Waveform Modulations and Techniques
2.4 NONLINEAR FREQUENCY MODULATED WAVEFORMS
2.4.5 Summary
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2.4 Nonlinear Frequency Modulated Waveforms 57
−37 dB; the goal was−40 dB. The peak sidelobe levels are close to the designed-to level. The waveform’s time–bandwidth product does have a direct bearing on sidelobe performance. In general, the higher the time-bandwidth product the closer one gets to the design-to level. Time–bandwidth products greater than 100 are generally required.
For an LFM waveform with 500 MHz swept bandwidth, the time resolution, measured at the –4 dB width, is 0.002μsec. A close examination of the response in Figure 2-20 reveals a−4 dB width of 0.0029μsec. The modulation has degraded the resolution by a factor of 1.45 over that achieved by an LFM swept over the same bandwidth. Of course, a similar degradation in resolution occurs when an amplitude taper is applied to an LFM 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, andωis 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
|X(ω)| =
sin N
2
ω−ωR0
sin
ω−ωR0
2
(2.114)
whereωR0 = 2π2R0
c f ,ω = 2π2R
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
|X(ω)| =
sin N
2ω
sin ω
2
(2.115)
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 2πinω; 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 forω
δω= 2π
N (2.116)
whereδωis 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
Rk = c
2f Mk k =0, . . . , (M−1) (2.120) Inserting equation (2.119) into (2.113) yields the DFT
X(k)=
N−1
n=0
x(n)exp
−j2πk M n
M ≥N (2.121)
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
|X(k)| =
sin N
2 2π
Mk−ωR0
sin 2π
Mk−ωR0
2
k=0, . . . , (M−1) (2.122) 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].