Feature Article:

Application of Waveform Weighting for a Frequency-Invariant Transmit BeampatternFaruk Uysal, University of Oklahoma, Advanced Radar Research Center, Norman, OK, USA and Microwave Sensing, Systems and Signals, Delft University of Technology, Delft, Netherlands Zachary Dunn and Mark Yeary, University of Oklahoma, Advanced Radar Research Center, Norman, OK, USA. Rafael Rincon, NASA, Goddard Space Flight Center, Greenbelt, MD, USA

DOI . NO. 10. 1 109/MAES.2016. 150201

INTRODUCTION

Advanced radar systems rely on multichannel, wideband opera-tion. NASA’s Ecosystem SAR (EcoSAR) and second-generation digital beamforming SAR [1]–[3] use active array architectures, where an active array radar system contains independent transmit and receive chains with unique amplifiers for each antenna ele-ment in the multichannel system. These radar systems have an in-dependent arbitrary waveform generator for each channel and are designed to address wideband transmit beamforming. Unlike con-ventional narrowband beamforming, which only require a unique set of beamsteering coefficients per look angle, wideband beam-forming requires beamsteering coefficients that vary with both look angle and frequency.

Transmitter and receiver beamforming for wideband signals has been previously studied in the literature. In receiver process-ing, frequency-dependent steering vectors are separated using a Bessel function expansion and used to focus data vectors to a sin-gle frequency component in [4]. Subband decomposition methods with different filter banks for wideband beamforming are proposed in [5], [6], [7]. Although frequency-dependent beampatterns can be focused through receiver postprocessing, the actual echoes re-flected from the target area have an inherent element of destructive interference due to the frequency-dependent nature of the transmit-ted waveforms. Thus, intelligent beamforming of the transmitted wideband waveforms is necessary to minimize undesired effects.

The direct optimization of the resulting wideband array pattern through the design of finite-impulse response (FIR) filters for use on the transmitted waveform is proposed in [8], [9], [10]. Scholnik and Coleman were able to achieve low sidelobes in the processed beampattern while preserving constant power over the mainbeam; however, their examples show that the optimized design produces high sidelobes (or grating lobes) outside the bandwidth of inter-est [8]. Neinhus and Solbach experimentally verified a FIR filter-based beamforming network, achieving a nearly frequency-inde-pendent radiation pattern; however, the maximum sidelobe level was reported as −10 dB [10]. Optimization of the cross-spectral density matrix to generate a desired spatial beampattern, subject to transmitter power constraints, was first introduced in [11]. It was proposed by [11] that even though it is possible to achieve a desired beampattern through optimization, it may be challenging to synthesize desirable waveforms sets that also maintain very low peak-to-average power ratios. To address this issue, He et al. intro-duced an iterative technique for wideband beampattern formation that constrains the peak-to-average power ratio to achieve wave-forms that can be efficiently generated with current hardware [12]. The resulting synthesized waveforms satisfy unit-modulus or low peak-to-average power ratio constraints, but they do not clearly address the formation of a frequency-invariant beampattern. The linear least-squares-based beampattern optimization has been stud-ied in multiple papers.

Differing from previous methods, the approach in this article proposes to design and synthesize a multichannel waveform set that successfully performs wideband beamforming, while preserv-ing a frequency-invariant beampattern, producing low beampat-tern sidelobes, and creating an effective waveform with a constant modulus main beam power envelope over the target area. In ad-dition to the design study, trade-offs of the resulting optimized waveform are addressed, including power efficiency and pulse compression ratio.

BACKGROUND AND TERMINOLOGY

NARROWBAND BEAMPATTERN

The far-field narrowband beampattern of a horizontal uniform lin-ear array (ULA) is given by

Authors’ current addresses: F. Uysal, Microwave Sensing, Systems and Signals (MS3), Delft University of Technology, 2628CD, Delft, Netherlands. E-mail: f.uysal@tudelft.nl. Z. Dunn and M. Yeary, University of Oklahoma, Advanced Radar Research Center, 3190 Monitor Avenue, Norman, Oklahoma 73019, USA. R. Rincon, NASA, Goddard Space Flight Center, Code 555, Greenbelt, MD, 20770, USA. This work is partially supported by NASA grant NNX13A-D37A. Any opinions, findings, conclusions, or recommenda-tions expressed in this material are those of the authors and do not necessarily reflect those of NASA. Manuscript received September 14, 2015, revised April 4, 2016, and ready for publication May 11, 2016. Review handled by S. Goldstein. 0885/8985/$26.00 © 2016 IEEE

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(1)

where N is the number of sensor elements, q is the azimuth angle (–p/2 £ q £ p/2), d is the interelement spacing, and l is the operat-ing wavelength.

