Radar emission spectrum engineering Shannon D. Blunt 1 , John Jakabosky 1
1.2 Polyphase-coded FM
In Section 1.4, another recent emission implementation/optimization scheme is described that relies on the relationship between the PSD and autocorrelation of the waveform to realize pulsed [36] and continuous wave (CW) [37] modalities, respectively, that achieve both good spectral containment and low range sidelobes with little to no SNR loss. As with PCFM-based approaches, experimental measurements demonstrate the efficacy of these spectral shaping methods. This formulation also permits joint optimization of the waveform with a low-loss amplitude taper, and it is likewise shown how in-band spectral gaps (avoidance regions) can be generated without incurring the sensitivity degradation that other- wise generally arises.
realizable. In contrast, awaveformis a continuous physical signal, such as the well- known LFM chirp [1, pp. 57–61]. Finally, theemissionis the transmitter-distorted version of the waveform that is launched into free space, including effects such as rise/fall-time ringing and distortion caused by undesired conversion of phase modulation (PM) into AM.
The purpose of the PCFM implementation is to convert an arbitrary polyphase code into a continuous FM waveform that is amenable to the bandlimiting/
non-linear effects of the transmitter, the deleterious impact of which can be reduced but not eliminated. By extension, the selection of an underlying code implemented with the PCFM framework can subsume the transmitter distortion effects so as ultimately to optimize the physical radar emission launched into free space. In doing so, the spectral content may likewise be addressed for this physical emission, to the degree that the transmitter-distorted spectral response can be controlled by the underlying code. Total control over the emission necessitates joint design of the waveform and transmitter [2].
1.2.1 PCFM implementation
To provide a suitable implementation for the length Nþ1 polyphase code q0 q1 qN
½ T, a modified version of the CPM framework [27] was proposed for the radar application [26]. Figure 1.2 illustrates this scheme where a train of N consecutive impulses with time separationTpare formed to yield a total pulse- width T¼NTp. The nth impulse is weighted by an; which is the phase change between successive chips of the polyphase code as determined by
an¼Yð Þ ¼a~n a~n
~
an2psgnð Þ~an
( if j j a~n p
if j ja~n >p (1.1) where
~
an¼qnqn1 for n¼1;. . .;N (1.2)
and sgn(•) is the sign operation.
The shaping filter g(t) in Figure 1.2 is the same as that used for communica- tions, with the most common examples being rectangular and raised cosine [27].
The requirements on the shaping filter are: (1) thatg(t) integrates to unity over the
∑and(t – (n–1)Tp) (·)
∫ dt + q0 exp{ j(·)} s(t;x) x =
g(t)
N
t 0 n=1
a1 a2 ... aN T
Figure 1.2 Modified CPM implementation to generate polyphase-coded FM (PCFM) radar waveforms (2014 IEEE, reprinted with permission from [26])
real line and (2) that g(t) has time support on [0, Tp]. The integration stage in Figure 1.2 is initialized toq0and the sequence of phase changes are collected into the vector x¼½a1 a2 aNT, which parameterizes the complex baseband PCFM waveform
sðt;xÞ ¼exp j Z t
0
gðtÞ XN
n¼1
and t ðn1ÞTp
" #
dtþq0
!
( )
(1.3) Note that (1.1) and (1.2) provide for the conversion of an existing polyphase code
q0 q1 qN
½ T into the ‘phase change’ code x parameterizing the PCFM waveform. However, optimization of a PCFM waveform can be performed directly by selection of the values in x, for p an þp and with q0 from (1.3) an arbitrary phase offset that does not affect the goodness of the waveform. The selection of the shaping filterg(t) also impacts the waveform and it could likewise be made a free parameter for optimization (though such is not considered here).
To demonstrate the physical attributes of the PCFM scheme, consider the four waveforms described in Table 1.1 that represent different implementations of a P4 code [1, Section 6.2] withN¼64. A discretized version of each waveform is represented with 150 samples/chip and a pulsewidth of 64ms that is then loaded onto an arbitrary waveform generator (AWG) for measurement in a loopback configuration (the transmitter connected directly to the receiver) using an S-band testbed. The testbed includes a mixer, preamplifier, bandpass filter and a class AB solid-state GaN PA.
