Signal design and processing for noise radar

An efficient and secure use of the electromagnetic spectrum by different telecommu-nications and radar systems represents, today, a focal research point, as the coexistence of different radio-frequency sources at the same time and in the same frequency band requires the solution of a non-trivial interference problem. Normally, is with diversity in frequency, space, time, polarization, or code. In some radar applications, a secure use of the spectrum calls for the design of a set of transmitted waveforms highly resilient to interception and exploitation, i.e., with low probability of inter-cept/exploitation capability. In this frame, the noise radar technology (NRT) transmits noise-like waveforms and uses correlation processing of radar echoes for their optimal reception. After a review of the NRT as developed in the last decades, the aim of this paper is to show that NRT can represent a valid solution to the aforesaid problems.

a Slow time/fast time data matrix ( L = 10 range bins and N = 4 PRI's). b Radar echoes in time of two targets: the first (green) is stationary; the second (red) is moving. c A typical data matrix orders of magnitude less than cavity magnetrons (and the ensuing power amplifier tubes). The reduced peak power has forced to use long pulses with duty cycles in the order of 10%. Hence, pulse compression techniques are needed to achieve the requested range resolution and to avoid the blind ranges. Pulse compression can be implemented either using deterministic waveforms, the most popular being the frequency-modulated "chirp" signal [8,9], or using random (noise) waveforms, as shown in the next section.

Deterministic waveforms
Let g(t) = a(t) · e jφ(t) be the complex envelope of a signal with bandwidth B and duration T. The transmitted signal is modulated both in amplitude (AM) and in phase (PM) by the functions a(t) and φ(t), respectively. Let us denote by φ ′ (t) and φ ′′ (t) the first and the second derivative of φ(t) . When the time-bandwidth product B T is much greater than 1 , according to the stationary phase principle [8,9] the spectrum of the signal for a given instantaneous frequency f t = 1 2π φ ′ (t) is approximated by: S f t 2 ∼ = 2π a 2 (t) |φ ′′ (t)| . Hence, the amplitude a(t) is related both to the spectrum and to the phase φ(t) by: a(t) ∼ = 1 2π |S(φ ′ (t))| 2 · |φ ′′ (t)| . In a linear frequency-modulated (LFM) signal, φ ′′ (t) is constant and, if also a(t) is constant, the spectrum is approximately uniform (for BT ≫ 1) inside the band B, obtaining the widely known chirp signal. The autocorrelation function (ACF) of a LFM chirp has a main lobe width of 1/B and a side lobe level of 13 dB below the peak, which is too high for many practical applications. To attenuate the sidelobes, it is possible to shape the amplitude of the pulse according to a suited time window [10], leading to some loss in SNR because of the reduced transmitted power due to the amplitude modulation. Moreover, the time window might lead to mismatching losses if weighting is not applied also at the receiver, i.e., if the matched filter is not fully implemented. Another way to lower the sidelobe level of the signal is to maintain a constant amplitude (with full exploitation of the transmitted power) and to pre-distort (i.e. to shape) its frequency law φ ′ (t) , weighting its spectrum ( φ ′′ (t) being no longer constant). Shaping the frequency law leads then to a nonlinear frequency-modulated signal (NLFM), able to reach very low sidelobe levels (typically less than − 40 dB) even when a(t) is kept constant, i.e., when the spectrum is shaped through φ ′′ (t) only [11]. The historical most famous NLFM signal is the Millett waveform [12], i.e., a "cosine squared on a pedestal" weighting, with a theoretical peak sidelobe level of − 42 dB for large BT(> 1000) . When a low BT is required, as in various applications like air traffic control radar and the last-generation coherent marine (or navigation) radar, the stationary phase principle is no longer applicable. Theoretically, very low sidelobe levels in the ACF can be obtained by a "hybrid" approach, i.e. combining a proper choice of the phase law with a suited amplitude modulation to reduce the SNR loss. Using the hybrid nonlinear frequencymodulated (HNLFM) waveforms [details are found in [13][14][15], it is possible to attain very low sidelobe levels, less than − 60 dB, already for BT ≅ 100, with very low SNR loss (− 0.58 dB only). However, to reach this performance, HNLFM requires high accuracy in the a(t) which is a hardly achievable task if high-power amplification is required. The examples shown in Figs. 3 and 4 are referred to the illustrative case of a marine/ coastal radar operating at X-band (9.3-9.5 GHz) around a central frequency of 9.4 GHz (wavelength of about 3.2 cm). The bandwidth is 50 MHz, with a waveform duration from 2 to 2000 μs (i.e., BT from 10 2 to 10 5 ). Figure 3 shows the normalized ACF of a HNLFM with BT values ranging from 10 2 to 10 5 for a stationary target (i.e., without Doppler shift). In order to take the radial velocity (i.e., the Doppler effect) into account, it is necessary to introduce the ambiguity function (AF) which represents the output of the matched filter when the echo is affected by a Doppler shift f [16,17]: Figure 4 shows the AF of a HNLFM signal with unit energy and BT = 1000. For more than 50 years, chirp signals have dominated the radar scenes although many other waveforms have been developed, such as Golomb sequences [18], Barker [19], Frank [20], Costas [21], complementary codes [22,23] and more [24]. In the frame of frequency-modulated signals, a Costas waveform of length T = Mt b (with M being an integer) is: It is obtained dividing the time-frequency plane in M sub-elements (chips) of equal duration t b (chip time) and band f = 1 t b with f m = a m f (m = 1, 2, . . . , M) the carrier frequency of the mth chip. The code is defined by a = [a 1 , a 2 , . . . , a M ] , the sequence of integers between 1 and M (hopping sequence, Fig. 5). In Eq. (2), rect t b (t) is equal to 1 for 0 ≤ t < t b and zero elsewhere. The band of the Costas signal is B = M · f , and the compression ratio is M 2 . Figure 6 shows the ambiguity function of a Costas code with M = 40 and T = 16 μs (BT = 1600) having unit energy. A drawback of these codes is the limited number of available waveforms.
Generalizing this type of waveforms, the need for low probability of intercept (LPI) radar has pushed the "randomization" of the transmitted frequency (and sometimes of the pulse repetition interval) of military radar since the WWII [25], leading to "frequency agile" and "random PRT stagger" operation [26]. The pertaining ambiguity function has been mathematically analyzed in [27]. Summing up, all the classical deterministic radar waveforms, used since the early days of radar, have intrinsic problems: high sidelobe level for high Doppler shifts and/or limited availability of orthogonal modulations (implemented as "up" and "down" direction of the frequency modulation in LFM/NLFM/HNLFM chirps) that make them suited to defined applications.
In principle, noise (random) waveforms, as analyzed in the following, do not suffer from the above disadvantages since, by definition, they have a "thumbtack-like" ambiguity function without the need for a complex generation algorithm. Moreover, being each waveform a single realization of an endless random process, their number is theoretically unlimited.
Finally, we believe the bank of Doppler filters to be the best solution for range-Doppler processing in noise radar surveillance, similar to the well-known and widely used moving target detector.
Considering the intrinsically low Doppler tolerance of the random codes used in all noise radars, a bank of Doppler filters is best solution for range/Doppler processing for the surveillance function using NRT. This approach is called range filter bank (see Sect. 6.3) and extends the widely used moving target detector to NRT.

