sensors

Article Introduction to Noise Radar and Its Waveforms

Francesco De Palo 1, Gaspare Galati 2 , Gabriele Pavan 2,* , Christoph Wasserzier 3 and Kubilay Savci 4

1 Department of Electronic Engineering, Tor Vergata University of Rome, now with Rheinmetall Italy, 00131 Rome, Italy; [email protected] 2 Department of Electronic Engineering, Tor Vergata University and CNIT-Consortium for Telecommunications, Research Unit of Tor Vergata University of Rome, 00133 Rome, Italy; [email protected] 3 Fraunhofer Institute for High Frequency Physics and Radar Techniques FHR, 53343 Wachtberg, Germany; [email protected] 4 Turkish Naval Research Center Command and Koc University, Istanbul, 34450 Istanbul,˙ Turkey; [email protected] * Correspondence: [email protected]



Received: 31 July 2020; Accepted: 7 September 2020; Published: 11 September 2020


Abstract: In the system-level design for both conventional radars and noise radars, a fundamental element is the use of waveforms suited to the particular application. In the military arena, low probability of intercept (LPI) and of exploitation (LPE) by the enemy are required, while in the civil context, the spectrum occupancy is a more and more important requirement, because of the growing request by non-radar applications; hence, a plurality of nearby radars may be obliged to transmit in the same band. All these requirements are satisfied by noise radar technology. After an overview of the main noise radar features and design problems, this paper summarizes recent developments in “tailoring” pseudo-random sequences plus a novel tailoring method aiming for an increase of detection performance whilst enabling to produce a (virtually) unlimited number of noise-like waveforms usable in different applications.

Keywords: noise radar technology; noise waveforms; peak sidelobe level; ambiguity function; waveform design; peak-to-average-power-ratio

1. Introduction Noise radar technology (NRT) is based on the transmission of random waveforms as opposed to the classical, often sophisticated, deterministic radar signals [1,2]. Both NRT and “deterministic radar” technology use the “matched filter”, or an approximation of it, to maximize the signal-to-noise ratio (SNR). In a conventional radar, a single waveform (or a finite set of waveforms), is used for transmission and reception with matched filtering. Conversely, NRT is able to transmit a—virtually unlimited—set of realizations (i.e., “sample functions”) of a random process, with the pertaining matched filter set up in real time to implement the well-known “correlation receiver”. NRT was conceived long ago [3], but its practical interest revamped around the 2000s with the advancements in digital acquisition, processing, and generation of signals plus, in the military sector, with the increasing need for low probability of intercept (LPI) radars. The main advantages of NRT are recalled here.

Hard, if not impossible, exploitation of the transmitted signals by an enemy party aimed to • intercept, classify, identify (and possibly jam) the radar. The low probability of intercept, i.e., the reduced visibility of radar signals by ESM (electronic support measures) and ELINT (electronic intelligence) systems [4] is significantly enhanced by the random nature of NRT signals. The low

Sensors 2020, 20, 5187; doi:10.3390/s20185187 www.mdpi.com/journal/sensors Sensors 2020, 20, 5187 2 of 26

probability of intercept and low probability of exploitation by a hostile counterpart will be analysed in more detail in a companion paper to be submitted on the counter-interception and counter-exploitation features of NRT. Limitation of the mutual interferences to/from many radars located in a crowded environment • and operating in a limited frequency band. In fact, using NRT the interfered radar perceives and treats the interference as a kind of noise [5,6]. This feature will be more and more important in the developments of marine radars, due to their use of solid-state transmitters, having much greater “duty cycle” (longer pulses) than the legacy magnetron [7], as well as in automotive applications, [8] because of the more and increasing number of used automotive radars, causing a high spatial density of these particular radar sets. Favourable shape of the “ambiguity function”. The ambiguity function [1,2] of a radar waveform • defines its “selectivity” both in distance, i.e., in radar range (proportional to the time delay) and in radial velocity (proportional to the frequency shift due to the Doppler effect). The ambiguity function of a conceptual waveform made up by an infinitely long realization of “white noise” (i.e., with duration T and bandwidth B both going to infinity) tends to a Dirac’s “delta-shape” function with an ambiguity level going to zero. However, in all real applications both B and T are limited, and the peak side lobe ratio (PSLR) of the ambiguity function has a mean value close to the “processing gain” B T (hereafter, BT). Both bandwidth and duration are practically · limited. The limitations on B come from many factors, including the constraints on the usable portion of the electromagnetic spectrum [9]. Those on T depend (i) on the dynamic behaviour (“decorrelation time”, TC) of the radar cross section of most targets and (ii) on the dwell time, TD, and the pertaining type of radar processing. Echoes from moving targets (vehicles) have some typical decorrelation time TC (related to the signal fluctuation) of the rough order of hundreds of milliseconds. Doppler effect may pose a further limitation on the coherent signal integration time T, imposed to be smaller than both TC and TD, and typically in the order of tens or hundreds of milliseconds. With an exemplary bandwidth of 50 MHz and an exemplary value of T = 20 ms, the processing gain of a continuous-emission (CE) noise radar is of the order of one million. The resulting average PSLR is about 60 dB, which is acceptable in many radar applications, when the “fourth power of distance” term does not affect too much the needed dynamic range.

A comparison between the ambiguity function of a typical application of the NRT (Figure1a) and of two deterministic radar signals (non-linear frequency modulation (NLFM), Figure1b, and 13-elements Barker code [2], Figure1c) are shown in Figure1. Concerning practical radar implementations, the disadvantages of NRT are mainly due to the use of a real-time correlation receiver/processor, with a burden, for computation and analog-to-digital conversion, significantly greater than a classical frequency modulated, continuous wave (FM-CW) radar. The sampling rate of digital correlation in NRT needs to cover the full frequency band, e.g., 50 MHz when a resolution of 3 m is required. On the other side, FM-CW signals are treated by the de-ramping (or homodyne) receiver principle [10], where the correlation is performed by “de-chirping”, followed by a bank of video filters. This operation is based on beating the received signal with a replica of the transmitted one (a frequency ramp), extracting the baseband spectrum via analog filtering and finally obtaining the correlation; hence, the “Range” of targets, via fast Fourier transform (FFT). This procedure permits the usage of analog-to-digital converters (ADCs) with low sampling rates and high dynamic range, as not the full RF bandwidth but a much narrower base-band converted signal [11] has to be sampled, with a relatively low maximum frequency related to the maximum target distance. Sensors 2020, 20, 5187 3 of 26 Sensors 2020, 20, x FOR PEER REVIEW 3 of 25

Figure 1. Ambiguity function function (values (values in in dB dB for ( a,b)) and in linear scale for (c)): ( a) Noise waveform (BT(BT = 100);100); ( (bb)) non-linear non-linear frequency frequency modulated modulated (NLFM) (NLFM) “chirp” signal (BT = 100),100), and and ( (cc)) 13-elements 13-elements Barker code. The The low low and impractical BT values are here chosen for the clarity of display.

In mostmost FM-CWFM-CW radars radars the the duration durationT of the of rampthe ramp is related is related to the maximumto the maximum delay T maxdelay= 2 Rmax/=c 2(compatible/ (compatible with the maximumwith the maximum range Rmax range) by T max )T /byβ with β≤/ > 1. Thewith related >1 energy. The related loss at ≤ energyRmax is loss equal at toL = 10 is equallog10( to1 =10∙1/β). To limit(1L −below 1/). one To limit dB, β must below be notoneless dB, than must five be (for notβ =less5, · − thanL  0.97 fivedB; (for for =5β = 6, ,L ≅ 0.970.8 dB). dB;However, for =6 once, ≅0.8 fixed dB). both However,B (by the internationalonce fixed both regulations (by the on internationalthe electromagnetic regulations spectrum) on the and electromagneticTmax (Rmax) by spectrum) the application, and an ( increase) by of theβ; application, hence, of T foran increasethe prescribed of ; Thence,max, implies of T for a reduction the prescribed of the slope B, /impliesT of the afrequency-vs-time reduction of the law.slope The / resulting of the frequency-vs-timeincrease of the unwanted law. The Range–Doppler resulting increase coupling of the makes unwanted the FM-CW Range–Doppler radar less coupling flexible inmakes terms the of FM-CWchoice of radarT than less a continuousflexible in terms emission of choice (CE) noiseof T than radar. a continuous emission (CE) noise radar. For a range range resolution resolution of of 33 m, i.e., a height height of of the the ramp ramp =50B = 50 MHz and a a duration duration =1T = 1 ms

(compatible with with a a maximum maximum range range Rmax =30= 30 km,km, i.e., β=5= 5),), thethe beatbeat frequencies,frequencies, in the FM-CW case, are are between between 0 andand 1010MHz, MHz, i.e., i.e., the the sampling sampling raterate isis fivefive timestimes lessless thanthan the NRT, with and a much cheaper processing than a full correlation. This example clarifies clarifies that noise radar technology can be preferred (for example, over the the simple simple FM-CW FM-CW solution) solution) when when there there are are precise precise requirements requirements concerning eithereither a a high high resolution resolution in Range in Range and inand Doppler in Doppler (calling (calling for a “thumbtack-like” for a “thumbtack-like” ambiguity ambiguityfunction of function the pseudorandom of the pseudo signals),random or thesignals), radiation or the of manyradiatio “nearly-orthogonal”n of many “nearly-orthogonal” signals with signalsminimal with “cross-correlation” minimal “cross-correlation” and LPI/LPE and features. LPI/LPE features. Having defineddefined thethe mainmain signalsignal parametersparameters (duration,(duration, bandwidth,bandwidth, ...…),), another remaining flexibilityflexibility in the system design of a noise radar is the possibility to select the peak-to-average power ratio (PAPR) (PAPR) of each radiated radiated waveform waveform ()g(t), here defineddefined by its its samples samples g[[k]]with withk =1,2,…,= 1, 2, ... , N,, being fs the the sampling sampling frequency frequency and andN == fs T∙: : · max|[ ]| 2 = max g[k] 1 k (1) PAPR =∑| [ ]| (1) 1PN 2 N k=1 g[k] The FM-CW signals and the other deterministic waveforms (e.g., chirp, Barker, Polyphase codes, etc. [1,2]) generally have = 1, i.e., exhibit phase/frequency only modulation. Because of their unitary modulus, they can be named “unimodular”. Of course, an arbitrary, or noise, waveform (unless hard-limited, i.e., amplitude saturated) has a greater than the unity, which depends

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The FM-CW signals and the other deterministic waveforms (e.g., chirp, Barker, Polyphase codes, etc. [1,2]) generally have PAPR = 1, i.e., exhibit phase/frequency only modulation. Because of their unitary modulus, they can be named “unimodular”. Of course, an arbitrary, or noise, waveform (unless hard-limited, i.e., amplitude saturated) has a PAPR greater than the unity, which depends on its statistics. Amplitude saturation upstream the matched filter reduces the signal energy; hence, causing a detection loss: Table1 shows the signal-to-noise ratio (SNR) losses as related to the PAPR, i.e.: Loss[dB] = 10 log (PAPR). Amplitude saturation losses are not present when using a unimodular − · 10 noise waveform, i.e., when saturation/hard limiting is applied in the waveform generation (i.e., choosing a “phase-only modulation”). However, a unimodular waveform has a reduced number of degrees of freedom, i.e., a number BT of phase values in place of 2BT “real” degrees of freedom, i.e., amplitude and phase pairs, or equivalently, I and Q (in-phase and quadrature) pairs.