Figure 1.3 shows the pulse shapes for each of the four loopback measurements.
The waveform denoted as ‘Ideal Chip’ represents the closest approximation possible to an idealized code given the sampling rate of 150 samples/chip. This waveform exhibits an amplitude null each time a chip transition occurs. Such significant deviations from constant amplitude can produce problems for a PA that is operated in saturation, including voltage-standing-wave-ratio (VSWR) fluctuations, increased phase noise and possibly even damage to components since a significant portion of the delivered power may not be radiated.
The ‘10% Transition’ waveform partially alleviates this problem by performing a linear phase interpolation between adjacent chips, though the amplitude nulls are still clearly visible. Interestingly, the PCFM implementation using a rectangular (RECT)
Table 1.1 Waveform implementations and their characteristics
Implementation Waveform characteristics
Ideal chip Fastest phase transition possible given AWG limitations
10% Transition Linearly interpolated phase transitions over 10% of each chip width
PCFM-RECT Uses a rectangular filter forg(t); approximates LFM with piecewise linear phase transitions PCFM-RC Uses a raised-cosine filter forg(t)
filter forg(t), which is a piece-wise linear phase approximation of the LFM waveform from which the P4 code is derived [1, Section 6.2], can be viewed as extending the 10% Transition case to a full 100% Transition with linear interpolation. Of course, instead of just being an ad hoc fix to the limitations of generating codes, the PCFM implementation provides a new design framework via the determination of thean
parameters and the flexibility to select different shaping filters such as demonstrated by use of the raised-cosine (RC) filter in Figure 1.3.
DenotingAWG waveformas the version of each of the four waveforms that is loaded onto the AWG andloopback emissionas the resulting version captured by the receiver of the S-band testbed, Figures 1.4–1.7 illustrate this ‘before-and-after’
spectral content for each of the four implementations. The distortion induced by the transmitter is most clearly evident for the Ideal Chip and 10% Transition wave- forms in Figures 1.4 and 1.5, respectively. While it might appear for these cases that the transmitter is providing needed improvement in spectral containment, this distortion actually produces mismatch effects that translate into SNR losses on receive [26]. In Figures 1.4 and 1.5, the mismatch starts to appear just below10 dB relative to the peak.
0 10 20 30 40
Ideal chip
10% Transition
PCFM-RECT
PCFM-RC
50 60
0 10 20 30 40 50 60
00 0.005 0.01 0.015 0 0.005
Amplitude (V)
0.01 0.015 0 0.005 0.01 0.015
10 20 30 40 50 60
00 0.005 0.01 0.015
10 20 30
Time (µs)
40 50 60
Figure 1.3 Pulse shapes for four loopback emissions after transmitter distortion [26]
–60–20 –50 –40 –30 –20
Normalized power (dB)
–10 0
–15 –10 –5 0
Frequency (MHz)
5 10 15 20
AWG waveform Loopback emission
Figure 1.4 Spectral content of Ideal Chip waveform before (top) and after the transmitter (bottom) (2014 IEEE, reprinted with permission from [26])
–60–20 –50 –40 –30 –20
Normalized power (dB)
–10 0
–15 –10 –5 0
Frequency (MHz)
5 10 15 20
AWG waveform Loopback emission
Figure 1.5 Spectral content of 10% Transition waveform before (top) and after the transmitter (bottom) (2014 IEEE, reprinted with permission from [26])
–60–20 –50 –40 –30 –20
Normalized power (dB)
–10 0
–15 –10 –5 0
Frequency (MHz)
5 10 15 20
AWG waveform Loopback emission
Figure 1.6 Spectral content of PCFM-RECT waveform before (top) and after the transmitter (bottom) (2014 IEEE, reprinted with permission from [26])
–60–20 –50 –40 –30 –20
Normalized power (dB)
–10 0
–15 –10 –5 0
Frequency (MHz)
5 10 15 20
AWG waveform Loopback emission
Figure 1.7 Spectral content of PCFM-RC waveform before (top) and after the transmitter (bottom) (2014 IEEE, reprinted with permission from [26])
In contrast, the transmitter-induced mismatch for the two PCFM implementa- tions (Figures 1.6 and 1.7) starts to appear below 30 dB relative to the peak, meaning these waveforms are much more amenable to the physical attributes of the transmitter. Also note that between the two PCFM waveforms, the one using the rectangular (RECT) shaping filter demonstrates better spectral containment, where the RC shaping filter produces a pair of close-in spectral sidelobes.