Pseudo-random waveforms
In noise radar technology (NRT), the transmitted signals are random or, strictly speaking, pseudo-random waveforms (i.e., realizations of a random process) and theoretically uncorrelated with each other.
Since the 2000s, the advancement of technology has offered the ability to produce and process noise waveforms using modern computation elements and high-speed analog-to-digital (ADCs) and digital-to-analog (DACs) converters. Along with FPGAs (field-programmable gate arrays), these building blocks permit the generation of highperformance digital noise waveforms. The digitalization of the radar signal generation and processing allows us to design waveforms with novel approaches, driven by signal processing principles rather than by the capabilities of the hardware. However, one must remember what Cook and Bernfeld stated [28]: "in the extreme case, all signals (waveforms) are equally good (or bad) as long as they are not compared against a specific radar environment. " Hence, it is very important for the waveform radar designer to know where the radar is going to operate and which requirements are needed for the specific application. This suggests that NRT is a powerful tool when it is "tailored" to the specific need, but, of course, that no single waveform (including NRT) is to be considered a kind of universal solver.
Concerning the "randomness" of the NRT waveforms, while in principle it is possible to use an analog noise generator [29], most recent trends rely on more effective and manageable pseudo-random number (PRN's) generators [30,31].

Advantages of random waveforms
There are three main advantages of noise waveforms in comparison with the deterministic ones: first, a desirable ambiguity function, i.e., a "thumbtack-like" one, able to guarantee high range/Doppler resolution and the absence of range-Doppler coupling; second, the potential for coexistence with other radars and/or other sources, thanks to the possibility to generate a very large number of orthogonal waveforms; finally, the possibility to design waveforms with low probability of intercept and exploitation (LPI/LPE) features [32].

Ambiguity function
A realization of an infinitely long (B and T both tending to infinity) "white noise" has an AF going to a delta Dirac function, hence with a level of ambiguity decreasing with BT increasing. However, in real applications both B and T are limited, and the peak side level (PSL) of the AF has a fluctuating value with a mean value close to the "processing gain" BT (see Sect. 2.4.1). Figure 7 shows the exemplary ambiguity function of a noise waveform with unit energy and BT = 1000.