Table 1. Signal-to-noise ratio (SNR) loss (dB) versus peak-to-average power ratio (PAPR).

PAPR Loss (dB) 1.0 0.0 1.25 0.97 1.5 1.76 1.75 2.43 2.0 3.01 3.0 4.77 5.0 6.99 7.0 8.45 10.0 10.0

An effective noise radar operation requires a large enough BT granting: (i) the radiated signals to appear “noise-like”, i.e., random to any hostile counterpart, (ii) high range resolution, and (iii) detectability of small targets in the presence of the Range (and possibly, Doppler) sidelobes of the response of stronger targets at different ranges. These “interfering” targets may have not only a large radar cross section, but also shorter ranges with respect to the wanted targets, causing very strong disturbing returns. Hence, spectrum and time resources have to be fully exploited, leading to a continuous emission (CE) [12] preferred architecture, as shown in Figure2a,b. Note that in Figure2 the unavoidable leakage due to antenna coupling has been shown. The architecture shown in Figure2b circumvents the problem of the non-ideal transmitting channel but implies the cost of a wideband reference channel and of an additional fast analog to digital convertor (ADC). A final consideration is that continuous emission is needed to best exploit the LPE feature: the absence of the edges of pulses (as well as the random modulation) makes it difficult to extract the signal features. This relevant topic, beyond the scope of this introductory paper, will be treated in an ensuing work. As anticipated, the “leakage” from the transmitting to the receiving path due to the antenna coupling (Figure2) is a critical problem with CE operation. This e ffect is someway similar to a strong target very close to the radar and depends on the antenna system and the RF connections. Coupling affects the radar operation at close or medium distance due to the sidelobes of the autocorrelation function of the radiated waveform. In addition to a careful antenna design, an effective way to counteract this damaging phenomenon is to transmit waveforms with low Range sidelobes. Sensors 2020, 20, 5187 5 of 26 Sensors 2020, 20, x FOR PEER REVIEW 5 of 25

(a)

(b)

FigureFigure 2. 2.Basic Basic Block Block Diagram. Diagram. ((aa)) TheThe reference is is the the digital digital record record of of the the transmitted transmitted code; code; (b) ( bthe) the referencereference is theis the record record of the of transmittedthe transmitted signal sign at theal at antenna the antenna port; ADC port;= ADCAnalog-to-Digital-Converter; = Analog-to-Digital- DACConverter;= Digital-to-Analog-Converter. DAC = Digital-to-Analog-Converter.

2.2. Noise Noise Waveforms Waveforms Properties Properties

2.1.2.1. Deterministic Deterministic Waveforms Waveforms and and Noise Noise Waveforms Waveforms ConcerningConcerning the the propertiesproperties ofof thethe NRTNRT waveforms, it it is is useful useful to to clarify clarify the the significance significance of of “randomness”“randomness” in in this this context. context.Of Of course,course, if randomness is is used used in in its its meaning meaning of of“unpredictability”, “unpredictability”, onlyonly “pseudorandom” “pseudorandom” signals signals may may bebe generatedgenerated and transmitted. transmitted. A A pseudorandom pseudorandom sequence sequence [13] [13 is] is defineddefined (in (in cryptology, cryptology, for for example), example), as as oneone whichwhich passe passess some some statistical statistical tests tests suited suited to tothe the particular particular application.application. In In the the military military field, field, suchsuch aa signalsignal is unknown to to the the enemy enemy and and unpredictable unpredictable by by it. it. InIn principle, principle, noise from from physical physical phenomena phenomena (e.g., (e.g., “thermal “thermal noise” noise”in amplifiers in amplifiers or Zener diodes) or Zener diodes)can be canused be as used a random as a random source. However, source. However, ordinary analogue ordinary noise analogue generators noise suffer generators from important suffer from importantlimitations limitations in accuracy/resolution in accuracy/resolution and dynamic and dynamic range, often range, making often makingtheir use their impractical use impractical and andgenerally generally not not complying complying with with most most applications. applications. Hence, Hence, as asit happens it happens in inmost most cryptographic cryptographic applications,applications, the the preferred preferred implementationimplementation of the NRT NRT waveforms waveforms is is the the “digital” “digital” one, one, i.e., i.e., using using a a pseudo-random-number-generatorpseudo-random-number-generator (PRNG) (PRNG) [ 13[13].]. AsAs the the desirable desirable LPILPI/LPE/LPE capabilitiescapabilities ofof NRTNRT relyrely on on the unpredictability of the signals, in principle this feature would be compromised if pseudo- the unpredictability of the signals, in principle this feature would be compromised if pseudo-random random sequences should be used instead of “pure noise”. However, nowadays it is easy to generate sequences should be used instead of “pure noise”. However, nowadays it is easy to generate pseudo-random signals of such complexity, and with a repetition period [13,14], so long as to make pseudo-random signals of such complexity, and with a repetition period [13,14], so long as to make signal analysis and reproduction practically unfeasible. signal analysis and reproduction practically unfeasible. However, a “pure digital noise”, e.g., a Gaussian noise sequence with some suitable spectral densityHowever, in the a radar “pure band, digital is noise”,not always e.g., a Gaussiangood choice noise for sequenceNRT systems with because some suitable of two spectralmain densityproblems. in the First, radar a band,“PAPR is problem”: not always most a good radar choice tran forsmitters NRT systemsoperate becausein saturation, of two i.e., main emit problems. with First, a “PAPR problem”: most radar transmitters operate in saturation, i.e., emit with constant

Sensors 2020, 20, 5187 6 of 26 envelope; the penalty in the radar power budget for transmitting noise waveforms with a relatively large PAPR is shown Table1 and may be too large for many applications. Secondly, in noise radar the correlation processing of echoes of duration T and bandwidth B generates the well-known “random fluctuations” of the sidelobes as described in the following. In any ordinary radar using a “deterministic” waveform g(t) the peak sidelobe ratio (PSLR) is defined as the ratio between the highest sidelobe of the autocorrelation function (ACF) of g(t) and the maximum value of the ACF. In noise radar, due to the “stochastic” nature of the sidelobes (Section 2.2), we have to define a (statistical) Peak Sidelobe Level PSL, which is not exceeded with a given probability. Such a PSL depend on the particular type of noise waveform and, for the reasons explained in the following, can be estimated as

PSL = 10 log (BT) + K (2) dB − · 10 with K being a constant value, depending on the chosen probability and generally in the range 10–12 dB. The statistics of PSL are analysed in Section 2.2. Nowadays, interest is growing in “tailored” pseudorandom waveforms able to suppress the range sidelobes of the autocorrelation function in a particular delay interval. Powerful algorithms, in particular those said CAN (cyclic algorithm new), are analysed in [15]. Their main limitations are mostly related to the bandwidth constraints often associated to each operating environment. To mitigate these limitations a novel algorithm, named band limited algorithm for sidelobes attenuation (BLASA), has been presented in [16]. A much simpler technique to suppress the sidelobes in a range interval of interest will be described in Section3 with some results. Its main advantage over CAN and BLASA is their low computational burden, especially for high BT values.

2.2. Statistical Properties of Autocorrelation Function, Sidelobes, and Power Spectrum Let g(t) = X(t) + jY(t) be the complex envelope of a realization of an ergodic, stationary, and band-limited random process. The autocorrelation function (ACF) of g(t) is defined as

+ Z ∞ R(τ) = g∗(t)g(τ + t)dt (3)

−∞ The power spectral density (PSD) of the complex envelope, i.e., the Fourier transform of R(τ), is supposed as symmetrical around f = 0, with bilateral band B, i.e., from B to + B . Hence, the real − 2 2 part X(t) and the imaginary part Y(t) of g(t) give an equal (50%) contribution to R(τ), which is a real function. After a maximum delay T/2 we may suppose that R(τ) practically vanishes, being T the “coherent processing interval”, whose value is related to the “time on target” (depending on the antenna rotation rate, and/or on the target velocity) and to the echo decorrelation. In practice T is much greater than the delay of the farthest target Tmax, which defines the maximum range of interest cTmax Rmax = 2 . def We consider a sampled version of the ACF, i.e., R(kTs) = R[k] obtained by sampling g(t) at frequency F = k 1 = k B where k is the oversampling factor. In practical applications it is set k > 1, s F Ts F F F but for the sake of simplicity, we consider here the ideal case kF = 1. Hence, each waveform at hand is def described by a number NF = int(B T) 1 of (theoretically) uncorrelated samples. These NF samples ·  barely contain the information embedded in the random process g(t), i.e., its PSD S( f ) and the related Fourier transform R(τ) in the interval T < τ < + T . − 2 2 The sampled version of R(τ) is

+NF/2 X NF NF R[k] = g∗ g + k + (4) i · i k − 2 ≤ ≤ 2 i= N /2 − F Sensors 2020, 20, 5187 7 of 26

 with max R[k] = R[0]. Let us consider the ratio: k

PNF/2 [ ] g∗ gi+k def R k i= NF/2 i · ρk = = − (5) PN /2 2 R[0] F g i= N /2 i − F which of course represents the sample correlation coefficient of the process. For a real process, ρk is real with values in the range 1 ρ +1; the statistical distribution of ρ has been studied by − ≤ k ≤ k Fisher [17], who showed that when NF 1, ρ is approximately Gaussian-distributed with expected  k value ηρk  ρTrue (i.e., the true correlation coefficient of the generating random process; we do not k 2 1 mention its dependence on for the sake of simplicity) and variance σρk  N 3 . Note that ρTrue is the F− “true value” of the correlation coefficient, while ρk is the “sample” correlation coefficient.

