Radar emission spectrum engineering Shannon D. Blunt 1 , John Jakabosky 1
1.4 Spectrally shaped optimization
enhanced control over radar emissions, as a means to realize better spectral con- tainment and to start realizing the practical potential of waveform diversity.
Given an initial waveformp0ðtÞ, a desired PSD |G(f)|2, and a desired ampli- tude taperw(t) to serve as a soft constraint to minimize SNR loss, the first stage of the optimization process [36] involves the iterative application of
riþ1ðtÞ ¼ F1fjGðfÞjexpðjffFfpiðtÞgÞg (1.16) and
piþ1ðtÞ ¼wðtÞexpðjffriþ1ðtÞÞ; (1.17) whereF is the Fourier transform, F1 is the inverse Fourier transform and ∠() extracts the phase of the argument. These steps are repeatedItimes to generate the first stage output waveformpIðtÞthat possesses both FM and AM attributes.
Setting q0ðtÞ ¼pIðtÞ as the initialization for the second stage, the iterative application of
qkþ1ðtÞ ¼xðtÞF1fjGðfÞjexpðjff FfqkðtÞgÞg (1.18) is then performed Ktimes to produce the final waveform qKðtÞ, withx(t) a rec- tangular window having the same length as the pulsewidthTwhich serves to limit the temporal extent ofqkþ1ðtÞ:Both stages can be efficiently implemented using FFT and IFFT processing on a GPGPU.
As an example, a joint waveform/taper optimization is performed forBT¼128, where bandwidth B¼80 MHz and pulsewidth T¼1.6ms. The PSD |G(f)|2 is a Gaussian shape and the initial taper w(t) is a Tukey taper with amplitude roll-off occurring during the first and last 50 ns. The waveform/taper is optimized for I¼K¼5,000 iterations in each of the two stages. The initial waveformp0ðtÞis the (L¼8,M¼2) over-coded waveform from Section 1.4 havingBT¼64 that has been interpolated up toBT¼128 using polynomial fitting. Note that this initial waveform from [45] is still rather chirp-like, albeit with the time-frequency dithering that arises when optimizing physical FM waveforms.
Figure 1.21 depicts the resulting amplitude envelope after joint waveform/
taper optimization in which the associated SNR loss is only 0.26 dB. The trade-off for this small SNR loss can be observed in Figure 1.22 in which the autocorrelation is shown for the intermediate stage as well as the final stage, with and without inclusion of the taper. While the intermediate stage achieves a PSL of59.9 dB, the joint waveform/taper optimization attains a PSL of108.1 dB, far exceeding what would often be needed for many sensing applications. Interestingly, if the taper is removed the PSL degrades by more than 60 dB. Further, Figure 1.23 shows that removal of the amplitude taper likewise results in significant degradation in terms of spectral containment, thus emphasizing the important linkage between waveform performance for sensing and the spectral containment of the radar emission.
Finally, Figure 1.24 illustrates the matched filter response for a loopback mea- surement when this jointly optimized emission is implemented on the radar testbed used in the previous sections. This response includes the effect of a Least-Squares based compensation of linear artifacts that were generated by the spectrum analyzer and LINC architecture. The loopback measurement without using LINC yields a PSL
Time Initial envelope
0 –2 –1.5 –1 –0.5 0 0.5
0.25T 0.5T
Final envelope
0.75T T
dB
Figure 1.21 Pulse amplitude envelope after optimized joint waveform/taper optimization with BT¼128 (2015 IEEE, reprinted with permission from [36])
Delay Final w/o taper
–0.5T
–T 0 0.5T
Final w/ taper
T
dB
–20
–40
–60
Intermediate
–80
–100
–120 0
Figure 1.22 Waveform autocorrelation of intermediate design stage, and with and without jointly optimized amplitude taper (2015 IEEE, reprinted with permission from [36])
Normalized frequency
–B –0.5B 0 0.5B B
w/ Taper w/o Taper
dB
–20 –10
–40 –30
–60 –50
–70 –80 0
Figure 1.23 Spectral content of waveform with and without jointly optimized amplitude taper. The case that includes the taper closely matches the desired Gaussian PSD (2015 IEEE, reprinted with
permission from [36])
Delay
–T –0.5T 0 0.5T T
Measured
Measured w/ LINC
Optimized
dB
–20
–40
–60
–80
–100
–120 0
Figure 1.24 Matched filter response using an experimental loopback
measurement (2015 IEEE, reprinted with permission from [36])
of83.2 dB while the use of LINC realizes a PSL of81.8 dB, thus demonstrating the potential to implement these AM effects at high power. The difference between these measured results and simulated performance is most likely due to the 10-bit resolution of the AWG. The measured waveform also maintains the same spectral containment as demonstrated in Figure 1.23. In [36] it is shown how this ultra-low sidelobe emission provides dynamic range greater than 70 dB for free-space measurements.