Coexistence with other radars and other sources
The increasing scarcity of the electromagnetic spectrum (which forms the basic, nonrenewable resource for radio communications, radio navigation and radar) calls for an organized coexistence of the radio-based services. This issue might be addressed by the communication and radar spectrum sharing (CRSS), a complicated problem, whose practical solution, today, can be based on the use of artificial intelligence (AI) with its machine learning/deep learning tools [33]. Moreover, orthogonality between waveforms plays a fundamental role in many applications, particularly in MIMO (multiple-input, multiple-output) radars [34] and, potentially, in modern civil marine radars [35]. Novel modulations for marine radars are needed, mainly to improve the spectral efficiency and to allow continuous operation of multiple radars with solid-state transmitters in the same frequency band and in the same marine area [35]. Using NRT, mutual interferences to/from many radars located in the same area and operating in the same limited frequency band are reduced. Equation (3) defines the cross-ambiguity function (CAF) between two signals g(t) and s(t): For zero-Doppler (i.e., f = 0 ), Eq. (3) becomes the cross-correlation function (CCF), which can be used to evaluate the orthogonality between two waveforms. Figure 8 compares the ACF (in blue) of a noise waveform with the CCF (in red) of two noise waveforms with the same bandwidth and duration (BT = 1000). In the sidelobes region, both curves show a similar behavior.

Low probability of intercept and of exploitation
The acronym LPI (low probability of intercept [32]) indicates the feature of a radar to make difficult its detection by an opponent using means of passive interception such as electronic support (ES), radar warning receivers (RWR), or electronic intelligence (ELINT) receivers with the final aim to create a suited radar jamming. Today, both electronic warfare (EW) [36][37][38][39][40][41][42][43][44][45][46] and radar systems are exploiting more and more "intelligence" thanks to the convergence of computer science, big data analysis and communications. There are many methods in the literature for intercepting LPI radars. In [47,48], filter banks with higher-order statistics and wavelet transform are proposed. Some methods are based on time-frequency transforms such as short-time Fourier transform and Wigner-Ville transform [49,50]. In [51], quadrature mirror filter banks are used, while [52] uses cyclostationary processing. An innovative method is described in [53] and [54], based on the use of two receivers on board of a fast-moving platform (e.g., an airplane or a satellite). Recently, a new waveform recognition technique (WRT), based on a convolutional neural network (CNN), has been proposed in [55].
In the frame of a specific emitter identification (SEI) capability in modern ES/ELINT systems [56] introduces a classification technique based on some suitable features evaluated from the cumulants of the signal emitted by the radar system.
Noise radar is probably a very effective answer to the increasing demand for operational LPI radars [32], a feature that improves if coupled with low power, wide bandwidth, frequency variability, or other suited design attributes. As a matter of fact, modern interception methods such as the two-antenna correlation receiver [57], seems effective against sophisticated radar waveforms including the pseudo-random ones.

Limitations of random waveforms
There are two main limitations when using noise waveforms: First, the peaks of the side lobes in the autocorrelation function are random with a mean value related to the product BT [1]. Second, the scarce exploitation of the power amplifier when the peak-toaverage power ratio (PAPR) is significantly greater than the unity.

Random sidelobes and peak sidelobe level
For a noise waveform g(t) of duration T and bandwidth B , the correlation processing (matched filtering) generates random fluctuations in the peak sidelobes level (PSL) of the ACF; see Fig. 9. Normally, a delay τ * divides the time axis into two parts (in Fig. 9, τ * ≅ 0.1 μs). It is defined as follows: For delay less than τ * , all different realizations of the ACF are very close to each other both in the main lobe and in the first sidelobes. When the delay is greater than τ * (i.e., in the "random sidelobes region"), the sample mean of the ACF converges to the theoretical mean π 4BT , as shown in [1], where the distribution of the PSL has been deduced, which permits to estimate a PSL level, that is not exceeded for a given probability δ . Such a level depends on the product BT and on the particular type of noise spectrum. It can be estimated as: where K is a constant, typically in the range 10-13 dB, depending on the chosen probability δ , as shown in Fig. 10 for different values of BT.

Limited exploitation of the power amplifier
In the design of noise radar waveforms, in addition to the bandwidth B and duration T, another relevant parameter is the peak-to-average power ratio (PAPR), i.e., the (4) PSL dB = −10 · log 10 (BT) + K , with k = 1, 2, . . . , N , are the samples of the signal taken at the Nyquist (or smaller) interval.
Normally, deterministic waveforms (chirp, Barker, polyphase codes, etc.) have unitary PAPR (i.e., the signals are "unimodular" with phase or frequency modulation), while an arbitrary noise waveform (unless hard-limited) has a PAPR greater than one with a maximum (said "natural" PAPR in the following) around 10.5 to 12.5 for a probability δ = 0.01 and BT between 10 3 and 10 6 ; see Fig. 10. In this situation, the power transmitter is less efficient than one that could work in saturation; hence, a loss in SNR arises equal to −10 · log 10 (PAPR) [dB]. Figure 11 shows this loss versus the PAPR.
Of course, the advantage to use unimodular noise signals (PAPR = 1) is the absence of such a loss, while the drawback is that the number of degrees of freedom of the waveform is halved, passing from 2BT real degrees of freedom (amplitude and phase pairs, or real and imaginary parts of the signal) to BT. This point is important in the identification and exploitation of the signal and calls for a trade-off between the transmitted power exploitation, the number of degrees of freedom and the PAPR.