Defining the absolute value r = ρ , when considering the case of interest in which ρTrue 1, i.e., k  the sidelobes region of the autocorrelation function, simple computations show that, for a real signal, q 2 1 the random variable r has an expected value of ηr = . π · NF

For a complex process g(t), when ρTrue  1, it results Im[ρk]  0 having supposed the PSD symmetrical around f = 0 and NF 1. Decreasing ρTrue , Im[ρk] increases and when ρTrue  0  (sidelobes region of the ACF) the real and the imaginary part of ρk give a nearly equal contribution to ρk , being both, approximately, Gaussian-distributed with zero mean value (ρ  0) and same variance True 2 2 1 σ = σ  . Hence, for ρTrue  0, the sidelobes have a random behaviour and the output of Re[ρk] Im[ρk] 2NF the correlation filter (we assume that matched filtering is used) is a complex signal similar to receiver noise, i.e., the aforementioned normalized sidelobe level r = ρk becomes a Rayleigh-distributed random variable with the well-known “probability density function” (PDF) f (r) and the “cumulative distribution function” (CDF) F(r):   2 r  r  f (r) = exp U(r) (6) 2 − 2  aRay 2aRay    2   r  F(r) = 1 exp U(r) (7)  − − 2  2aRay

2 def 1 where U( ) is the unitary step and a = . In the following, we substitute NF with the more · Ray 2NF significant product BT; hence, the mean value and the standard deviation of r result: r r r π 1  π 4 ηr = σr = 2 = ηr 1 (8) 4BT 2BT − 2 π −

Now, we may rewrite the CDF of r, or its complement Q(r) = 1 F(r), as a function of the − parameter K (in dB) introduced in Equation (2) and rewritten here:

rK(dB) = 10 log (BT) + K (9) − · 10 Equation (9) in linear scale becomes

rK(dB) 1 0.05 K rK = 10 20 = 10 · (10) √BT · and the resulting probability P r > rK = Q(rK) is { }

r2 k 2a2 2 0.1 K − Ray r BT 10 · Q(rK) = e = e− k · = e− (11) Sensors 2020, 20, 5187 8 of 26

 For K = 0 dB, it results P r(dB) > 10 log (BT) = e 1  0.3679. Varying K from 0 dB up to 14 dB, − · 10 − EquationSensors (11)2020, is20, drawn x FOR PEER as in REVIEW Figure 3. 8 of 25

Figure 3. Probability of the event r > rK , rK given by Equation (11), versus K in dB. Figure 3. Probability of the event{ { >} }, given by Equation (11), versus in dB. From Equation (11), the Rayleigh CDF of r results: From Equation (11), the Rayleigh CDF of results:

0.1.∙K 10 · F((rK)) ==1−1 e− (12)(12) − Hereafter we will assume ≫ 1 and in the sidelobes region (i.e., for | | ≅0) the PSL, i.e., Hereafter we will assume BT 1 and in the sidelobes region (i.e., for ρTrue  0) the PSL, i.e., ≝max{ } 1− def , can be defined as the one having a prescribed probability (hereafter named ) of rp = max rK , can be defined as the one having a prescribed probability (hereafter named 1 δ) of not not being{ }exceeded in the whole sidelobe region of the ACF (with about ≅ sidelobes).− Outside being exceeded in the whole sidelobe region of the ACF (with about NF  BT sidelobes). Outside the the main lobe region, the CDF of the maximum sidelobe , Equation (12), becomes main lobe region, the CDF of the maximum sidelobe rp, Equation (12), becomes .∙ ()=[()] =1− (13) h 0.1 K iNp Np 10 · where = −1. Fp(rK) = [F(rK)] = 1 e− (13) − For =0,n o() = [1− ] ≅0 being ≫1 ( ≫ 1) and the probability that = BT whereexceedsNP −10int ∙2 1 (). is very close to the unit. − h iN ( ) =1− 1 p ForPosingK = 0, Fp(r ) = 1 e and solving 0 being EquationNp (13),1 (BT the corresponding1) and the probability value of that (inrp dB)exceeds is 0 − −   10 log10(BT) is very close to the unit. − · =10∙ − 1 − √1− (14) Posing Fp(rK) = 1 δ and solving Equation (13), the corresponding value of K (in dB) is − Figure 4 shows Equation (14) varying from 10 to 10 and from 10 to 0.9 . h  Np i K = 10 log10 ln 1 √1 δ (14) Probability values in the order of 0.05· to 0.2− justify− in −Equation (2) the values of in the range 10 − 12 dB to estimate the for the, usually large, values of (order of 10 −10). 3 6 3 Figure4 shows Equation (14) varying BT from 10 to 10 and δ from 10− to 0.9. Probability values δ in the order of14 0.05 to 0.2 justify in Equation (2) the values of K in the range 10–12 dB to estimate the PSL for the, usually large, values of BT (order of 104–106). The sidelobes13 structure depends on the power spectral density of the generating noise. In the 6 following it will be analyzed first in the exemplary case of uniformBT = 10 spectrum, then in the case of 12 spectral shaping. 11 BT = 106 5 K (dB) 10 BT = 5x10 BT = 105 4 9 BT = 5x10 BT = 104 BT = 103 BT = 5x103 8 BT = 2x103 BT = 103 7 0.001 0.002 0.005 0.01 0.02 0.03 0.05 0.1 0.2 0.3 0.5 0.9 = P[r (dB) > -10log (BT) + K] p 10 Figure 4. Parameter (dB), as defined in Equation (2), versus with BT varying, Equation (14).

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Figure 3. Probability of the event { > }, given by Equation (11), versus in dB.

From Equation (11), the Rayleigh CDF of results:

.∙ () =1− (12)

Hereafter we will assume ≫ 1 and in the sidelobes region (i.e., for || ≅0) the PSL, i.e., ≝max{}, can be defined as the one having a prescribed probability (hereafter named 1−) of not being exceeded in the whole sidelobe region of the ACF (with about ≅ sidelobes). Outside the main lobe region, the CDF of the maximum sidelobe , Equation (12), becomes

.∙ ()=[()] =1− (13) where = −1. For =0, () = [1− ] ≅0 being ≫1 ( ≫ 1) and the probability that exceeds −10 ∙ () is very close to the unit. Posing () =1− and solving Equation (13), the corresponding value of (in dB) is

=10∙ − 1 − √1− (14)

Figure 4 shows Equation (14) varying from 10 to 10 and from 10 to 0.9 . SensorsProbability2020, 20 ,values 5187 in the order of 0.05 to 0.2 justify in Equation (2) the values of in the 9range of 26 10 − 12 dB to estimate the for the, usually large, values of (order of 10 −10).

14

13 BT = 106 12

11 BT = 106 5 K (dB) 10 BT = 5x10 BT = 105 4 9 BT = 5x10 BT = 104 BT = 103 BT = 5x103 8 BT = 2x103 BT = 103 7 0.001 0.002 0.005 0.01 0.02 0.03 0.05 0.1 0.2 0.3 0.5 0.9 = P[r (dB) > -10log (BT) + K] p 10 FigureFigure 4. 4. ParameterParameter K (dB), (dB), asas defineddefined inin EquationEquation (2),(2), versusversus δwith with BT BT varying, varying, Equation Equation (14). (14). 2.2.1. Noise Waveforms with Uniform Power Spectrum h i First, we consider a noise with bandwidth B and “uniform” power spectrum between B , + B : − 2 2 1  B S ( f ) = rect f (15) 0 B − 2

B2 The variance (mean square frequency value) of S0( f ) is 12 and the corresponding ACF, equal to the correlation coefficient, is sin(πBτ) ρ (τ) = (16) 0 πBτ

A “transition delay” τ∗ may be defined equating the sidelobes envelope, given by the denominator of Equation (16), to the average random sidelobes level given by Equation (8): r π . 1 ηr = = (17) 4BT πBτ∗ q corresponding to = 4T . This delay divides the time axis in two intervals: the first (0, ) in τ∗ π3B τ∗ τ∗ which the different realizations of the normalized ACF are very close to each other, in the main lobe and, possibly, in the first sidelobes; the second: τ > τ∗ (sidelobes region, after the transition zone), where the p π estimated mean value of the normalized ACF converges to the theoretical one, i.e., 4BT . In fact, for a single realization of the ACF, the sidelobes fluctuate according to the Rayleigh law, Equation (6). Some computer simulation, shown in the following, highlight the above considerations. Let us consider an exemplary transmitted signal with Gaussian statistics, duration T = 100 µs and rectangular power spectrum with B = 50 MHz (BT = 5000). In the sidelobes region, the theoretical mean value of r p π results, from Equation (8), ηr =  38.04 dB and τ B  25.4, i.e., there are 24 sidelobes out of the 4BT − ∗ main lobe zone in time interval (0, τ∗). Figure5 shows the result of a simulation of 1000 trials with a comparison between the correlation coefficient r of a single realization, the sample mean ηr and the theoretical value of the ACF, Equation (16). The dashed horizontal line represents the theoretical mean value ( 38.04 dB) in the sidelobes region. − In the range 50 < τB 2500 of a simulated autocorrelation, a sample of 2450 realizations of r has been ≤ recorded. To get a quantitative evaluation of the simulative results, we have applied to this data-set the “Chi-squared-test” with a significance level α = 0.05. Sensors 2020, 20, 5187 10 of 26 Sensors 2020, 20, x FOR PEER REVIEW 10 of 25

Noise, B(MHz) =50, T( s) =100, BT =5000 0 Mean on 1000 Trials Theoretical Single realization -10 Mean in Sidelobe Region

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-50 0 5 10 15 20 25 30 35 40 B Figure 5.5.Normalized Normalized autocorrelation autocorrelation of aof single a single realization: realization: comparison comparison between thebetween single realization,the single therealization, sample mean,the sample and the mean, theoretical and the value. theoretical value.

The “null” hypothesis: In agreement with Equations (6) and (7), the probability density function of the PSL (i.e., of ) def is: H0 = {the random variable r is Rayleigh-distributed with PDF given by Equation (6)} is not rejected with a high p-value of 0.9. () = ∙[()] ∙ () = ∙1− ∙() (18) In agreement with Equations (6) and (7), the probability density2 function of the PSL (i.e., of rp) is:

By simulation on 10,000 trials with = 2000, applyingr2 theNp Chi-squared-test1 r2 with =0.05, the Np 1 − r fp(r) = Np [F(r)] − f (r) = Np 1 e− 4BT e− 4BT U(r) (18) null hypothesis: · · · − 2BT · ≝ {the random variable is distributed with PDF given by Equation (18)} is not rejected By simulation on 10,000 trials with N = 2000, applying the Chi-squared-test with α = 0.05, with a high p-value of 0.6. p the null hypothesis: def 2.2.2.H Noise0 = {the Waveforms random variablewith “Shaped”rp is distributed Power Spectrum with PDF given by Equation (18)} is not rejected with a high p-value of 0.6. In most NRT applications the time delays of interest respect the inequality ≪, and the 2.2.2.“uniform” Noise spectral Waveforms density with “Shaped”() in Equation Power Spectrum(15) may be made more convenient in the region of interest using an “m-order convolution” to shape the spectrum in order to reduce the PSL: In most NRT applications the time delays τ of interest respect the inequality τ T, and the 1 1  “uniform” spectral densityS ( f ) in Equation (15) may be made more convenient in the region of (0) = − ⨂ − (19) interest using an “m-order convolution” to shape2 the spectrum in order2 to reduce the PSL: leading to normalized ACF:   (  )    WB 1 sin B 1 B S ( f )= (rect) =f m rect f (20)(19) m B −2 ⊗ B − 2 leadingThis to generalization, normalized ACF: hereafter named “” for wideband, assumes an unlimited availability of " #m+1 spectrum. In fact, with increasing,WB the powersin (spectrumπBτ) tends to a Gaussian shape with an ρ (τ) = (20) increasing variance equal to ( + 1)m , i.e., to a widebandB noise, corresponding to a narrower and π τ narrowerThis generalization,main lobe of the hereafter autocorrelation named “function.WB” for wideband, assumes an unlimited availability of spectrum.Hence, In fact,the withfollowingm increasing, evaluations the power consider spectrum a more tends realistic to a Gaussian generalization shape with named an increasing “” variance(normalized-variance) equal to 1 (m reduces+ 1)B2, i.e.,the tospectrum a wideband variance noise, (back corresponding to the “original” to a narrower value and) applying narrower to 12 mainthe frequency lobe of the axis autocorrelation the scaling factor: function.