WIDEBAND BEAMPATTERN

Bandpass signals whose complex envelopes satisfy

(2)

are defined as narrowband signals, where B refers to signal band-width and c is the speed of light [13]. Equation (1) is valid when a signal’s bandwidth is narrow enough to satisfy the assumption of constant wavelength. Depending upon the interelement spacing d, this assumption may not be valid in wideband signals. In this case, it is appropriate to represent the array pattern as a two-dimensional function of frequency and azimuth angle. Thus, the frequency-de-pendent wideband array pattern for a beam that is steered toward a desired azimuth angle qt is given by

(3)

where f is the frequency.Figure 1 depicts the two-dimensional wideband beampattern

for an eight-element ULA (for qt = 0°), displaying some inter-esting properties that highlight the problem to be solved. In this example, the bandwidth is 200 MHz centered at 435 MHz. The antenna elements are spaced by half wavelengths, as calculated at the smallest wavelength within the signal bandwidth (i.e., the high-est frequency fH) [14]. The physical size of the array is fixed, but the electrical size varies with frequency. At the highest frequency, the array’s electrical size is at its largest; hence, the beampattern at this frequency has the narrowest mainlobe and the most nulls. However, as frequency decreases, the wavelength increases, and the electrical length of the array becomes smaller. Therefore, the beampattern widens, and we observe the necessity of the two-di-

mensional frequency-dependent pattern. From the perspective of a single azimuth angle, we see that the illumination pattern varies over the bandwidth of the waveform. For some azimuth angles, the transmit array has a null at some frequencies within the signal bandwidth but not at other frequencies.

It will be useful to define a metric that reduces the two-dimen-sional wideband beampattern to a one-dimensional pattern that can be used as a reference or designed to meet a desirable behavior. Two different metrics appear reasonable for this purpose. First, we could take the maximum of the wideband beam power pattern over all frequencies at a given angle [14], such that

(4)

Alternatively, we can use the total energy of the beampattern over all frequencies at a given angle [11], [14], [15], such that

(5)

In this paper, we use the total energy definition, which seems to be the more commonly used metric in the literature. Figure 2 compares these two metrics, along with the narrowband pattern resulting from the highest frequency present, for the eight-element wideband pattern shown in Figure 1. As seen in the figure, both metrics result in the deep nulls disappearing relative to any nar-

Figure 1. Wideband beampattern angle versus frequency.

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rowband beampattern within the same frequency range due to the frequency-dependent nature of the wideband beampattern. In some radar applications, such as target detection [4], it may be preferred for the signal to have a frequency-independent beampattern (i.e., a pattern with aligned nulls and main beam). In the next sections, we introduce optimization-based techniques to find complex waveform-weighting matrices to achieve frequency-independent wideband beampatterns.

BEAMPATTERN WEIGHTING

For a given ULA with an active array architecture, the array pattern can be controlled directly by altering the waveforms generated in each transmit chain’s digital-to-analog converter. Although amplifiers exhibit nonlinear behavior across their input power range and experience compression at high input powers, radar systems are usually operated in compression, as this results in the maximum power-added efficiency. There exist some approaches for linearly reproducing the desired waveform, including digital predistortion (as described in the “Digital Predistortion” section), which allows the system to linearly reproduce the desired waveform while still operating in the amplifier’s compression region, but in this section, we assume all amplifiers in the system are operating in their linear region and are not in compression. Amplification is assumed to be linear in this section so that the design of the waveform weights can be calculated assuming ideal unweighted wave-forms. This assumption is also valid in a physical system with digital predistortion implemented, as each channel uses its in-dividual weighted ideal waveform as the predistortion input signal, thus leading to accurate physical reproductions of the modeled waveforms.

Under these assumptions, it is possible to find the frequency-dependent [N × F] complex weighting matrix W for a [L × F] given the desired (template) beampattern matrix Y set at target azimuth angle qt. Note that F is the number of frequency bins, N is the number of array elements, and L is the number of beampat-tern azimuth angles q to be evaluated. Each [N × 1] column of the complex weighting matrix W can be represented by the vector wf. Similarly, each [L × 1] column of the template beampattern matrix Y can be represented by the vector y. Both wf and y represent indi-vidual frequencies within W and Y, respectively. The full complex weighting matrix W can be assembled one column (frequency) at a time by minimizing the cost function

(6)

where the [L × N] array steering matrix S is calculated for each frequency f. The nth column (array element) of S is calculated as

(7)

This optimization will result in the set of array weights wf that most closely reproduce the desired one-dimensional beampattern y at each frequency.

