Waveform Modulations and Techniques
2.3 STEPPED CHIRP WAVEFORMS
2.3.4 Processing Options
The objective is to create, within the signal processor, a composite waveform that ex- hibits a range resolution commensurate with composite waveform’s transmit bandwidth by properly combining the Nsc baseband returns in equation (2.59). A means for inter- polating in time and shifting in frequency the received pulses is required. In addition, careful consideration must be given to coherently combining the returns and accounting for phase differences between pulses. Both time-domain (TD) and frequency-domain (FD) approaches are described in the literature [7, 8, 10, 11] and are covered in subsequent sec- tions. An example of a stepped chirp waveform processed in the frequency domain is also presented. The analysis focuses on the return from a point target, but the process is linear and is applicable to a superposition of returns at different delays.
2.3.4.1 Time Domain
The TD approach for processing a stepped chirp waveform is discussed in [7, 8], and the steps are outlined here for completeness:
1. When implementing the TD approach, it is assumed that the frequency step size is equal to the pulse bandwidth (i.e.,f =β).
2. In the signal processor, the first step is to up-sample (i.e., interpolate) the baseband signals in equation (2.59) by a factor of Nsc. The interpolation may be implemented using a finite impulse response (FIR) filter. The interpolated signal is required to support the composite waveform bandwidth and is denoted yn (m). The interpolated pulses have a frequency support defined by−πNscFs ≤≤πNscFs.
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2.3 Stepped Chirp Waveforms 43
3. A frequency shift is applied to each pulse by mixing the pulse with a complex sinusoid having a linear phase response. The mixer product is
zn(m)=yn (m)exp
j 2πpf m NscFs
m =0, . . . ,
NscFs
τ+2R c
(2.62) 4. Next a phase correction is applied to each pulse to force a continuous phase transition between pulses. For an odd number of pulses, the phase correction applied to the n-th pulse is
φn = 2πβτ((Nsc−1)/2)−n
i=1 i for 0≤n ≤((Nsc−1)/2)−1 (2.63) and
φn =2πβτ
n−(Nsc−1)/2 i=1
(i −1) for((Nsc−1)/2+1)≤n≤(Nsc−1) (2.64) To reduce the number of operations, the phase correction may be applied prior to the interpolation step.
5. The frequency-shifted and phase-corrected pulses are then time aligned via
z(m)=
Nsc−1 n=0
zn(m−n NscFsτ) (2.65) The time alignment places a requirement on the product Fsτto be an integer. Note that zn(m)=0 for m<0 and for m>[Fs(τ+2R/c)]
2.3.4.2 Frequency Domain
The FD approach is defined in [10, 11]. In most instances, the frequency step size is chosen to be less than or equal to the pulse bandwidth (i.e.,f ≤β). A continuous or overlapped frequency response prevents the introduction of gaps in the composite waveform spectrum, which would increase the range sidelobes.
The FD approach is introduced starting with the pulses received from a point target at time delay td. The pulses have been mixed to baseband and sampled. The spectrum of the sampled signal in equation (2.59) is
Xn(ω)=exp(−j 2πf0td)exp(−j 2πpf td)X(ω)exp(−jωFstd) (2.66) where−π ≤ω≤π and
X(ω)=DTFT
exp
jπB τ
m Fs − τ
2 2
0≤m ≤[FSτ] (2.67) The filter matched to a pulse with spectrum X(ω)is X∗(ω), where the asterisk denotes the conjugate operator. Applying the filter to the pulses in equation (2.66)
Yn(ω)=exp(−j 2πf0td)exp(−j 2πpf td)|X(ω)|2exp(−jωFstd) (2.68) Equation (2.68) represents the match filtered spectrum for the n-th pulse. On receive, each pulse is mixed and centered at baseband. To reconstruct the waveform, the pulses
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44 C H A P T E R 2 Advanced Pulse Compression Waveform Modulations
are shifted by their respective frequency offsets, pf . Shifting the n-th spectrum by 2πpf/Fs
Yn
ω− 2πpf Fs
=exp(−j 2πf0td)exp(−j 2πpf td) X
ω−2πpf Fs
2
exp(−j(ωFs−2πpf)td) (2.69) and after reducing terms
Yn
ω− 2πpf Fs
=exp(−j 2πf0td) X
ω− 2πpf Fs
2exp(−jωFstd) (2.70) where(−π+2πpf/Fs)≤ω≤(π +2πpf/Fs). Note that the phase term 2πpf td in equation (2.68) has been removed by the shifting operation. Its removal is necessary to coherently combine the pulses. The match filtered and frequency shifted spectra contain a constant phase term that is a function of f0and td and a linear phase term proportional to td. The linear phase term positions the response in time after the inverse transform is applied.
The spectra in equation (2.70) are a result of applying the DTFT. To compute the spectra inside a signal processor, the DFT is applied. The K -length DFT is constructed by sampling the spectrum in equation (2.70) atωk = 2πk/K,−K/2≤k≤K/2−1 to yield
Yn
2π k
K − 2πpf Fs
=exp(−j 2πf0td) X
2π k
K − 2πpf Fs
2exp
−j 2π k KFstd
(2.71) To align the pulse spectra, the frequency step size must be an integer multiple of the DFT bin size. The size of a DFT bin is found by computing
δfD F T = Fs
K (2.72)
The frequency step size,f , is therefore constrained to be
f = PδfD F T (2.73)
where P is an integer. Substituting equations (2.72) and (2.73) into equation (2.71) Yn
2π k
K −2πp P K
=exp(−j 2πf0td) X
2π k
K −2πp P K
2exp
−j 2π k K Fstd
(2.74) where (−K/2+ p P) ≤ k ≤ (K/2−1+ p P). Shifting the spectra has increased the size of the DFT to K+(Nsc−1)P, and the effective sample rate is FsK +(Nsc−1)P
K .
Zero padding may be applied in the frequency domain to force a power of 2 size FFT or to further interpolate the match filtered time-domain response.
At this point, the spectra have been shifted, and a matched filter has been applied to each pulse. The next step is to properly stitch together the shifted spectra to form a composite spectrum that achieves the desired main lobe and sidelobe response when the inverse transform is applied. The stitching process is illustrated in Figure 2-9 using three
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2.3 Stepped Chirp Waveforms 47
−400 −300 −200 −100 0 100 200 300 400 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
frequency (MHz)
magnitude
FIGURE 2-12 The overlapping spectra are stitched together to form a composite spectrum.
create a smooth composite spectrum that resembles a rectangle. The stitching process is described in Section 2.3.4.2. Next, the range compressed response is obtained by taking the inverse DFT. A portion of the compressed response is shown in Figure 2-13, and the main lobe and adjacent sidelobes are plotted in Figure 2-14. Using the composite bandwidth defined in equation (2.75) andδR = c/2βsc, the computed resolution is 0.192 meters, which is very close to the Rayleigh resolution achieved by the stepped chirp waveform in Figure 2-14. If desired, an amplitude taper may be applied to the composite spectrum in Figure 2-12 to reduce the range sidelobes.