/ =(+1) (21) The resulting spectrum and the corresponding normalized ACF are

1 () = (22)

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Hence, the following evaluations consider a more realistic generalization named “NV” B2 (normalized-variance) reduces the spectrum variance (back to the “original” value 12 ) applying to the frequency axis the scaling factor:

1/2 αm = (m + 1)− (21)

The resulting spectrum and the corresponding normalized ACF are

Sensors 2020, 20, x FOR PEER REVIEW ! 11 of 25 NV 1 WB f Sm ( f ) = Sm (22) αm αm ( ) sin" #m+1 NV() = sin(πBαmτ) (23) ρm (τ) = (23) πBαmτ −3 −6 −10 −20 −40 InIn Table Table2 the2 the normalized normalized bandwidths bandwidths (wrt (wrtB ) at) at 3, , 6, ,10 , 20, , and, and40 dB are dB shown are shown for = 1, 2, 3, 4, 5, and 10 − − − − − mfor= 1, 2, 3, 4, 5, and 10. They. They are are evaluated evaluated by applyingby applying Equations Equations (19) (19) and and (22). (22). We observeWe observe that forthat for ≥2 the bandwidth ≅, highlighted in bold face in Table 2. Figure 6 shows, for = m 2 the bandwidth B 6  B, highlighted in bold face in Table2. Figure6 shows, for m = 1, 2, and 5 1,≥ 2, and 5 − by simulation by ofsimulation 1000 trials, ofthe 1000 mean trials, values the mean of the values spectrum. of the spectrum.

TableTable 2. 2.Normalized Normalized bandwidth bandwidth ( (3,−3, 6,−6, −10,10, −20,20, andand −4040 dB)dB) ofof thethe spectrumspectrum forfor “NV”“NV” waveforms.waveforms. − − − − − “NV” Waveforms “NV” Waveforms m B−3/B B−6/B B−10/B B−20/B B−40/B m B 3/BB 6/BB 10/BB 20/BB 40/B 1 0.67 − 1.06− −1.27 − 1.34 − 1.41 2 10.73 0.67 1.021.06 1.271.28 1.341.42 1.41 1.72 2 0.73 1.02 1.28 1.42 1.72 3 30.72 0.72 1.001.00 1.261.26 1.421.42 1.93 1.93 4 40.70 0.70 0.980.98 1.251.25 1.681.68 2.06 2.06 5 50.71 0.71 0.990.99 1.241.24 1.411.41 2.15 2.15 10 100.69 0.69 0.980.98 1.251.25 1.411.41 2.32 2.32

FigureFigure 6. 6.Spectrum Spectrum of of the the “Normalized “Normalized Variance”, Variance”, i.e. “NV”, i.e. “NV”, noise noise waveform waveform (estimated (estimated on 1000 on trials), 1000 BTtrials),= 5000. BT Dashed-line= 5000. Dashed-lin showse the shows Gaussian the Gaussian spectrum, spectrum,m = 1, 2, m 5. = 1, 2, 5.

AsAs one one could could expect expect because because of theof the “central “central limit limit theorem”, theorem”, for m forequal equal to (or greaterto (or greater than) some than) B2 quitesome low quite value low (e.g.,value for (e.g.,m for5 ),≥5 the spectrum), the spectrum is very is similar very similar to the to Gaussian the Gaussian one with one with variance variance. ≥ 12 Figure 7 shows the corresponding mean values of the normalized autocorrelation. We note that . Figure 7 shows the corresponding mean values of the normalized autocorrelation. We note that (i)(i) thethe main main lobe lobe width width is is ≅11/B⁄; and; and q π p π (ii) the mean value of the PSL is ηr  (regardless of m), a value very close to corresponding (ii) the mean value of the is 4B6T≅ (regardless of ), a value very4BT close to to m = 0. corresponding to =0.

Sensors 2020, 20, 5187 12 of 26

A third generalization of the spectrum considers situations in which the spectral occupancy is h i “severely-controlled” (hereafter named “SC”), i.e., the whole PSD has to strictly stay in the B , + B − 2 2 interval. In this case the scaling factor, corresponding to the one of Equation (21), becomes

1 βm = (m + 1)− (24)

The resulting spectrum and the corresponding normalized ACF are ! SC 1 WB f Sm ( f ) = Sm (25) βm βm

" #m+1 SC sin(πBβm τ) ρm (τ) = (26) πBβm τ Table3 reports the normalized bandwidths. Increasing m, the bandwidth is severely reduced (Figure8) and the main lobe-width and the mean values of the normalized ACF in the sidelobes region

Sensorsincrease 2020 as, 20 shown, x FOR in PEER Figure REVIEW9. 12 of 25

Figure 7. Normalized autocorrelationautocorrelation ofof “NV” “NV” noise noise waveforms waveforms for for BT BT= 5000= 5000 (values (values are are estimated estimated on 1000on 1000 trials), trials),m = m1, = 2, 1, and 2, and 5. 5.

A third generalization of the spectrum considers situations in which the spectral occupancy is Table 3. Normalized bandwidth ( 3, 6, 10, and 20 dB) of the spectrum for “SC” waveforms. “severely-controlled” (hereafter named− “”− −), i.e., the− whole PSD has to strictly stay in the − ,+ “SC” Waveforms interval. In this case the scaling factor, corresponding to the one of Equation (21), becomes m B /BB /BB /BB /B 3 6 10 20 − =(+1)− − − (24) 1 0.50 0.75 0.90 0.95 The resulting spectrum and2 the corresponding0.42 0.59 normalized0.74 ACF 0.82are

3 0.36 10.50 0.63 0.71 () = (25) 5 0.29 0.40 0.51 0.58

10 0.21 (0.29 ) 0.38 0.43 sin () = (26) Table 3 reports the normalized bandwidths. Increasing , the bandwidth is severely reduced (Figure 8) and the main lobe-width and the mean values of the normalized ACF in the sidelobes region increase as shown in Figure 9.

Table 3. Normalized bandwidth (−3, −6, −10, and −20 dB) of the spectrum for “SC” waveforms.

“SC” Waveforms m B−3/B B−6/B B−10/B B−20/B 1 0.50 0.75 0.90 0.95 2 0.42 0.59 0.74 0.82 3 0.36 0.50 0.63 0.71 5 0.29 0.40 0.51 0.58 10 0.21 0.29 0.38 0.43

Figure 8. Spectrum of the “SC” noise waveforms (estimated on 1000 trials), BT = 5000, =1,2, 5.

Sensors 2020, 20, x FOR PEER REVIEW 12 of 25

Figure 7. Normalized autocorrelation of “NV” noise waveforms for BT = 5000 (values are estimated on 1000 trials), m = 1, 2, and 5.

A third generalization of the spectrum considers situations in which the spectral occupancy is “severely-controlled” (hereafter named “”), i.e., the whole PSD has to strictly stay in the − ,+ interval. In this case the scaling factor, corresponding to the one of Equation (21), becomes

=(+1) (24) The resulting spectrum and the corresponding normalized ACF are

1 () = (25)

( ) sin () = (26) Table 3 reports the normalized bandwidths. Increasing , the bandwidth is severely reduced (Figure 8) and the main lobe-width and the mean values of the normalized ACF in the sidelobes region increase as shown in Figure 9.

Table 3. Normalized bandwidth (−3, −6, −10, and −20 dB) of the spectrum for “SC” waveforms.

“SC” Waveforms m B−3/B B−6/B B−10/B B−20/B 1 0.50 0.75 0.90 0.95 2 0.42 0.59 0.74 0.82 3 0.36 0.50 0.63 0.71 Sensors 2020, 20, 5187 5 0.29 0.40 0.51 0.58 13 of 26 10 0.21 0.29 0.38 0.43

Sensors 2020, 20, x FOR PEER REVIEW 13 of 25 SensorsFigure Figure2020, 8.20 8.,Spectrum x SpectrumFOR PEER of ofREVIEW the the “SC” “SC” noise noise waveformswaveforms (estimated on on 1000 1000 trials), trials), BT BT = =5000,5000, =1,2m = 1,, 5. 2,13 5. of 25

Figure 9. Normalized autocorrelation of “SC” noise waveforms, BT = 5000, m = 1, 2, and 5 (values are FigureFigure 9. 9.Normalized Normalized autocorrelation autocorrelation ofof “SC”“SC” noise waveforms, BT BT == 5000,5000, mm = =1, 1,2, 2,and and 5 (values 5 (values are are estimated on 1000 trials). estimatedestimated on on 1000 1000 trials). trials).

VeryVery preliminary preliminary experimental experimental resultsresults using using noise waveformwaveform “NV” “NV” with with m m = =1, 1,2, 2, and and 5 are5 are publishedpublished in in[18]. [18 [18].]. A A classical classical approachapproach approach to to strictlyto strictly strictly control controlcontrol the thethe bandwidth bandwidth uses usesuses spectral specspectral windowstral windows windows [19 ][19] to [19] suitably to tosuitably suitably shape shapetheshape spectrum the the spectrum spectrum in order in in to order order have to ato lowhave have level a a low low of level thelevel sidelobes of the sidelobessidelobes of the related of of the the ACF.related related Figure ACF. ACF. 10Figure Figure shows 10 10 theshows shows mean thespectrumthe mean mean spectrum (simulated spectrum (simulated (simulated on 1000 trials) on on 1000 1000 of a trials) trials) noise ofof waveform a noise waveformwaveform shaped by shaped shaped three by di byff threeerent three different spectral different spectral windows: spectral windows:Rectangular,windows: Rectangular, Rectangular, Hamming, Hamming, andHamming, Blackman-Nuttall. and and Blackman-Nuttall. Blackman-Nuttall. Spectrum [dB/MHz] Spectrum [dB/MHz]

Figure 10. Shaped spectrum of a noise spectrum by Rectangular, Hamming, and Blackman-Nuttall Figurewindow 10. Shaped(valuesShaped arespectrum spectrum estimated of of ona noise 1000 trials).spectrum by Rectangular, Hamming, and Blackman-Nuttall window (values are estimated on 1000 trials). Theoretically, the Blackman-Nuttall spectral weighting applied to deterministic waveforms permitsTheoretically, to reach verythe Blackman-Nuttalllow sidelobe levels spectral(< −100 dB) weighting in the normalizedapplied to ACF.deterministic However, waveforms such an permitsexcellent to reachperformance very low is notsidelobe achieved levels with (< nois −100 dB)e waveforms in the because normalized of the ACF.“random However, fluctuations” such an excellentof the sidelobes performance as discussed is not achieved above. An with example noise is waveforms shown in Figure because 11, ofwhere the “randomthe mean normalizedfluctuations” of ACFthe sidelobes (evaluated as with discussed 1000 Monte above. Carlo An trials)example is comp is shownared with in Figure the theoretical 11, where one. the The mean constant normalized level (dashed line in Figure 11) corresponds to the estimated mean value (sample mean) in the sidelobes ACF (evaluated with 1000 Monte Carlo trials) is compared with the theoretical one. The constant level region, i.e., ̂. Inverting Equation (8) we may define an equivalent band: = < for a noise (dashed line in Figure 11) corresponds to the estimated mean value (sample mean) in the sidelobes region,signal, i.e., corresponding ̂ . Inverting to Equation a waveform (8) with we may uniform define spectrum an equivalent and sidelobe band: levels of= the ACF< Rayleigh- for a noise distributed with parameter = . By simulation, applying the Chi-squared-test with = signal, corresponding to a waveform with uniform spectrum and sidelobe levels of the ACF Rayleigh- distributed0.05, the nullwith hypothesis: parameter ≝ {the= random. By variablesimulation, r is Rayleigh-distributed}applying the Chi-squared-test is not rejected with with = − value = 0.90 and 0.42 for Hamming and Blackman-Nuttall, respectively. This example shows 0.05, the null hypothesis: ≝ {the random variable r is Rayleigh-distributed} is not rejected with that the “random fluctuations” of sidelobes in NRT may lead to prefer “less performing” windows, − value = 0.90 0.42 and for Hamming and Blackman-Nuttall, respectively. This example shows that the “random fluctuations” of sidelobes in NRT may lead to prefer “less performing” windows,