1.2.2 PCFM optimization
Inclusion of the PCFM implementation from Figure 1.2 into a comprehensive design framework permits optimization of the actual continuous waveform, and ultimately the physical free space emission inclusive of any distortion by the transmitter. Figure 1.8 provides a notional representation of this all-inclusive optimization paradigm.
The PCFM implementation defined in (1.3) and Figure 1.2 can be expressed compactly as the operator
sðt;xÞ ¼TPCFMf gx (1.4)
that generates the continuous-time PCFM radar waveform associated with the phase-change code x. The distortion induced by the transmitter may likewise be represented by the operation
uðt;xÞ ¼TTx½sðt;xÞ ¼TTx½TPCFMf gx (1.5) where the resulting signaluðt;xÞis the physical emission launched from the radar.
For an idealized transmitter (no distortion) this latter operation simply yields idealized Tx: uðt;xÞ ¼TTx½sðt;xÞ ¼sðt;xÞ (1.6) The most common metrics for waveform (or here emission) optimization are peak sidelobe level (PSL) and integrated sidelobe level (ISL). Spectral containment to
Code selection
Code-to- waveform implementation
Physical effects:
mixer, filter, amplifiers, antenna/platform
Assess emission performance
Physical domain Continuous domain
Figure 1.8 Notional representation of the optimization of physical radar emissions [33]
minimize the transmission of out-of-band spectral content and the incorporation of in-band spectral avoidance regions are emerging metrics driven by growing spec- tral congestion. DefineF½uðt;xÞas the generic evaluation of the physical emission according to some metric.
The optimization problem then becomes one of determining the parameters in x¼½a1 a2 aNT that yield a sufficiently optimal solution, noting that the rather high dimensionality of the problem for a useful time-bandwidth product (BT) likely precludes determination of global optimality.
The goodness of a waveform (emission) is generally based on the delay- Doppler ambiguity function defined as
cðt;wÞ ¼ Z þT
t¼T
ejwtuðt;xÞuðtþt;xÞdt (1.7) wheretis delay,wis Doppler andTis the pulsewidth. Clearly thew¼0 cut of this function is the waveform autocorrelation, which is worth noting, relates directly to the PSD of the waveform through a Fourier transformation.
The PSL [1] is thus usually defined for the zero-Doppler cut as FPSL½cðt;w¼0Þ ¼max
t
cðt;0Þ cð0;0Þ
(1.8)
for t2½tm;T;in which the interval ½tm;tmcorresponds to the autocorrelation mainlobe and½T;Tis the time support ofcðt;0Þdue to finite pulsewidth. The PSL metric provides a worst-case perspective on the sidelobe interference induced by a waveform/matched filter pair. Likewise, the ISL metric [1] for the zero- Doppler cut can be defined as
FISL½cðt;w¼0Þ ¼ RT
tmjcðt;0Þjdt Rtm
0 jcðt;0Þjdt (1.9)
In contrast to PSL, the ISL metric provides a measure of the aggregated sidelobe interference that could be encountered due to distributed scattering such as clutter.
Given the trend toward tighter restrictions on out-of-band radar spectral con- tent, frequency-domain metrics are of growing importance. While it might at first seem that such a metric may be in conflict with the commonly used PSL and ISL metrics that are used to optimize for low range sidelobes, the relationship between PSD and autocorrelation reveals the fact that a PSD that decreases towards the band edges corresponds to an autocorrelation with low range sidelobes [12]. As such, it is possible to achieve both good spectral containment and low range sidelobes by employing an appropriate frequency-domain metric.