Trade-off related to the peak-to-average power ratio
As shown in the previous section, for a noise waveform the "natural" PAPR is around 10-12 implying some high loss in SNR (order of − 10 dB, see Fig. 11). To reduce this loss, the PAPR can be forced, using a suited nonlinear waveform transformation, to a much lower value (e.g., to PAPR = 1.5 which reduces the loss to − 1.76 dB only). An algorithm, called "Alternating Projection" [58], forms a robust and computationally efficient method to control the PAPR and to design waveforms with given structural properties [59].
However, a nonlinear transformation aimed to the PAPR reduction modifies the underlying random process and destroys its original Gaussian statistics. To measure the distance of a zero-mean random variable X of variance σ 2 from the Gaussian one, the negentropy J (X) is used [60]: where h G (X) = 1 2 ln 2π eσ 2 is the entropy of a Gaussian random variable of variance σ 2 . It is well known that the Gaussian has the maximum entropy among all distributions with the same variance. Then, J (X) > 0 is a measure of the distance of X from the Gaussian distribution.
Often, the evaluation of the negentropy is very difficult when the probability density function is unknown. Figure 12 shows the estimated negentropy of the real part of a single noise waveform with uniform spectrum (B = 50 MHz, σ 2 = 0.5 ) varying the PAPR, where h G (X) = ln √ π e ∼ = 1.0724 , while h(X) is estimated using the histogram approximation of the probability density function. With the PAPR decreasing up to a value of 4σ ∼ = 2.8 , the underlying process "starts" to be non-Gaussian, and the SNR loss also decreases (Fig. 11), i.e., the loss of Gaussianity is the price paid for the transmitted power efficiency.

Historical overview of noise radar technology
Noise radar was introduced for the first time in 1959 by Horton for a system able to measure the distance of a target [61]. Historically, the generation of noise signals was first implemented using analog sources [62]. For instance, in the 1990s, methods for an efficient generation of "chaotic" signals from an analog source at W-band were developed and tested in Ukraine [29,63]. In China, studies of random signals radar (RSR) date back to the 1990s; they concerned the design of noise FM-CW radar and its implementation for collision warning of vehicles and for battlefield surveillance [64,65]. In the USA, at the end of 1990s, noise radar technology was applied to design a coherent ultra wide-band (UWB) random noise system for Doppler estimation [66,67] and airborne imaging [68], and a new technique for the range sidelobe suppression of UWB random noise radar was proposed and analyzed. Since the early 2000s, a constant, experimental research activity with numerous field trials takes place in Ukraine, where in 2002, the First International Workshop on Noise Radar Technology (NRTW 2002) was held [69,70]. Theoretical analysis was developed in [71][72][73] regarding noise radar using random phase/frequency modulation. Concerning the analysis carried out in [72,73], it has to be noticed [74] that a main drawback (which seems to "propagate" among different researchers) is the wrong assumption that the phase θ (t) is a Gaussian random process, neglecting the fact that the phase is folded in the range −π to +π. In Poland, at the Warsaw University of Technology, a noise radar demonstrator was implemented (starting from 2010 and using commercial hardware) for the detection of moving targets at short ranges. The signal processing included adaptive filtering for clutter removal. The processing was validated with real-life experiments, during which cars at ranges of hundreds of meters and aircraft at ranges of a few kilometers were detected [75]. During the experiments, the main drawback was the occurrence of the masking effect (i.e., a weak target is masked by the sidelobes of a strong one). Hence, in [76], a filter-based method was introduced to create noise-like waveforms with very low sidelobes in certain range and Doppler intervals, and more publications followed.
From 2005, a considerable effort on NRT pre-competitive and unclassified research is being developed in the frame of the NATO Science and Technology Organization . From December 2020, the RTG SET-287 "Characterization of Noise Radar" is active. In SET-101 and SET-184, the objectives were to identify the areas where noise radar would provide significant advantages over the classical radar. The conclusions were an evaluation of architectures, technologies and components for the development of noise radar technology with emphasis on implementation of software-defined radar, pseudo-random waveforms design/generation and radar returns processing using arbitrary waveform generators (AWG), FPGA-based signal processing and multicore/parallel computing.
In SET-184, it was shown that noise radar technology is capable of providing new electronic protection capabilities to short-range operational tactical radars [77][78][79].
In SET-225, it was demonstrated that noise radars can be developed to fulfill useful military applications. This was done by cooperatively building and field testing two radar transportable demonstrators at the Fraunhofer FHR Institute in Wachtberg (Germany) and at the Turkish Naval Research Center Command (TNRCC) in Istanbul (Turkey), respectively. In the summer of 2018, field trials of the first demonstrator were held at the FHR site in Wachtberg [80]. Both static targets (corner reflectors) and moving targets (cars, aircraft) have been used to check detection capabilities in the presence of clutter, multipath and antennas' coupling, as well as to test radar resolution. The TNRCC demonstrator was tested in the coastal environment of Marmara Sea (Turkey) [81]. In both sets of field trials, the importance of the range sidelobes from close-in targets and from the transmit-receive leakage in affecting the practical sensitivity of the radar has been clearly highlighted. A large amount of work, that has been carried out by the members of the NATO groups, particularly SET-225, is described in [2]. Active since December 2020, the unclassified RTG SET-287 "Characterization of Noise Radar" aims to produce recommendations for the specification and characterization of noise radars. The main topics of this research are the following: • Control and characterization of direct leakage between transmitter and receiver. • Control of sensitivity floor due to range sidelobes of large targets. • Minimization of transmitter power consumption and processing load. • Development of low probability of intercept (LPI) and interference rejection features. • Effective use of RF spectrum by exploiting the flexibility of transmitted waveforms.
In this frame, the evolution of the previously introduced demonstrators was tested in different (land and marine) environments showing their suitability for short-range, high-resolution surveillance [3]. The experiments confirmed the need for tailoring the transmitted waveform to improve the receiver dynamic range and to counteract the damaging effects of leakage and of nearby clutter and strong targets.
The interested reader can find a signal processing viewpoint introduction to noise radar, in [82], while an overview of the Italian academic contribution to radar (and to NRT) is available in [83].