1.4.2 Non-recurrent nonlinear FMCW emissions
The spectrally shaped design approach from the previous section addressed joint waveform/taper optimization to obtain a particular ultra-low sidelobe radar emis- sion that is repeated during a coherent processing interval (CPI). Here we consider a similar design approach but focus instead on the development of a non-repeating structure for which sidelobes do not coherently combine [37]. As such, the following can be viewed as a form of FM noise radar [53] that is designed on a segment-wise basis.
Denoted as pseudo-random optimized (PRO) FMCW, this formulation relies on the same strategy as (1.16) and (1.17), with two important differences. First, because a constant amplitude is desired to simplify transmitter requirements and mitigate SNR loss,w(t) in (1.16) is now a rectangular window of lengthT, which corresponds to the length of each waveform segment. Second, each segment is initialized with a new randomly generated FM waveform (generally not chirp-like) that, via iterative application of (1.16) and (1.17), is imbued with the desired spectrum shape (Gaussian is still used). Themth optimized waveform segment is then phase rotated as
smðtÞ ¼expjfend;m1
pK;mðtÞ (1.19)
wherefend;m1 is the ending phase value for the (m1)th segment, to avoid phase discontinuities. As such, each segment is optimized to provide low autocorrelation range sidelobes, yet does not repeat, thereby avoiding range ambiguities via low crosscorrelation with other segments. The varying structure also prevents sidelobes from combining coherently when performing subsequent Doppler processing across the segments on receive.
As an example, consider a bandwidth ofB¼80 MHz and a total length of 200 ms comprised of 104segments of 20ms each. Thus, each segment hasBT¼1,600 and the total processing gain is 1.6107, or 72 dB. Each segment was optimized forK¼500 iterations, with the initialization a random sequence of 1,600 phase values drawn from a uniform distribution on [p, p] and subsequently implemented using the PCFM framework of [26].
Figure 1.25 depicts the RMS average autocorrelation responses for the initi- alization and optimized measurements across the 104 segments. On average, the PSL per segment is improved by 10 dB by using the spectral shaping approach.
In contrast, the RMS average crosscorrelation between adjacent segments increa- ses by 1.5 dB (see Figure 1.26) for the optimized measurement relative to the initialization. This result is to be expected since the optimization imposes greater
Delay
–T –0.5T 0 0.5T T
Initial
Optimized
dB
–20 –10
–40
–50 –30 0
Figure 1.25 RMS average autocorrelation responses for initialization and measurement of optimized (2015 IEEE, reprinted with permission from [37])
Delay
–T –0.5T 0 0.5T T
Initial Optimized
dB
–40
–50 –30
–35
–45
–55
Figure 1.26 RMS average crosscorrelation response for adjacent segments using initialization and measurement of optimized (2015 IEEE, reprinted with permission from [37])
spectral containment and thus naturally introduces somewhat greater similarity between different optimized segments (Figure 1.27).
In Figure 1.27, the desired PSD is again a Gaussian spectrum. The alternating projection optimization procedure is found to attain this desired PSD fairly well given the initial random FM waveform segments (that possess a greater bandwidth due to being unconstrained). Note that at the edges (just beyondB) the measured optimized spectral response is truncated by the 160 MHz analysis bandwidth of the spectrum analyzer.