Sensors 2020, 20, 5187 14 of 26

Theoretically, the Blackman-Nuttall spectral weighting applied to deterministic waveforms permits to reach very low sidelobe levels (< 100 dB) in the normalized ACF. However, such an − excellent performance is not achieved with noise waveforms because of the “random fluctuations” of the sidelobes as discussed above. An example is shown in Figure 11, where the mean normalized ACF (evaluated with 1000 Monte Carlo trials) is compared with the theoretical one. The constant level (dashed line in Figure 11) corresponds to the estimated mean value (sample mean) in the sidelobes π region, i.e., ηˆr. Inverting Equation (8) we may define an equivalent band: Beq = 2 < B for a 4ηˆr T noise signal, corresponding to a waveform with uniform spectrum and sidelobe levels of the ACF Rayleigh-distributed with parameter a2 = 1 . By simulation, applying the Chi-squared-test with Ray 2BeqT def Sensorsα = 0.05 2020, the, 20, null x FOR hypothesis: PEER REVIEWH0 = {the random variable r is Rayleigh-distributed} is not rejected14 withof 25 p-value = 0.90 and 0.42 for Hamming and Blackman-Nuttall, respectively. This example shows that unlessthe “random sidelobes fluctuations” suppression of sidelobesmethods in“are NRT applied may lead to each to prefer generated “less performing” waveform”, windows, as explained unless in Sectionsidelobes 3. suppression methods “are applied to each generated waveform”, as explained in Section3.

B = 50 MHz, T = 100 s, BT = 5000 0 Mean ACF (1000 Trials) Mean in sidelobes region -10 Theoretical Black-Nuttal ACF

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-60 012345678 B (a) (b)

Figure 11. Normalized autocorrelationautocorrelation functionfunction (ACF):(ACF): (a(a)) Hamming Hamming noise noise waveform waveformηˆr = =36.69 36.69dB; = dB;(b) Blackman-Nuttall (b) Blackman-Nuttall noise noise waveform waveformηˆr 35.08 =35.08 dB. Case dB (.a Case) grants (a) grants a better a rangebetter resolution.range resolution. 2.2.3. The Range Transition Zone 2.2.3. The Range Transition Zone In the previous analysis, it has been shown that the ACF has two main intervals: one, around zero In the previous analysis, it has been shown that the ACF has two main intervals: one, around delay, in which the deterministic shape, related to the PSD, dominates, and another, at larger delays, zero delay, in which the deterministic shape, related to the PSD, dominates, and another, at larger where random sidelobes dominate with a level depending on the BT product. A transition zone is in delays, where random sidelobes dominate with a level depending on the BT product. A transition between. Considering the cases analysed in Section 2.2.2, for the “NV” case the bandwidth and the zone is in between. Considering the cases analysed in Section 2.2.2, for the “NV” case the bandwidth mean PSL show a weak dependence on the parameter m. Hence, similar to Equation (17) it is possible and the mean show a weak dependence on the parameter 1 . Hence, similar to Equation (17) it to define a “transition lag zone” with central value τ and width (the nominal duration of a sidelobe is possible to define a “transition lag zone” with central∗ value ∗ Band width (the nominal duration of the ACF), after which the peaks of the “nominal” autocorrelation function are embedded in the of a sidelobe of the ACF), after which the peaks of the “nominal” autocorrelation function are random fluctuations. embedded in the random fluctuations. The value of τ∗∗ is obtained equating the denominator of Equation (23) with the mean value ηr: The value of is obtained equating the denominator of Equation (23) with the mean value : + r  1 m 1 π 1 1 = 1 = ∙ (27)(27) πBαm∗ τ∗ 4 ·B T 4 ∙· ∗ Normalizing to the waveform duration , i.e., using = : 2 4 = √+1 (28) In the ordinary condition of ≫ 1, from the above formulas and from simulation it results that the delay interval in which the “true” autocorrelation function dominates is a very small fraction of . In the ideal case of a perfectly flat spectrum constrained in the considered bandwidth B (i.e., =0), typical values of the “transition range” ∗ = ∗, varying (20– 100 MHz) and (0.2– 20 ms), are of the order of hundreds of meters with a maximum around 1.6 km (see Figure 12). The corresponding autocorrelation (see for instance Figure 5) has the highest sidelobes, with a maximum −13.2 dB level below the main peak. In the practical case of lowering these “deterministic” sidelobes by frequency-domain weighting, i.e., smoothing the spectrum (≥1), ∗ would be reduced. Some exemplary values of ∗ for =1 and =5 are shown in Figure 13.

Sensors 2020, 20, 5187 15 of 26

τ∗ Normalizing to the waveform duration T, i.e., using γ = T :

 2   2m+1 2 4 − 2m+2 γ = √m + 1 BT (28) π π In the ordinary condition of BT 1, from the above formulas and from simulation it results that  the delay interval in which the “true” autocorrelation function dominates is a very small fraction of T. In the ideal case of a perfectly flat spectrum constrained in the considered bandwidth B (i.e., m = 0), c typical values of the “transition range” R∗ = 2 τ∗, varying B (20–100 MHz) and T (0.2–20 ms), are of the order of hundreds of meters with a maximum around 1.6 km (see Figure 12). The corresponding autocorrelation (see for instance Figure5) has the highest sidelobes, with a maximum 13.2 dB level − belowSensors the2020 main, 20, x FOR peak. PEER REVIEW 15 of 25

Figure 12. Level curves (in meters) of the transition Range R* varying B and T in the ideal case of flatflat spectrumspectrum ((mm == 0). In the practical case of lowering these “deterministic” sidelobes by frequency-domain weighting, i.e., smoothing the spectrum (m 1), R would be reduced. Some exemplary values of R for m = 1 ≥ ∗ ∗ and m = 5 are shown in Figure 13. With m = 1, the transition region is of the order of tens of meters, while for m = 5 it is of the order of meters. Note that these are the typical delay intervals of the above-discussed critical phenomena, i.e., the antenna leakage (Section1) and the close-in clutter. To cope with this “short range” problem, a method to suppress the sidelobes of the autocorrelation function in a limited delay interval is introduced in the next section. Summing up, to mitigate the sidelobes when a relatively large BT is needed (order of hundred of thousand or more) the most practical way seems to act not only on the “random process” (i.e., the overall set of waveforms) but also on its “realizations” (i.e., on each waveform), paying the price of a potential reduction of LPI/LPE. The topic of Range sidelobe attenuation has been widely published in the literature for both deterministic radar signals [1,2] and noise radar signals [15,16,20–23]. A computationally simple, effective method, suited to CE noise radars, is presented in Section3.

Figure 13. Level curves (in meters) of the transition Range R* for “NV” varying B and T, for m = 1 and m = 5.

With =1, the transition region is of the order of tens of meters, while for =5 it is of the order of meters. Note that these are the typical delay intervals of the above-discussed critical phenomena, i.e., the antenna leakage (Section 1) and the close-in clutter. To cope with this “short range” problem, a method to suppress the sidelobes of the autocorrelation function in a limited delay interval is introduced in the next section. Summing up, to mitigate the sidelobes when a relatively large is needed (order of hundred of thousand or more) the most practical way seems to act not only on the “random process” (i.e., the overall set of waveforms) but also on its “realizations” (i.e., on each waveform), paying the price of a potential reduction of LPI/LPE. The topic of Range sidelobe attenuation has been widely published in the literature for both deterministic radar signals [1,2] and noise radar signals [15,16,20–23]. A computationally simple, effective method, suited to CE noise radars, is presented in Section 3.

Sensors 2020, 20, x FOR PEER REVIEW 15 of 25

Sensors 2020Figure, 20, 12. 5187 Level curves (in meters) of the transition Range R* varying B and T in the ideal case of flat 16 of 26 spectrum (m = 0).