Noise radar architectures
The basic elements of many noise radar designs are similar. In the preferred, continuous emission (CE) architecture, the transmission and the reception channels are implemented with a specified mutual isolation to avoid leakage due to antenna coupling as shown in Fig. 13. To reduce the damaging leakage on target detection, in principle there are three possible approaches: first, acting on the apparatus, i.e., increasing the antenna isolation (the state of the art for co-located antennas, X-band, is around 90 dB); second, using adaptive power cancellers; and third, acting on the transmitted waveform generator, i.e., improving the peak to sidelobes level PSL. Figure 14 shows a simple block diagram of a continuous emission noise radar, where a general functional separation into a digital back end and an analog front end is shown. The front end has the simple function to convert the radar signals between an intermediate frequency (IF) band and a desired radio frequency (RF) band. The ports of the front end are labeled by S (received or surveillance signal), W (waveform), RX and TX. Let us consider the transitions between these ports. The up-conversion is performed from port W, the waveform input, to the transmitting antenna at port TX. The back end embodies most of the typical CE noise radar functionalities, including the correlation (more generally, the cross-ambiguity function (CAF)) of the received signal with the pertaining  reference. The latter represents a replica of the transmitted waveform, and its acquisition distinguishes two basic approaches: solution (a) of Fig. 14 and solution (b) of Fig. 15. Solution (a) is the simplest implementation of a CE noise radar, which obtains the reference signal from a digital copy of the waveform samples that are provided to the digital-to-analog converter (DAC). Its advantage is the simple RF front end. Its drawback is that the reference signal might deviate from the actual transmission when the all-digital reference acquisition does not respect the transfer function of the front end.
Consequently, Fig. 15 shows the alternative approach, where a coupler from the RF transmission path supplies the reference signal which is down-converted along with the surveillance signal and digitized by a separate analog-to-digital converter (ADC).
This implementation, with the cost of an ADC, an additional channel signal to the additional front-end port R, delivers a more robust reference signal and, more important, makes the waveform generation independent of the implementation of the signal processing. In fact, the waveform might be created by any source, not necessarily digital, complying with the signal specifications of the port W, where the reference signal is acquired by the front end.