To accurately produce the required frequency range with the narrowest mainlobe beamwidth and minimum number of elements (to minimize hardware costs), the interelement spacing for the ULA should be selected as d = c/(2fH), where fH = (fc + B/2) is the highest in-band frequency and fc is the carrier (center) frequency of the desired signal set [11]. This ensures that no grating lobes ap-pear in the beampattern. In this paper, the primary goal is to obtain a set of waveforms that results in a beampattern that is frequency invariant for the intended platform EcoSAR [16]. The secondary goal is to minimize the mainlobe beamwidth of the beampattern. A template beampattern must be defined such that these goals are physically achievable. For a given ULA antenna spacing, the low-est in-band frequency fL = (fc – B/2) results in the widest mainlobe [14]. Therefore the frequency-invariant template beampattern Y should have a mainlobe equivalent to the lowest in-band frequency case to minimize the mainlobe beamwidth, while still allowing a frequency-invariant solution. The array excitation can always be tapered at higher frequencies to produce a wide pattern, but if the template pattern is defined at the highest frequency in the band-width, it may be impossible for the array to achieve the desired beamwidth at lower frequencies. Using the unweighted beampat-tern at the lowest in-band frequency and half element spacing at the highest in-band frequency, the template beampattern can be written as [17]

(8)

To reduce the computational burden and strictly limit the weighting optimization to the frequency bins of interest, we pro-pose to only compute the weights inside the desired bandwidth and to hard set the weights corresponding with frequency bins outside the desired bandwidth. This approach is shown as

(9)

where e is a very small value in relation to the calculated weights.Note that this approach effectively defines the transmit

waveforms by obtaining a set of frequency-dependent weights. In other words, the resulting set of frequency-dependent weights for a particular antenna element wn gives the frequency-domain

Figure 2. Narrowband beampattern compared with one-dimensional wideband beampattern visualization metrics.

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specification of the waveform for that element. A time domain signal set can be achieved by taking the inverse Fourier trans-form of the weights. Unfortunately, it has been observed that the inverse Fourier transform of the weights suffers from the Gibbs phenomenon, which creates high ripples in the time do-main because the resulting weights are strictly band limited, and there is no transition band in the computed weights (the template beampattern is also strictly band limited). This ap-proach can yield waveforms that may not be appropriate for some radar applications. However, these weights can be used to correct the frequency dependency of a selected signal set. Thus, a time domain frequency-independent (weighted) signal set can be achieved by taking the inverse Fourier transform

, where xn is the Fourier domain representation of the desired time domain signal set The proposed approach to achieve a time domain signal set that will result in the cre-ation of a frequency-independent beampattern is highlighted in Table 1.

For demonstration purposes, a 20-μs windowed linear frequency-modulated (LFM) chirp waveform is used as the desired transmit signal. The window used is equivalent to a Hanning window convolved with a rectangular window to smoothly transition the beginning and ending portions of the waveform, while preserving the central portion of the wave-form. The bandwidth of the waveform is selected as 200 MHz, the carrier frequency fc is set to 435 MHz, and the number of frequency bins is selected as 215. Figure 3 shows the resulting weighted beampattern for an eight-element ULA. Note that the waveform, system, and antenna param-eters were chosen due to their similarity or applicability to the NASA EcoSAR platform. Note the similarity in general appearance of Figures 1 and 3, except the latter is nearly uniform across the frequency domain, thus demonstrating the frequency-independent nature of the optimization. Figure 4 depicts the weighted beampattern for the same system fre-quency parameters when the number of array elements is in-

creased to 16, and it can be seen that the frequency-indepen-dent nature of the weighted beampattern is further improved as the number of antenna elements is increased. Notice that due to the proposed in-band optimization approach shown in Equation 9 the achieved beampattern is limited to only in-band frequencies. Unlike the other approaches mentioned in the “Introduction” section, no unwanted lobes outside the desired bandwidth are produced.