One such metric was recently defined as the frequency template error (FTE) in [33] as
FFTE½Uðf;xÞ ¼ 1 fHfL
Z fH
fL
jUðf;xÞjpjWðfÞjpqdf (1.10)
where fL and fH denote the ‘low’ and ‘high’ edges of the frequency interval of interest,Uðf;xÞis the frequency response of the emission andWðfÞis a frequency weighting template (e.g., a Gaussian window). The valuespandqpermit control over the degree of emphasis placed at different frequencies. Forp¼1 andq¼2, (1.10) realizes a frequency-domain mean-square error metric. Alternatively,p>1 overly emphasizes in-band (higher power) frequencies while p<1 overly empha- sizes out-of-band (lower power) frequencies.
The notional emission design framework of Figure 1.8 is formalized in Figure 1.9 specific to the case of PCFM waveforms defined in (1.1)–(1.3). Alter- natively, with small modifications one could likewise specify the design scheme for binary codes implemented with DPSK or MSK [20,21] or even orthogonal fre- quency division multiplexing (OFDM) [41], though the latter is not necessarily recommended for radar applications due to the significant AM exhibited by OFDM.
The design problem exemplified by Figure 1.9 is of particular interest because polyphase codes provide greater freedom than binary codes and the associated PCFM implementation yields FM waveforms that are spectrally well contained and that are amenable to a high-efficiency, PA. The design problem thus becomes one of determining the sequence x¼½a1 a2 aNT that provides a subsequent physical emission to achieve a specified degree of performance according to a specified metricF.
If each phase-change parameteranforn¼1, 2, . . . ,Ncan be one ofLpossible values from a discrete, equally spaced constellation on [p,þp], then there exist LNdifferent coded emissions that could possibly be generated via (1.3) for a given shaping filter g(t). Clearly L andNneed not be very large before the number of possibilities becomes too unwieldy to determine the global optimum via evaluation of all LNcandidate waveforms. As such, a search strategy is needed to identify locally optimum waveforms that are ‘good enough’.
Code selection
a1 a2 aN
Code Waveform
Physical emission X
Code-to-waveform implementation
TPCFM{X} TTx[s(t;X)]
Φ[u(t;X)]
u(t;X) s(t;X)
Emission assessment
Transmitter effects
Figure 1.9 Formal representation of the optimization of physical radar emissions resulting from PCFM waveforms (2014 IEEE, reprinted with permission from [33])
Many search strategies exist for this type of problem (e.g., Tabu search, simulated annealing, genetic algorithms, particle swarm optimization). Such strategies can generally be classified as either a single point (or local) search or as a population-based search [42]. The former rely on various heuristics to avoid local minima during the search while the latter employ a distributed sampling of the search space that is perturbed to discover new regions of the search space.
Within the context of radar waveform design, consider that the range-Doppler ambiguity function integrates to a constant for an arbitrary continuous, constant amplitude waveform [1]meaning that sidelobes can only be moved around but not eliminated. As such, the consolidation of ambiguity into the range-Doppler ridge, as realized by chirp-like waveforms, provides an excellent initialization from which to search for optimal waveforms since the existence of the ridge effectively
‘absorbs’ much of the ambiguity. In other words, one can expect to obtain quite low range sidelobe levels by performing a local search onxin the vicinity of phase- change codes whose associated PCFM waveforms possess a chirp-like structure.
The difficulty with a local search is the prospect of ‘getting stuck’ in a local minimum. Well-known heuristics such as simulated annealing and Tabu search [42] were thus developed as means to escape from these local minima in the hopes of discovering the global minimum (or at least a better local one). For the problem of optimizing a physical radar waveform, a new approach was devised in [33] that relies on the complementary nature of the waveform metrics of PSL, ISL and FTE as defined in (1.8), (1.9) and (1.10), respectively. Denoted asperformance diver- sity, this search strategy exploits the fact that each of these metrics differently measures the same fundamental property of ‘how large are the range sidelobes relative to the mainlobe’ for the zero Doppler cut (with possible simple extension to non-zero, yet small, Doppler as well). As such, an improvement in one metric tends to be an improvement in the others. An important exception, however, is that local minima are not necessarily the same across these different metrics, thus providing a way to escape some local minima by simply changing the metric being evaluated.
While a local minimum shared by all three is certainly possible (though more rare), it is also likely to be a rather good local minimum, thereby likely achieving the design goal.
