π in ω; therefore, the range compressed response is also periodic with periodicities spaced by c/2f . An implication is that range measured at the output of the DTFT is relative to the range gate and not absolute range.

Absolute range is defined by the time delay to the range gate and the relative range offset within the profile.

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62 C H A P T E R 2 Advanced Pulse Compression Waveform Modulations FIGURE 2-25 A

SF waveform’s compressed response is a digital sinc function.

0

−10

dB −20

−30

−25

−3 −2 −1 0 1 2 3

−40

−35

ω

−15 5

−5

The waveform’s Rayleigh resolution is found by setting the argument of the numerator in equation (2.115) equal toπ and solving for ω

δω = 2π

N (2.116)

whereδω is the resolution in terms of digital frequency. To convert the frequency resolution in equation (2.116) to a range resolution, consider two point targets located at ranges R₁ and R₂and separated in range byδR = |R2− R1|. The difference in their phase rotation rates is 2π2δR

c f . Equating the rate difference to the frequency resolution defined in equation (2.116)

2π2δR

c f = 2π

N (2.117)

and solving for the range difference yields the Rayleigh resolution

δR = c

2Nf (2.118)

The range resolution achieved by the SF waveform is inversely proportional to the wave-form’s composite bandwidth Nf . In this case, the Rayleigh resolution is equivalent to the main lobe’s –4 dB width.

2.5.4.2 Discrete Fourier Transform

The DTFT is defined over continuous frequency; however, to realize the compressed response in digital hardware requires one to evaluate the DTFT at a finite number of frequencies over the interval [0, 2π). It is common to evaluate the DTFT at equally spaced frequencies

ωk= 2π k

M k = 0, . . . , (M − 1) (2.119)

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2.5 Stepped Frequency Waveforms 63

where M ≥ N. This is equivalent to evaluating the response at equally spaced ranges defined by

R_k = c

2f Mk k = 0, . . . , (M − 1) (2.120) Inserting equation (2.119) into (2.113) yields the DFT

X(k) =

which is often implemented using an FFT for computational efficiency. For a point target located at range R0, the compressed response is a sampled instantiation of the digital sinc in (2.114) or

The DFT is a linear operator; thus, the range profile associated with multiple scatterers is simply the superposition of the individual responses.

Zero padding is often used in conjunction with a DFT to decrease the filter spacing, which reduces straddle loss. For M > N, the sequence is said to be zero padded with M− N zeros. For M = N, the filters or range bins are spaced by the nominal waveform resolution c/2Nf .

2.5.4.3 Range Sidelobe Suppression

Range sidelobes associated with large RCS targets may mask the presence of smaller targets. As noted in Figure 2-24, sidelobes are suppressed by applying an amplitude weighting to the complex samples collected at a range gate. When selecting an ampli-tude taper, the associated reduction in resolution and loss in SNR should be taken into account [36].