Waveforms design
In NRT, the transmitted signal could be derived from physical phenomena, as thermal noise in amplifiers or Zener diodes [29]. However, this approach is often impractical due to accuracy, resolution and dynamic range limitations. Hence, digital implementation, as in most cryptographic applications, is preferred. In this frame, the design of noise waveforms is based on the computer generation of sequences of pseudo-random numbers (PRNs). Starting from an initial value, called "seed", which defines the whole sequence, when the machine returns to the initial point, the sequence repeats itself. Therefore, one should speak of pseudo-randomness, meaning that these numbers appear random to an external observer as they pass some statistical tests [84]. The aforementioned periodicity problem is made immaterial by generation algorithms such as the "Mersenne Twister" [30], a family of PRNGs (pseudo-random number generators) based on F2-linear maps [85], whose period is the huge 2 19937 − 1 for 32-bit integers. In practice, the period of the sequence is a minor problem with respect to the low statistical quality of some widely used PRNGs [86].
For constant amplitude noise signals (PAPR = 1), a simple way to generate noise waveforms is to introduce the randomness into the phase (or frequency) as previously discussed [71,72]. In [87,88], noise signals are generated adding noise to the wellknown deterministic LFM chirp signals. In [89], starting from the classical approach shown in [90], waveforms are generated using a white Gaussian process with its independent, identically distributed (IID) in-phase (I) and in-quadrature (Q) components which is filtered (spectral shaping) in the requested frequency band. To reduce the PSL [91] and to control the spectral width, an iterative algorithm has been proposed in [92] starting from a zero-mean white complex Gaussian process to be band limited in the frequency domain (FFT) by a low-pass filter, then by IFFT normalized (amplitude limited) in the time domain. Iterating this time/frequency projection, the PSL, after some tens of iterations, converges to a minimum. Figure 16 shows a block diagram of a digital noise generator including the PAPR control and the sidelobes suppression in a specific radar range interval [1].
Setting the duration T of the noise waveform and the sampling frequency F s , the steps are: (i) A PRNG generates F s · T independent and identically distributed Gaussian pairs of samples to form the complex sequence with uniform spectrum. By filtering with H f = S f , the initial power density spectrum is "shaped" as S f inside the band B. The number of independent real samples 2BT represents the "degrees of freedom" of the waveform. (ii) The PAPR control block sets, by the alternating projection algorithm [58], the PAPR to a desired value (note that before the spectral shaping, the PAPR has its "natural value" of 10-12 or greater). Normally, the PAPR is forced to lower values (e.g., in the range 1.5-2.0), to reduce the SNR loss typically below 3 dB (see Fig. 11). The output is the signal g 1 (t) , see Fig. 16, i.e., a non-Gaussian sequence with the assigned PAPR. (iii) The sidelobes suppression function improves the PSL in a specific range interval, as shown in Fig. 17a. This method is based on the inverse filtering approach (well known in the literature [93]) applied in Fig. 17b to the signal g n (t) . Due to the folding of the spectral components into −B 2, +B 2 , the output must be iteratively processed by the algorithm (see Fig. 17c, starting from n = 1 up to n iter ) to obtain the desired range zone suppression.
The "tailored" signal at the end of the procedure is used as transmitted waveform.
In [94], a different approach is proposed, called COSPAR (combined spectral shaping and peak-to-average power ratio reduction). It uses a pseudo-random sequence θ of Uniform in (−π , +π) phases to generate the noise waveform. In the frequency domain, the term e jθ multiplies the square root of the desired spectral density S (see Fig. 18).
Being the length of sequence, normally much greater than one, applying the central limit theorem to the IFFT of Z = √ S · e jθ , this transformation makes the components I and Q Gaussian-distributed in the time domain. The peculiarity of this method is that all pseudo-random sequences have the same (deterministic) power spectrum S, chosen so that the autocorrelation function has a deterministic PSL. In [94], the autocorrelation function comes from a Taylor window with 50 dB of PSL (all sidelobes having the same PSL).

Loss in the optimal processing
Let g(t) be the transmitted radar signal, i.e., a pseudo-random waveform of duration T (often called coherent processing interval-CPI) and bandwidth B around the central frequency f 0 , and the received echo signal s(t) , a delayed copy of the transmitted signal modulated by the Doppler frequency f , respectively. The digital record of g(t) is the input to the reference channel, while the digital record of s(t) is the input to the surveillance channel of the radar processor (see Figs. 14 and 15).
The optimum receiver (matched filtering), for a given pair of delay τ and Doppler frequency f, has the well-known non-casual impulse response h(t) = g * (−t) · e −j2π ft the output being the convolution between h(t) and s(t) , whose amplitude is the well-known CAF, Eq. (3), that, for s(t) = g(t) , equals the AF of the transmitted signal, Eq. (1).
In practice, the durations of the surveillance and reference signals are both finite, equal or less than the CPI. Introducing the rectangular window p T (t) , i.e., a unit pulse of duration T, the output Y T τ , f of the matched filter is a random process, whose expected amplitude is [95]: with R sg (τ ) the cross-correlation function between the random processes s(t) and g(t) , while AF p T τ , f is the ambiguity function of p T (t) . For zero-Doppler, the energy of the received signal is E T = R sg (0) · T , that is fully recovered at a near zero distance. Increasing the delay τ in Eq. (6), the finite duration of p T (t) causes a loss L R = 10 · log 10 1 − τ T in [dB]. Note that L R is negligible when T ≫ τ , with τ = 2R c and R the distance of the target of interest. At zero-delay, a moving target with radial velocity v = 0 produces a Doppler shift f = 0 , which causes a loss L D = 10 · log 10 sin (πfT)

Sidelobes Suppression Algorithm
Initial value

Sidelobes Suppression Algorithm
No Yes Fig. 17 a The sidelobes suppression interval ( τ 1 , τ 2 ) as defined by q(t). b The inverse filtering method to suppress the sidelobes in a specific range zone. c The iterative algorithm to improve the suppression of the sidelobes Combining the delay and frequency effects τ = 0, f = 0 , normalizing the time and the frequency axis, i.e., using τ n = τ T and f n = fT , the loss in [dB] is equal to: L = 10 · log 10 AF p T τ n , f n . Figure 19 shows L with ( τ n , f n ) varying. For τ n < 0.1 and f n < 0.25 , the loss may be neglected (being less than − 0.8 dB).