For example, using a phase-change code that closely approximates an LFM chirp as the initialization and assuming an idealized transmitter (i.e., no distortion), Figure 1.10 illustrates the autocorrelation obtained after optimization using the individual metrics of PSL, ISL and FTE, along with the result from using all three combined within this performance diversity paradigm. Table 1.2 quantifies the per- formance for each according to how the emission was optimized. Interestingly, the performance diversity approach yields the best performance for all three metrics, which stands to reason since this approach is not a multi-objective optimization in which competing goals are sought, but instead exploits the fact that these metrics are actually complementary to one another.
Relative to the LFM waveform, it is also observed in Table 1.2 and Figure 1.11 that all of the optimized waveforms involve a small degradation in range resolution.
This effect is to be expected for NLFM waveforms since the more gradual spectral
roll-off (relative to LFM), which provides for the achievement of lower range sidelobes, also slightly reduces the 3 dB bandwidth. It is interesting to note that the better performing NLFM waveforms in terms of sidelobe reduction realize a slightly greater trade-off in resolution (also to be expected given the conservation of range-Doppler ambiguity). Finally, these waveforms retain the chirp-like struc- ture and thus also retain much of their Doppler tolerance [33].
In short, it is possible to optimize an FM radar waveform by using a para- meterized search in the neighbourhood of chirp-like waveforms. In so doing, the goals of spectral containment and low-range sidelobes can both be met.
Now consider how the distortion induced by the radar transmitter can likewise be addressed.
–500 –40 –30 –20
Power (dB)
–10 0
ISL PSL
FTE
0.25 T 0.5 T
Delay
0.75 T T
Performance diversity
Figure 1.10 Autocorrelation of the optimized ideal emissions using the PSL, ISL and FTE metrics individually and the performance diversity paradigm (2014 IEEE, reprinted with permission from [33])
Table 1.2 Quantified performance for optimization of emissions for an idealized transmitter (2014 IEEE, reprinted with permission from [33])
Optimization metrics
LFM PSL ISL FTE Performance
waveform only only only diversity Final PSL value (dB) 13.5 21.1 27.0 31.3 40.2 Final ISL value (dB) 9.8 6.7 20.0 15.6 24.9 Final FTE value (dB) 17.0 16.6 22.4 30.1 32.1 Relative 3 dB resolution 1.00 1.06 1.11 1.26 1.28 Relative 6 dB resolution 1.00 1.08 1.13 1.30 1.33
1.2.3 Optimizing physical radar emissions
With the establishment of a framework to optimize the physical attributes of a waveform, it is possible to now consider the inclusion of transmitter distortion effects to optimize the physical emission launched from the radar. Generally speaking, there are two approaches one may take: model-in-the-loop (MiLo) opti- mization in which the linear/non-linear distortion is represented as a mathematical model in simulation and hardware-in-the-loop (HiLo) optimization in which the actual physical transmitter is used in the optimization process [33]. The MiLo approach has the benefit of permitting a fast search of this high-dimensional space, such as by using general purpose graphics processing units (GPGPU), though even the best model cannot exactly characterize the hardware it is intended to represent.
In contrast, the HiLo approach optimizes the emission specific to the very trans- mitter that will use it, though the latency involved with uploading a waveform onto an AWG and subsequently capturing it for evaluation can slow down the search process considerably.
Figure 1.12 again depicts the autocorrelation of the performance diversity wave- form from Figure 1.10 that was optimized under the idealistic transmitter condition (distortion free). In addition, Figure 1.12 reveals the autocorrelation of this waveform after being distorted by the solid-state amplifier model from [43]. Relative to the idealistic case, the distorted waveform results in 4.6 dB degradation in PSL and 2.8 dB degradation in ISL [33]. However, when MiLo optimization is used (Figure 1.12), 4.4 and 2.2 dB of lost PSL and ISL sensitivity is recovered, respectively.
–10
–T/N –0.5 T/N 0
PSL ISL
LFM
Performance diversity
FTE
Delay
0.5 T/N T/N
–8 –6 –4
Power (dB)
–2 0
Figure 1.11 Autocorrelation of the optimized ideal emissions using the PSL, ISL and FTE metrics individually and the performance diversity paradigm (mainlobe zoom-in) (2014 IEEE, reprinted with permission from [33])