SIDELOBE REDUCTION (SR)

In addition to the goal of obtaining a waveform set that results in a frequency-invariant beampattern with a minimal mainlobe beam-width, the additional goal of minimizing the peak sidelobe level can also be implemented. It is possible to reduce the sidelobes of the designed beampattern by constraining the sidelobes of the tem-plate pattern [17]. For example, one such definition is

(10)

where T is the power threshold value and e is a very small value compared to the peak power. The new SR template pattern ySR is then used in place of y in Equation 9. The idea behind the thresh-

Figure 3. Wideband beampattern of weighted LFM signal set for an eight-element ULA.

Figure 4. Wideband beampattern of weighted LFM signal set for a 16-element ULA.

Table 1.

Algorithm of Frequency-Independent Beampattern Design

Input xn,y

for each frequency bin f

1. Compute S

2. Compute weight vector wf (columns of W)

end

Compute time domain waveform for each sensor −1 {xnwn} F

−1 {xnwn}

xn is a (1 × L) vector and represents the Fourier domain representation of the desired signal set.

wn is the (1 × L) weight vector for a particular antenna that is a single row of W.

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olding approach is to reduce the sidelobes, while using the same unconstrained optimization scheme. This approach can be replaced with a constrained optimization or min-max design (not covered in this paper) to achieve the exact peak-to-sidelobe ratios (PSLR).

For an eight-element ULA, the beampatterns of the original wideband waveform case, the weighted wideband waveform case, and the weighted wideband waveform with the SR case are shown in Figure 5, using the total energy metric described in the “Wide-band Beampattern” section, where the beam power is adjusted to the same level for comparison purposes. The power threshold value T in Equation (10) for SR is selected as 11 dB lower than the peak power to preserve the main beam, while suppressing the sidelobes. The parameter e is set to a value of 2−52 (IEEE Standard 754 for dou-ble precision). Note that the weighted wideband beampattern with-out SR processing has the same null structure as the beampattern of a narrowband waveform at the lowest in-band frequency [17], while the weighted waveform with SR processing has a slightly widened first peak, with more of the beampattern’s overall power inside the main peak. In Figure 5, the PSLR of the averaged beampatterns of the original wideband waveform, the weighted wideband waveform (referred to as SR-Off), and the weighted wideband waveform with beampattern SR (referred to as SR-On) are computed as 14.31 dB, 12.64 dB, and 23.54 dB respectively.

DIGITAL PREDISTORTION

As mentioned in the beginning of this section, some approaches exist for linearly reproducing desired waveforms in hardware exhibiting nonlinear output characteristics. The sets of wideband waveforms formed through weighting both with and without SR exhibit noncon-stant envelopes and require high-fidelity generation in both ampli-tude and phase, making the wideband waveforms ideal candidates for digital predistortion as presented in [18]. The main purpose be-hind digital predistortion is to make possible high-quality synthesis of desired waveforms at high amplifier output powers. This is accom-plished by carefully remapping given input signals to different input signals, so when altered by the nonlinear distortion of the high power amplifier (HPA), the desired ideal output signals result. This also al-lows the system to minimize spectral regrowth, while still operating in the compression region of the amplifier, thus maximizing power added efficiency. In practice, this is accomplished through collecting a set of calibration data for the transmit chain that captures the behav-ior of the system over the amplitude and frequency range to be used. Using a memory polynomial (MP) model with predetermined order of memory and nonlinearity to represent the wideband predistortion

function, the calibration data is used as a set of training data to solve for the set of coefficients satisfying the least-squares solution to the MP model. These coefficients are then used by the MP model to pre-distort the desired beamforming waveforms, thus maximizing power added efficiency and minimizing spectral spreading upon amplifica-tion. A high-level flowchart of the digital predistortion methodology is presented in Figure 6. Further explanation of the wideband digital predistortion method used can be found in [18], [19].

DESIGN TRADE-OFFS

EFFICIENCY

Figure 7 depicts the power consumption of the system when weight-ed waveforms are used. The power consumption of the system is defined as 100% for original waveforms, and the weighted wave-forms are all linearly scaled so that the maximum output voltage out of all of the weighted waveforms equals the constant output voltage of the original waveforms. This rescaling results in a decrease in system power utilization for any set of weighted waveforms, where the output power is not constant. As seen from the figure, there is a significant transmit power drop associated with the weighted wave-forms. The maximum power drops are computed as 7.14 dB (at ±12°) and 20.87 dB (at ±30°) for weighted waveforms (SR-Off) and weighted waveforms with SR (SR-On), respectively. At some angles, such as ±20°, the power difference between SR-Off and SR-On waveforms is 1.28 dB, while the PSLR difference between SR-Off and SR-On waveforms is 11.48 dB. Therefore, around 0° and ±20° SR-On waveforms may be preferred over SR-Off waveforms.