Some considerations on the implementation of the correlation filter
To guarantee a high processing gain BT, the duration T is in the order of the CPI, i.e., in the typical range of 10-20 ms. Hence, T results much greater than τ max , i.e., the delay pertaining to the maximum expected distance, and in the evaluation of the CAF, Eq. (3), a large amount of immaterial data, beyond the maximum range of interest R max = c·τ max 2 ( c = 3 · 10 8 m/s), significantly increases the processing load. If the CPI is divided into P contiguous sub-intervals T r (i.e., one sets T = P · T r ), the reference and surveillance signals can be written as: g(t) = P−1 p=0 g p (t) and s(t) = P−1 p=0 s p (t) , where g p (t) and s p (t) are the signals in each sub-interval p . At zero-Doppler, R sg (τ ) is: where ⊗ denotes the correlation operator. The envelope of R sg (τ ) is the well-known triangle of base 2PT r and height PT r . Note that for the delay τ max of the farthest target, it results τ max ≪ T and we can set τ max = M · T r , where the duration T r is short enough to make the related Doppler shift negligible along the time T r and the integer M is such that the maximum range of interest (distance of the farthest expected target) is cMT r 2 . Imposing a maximum Doppler shift of �ϕ = 2π fT r < π 16 , the maximum T r is equal to  Table 1 shows possible values of T r for different Doppler velocities of the target.

Range Filter Bank (RFB)
The aim of the algorithm named range filter bank (RFB) [96] is to compute the CAF, Eq. (3) in a limited delay and Doppler interval of interest, avoiding any unnecessary processing. The RFB is made up by M correlation filters with their impulse responses related to the reference signal g(t) as shown in Fig. 20. In reception, the surveillance signal s(t) is divided into P contiguous sections of duration T r : The first section s 0 (t) is sent as input to the RFB. The filter's outputs are coherently added creating a first single range profile. This process is repeated for the second section s 1 (t) ; being the reference signal delayed by T r , a second range profile is generated. At the pth step, it results: sg (t) , p = 0, 1, . . . , P − M , represents a correlation function whose amplitude envelope is an isosceles trapezoid with major base (M + 1)T r , minor base (M − 1)T r and height T r . For p = 1, . . . , P − M , each trapezoid is delayed by T r with respect to the previous one. In order to cope with the maximum range requirement equal to cMT r 2 , each R (p) sg (t) , for 1 < p ≤ P − M , is realigned in time by compensating for the delay pT r , and the coherent sum becomes: sg (t) = s p (t) ⊗ g p (t) + g p+1 (t) + . . . + g p+M−1 (t) The envelope of the sum results in an isosceles trapezoid with major base (M + 1)T r , minor base (M − 1)T r and height (P − M + 1)T r . The loss, in dB, with respect to the peak of the triangle ( PT r ) is L RFB = 10 · log 10 1 − M−1 P [dB] . To guarantee L RFB < 1 dB, P and M has to be: M ≤ P 5 + 1 . Normally, due to the practical value of the maximum range of interest, M ≪ P and the L RFB is negligible. The maximum range of the RFB is determined by the minor base of the trapezoid: R max = (M − 1) cT r 2 , while the correlation envelope reaches zero at R tot = M cT r 2 , as shown in Fig. 21 for P = 10 and M = 3 , where the abscissa reports the range normalized to R 0 = cT r 2 . Note that a fading effect in the Mth range filter occurs with constant length of Each term of the sum in Eq. (9) represents a fast time processing with the aim of extracting range data. A data reorganization follows the processed data (realigning in time of the range profiles). An ensuing slow time processing has the aim to provide a further processing gain and to extract the Doppler radial velocity. In such a way, the only remaining losses are those due to the "Doppler mismatch" (constant phase in place of linear phase) within each interval of duration T r , readable on the y-axis of Fig. 19, and made negligible with a suited choice of T r . In [96], the parameter P is factorized in the product of the two integers N and N F , i.e., P = N · N F . The first term N represents the number of coherent sums of range profiles (prior of the FFT operation). When N � > 1 , the number of FFT points ( N F ) is reduced accordingly for each range cell.
Considering the MTD processing as described in the Introduction, the RFB shows a high similitude with the MTD. Both processing methods use fast time/slow time ( L × N ) data matrix to extract range and Doppler parameters. With the same CPI and sampling interval of 1 B (with B the bandwidth of the signal), in RFB, N = P M and L = M · T r · B.

Noise radar demonstrator
A noise radar demonstrator was developed at FHR during the past few years, which operates in the same frequency region of the X-band as standard marine navigation radars, i.e., between 9300 and 9400 MHz. This choice was made in order to enable experiments that include the coexistence of noise radars and the various emissions of commercial radars. The noise radar demonstrator follows the design criteria as described in Sect. 4 and, in principle, is capable of exploiting both the aforementioned choices for the reference signal. However, one has to notice that in most FHR experiments an analog reference signal taken at the X-band is preferred and that [3] makes use of the system's capability to compare both choices of reference signal on the same set of surveillance data. An FPGA, a mass storage device and appropriate A/D and D/A converters form the digital core element of this demonstrator. A picture of this ensemble is displayed on the left-hand side of Fig. 22. The full demonstrator system (excluding the antennas) is shown on the right-hand side of the same figure. The upper tray of the displayed flight case contains the digital processor. The front end, whose principles and ports are described in Sect. 4, is located in the middle tray with ports W, S, R connected to the digital back end and ports Rx, Tx connected to the antennas by green, low-loss cables. The lower tray holds the local oscillator signal that is used by the mixers in the RF front end.