Sensors 2020, 20, x FOR PEER REVIEW 16 of 25

Sensors3. RangeFigure 2020Figure Sidelobes, 20 13. ,13. xLevel FOR Level PEERSuppression curves curves REVIEW (in (in meters) meters) for the ofof thethe Auto transitioncorrelation Range FunctionR* R* for for “NV” “NV” of varying varyingNoise BWaveforms Band and T,T, for for mm = 1= and1 and 16 of 25 m =m5. = 5. 3.3.1. Range The FMethSidelobes (Filtering Suppression Method) for Algorithm: the Autocorrelation General Description Function of Noise Waveforms 3. RangeWith Sidelobes =1, the Suppression transition region for the is Autocorrelation of the order of tens Function of meters, of Noise while Waveformsfor =5 it is of the orderWe ofconsider meters. continuous Note that theseemission are noisethe typical radar delayfor the intervals rationales of describedthe above-discussed in the Introduction. critical 3.1. The The FMeth (Filtering(Filtering Method) Method) Algorithm: Algorithm: General General Description Description Thephenomena, emission hasi.e., thecoherent antenna time leakage intervals (Section with 1) duration and the close-in (Figure clutter. 14), Towith cope ≤ with , thiswhere “short is equalrange”We to consider orproblem, smaller continuousa thanmethod the to decorre emission suppresslation thenoise timesidelobes radar (or the for of thedwell the autocorrelation rationales time). described function in inin the thea limited Introduction. Introduction. delay ≤ Theinterval emissionemission is introduced hashas coherent coherent in the time ti nextme intervals intervalssection. with with duration durationT (Figure (Figure 14), with14), withT TD, where , whereTD is equal is ≤ equalto or smaller toSumming or smaller than up, the than to decorrelation mitigate the decorre() the sidelobeslation time (or (time) thewhen (or dwell athe relatively time).dwell() time). large is( needed) (order of hundred of thousand or more) the =1most practical=2 way seems to =3 act not only. . . on the= “random process” (i.e., the overall set of waveforms) but also on its “realizations”() ((i.e.,) on each waveform), paying the price of a (T) T T (T) potential reduction of LPI/LPE. The topic of Range sidelobe attenuation has been widely published =1 =2 =3 . . . = in the literature for both deterministic radar signals [1,2] and noise radar signals [15,16,20–23]. A T Figure 14.T Coherent intervalsT in emission. T computationally simple, effective method, suited to CE noise radars, is presented in Section 3. Figure 14. Coherent intervals in emission. For each , () (i = 1, 2,Figure …, L 14.) defines Coherent a intervalsdifferent in realizationemission. of a band-limited (within B) random process. In almost all practical cases, the radar designer sets ≫ , defined in term of For each i, g (t) (i = 1, 2, ... , L) defines a different realization of a band-limited (within B) random i maximumFor each range , () (=i = 1, 2,, in…, order L) defines to obtain a differenta large enough realization processing of a band-limited gain . (within B) process. In almost all practical cases, the radar designer sets T Tmax, defined in term of maximum randomIn theprocess. followingcTmax In almost we use all the practical scheme cases, of Figure the radar15, creating designer a periodic sets ≫ signal ,structure defined byin terma single of range Rmax = , in order to obtain a large enough processing gain BT. maximum range2 = , in order to obtain a large enough processing gain . realizationIn the following(), i.e., wea circular convolution use the scheme of Figure in the 15 ACF, creating computation. a periodic With signal ≫ structure and by a single ≫ 1 realizationthe approximationIn the followingg(t), i.e., aof wecircular the use ACF the convolution schemeand of ofthe inFigure spectrum the ACF 15, creating computation. of () a periodicwith With their signalT sampled Tstructuremax and and BTby periodical a single1 the   realizationapproximationversions are () quite of, i.e., the good. ACFa circular convolution We and assume of the spectruma sampling in ofthe ginterval (ACFt) with computation. equal their sampledto With with and an≫ periodical oversampling and versions factor ≫ are 1 ∙ 1 () quitethe approximation good. We assume of the a sampling ACF and interval of the equal spectrum to ofwith an with oversampling their sampled factor andkF > periodical1. >1. kF B versions are quite good. We assume a sampling interval· equal to with an oversampling factor ∙ >1. ()() () () . . . . . (T)(T) (T) (T) . . . . . FigureFigure 15.15. PeriodicPeriodic signalsignal inin emission.emission. T T T T TheThe signalsignal g(t(),) realization, realization (sample Figure(sample 15. function) function) Periodic ofsignal of the the in random rand emission.om noisenoise process,process, isis generatedgenerated startingstarting  2 fromfrom thethe designdesign powerpower spectral density (PSD) (PSD) of of the the process, process, i.e., i.e., E[| ()G( f )| ], being G()( f ) thethe FourierFourier transformThe signal of ((). )The, realization ACF of (sample(), i.e., function) the output of the of rand the omcompress noise process,ion filter is (matchedgenerated filter),starting is transform of g(t). The ACF of g(t), i.e., the output of the compression filter (matched filter), is (hereafter from(hereafter the design we use power the continuous-time spectral density non-causal (PSD) of the formalism process, i.e.,for simplicity):[|()|], being () the Fourier we use the continuous-time non-causal formalism for simplicity): transform of (). The ACF of (), i.e., the output of the compression filter (matched filter), is () =()⨂∗(−) (29) (hereafter we use the continuous-time non-causal formalism for simplicity): R(τ) = g(τ) g∗( τ) (29) where ⨂ is the convolution operator. The Fourier transform⊗ − of Equation (29) is () =()⨂∗(−) (29) () =() ∙∗() = |()| (30) where ⨂ is the convolution operator. The Fourier transform of Equation (29) is Let () =() ∙() be the desired autocorrelation obtained by multiplying () by () =() ∙∗() = |()| (30) 0 [− <<−]⋃[ <<] Let () =() ∙() be( )the= desired autocorrelation obtained by multiplying () by (31) 1 ℎ 0 [− <<−]⋃[ <<] in order to suppress the range() sidelobes= inside the time interval (, ), as described in Figure 16.(31) 1 ℎ in order to suppress the range sidelobes inside the time interval (,), as described in Figure 16.

Sensors 2020, 20, 5187 17 of 26 where is the convolution operator. The Fourier transform of Equation (29) is ⊗ 2 S( f ) = G( f ) G∗( f ) = G( f ) (30) · Let eR(τ) = R(τ) q(τ) be the desired autocorrelation obtained by multiplying R(τ) by · ( S 0 [ τ < τ < τ ] [τ < τ < τ ] q(τ) = − 2 − 1 1 2 (31) 1 elsewhere

Sensorsin order 2020 to, suppress20, x FOR PEER the rangeREVIEW sidelobes inside the time interval (τ1, τ2), as described in Figure17 16 of. 25

FigureFigure 16. 16. TheThe sidelobes sidelobes suppression suppression interval interval as as defined defined by by q((τ),, redred line.line.

The product R(τ) q(τ) corresponds, in the frequency domain, to the convolution of the spectrum The product (·) ∙() corresponds, in the frequency domain, to the convolution of the S( f ) with the Fourier transform of q(τ). Denoting with the Fourier transform operator, it results in spectrum () with the Fourier transform of ().F Denoting{·} with ℱ{∙} the Fourier transform operator, it results in n o  eS( f ) = eR(τ) = S( f ) q(τ) (32) () =ℱ{F()} =()⨂ℱ⊗{() F } (32)  For how q(τ) is defined, the q(τ) contains spectral components outside the band (i.e., B/2 For how () is defined, theF ℱ{()} contains spectral components outside the band− (i.e., to +B/2) of g(t). Using the FFT to implement the Fourier transform, we should then “accept” the −/2 to +/2) of (). Using the FFT to implement the Fourier transform, we should then “accept” harmful effects introduced by the folding of the spectral components outside the band B, as shown in the harmful effects introduced by the folding of the spectral components outside the band , as the ensuing results. shown in the ensuing results. Let H( f ) be the frequency response of a filter such that Let () be the frequency response of a filter such that

2 (eS)(=f )|=( H)|( f )∙ (S() f ) (33) · Hence, from Equations (30) and (33), the frequency response of this sidelobes suppression filter is Hence, from Equations (30) and (33), the frequency response of this sidelobes suppression filter is () () = |(q)| (34) eS( f ) This approach, here named FMeth andH based( f ) = on the concept of “inverse filtering”, is well known(34) G( f ) in literature (see, for instance, [20]). The systematic description of the noise waveforms generation with Thissidelobes approach, suppression here named by FMeth FMeth is and shown based in onFigure the concept17a. of “inverse filtering”, is well known in literature (see, for instance, [20]). The systematic description of the noise waveforms generation with3.1.1. sidelobesPRNG and suppression Spectral Shaping by FMeth is shown in Figure 17a. Starting from a pseudo-random number generator (PRNG), a sequence of independent and identically distributed (i.i.d.) Gaussian complex samples with assigned power spectrum (spectral shaping) of a band-limited , is generated by filtering, see Section 2.2.2.

3.1.2. PAPR Setting by Alternating Projection After spectral shaping, the of the waveform is around 10 − 12 or greater (see the Introduction). To reduce the loss due to high (Table 1), the can be forced to a lower value (e.g., 1.5 − 2.0) using the “alternating projection” algorithm [24], which is a powerful and computationally efficient way to design waveforms with a given structural properties (e.g., prescribed norm, , etc. …). More specifically, the alternating projection method efficiently applies to the huge class of Inverse Eigenvalues Problems (IEP) aimed at creating a structured waveform or a structured “tight frame”. A tight frame is a generalization of an orthogonal system [25] and, according to [24], is described by its spectrum. Because of the spectral characterization of a tight frame, IEP is the most suitable solution for seeking the structured frame with prescribed spectral properties. The idea behind the alternating projection algorithm is to search for a point of intersection

Sensors 2020, 20, x FOR PEER REVIEW 18 of 25 between a matrix that satisfies a given spectral constraint and a matrix that satisfies a given structural constraint. In our case, the spectral constraint (matrix Z) is determined by the spectral occupancy of the generated waveform and the structural constraint (matrix ) is the desired PAPR. Hereafter we refer to a matrix (and not to a single waveform vector) since the IEP, of which alternating projection algorithm is an efficient solver, operates on tight frame [25]. Figure 17b depicts the alternating projection method in which two sets of matrixes (e.g., spectral constraint) and (e.g., structural constraint) are iteratively projected onto each other in order to find the intersection matrix that satisfies both constraints. The subscript “” represents the number of iterations and the initial matrix represents the starting noisy band-limited sequence. The algorithm details can be found in Appendix B.

3.1.3. FMeth Sidelobes Suppression The functional block diagram of the FMeth algorithm is shown in Figure 17c. The step-by-step description is summarized as follows. i. The sample autocorrelation () of the PAPR-regulated sequence is computed. ii. Zeroing the points of () between and by multiplication with (), Equation (31), the desired autocorrelation () =() ∙() is determined. iii. The spectrum () and the Fourier transform () of () are estimated by FFT, and, from Equation (34), the frequency response () is evaluated. iv. The random signal (), whose autocorrelation function shows attenuated sidelobes inside the

time interval (,), is generated by the following procedure: () =ℱ()=ℱ{() ∙()} (35) Due to the detrimental effects related to the use of the FFT, i.e., the folding of the spectral components of () into (−/2, +/2), the output () has to be iteratively processed by the

FMeth algorithm for a suitable number of times ( ) to improve the suppression sidelobes, as shown in Figure 17a.

At the end of the suppression procedure, i.e., after a number of iterations suited to satisfy the requirement of the , the “tailored” sequence () can be used as transmitted waveform. We underline that a possibly large number of iterations is not a practical problem being this Sensors 2020, 20, 5187 18 of 26 procedure designed for an offline-preparation of a large number of transmit signals.

PRNG Initial value b =1 c Sequence Spectral PAPR set by FMeth No of IID < () Shaping Alternating Sidelobes Samples Projection Suppression Yes

=+1

(a) FMeth Sidelobes Suppression

Set Y () () () PAPR () = . (35) Set Z constraint () Spectral … constraint Matched () () Filter output () . (29)

(b) (c)

FigureFigure 17.17. ((aa)) FunctionalFunctional blockblock diagramdiagram ofof thethe generation of a noise waveform. ( (bb)) The The “Alternating “Alternating Projection”Projection” method.method. ((cc)) FunctionalFunctional blockblock diagramdiagram ofof thethe FMethFMeth algorithmalgorithm toto suppresssuppress the the sidelobes. sidelobes.

3.1.1. PRNG and Spectral Shaping Starting from a pseudo-random number generator (PRNG), a sequence of independent and identically distributed (i.i.d.) Gaussian complex samples with assigned power spectrum (spectral shaping) of a band-limited B, is generated by filtering, see Section 2.2.2.