EFFECTIVE WAVEFORM AND MATCHED FILTER RESPONSE

In radar systems using active array architectures, the transmit array elements usually operate in a synchronized manner. If each ele-

Figure 6. High-level flowchart of digital predistortion methodology.

Figure 7. Power usage (percentage) for weighted waveforms.

Figure 5. Comparison of averaged wideband beampatterns using total energy metric.

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ment transmits a waveform xn at the same time, these waveforms will arrive at the (nonbroadside) target area at different times. Summation of these waveforms in the target area creates an effec-tive waveform that reflects from the target. The effective wave-form xe (t) in the far field can be defined as

(11)

where tn = n d sin qt/c. In the receiver, the collected data can have the template waveform applied as the matched filter to obtain the maximum signal-to-noise ratio. The matched filter response g(t) of the effective waveform with the template waveform is given as

(12)

It is observed that even though none of the individual transmitted waveforms match the template waveform, the effective waveform has similar envelop and phase characteristics as those of the given template waveform (see Figures 13 and 14).

Low Sidelobe Pulse Compression Waveforms

Once the effective waveform has been formed, the pulse compres-sion characteristics of the received waveform can be analyzed. The benefit of waveforms exhibiting low-pulse compression sidelobe behavior is now realized. The standard LFM chirp PSLR (about −13 dB) may be too large to be useful in some radar applications. The LFM chirp’s PSLR may be increased at the expense of a small amount of transmitted power through the application of a window-ing function. The PSLR may be further increased while maintain-ing the transmitted power and resolution by using a coded wave-form as the template waveform (such as a Frank or P4 code [20]) or by using a continuous nonlinear FM waveform as the template waveform, as in [21], [22], [23].

EXPERIMENTAL RESULTS

An experimental testbed, shown in Figure 8, was assembled to verify the weighted waveforms. The testbed consisted of a Tek-tronix AWG7122C Arbitrary Waveform Generator with a 12-GHz sample generation rate, a Tektronix DPO70604 Digital Phosphor Oscilloscope, with a 25-GHz sampling rate and 6-GHz instanta-neous bandwidth, a Mini-Circuits ZX60-V82-S+ wideband am-plifier used as a preamplifier, and a Specwave QBH-7-4012 high power amplifier (HPA).1 All proposed waveforms were created in MATLAB before being uploaded to the waveform generator, am-plified by the wideband amplifier, again amplified by the HPA, and then sampled by the oscilloscope. For all measurements, the measured signal for each channel was appropriately delayed in the digital domain to simulate the effective waveform at tar-get angle qt. The summation of all eight measured and delayed channels ultimately resulted in the same structure as the selected transmit signal x(t).

1 The actual EcoSAR hardware is different from the testbed shown here; for more details about the current EcoSAR system hardware, please see [16].

System parameters were selected identical to those used in simulation (an eight-channel ULA with 20-μs waveforms, a band-width of 200 MHz, 215 frequency bins, and a carrier frequency fc set to 435 MHz), and the optimization was executed with a main-beam steering angle of qt = 30° off broadside.2 Additionally, the maximum instantaneous output power at any point in the wave-form set was to +5.27 dBm to ensure that the HPA would operate in compression. Therefore, due to the relatively low carrier fre-quency and the large instantaneous bandwidth of the oscilloscope, the oscilloscope was able to capture the created signals at base-band without downconversion and these baseband measurements were downloaded into MATLAB for postprocessing. The generat-ed waveforms were digitally predistorted using the method intro-duced in the “Digital Predistortion” section to mitigate the system distortion, where applicable. The beamforming method was tested in hardware where the transmit signal x(t) was an LFM waveform, as well as where the transmit signal x(t) was a P4 polyphase-coded waveform. Due to the wide instantaneous bandwidth of the P4 polyphase-coded wavform, digital predistortion will result in a dramatic reduction in the spectral spreading experienced by the amplified waveform versus a nonpredistorted amplified version of the same waveform.

2 Note that the maximum allowable steering angle in EcoSAR is ±35° off broadside.

Figure 8. Experimental testbed setup.