Some trials
To describe a part of the FHR trials, an experiment is presented that makes use of the efficient computation capabilities of the range filter bank (RFB, Sect. 6.3) and his moving target detection capabilities.
In the setup, the demonstrator was assembled into a car with antennas oriented in the direction opposite to the normal motion of the platform. This arrangement provides an optimal exploitation of the Doppler effect while preserving a safe control of the car by the driver. A second person inside the car is required as a radar operator. This setup of a noise radar and a significantly moving platform, as a novelty, was first described in [97]. The measurement system is displayed in Fig. 23 showing, on the left-hand side, the installation of the digital and RF components to the trunk of the radar car. Next to these, a mobile power supply is contained. The right-hand side of Fig. 23 displays an outside view to the demonstrator arrangement with two standard horn antennas being mounted on each side of a microwave-isolating material block to the rear of the radar car. The following measurements were performed when driving along a straight highway, an optimal setup for Doppler investigations as all motions are along a radial direction to the radar. Hence, a direct mapping of the tachometer values with the indicated Doppler shift gives a simple indicator for the verification of the moving target detection by the RFB. The set of experimental data was recorded in a hilly area at a point where the radar car was driven on one uphill side of a slight valley, whereas some targets were located on the opposite uphill side. This configuration minimized the mutual shadowing effect that may affect the signals from targets of similar size and shape when recorded along their arrangement in a row.
The experimental environment includes static targets (trees and the guardrails of the highway). In the same direction in which the radar car drove, there were slower road users, but due to the higher traffic volume, no car was expected to drive faster than the radar car. On the other lane, driving in the opposite direction, the situation also consists of two groups of road users, a slower and a faster one. However, all cars of each group drove with approximately the same speed. Figure 24 shows a picture of a similar road situation photographed by the radar operator.
This arrangement of targets can be detected and displayed from the radar results as shown in Fig. 25. The range-Doppler map shows the distance to the targets on its x-axis and the Doppler frequency, converted to radial velocity of the targets on its y-axis. Here, the Doppler values face an offset due to the proper motion of the radar platform. When taking into account that the velocity of the radar platform was equal to v p = 130 km/h, all static targets are expected to show an indicated velocity of − 130 km/h and, indeed, the j-hook-shaped clutter returns converge against this value for grater ranges. Consequently, all targets at speeds similar to v p appear at relative velocity values close to zero, a region in the range-Doppler map that is highly affected by leakage effects between the transmitter and the receiver as explained in Sect. 4 and in [98]. Note that cars moving on the opposite lane produce distinct target echoes in the range-Doppler map that are neither affected by leakage nor by clutter effects. In Fig. 25, the slower group of road users is indicated by a dashed rectangular frame. Their relative velocity is measured to be − 212 km/h and when compensating for the platform speed v p , the actual velocity of this group is in the order of 80 km/h which is a typical speed of trucks in high traffic situations on German highways. Three other cars were measured at different speed levels. Their echoes are highlighted by solid circular frames in Fig. 25. The results indicate an overtaking process at the distance of 275 m where there are two reflections at different Doppler shifts. One belongs to the group of slower road user at 80 km/h, and the other one, at a higher Doppler level, corresponds to an overtaking car with an actual velocity of 120 km/h. These experiments evidentiate the high suitability of noise radar technology for moving target detection. In the past, the beneficial absence of range and Doppler ambiguities in noise radar was contradicted by the high computation load that the CAF requires. The latter may be replaced by the much more efficient range filter bank (RFB) algorithm enabling the use of the NRT in resource-limited applications. Please note that the SNR of the targets in this measurement was increased by reducing damaging effects of strong close-in clutter returns and direct leakage using the ECA-T [99] algorithm producing "white spots" in the range-Doppler cells, to which the ECA-T is applied.

Conclusions
This tutorial on noise radar technology has presented, in addition to the main historical elements and to the basic operating principles, some selected and recent scientific advancements in both the theoretical and the experimental arenas. The presentation might encourage further research in this stimulating area as well as the operational use of (pseudo) random signals in future radar systems. Having known and understood the specific capabilities of noise radar for many decades, science carries on to remove the practical hurdles and some preconceptions (biases) that still today impede the industrialization of noise radar technology in many practical applications. For example, one exemplary preconception is the sometime alleged "ten dB loss" due to the random modulation of the transmitted signal; however, it was shown to be possible to make this loss smaller and smaller by imposing a suitable PAPR to the "tailored" waveform.
The authors believe that the results presented in this paper will serve as enablers along this aspiring way.