3.1.2. PAPR Setting by Alternating Projection After spectral shaping, the PAPR of the waveform is around 10 12 or greater (see the Introduction). − To reduce the loss due to high PAPR (Table1), the PAPR can be forced to a lower value (e.g., 1.5 2.0) − using the “alternating projection” algorithm [24], which is a powerful and computationally efficient way to design waveforms with a given structural properties (e.g., prescribed norm, PAPR, etc. ... ). More specifically, the alternating projection method efficiently applies to the huge class of Inverse Eigenvalues Problems (IEP) aimed at creating a structured waveform or a structured “tight frame”. A tight frame is a generalization of an orthogonal system [25] and, according to [24], is described by its spectrum. Because of the spectral characterization of a tight frame, IEP is the most suitable solution for seeking the structured frame with prescribed spectral properties. The idea behind the alternating projection algorithm is to search for a point of intersection between a matrix that satisfies a given spectral constraint and a matrix that satisfies a given structural constraint. In our case, the spectral constraint (matrix Z) is determined by the spectral occupancy of the generated waveform and the structural constraint (matrix Y) is the desired PAPR. Hereafter we refer to a matrix (and not to a single waveform vector) since the IEP, of which alternating projection algorithm is an efficient solver, operates on tight frame [25]. Figure 17b depicts the alternating projection method in which two sets of matrixes Z (e.g., spectral constraint) and Y (e.g., structural constraint) are iteratively projected onto each other in order to find the intersection matrix X that satisfies both constraints. The subscript “j” represents the number of iterations and the initial matrix X0 represents the starting noisy band-limited sequence. The algorithm details can be found in AppendixB. Sensors 2020, 20, 5187 19 of 26

3.1.3. FMeth Sidelobes Suppression The functional block diagram of the FMeth algorithm is shown in Figure 17c. The step-by-step description is summarized as follows. i. The sample autocorrelation R(τ) of the PAPR-regulated sequence is computed. ii. Zeroing the points of R(τ) between τ1 and τ2 by multiplication with q(τ), Equation (31), the desired autocorrelation eR(τ) = R(τ) q(τ) is determined. · iii. The spectrum eS( f ) and the Fourier transform G( f ) of g(t) are estimated by FFT, and, from Equation (34), the frequency response H( f ) is evaluated. iv. The random signal gˆ(t), whose autocorrelation function shows attenuated sidelobes inside the time interval (τ1, τ2), is generated by the following procedure:

1n o 1 gˆ(t) = − Gˆ ( f ) = − H( f ) G( f ) (35) F F · Due to the detrimental effects related to the use of the FFT, i.e., the folding of the spectral components of q(t) into ( B/2, +B/2), the output gˆ(t) has to be iteratively processed by the FMeth − algorithm for a suitable number of times (niter) to improve the suppression sidelobes, as shown in Figure 17a. At the end of the suppression procedure, i.e., after a number niter of iterations suited to satisfy the requirement of the PSL, the “tailored” sequence gˆniter (t) can be used as transmitted waveform. We underline that a possibly large number of iterations is not a practical problem being this procedure designed for an offline-preparation of a large number of transmit signals.

3.2. Results for Zero Doppler–Sidelobes Level in the ACF

3.2.1. Sensitivity of the Sidelobe Attenuation to the Number of Iterations To analyse the effect related to the number of iterations in the FMeth algorithm, some exemplary results are obtained starting from a noise waveform, i.e., a realization of a complex Gaussian process with both a Blackman-Nuttall and a Hamming spectrum setting B = 50 MHz, T = 100 µs (BT = 5000), sampling frequency Fs = 16 B. To analyse the dependence of FMeth on the number of iterations (n ), · iter no constraint is applied to the PAPR (which “naturally” takes values around 10 12 or greater) and − the percentage of the suppression zone is set in the interval τ1B = 4, τ2B = 500 (in range R1 = 12 m, R2 = 1500 m) corresponding approximately to the 20% of the total duration. The results concerning the sidelobes attenuation are obtained for three values of niter : 1, 5, 100. Figure 18a,b shows the results for Blackman-Nuttall and Hamming noise waveforms, respectively. The sidelobes suppression is poor for a single iteration (the improvement is only 12 dB wrt the mean PSL). With five iterations, the PSL reaches ( 80, 78 dB) in mean and, for n = 100, the PSL decreases up to ( 95, 80 dB) for a single − − iter − − realization and ( 105, 93 dB) for the mean. − − Figure 19a,b shows the corresponding mean spectrum as compared with the original sequence. It is interesting to notice that both the distortion introduced by the non-linearity and the folding of the spectral components outside the band B look acceptable. Sensors 2020, 20, x FOR PEER REVIEW 19 of 25

3.2. Results for Zero Doppler–Sidelobes Level in the ACF

3.2.1. Sensitivity of the Sidelobe Attenuation to the Number of Iterations To analyse the effect related to the number of iterations in the FMeth algorithm, some exemplary results are obtained starting from a noise waveform, i.e., a realization of a complex Gaussian process with both a Blackman-Nuttall and a Hamming spectrum setting =50 MHz, = 100 μs ( =

5000), sampling frequency =16∙. To analyse the dependence of FMeth on the number of iterations (), no constraint is applied to the (which “naturally” takes values around 10 − 12 or greater) and the percentage of the suppression zone is set in the interval =4, = 500 (in range =12 m, = 1500 m) corresponding approximately to the 20% of the total duration. The results concerning the sidelobes attenuation are obtained for three values of :1,5,100. Figure 18a,b shows the results for Blackman-Nuttall and Hamming noise waveforms, respectively. The sidelobes suppression is poor for a single iteration (the improvement is only 12 dB wrt the mean PSL). With five iterations, the PSL reaches (−80, −78 dB) in mean and, for = 100, the Sensors 2020, 20, 5187 20 of 26 decreases up to (−95, −80 dB) for a single realization and (−105, −93 dB) for the mean. Normalized Autocorrelation [dB] Normalized

(a) Normalized Autocorrelation [dB] Normalized Autocorrelation

(b)

Figure 18. NormalizedNormalized ACFACF when when the the FMeth FMeth is used is used to suppress to suppress a sidelobes a sidelobes region region (varying (varying the number the numberof iterations: of iterations:niter = 1, 5,niter 100) = 1, for 5,a 100) single for realization a single realization and averaging and 100averaging trials. The100 PAPRtrials. assumesThe PAPR its “naturally”assumes its value“naturally” (10–12). value The suppression(10–12). The zone suppression is the 20% zone of the is total.the 20% (a) Blackman-Nuttallof the total. (a) Blackman- spectrum, Nuttalland (b) spectrum, Hamming and spectrum. (b) Hamming spectrum.

Sensors 2020, 20, x FOR PEER REVIEW 20 of 25

Figure 19a,b shows the corresponding mean spectrum as compared with the original sequence.

SensorsIt is interesting2020, 20, 5187 to notice that both the distortion introduced by the non-linearity and the folding21 of of 26 the spectral components outside the band look acceptable. Spectrum [dB/MHz]

(a) Spectrum [dB/MHz]

(b)

Figure 19. MeanMean spectrum spectrum (FMeth, (FMeth, 100 100 trials) trials) varying varying the the number number of ofiterations: iterations: nitern =iter 1, =5, 1,and 5, 100. and The 100. ThePAPR PAPR assumes assumes its “natural” its “natural” value value in the in range the range 10–12. 10–12. The Thesuppression suppression zone zone is the is the20% 20% of the of total. the total. (a) (Blackman-Nuttalla) Blackman-Nuttall spectrum, spectrum, and and (b) (Hammingb) Hamming spectrum. spectrum. 3.2.2. Sensitivity of the Sidelobe Attenuation to the PAPR and to the Width of the Suppression Zone 3.2.2. Sensitivity of the Sidelobe Attenuation to the PAPR and to the Width of the Suppression Zone The results shown in Section 3.2.1 are obtained for a Gaussian noise in the case of a linear system The results shown in Section 3.2.1 are obtained for a Gaussian noise in the case of a linear system with full dynamic range (no constraint on the PAPR) and supposing a 20% of suppression zone with with full dynamic range (no constraint on the ) and supposing a 20% of suppression zone with respect to the total range. If the PAPR is varied (i.e., decreased up to one) and the suppression zone is respect to the total range. If the is varied (i.e., decreased up to one) and the suppression zone increased (up to 100%), the effectiveness of the FMeth algorithm is significantly reduced. is increased (up to 100%), the effectiveness of the FMeth algorithm is significantly reduced. To quantify these effects, for BT = 5000 and niter = 100, we have set different PAPR values To quantify these effects, for = 5000 and = 100, we have set different values (9.0, 5.0, 2.0, 1.5, 1.0) and percentages of the suppression zone (20, 40, 60, 80, 95, and 100 per cent). (9.0, 5.0, 2.0, 1.5, 1.0) and percentages of the suppression zone (20, 40, 60, 80, 95, and 100 per cent). The “natural” PAPR of the starting noise waveform, i.e., g1(t) in Figure 17a, can be forced to The “natural “ of the starting noise waveform, i.e., () in Figure 17a, can be forced to a a different (lower) value (see Figure 17b). Being the FMeth generation based on a “mild” filtering, different (lower) value (see Figure 17b). Being the FMeth generation based on a “mild” filtering, which suppresses spectral components whose energy (residing in the sidelobes of the ACF) is a small which suppresses spectral components whose energy (residing in the sidelobes of the ACF) is a small fraction of total energy (mainly allocated in the main lobe of the ACF), one could guess that this fraction of total energy (mainly allocated in the main lobe of the ACF), one could guess that this sidelobes suppression does not significantly modify the PAPR as initially imposed, even when the sidelobes suppression does not significantly modify the as initially imposed, even when the suppression zone increases. Simulations have confirmed this hypothesis. suppression zone increases. Simulations have confirmed this hypothesis. Table4 (for the Blackman-Nuttall spectrum) and Table5 (for the Hamming spectrum) define a matrix whose columns (for a given PAPR set) report the mean PSL varying the percentage of the suppression zone, while the rows (for a percentage set) show the mean PSL varying the PAPR.

Sensors 2020, 20, 5187 22 of 26

Table 4. Mean peak side lobe (PSL) (dB) on 100 trials using FMeth varying the PAPR and the percentage of the suppression zone. Noise waveforms with Blackman-Nuttall spectrum.

Blackman-Nuttall: B = 50 MHz, T = 100 µs, BT = 5000 Suppression PAPR PAPR PAPR PAPR PAPR Zone [%] 9.0 5.0 2.0 1.5 1.0 20 103.9 100.8 89.3 75.9 57.9 − − − − − 40 104.3 101.1 79.6 66.3 54.5 − − − − − 60 102.5 95.2 71.1 60.7 52.3 − − − − − 80 99.7 88.6 63.0 56.9 51.5 − − − − − 95 89.1 78.3 58.8 55.1 50.6 − − − − − 100 83.1 69.5 56.5 53.8 49.9 − − − − −

Table 5. Mean PSL (dB) on 100 trials using FMeth varying the PAPR and the percentage of the suppression zone. Noise waveforms with Hamming spectrum.

Hamming: B = 50 MHz, T = 100 µs, BT = 5000 Suppression PAPR PAPR PAPR PAPR PAPR Zone [%] 9.0 5.0 2.0 1.5 1.0 20 92.3 93.6 89.9 76.8 59.3 − − − − − 40 90.7 93.3 80.3 67.5 56.0 − − − − − 60 88.4 89.1 71.7 61.9 54.0 − − − − − 80 84.0 82.9 64.3 58.3 52.9 − − − − − 95 79.2 74.9 60.2 56.4 52.2 − − − − − 100 73.1 68.3 58.2 55.0 51.4 − − − − −

These values can be explained by the following considerations. First, decreasing the PAPR the number of degrees of freedom is reduced (the amplitude variation is constrained). Second, a larger suppression zone, maintaining fixed the number of degrees of freedom, makes the suppression algorithm less efficient, being applied over a wider range. From the results, we can conclude that a good trade-off can be reached with, the Hamming spectrum, a suppression zone of the 20%, and a PAPR equal to 1.5, whose loss is still acceptable (see for the loss Table1).