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The beampattern resulting from the measured signal set using the LFM waveform transmit signal was created, and it was seen that the mainbeam, sidelobes, and nulls were well aligned in frequency. Therefore, the transmit gain pattern resulting from the weighted LFM waveform transmit signal set is, for all practical purposes, frequency independent. The same experiment was repeated for the weighted waveforms without and with the SR technique imple-mented. The resulting beampatterns are depicted in Figures 9 and 10, respectively. The power inside the mainbeam is preserved and experiences minimal changes across the frequency domain (see Figure 11), while the sidelobes are suppressed. Although all of the nulls are well aligned in the SR-Off weighted beampattern, SR-On processing results in a beampattern in which the first three nulls are aligned well and the other nulls are not well aligned in the angle domain due to the low sidelobe constraint. Figure 12 illustrates the comparison of the measured weighted waveforms both with and without SR implemented, and it can be clearly seen that while the PSLR is increased by SR processing, the peak power is reduced. The effective waveform in the far field and its matched filter re-sponse are computed, as described in the “Effective Waveform and Matched Filter Response” section. As seen from the figure, effec-tive waveform is in the shape of intended LFM signal. The real part of the effective waveform together with estimated envelop are depicted in Figure 13. Figure 14 shows the matched filter results

Figure 11. Angle cuts of beampattern from measured LFM SR-On signal set.

Figure 12. Averaged (total energy metric) wideband gain pattern of measured signals with weighted LFM waveform.

Figure 13. Real component and envelope of effective waveform at target angle (SR-On).

Figure 14.LFM-matched filter responses of effective waveforms with both SR-On and SR-Off.

Figure 15. Wideband gain pattern of measured signal set (SR-On) with P4-coded waveform.

Figure 9. Wideband gain pattern of measured signal set (SR-Off) with LFM waveform.

Figure 10. Wideband gain pattern of measured signal for reduced sidelobes (SR-On) with LFM waveform.

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for both the SR-Off and SR-On cases in which the matched filter outputs are practically identical in shape, with the only difference being the decreased magnitude in the SR-On case due to decreased power transmission, as addressed in the “Efficiency” section.

The beampattern of the measured signal set was calculated us-ing the P4 polyphase-coded waveform transmit signal. The sid-elobes were suppressed and the mainbeam, sidelobes, and nulls were well aligned in the frequency domain. However, there was noticeable roll-off in the mainbeam and sidelobe intensity as the instantaneous frequency departed from the carrier frequency. The P4 waveform was also weighted with the SR technique implement-ed, and the resulting beampattern is depicted in Figure 15. The mainbeam and sidelobe intensity experiences slight roll-off as the instantaneous frequency departs from the carrier frequency. These results demonstrate that weighted beampatterns resulting from coded waveforms, while not totally frequency independent, have significantly improved characteristics. The matched filter result of the effective waveform resulting from the measured weighted P4 waveform set with SR-On is shown versus the ideal P4-matched filter result in Figure 16. The resolution degradation is extremely small and the sidelobes, with a maximum of only 0.67 dB higher than the ideal P4-matched filter case, are much lower than those seen in the LFM waveform cases, shown in Figure 14. In con-junction with the measured LFM waveform results, the measured weighted P4 waveform results demonstrate that the primary trade-off in SR implementation is the reduction of power in exchange for increased PSLR.

CONCLUSION

The presented method for wideband beamforming has been shown, both in simulated and measured results, to create a set of optimally weighted transmit waveforms that results in a constant modulus beampattern in the target area that is, for all practical purposes, fre-quency invariant with low sidelobes at every other position. This proposed frequency weighting method results in a beampattern that has low sidelobes and constant power across the mainbeam, and it also does not exhibit grating lobes outside the bandwidth of interest and varies from very frequency invariant to completely fre-quency invariant, depending upon the initial waveform. The PSLR was shown to be further reduced through introduction of the SR technique at the expense of a small amount of peak power. For both beamforming optimization methods, the power utilization of the system is much lower than that of a similar nonweighted system. The decision to implement the weighted wideband beamforming

in a system must be made on a system-by-system basis, taking into account the implications of the trade-off between system power utilization and a frequency-invariant constant modulus waveform. In combination with digital predistortion, the weighted beamform-ing method was demonstrated through hardware to give effective results for both LFM and P4 polyphase-coded waveforms, show-ing that the presented wideband beamforming method has poten-tial application in any wideband active array system when a precise beampattern is desired.

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Figure 16. Normalized matched filter responses of ideal and measured P4 SR-On effective waveforms.

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12 IEEE A&E SYSTEMS MAGAZINE DECEMBER 2016

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