When Doppler processing is performed over the set of 104 segments, the range sidelobes do not combine coherently since the PRO-FMCW waveform does not repeat. Figure 1.28 illustrates the resulting zero-Doppler integrated auto- correlation response where it is observed that nearly 40 dB improvement is achieved relative to the RMS average autocorrelation. A similar improvement is observed is Figure 1.29 for the zero-Doppler integrated crosscorrelation response between adjacent segments. In [37], it is shown how this emission scheme pro- vides the means to detect moving targets more than 75 dB below the dominant scattering in free-space measurements.
1.4.3 Hopped spectral gaps
Given the relationship between the PSD of a waveform and the sidelobe level in the associated autocorrelation, one can draw the logical conclusion that the introduction of spectral gaps into the PSD will generally result in an increased sidelobe level for an otherwise optimized waveform (e.g., [54–56]). In [57],
Normalized frequency
–B –0.5B 0 0.5B B
Initial Optimized
Desired PSD
dB
–10
–30 –20 0 –5
–15
–25
Figure 1.27 Spectral content for initialization, desired PSD and measurement of optimized (2015 IEEE, reprinted with permission from [37])
the PRO-FMCW waveform [37] described above is modified to provide for gaps in the emitted spectrum to avoid other spectrum users or potentially to enable embedding of another RF function such as communications or navigation.
Figure 1.30 illustrates the mean PSD realized for cases involving 1 and 2 spectrum gaps that are stationary over the total waveform length of 200 ms
Delay
–T –0.5T 0 0.5T T
Initial
Optimized
dB
–40
–100 –20 0
–80 –60
Figure 1.28 Integrated autocorrelation response of PRO-FMCW emissions (2015 IEEE, reprinted with permission from [37])
Delay
–T –0.5T 0 0.5T T
Initial
Optimized
dB
–40
–100 –50 –30
–80 –70
–90 –60
Figure 1.29 Integrated crosscorrelation response of PRO-FMCW segments (2015 IEEE, reprinted with permission from [37])
(same bandwidthB¼80 MHz and 104segments as used in the previous section).
The simulation results in Figure 1.31 clearly show how the associated range side- lobe responses (zero-Doppler integrated over the 104 segments) have degraded relative to the previous measured result from Figure 1.28 due to the introduction of these stationary spectral gaps.
–80 –B –70 –60 –50 –40
Relative power (dB)
–30 –20 –10 0
–0.5B
2 gaps 1 gap
0
Normalized frequency
0.5B B
Figure 1.30 Mean PSD for 1 spectral gap and 2 spectral gap cases (2015 IEEE, reprinted with permission from [57])
–120–T –0.5T 0
Delay
0.5T T
–100 –80 –60
Relative power (dB)
–40 –20 0
1 gap 2 gaps
Figure 1.31 Doppler integrated autocorrelation responses for 1 gap and 2 gap cases (2015 IEEE, reprinted with permission from [57])
However, if the spectral gaps are hopped during the CPI, their deleterious impact on the PSD is significantly reduced. Figure 1.32 shows the mean PSD for each of 10 different spectral gaps that occur sequentially during the 200 ms interval.
Overall, this hopping produces the PSD in Figure 1.33 in which the impact of any
–80 –B –70 –60 –50 –40
Relative power (dB)
–30 –20 –10 0
–0.5B 0
Normalized frequency
0.5B B
Figure 1.32 Mean PSDs for each of 10 sequentially hopped spectral gaps (2015 IEEE, reprinted with permission from [57])
–30 –B –25 –20 –15
Relative power (dB)
–10 –5
10 sequential
100 random 0
–0.5B 0
Normalized frequency
0.5B B
Figure 1.33 Overall mean PSD for sequential and random hopped spectral gaps (2015 IEEE, reprinted with permission from [57])
single spectral gaps has been virtually eliminated, thus producing the improved Doppler integrated autocorrelation in Figure 1.34.
Further, when 100 spectral gaps are randomly hopped over the same time interval, the smoother overall PSD in Figure 1.33 is achieved which realizes the associated Doppler integrated autocorrelation in Figure 1.34 that is approaching the same performance as when no gaps are present at all. Ongoing work is inves- tigating how to implement these hopped spectral gaps when transmitter distortion induces spectral regrowth effects.