3.3. Doppler Sensitivity and Sidelobes Suppression for the Ambiguity Function In the presence of moving targets, the Doppler effect causes the well-known mismatching in the compression filter. The frequency shift produces two effects: (i) loss on the peak at the output of the matched filter (i.e., loss in SNR); and (ii) reduced effectiveness of the sidelobes suppression algorithm. At X-band (λ = 0.032 m) for noise waveforms generated starting from the Blackman-Nuttall spectrum (B = 50 MHz, BT = 5000), we have considered three values of the Doppler velocity: vD = 3, 10, 50 m/s. In Figure 20, the normalized ACFs are shown varying vD and compared with the zero Doppler case. No constraint is imposed on the PAPR and the suppression zone is set to 20% of the total duration. By simulation, for vD 20 m/s the loss in SNR is negligible (<0.1 dB), increasing vD ≤ up to 50 m/s the loss is still acceptable, i.e., 0.59 dB. The suppression algorithm was applied imposing − 100 iterations. Sensors 2020, 20, 5187 23 of 26 Sensors 2020, 20, x FOR PEER REVIEW 22 of 25 Normalized Autocorrelation [dB]

Figure 20.20. Normalized ACFACF withwith suppression suppression sidelobes sidelobes varying varying the the Doppler Doppler velocity velocity 3, 10, 3, and10, and 50 m 50/s. BTm/s.= BT5000, = 5000,niter =niter100. = 100. Mean Mean on 100on 100 trials. trials.

The range sidelobes suppression algorithm here introduced, as shown, is fairly sensitive to the Doppler shift. Hence, two solutions are possible. The first, first, and “classical”, one tries to “tailor” noise waveformswaveforms to yield low low sidelobes sidelobes of of their their “ambiguity “ambiguity function”, function”, i.e., i.e., both both in in“range” “range” (time) (time) and and in in“Doppler” “Doppler” (frequency). (frequency). Methods Methods to resolve to resolve this pr thisoblem problem are described are described in [21,22]. in [21 The,22]. second, The second, novel novelsolution, solution, most mosteffective effective from from the application the application point point of view, of view, uses uses the the “Range “Range filters filters bank” bank” (RFB) (RFB) as described in [26,27]. This solution has a “fast time” processing of sequences of duration T , which is described in [26,27]. This solution has a “fast time” processing of sequences of duration r , which is very slightly affected by the Doppler thanks to the choice of Tr, being Tr T, with a downstream very slightly affected by the Doppler thanks to the choice of , being ≪, with a downstream “slow“slow time”time” processingprocessing generatinggenerating aa bankbank ofof DopplerDoppler filtersfilters toto covercover thethe wholewhole velocityvelocity intervalinterval of interest,interest, inin aa wayway similarsimilar toto thethe well-knownwell-known movingmoving targettarget detectordetector (MTD)(MTD) [[28,29].28,29]. 4. Comments, Conclusions, and Short Reference to Future Experimental Work 4. Comments, Conclusions, and Short Reference to Future Experimental Work By definition, the transmitted waveforms form the key element of the NRT and the driver of its By definition, the transmitted waveforms form the key element of the NRT and the driver of its main advantages as described in this introductory paper. These advantages are best exploited in the main advantages as described in this introductory paper. These advantages are best exploited in the continuous emission configuration, which is the one considered here. continuous emission configuration, which is the one considered here. However, the simultaneous transmission and reception of a CE radar puts demanding requirements However, the simultaneous transmission and reception of a CE radar puts demanding both to the radar hardware, mainly in terms of the leakage from the transmitting antennas to the requirements both to the radar hardware, mainly in terms of the leakage from the transmitting receiving one, and to the radar waveforms. The latter have very demanding requirements on Range antennas to the receiving one, and to the radar waveforms. The latter have very demanding and Doppler sidelobes, on energetic effectiveness (PAPR) and on compatibility with the regulation for requirements on Range and Doppler sidelobes, on energetic effectiveness ( ) and on the use of the electromagnetic spectrum (see AppendixA). For military radars, more requirements compatibility with the regulation for the use of the electromagnetic spectrum (see Appendix A). For add in terms of robustness against “interception” and “exploitation” of the radiated signals by an military radars, more requirements add in terms of robustness against “interception” and adversary part. “exploitation” of the radiated signals by an adversary part. Hence, a novel design procedure for CE noise radar waveforms has been discussed with the Hence, a novel design procedure for CE noise radar waveforms has been discussed with the related analytical and simulative results. related analytical and simulative results. Ensuing papers of this special issue will tackle other facets of the NRT waveforms and present Ensuing papers of this special issue will tackle other facets of the NRT waveforms and present more results. In particular, a second paper will be submitted by the same team to treat in more practical more results. In particular, a second paper will be submitted by the same team to treat in more way the results of the waveforms design as checked by trials in different operating environments with practical way the results of the waveforms design as checked by trials in different operating the pertaining lessons learnt, while a third planned paper will deal with the counter-interception and environments with the pertaining lessons learnt, while a third planned paper will deal with the counter-exploitation features and the related analysis methods based on the information theory. counter-interception and counter-exploitation features and the related analysis methods based on the Authorinformation Contributions: theory. G.G. defined the main content, including synthesis methods, and the overall organization of the paper. G.P. developed the waveform analysis tools and obtained the related results. F.D.P. studied andAuthor implemented Contributions: the “AlternatingG.G. defined the Projection” main content, method including to control synthesis the PAPR. methods, K.S. and the C.W. overall studied, organization checked, andof the critically paper. reviewedG.P. developed the waveform the waveform generation analysis methods tools in and view obtained of their usethe related in future results. trials toF.D.P. operationally studied and test the results of this paper. All authors have read and agreed to the published version of the manuscript. implemented the “Alternating Projection” method to control the PAPR. K.S. and C.W. studied, checked, and critically reviewed the waveform generation methods in view of their use in future trials to operationally test the results of this paper. All authors have read and agreed to the published version of the manuscript.

Sensors 2020, 20, 5187 24 of 26 Sensors 2020, 20, x FOR PEER REVIEW 23 of 25

Funding:Funding: This research received no external funding. ConflictsConflicts of Interest: TheThe authors authors declare declare no no conflict conflict of of interest. interest.

AppendixAppendix A. Band Occupancy in NRT NRT Applications Applications TheThe m-m-orderorder convolution, convolution, defined defined by Equationby Equation (19), and(19), the and “frequency-scaling” the “frequency-scaling” factor of Equationfactor of (22),Equation allow (22), to disposeallow to of dispose a very of simple a very mathematical simple mathematical closed-form closed-form model formodel the for spectrum the spectrum of noise of signals,noise signals, with awith good a good approximation approximation of the of Gaussian the Gaussian PSD PSD when whenm is above is above the order the order of a veryof a very few units.few units. Increasing Increasingm, the, the40 −40 dB dB bandwidth bandwidth increases increases (see (see Table Table2) 2) and and for for m=4= 4 thethe bandwidthbandwidth − B40 ≅22B.. For radarradar applications,applications, ITU-R recommendations [[9]9] on thethe unwantedunwanted emissionsemissions inin thethe − “out-of-band”“out-of-band” (OoB) (OoB) domain domain (i.e., (i.e., the the adjacent adjacent frequency frequency region region employed employed by other by applications)other applications) define thedefine limits the oflimits the maskof the for mask the allocatedfor the allocated band, the band, OoB the and OoB the and spurious the spurious emission. emission. The most The recent most andrecent tight and recommendations tight recommendations assert that assert radar that systems radar should systems be designed should tobe meetdesigned the requirement to meet the of therequirement mask-roll-o of fftheat 40mask-roll-off dB per decade at 40 from dB per the decadeB 40 dB from bandwidth the to the dB bandwidth spurious level to the (Figure spurious A1), − aslevel shown (Figure in Figure A1), as 18 shown. in Figure 18.

FigureFigure A1. A1. DesignDesign objective objective for for radar radar systems. systems.

AppendixAppendix B. The “Alternating Projection” Projection” Algorithm Algorithm TheThe ideaidea behind behind the the alternating alternating projection projection algorithm algorithm is to searchis to search for a point for a of point intersection of intersection between abetween matrix describinga matrix describing a spectral a constraint spectral constraint and a matrix and describinga matrix describing a structural a structural constraint. constraint. In our case, In theour spectralcase, the constraint spectral constraint is determined is determined by the spectral by the occupancyspectral occupancy of the generated of the generated waveform waveform and the structuraland the structural constraint constraint is the desired is the PAPR.desired PAPR. InIn [[24],24], thethe alternatingalternating projectionprojection algorithmalgorithm isis wellwell described.described. Here wewe reportreport thethe mainmain stepssteps forfor thethe sakesake ofof simplicity.simplicity. LetLet usus suppose we have two set of matrices Y andand Z of of thethe samesame dimension,dimension, withwith thethe formerformer implementingimplementing spectralspectral constraint constraint and and the latterthe implementinglatter implementing the structural the structural constraint. constraint. The alternating The projectionalternating algorithm projection is algorithm described is as described follows: as follows: INPUTINPUT • AnAn (arbitrary)(arbitrary) initialinitial matrixmatrix Y • 0 • TheThe numbernumber ofof thethe iteration,iteration, J • OUTPUTOUTPUT • A matrix in and a matrix in A matrix Y in Y and a matrix Z in Z • PROCEDURE PROCEDURE (1) Initialize =0 (1)(2) FindInitialize a matrixj = 0 in such that

∈min − ∈

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(2) Find a matrix Zj in Z such that Zj argmin Z Yj F ∈ Z Z k − k ∈ Sensorsbeing 2020, 20, x FORoperator PEER REVIEW the Frobenius norm. 24 of 25 k · kF (3) Find a matrix Yj+1 in Y such that being ‖⋅‖ operator the Frobenius norm. (3) Find a matrix in such that Yj+1 argmin Y Zj F ∈ Y Y k − k ∈ ∈min − ∈ (4) Increment j by one. (4)(5) IncrementRepeat steps by 2–4 one. until j = J (5) Repeat steps 2–4 until = (6) Let Y = YJ and Z = ZJ 1 (6) Let = and =− To solvesolve minimization minimization in in step step 2 and 2 and step step 3 for 3 the for PAPR the PAPR structural structural constraint, constraint, the “matrix the nearness“matrix nearnessproblems” problems” [30] is used, [30] whose is used, algorithm whose algorithm is detailed is in detailed [24] (p. in 198). [24] (p. 198). To summarize the framework in which alternating projection algorithm operates is shown in the ensuing diagram, Figure A2A2..

STRUCTURED FRAME

solves

INVERSE EIGENVALUE PROBLEM (IEP)

solves

ALTERNATING PROJECTION

Step 1 Step 2 Step 3 Step 4 Step 5 Step 6

solves solves

NEARNESS MATRIX FigureFigure A2. Solution A2. Solution to the to Structured the Structured Frame Frame problem problem in inturn turn addressed addressed by by Alte Alternatingrnating Projection Projection and NearnessNearness Matrix Matrix problem. problem.

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