### 2.1 Deterministic waveforms

Let \(g\left( t \right) = a\left( t \right) \cdot e^{j\phi \left( t \right)}\) 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 \(a\left( t \right)\) and \(\phi \left( t \right),\) respectively. Let us denote by \(\phi^{\prime } \left( t \right)\) and \(\phi^{\prime \prime } \left( t \right)\) the first and the second derivative of \(\phi \left( t \right)\). 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 \(f_{t} = \frac{1}{2\pi }\phi^{\prime } \left( t \right)\) is approximated by: \(\left| {S\left( {f_{t} } \right)} \right|^{2} \cong 2\pi \frac{{a^{2} \left( t \right)}}{{\left| {\phi^{\prime \prime } \left( t \right)} \right|}}\). Hence, the amplitude \(a\left( t \right)\) is related both to the spectrum and to the phase \(\phi \left( t \right)\) by: \(a\left( t \right) \cong \sqrt {\frac{1}{2\pi }\left| {S\left( {\phi^{\prime } \left( t \right)} \right)} \right|^{2} \cdot \left| {\phi^{\prime \prime } \left( t \right)} \right|}\). In a linear frequency-modulated (LFM) signal, \(\left| {\phi^{\prime \prime } \left( t \right)} \right|\) is constant and, if also \(a\left( t \right)\) is constant, the spectrum is approximately uniform \(\left( {{\text{for}}\;{\text{BT}} \gg 1} \right)\) 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 \(\phi^{\prime } \left( t \right)\), weighting its spectrum (\(\left| {\phi^{\prime \prime } \left( t \right)} \right|\) 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 \(a\left( t \right)\) is kept constant, i.e., when the spectrum is shaped through \(\left| {\phi^{\prime \prime } \left( t \right)} \right|\) 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 \({\text{BT}} \left( { > 1000} \right)\). 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 frequency-modulated (HNLFM) waveforms [details are found in 13–15], 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 \(a\left( t \right)\) 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]:

$${\text{AF}}\left( {\tau ,f} \right) = \left| {\mathop \smallint \limits_{ - \infty }^{ + \infty } g^{*} \left( t \right)g\left( {\tau + t} \right)e^{j2\pi ft} dt} \right|$$

(1)

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 \(T = {\text{Mt}}_{b}\) (with *M* being an integer) is:

$$g\left( t \right) = \frac{1}{{\sqrt {{\text{Mt}}_{b} } }}\mathop \sum \limits_{m = 1}^{M} \exp \left( {j2\pi f_{m} t} \right) \cdot {\text{rect}}_{{t_{b} }} \left[ {t - \left( {m - 1} \right)t_{b} + \frac{T}{2}} \right]$$

(2)

It is obtained dividing the time–frequency plane in *M* sub-elements (chips) of equal duration \(t_{b}\) (chip time) and band \(\Delta f = \frac{1}{{t_{b} }}\) with \(f_{m} = a_{m} \Delta f\) \(\left( {m = 1, 2, \ldots ,M} \right)\) the carrier frequency of the *m*th chip. The code is defined by \(a = \left[ {a_{1} ,a_{2} , \ldots ,a_{M} } \right]\), the sequence of integers between 1 and *M* (*hopping sequence*, Fig. 5). In Eq. (2), \({\text{rect}}_{{t_{b} }} \left( t \right)\) is equal to 1 for \(0 \le t < t_{b}\) and *zero* elsewhere. The band of the Costas signal is \(B = M \cdot \Delta f\), 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.

### 2.2 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 high-performance 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].

### 2.3 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].

#### 2.3.1 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.

#### 2.3.2 Coexistence with other radars and other sources

The increasing scarcity of the electromagnetic spectrum (which forms the basic, non-renewable 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 \(g\left( t \right)\) and \(s\left( t \right)\):

$${\text{CAF}}\left( {\tau ,f} \right) = \left| {\mathop \smallint \limits_{ - \infty }^{ + \infty } g^{*} \left( t \right)s\left( {\tau + t} \right)e^{j2\pi ft} dt} \right|$$

(3)

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.

#### 2.3.3 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.

### 2.4 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-to-average power ratio (PAPR) is significantly greater than the unity.

#### 2.4.1 Random sidelobes and peak sidelobe level

For a noise waveform \(g\left( t \right)\) 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 \(\tau^{*}\) divides the time axis into two parts (in Fig. 9, \(\tau^{*}\) ≅ 0.1 μs). It is defined as follows: For delay less than \(\tau^{*}\), 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 \(\tau^{*}\) (i.e., in the *“random sidelobes region”*), the sample mean of the ACF converges to the theoretical mean \(\sqrt {\frac{\pi }{{{\text{4BT}}}}}\), 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 \(\delta\). Such a level depends on the product BT and on the particular type of noise spectrum. It can be estimated as:

$${\text{PSL}}_{{{\text{dB}}}} = - 10 \cdot \log_{10} \left( {{\text{BT}}} \right) + K$$

(4)

where \(K\) is a constant, typically in the range 10–13 dB, depending on the chosen probability \(\delta\), as shown in Fig. 10 for different values of BT.

#### 2.4.2 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 ratio \(\frac{{\mathop {\max }\limits_{k} \left| {g\left[ k \right]} \right|^{2} }}{{\frac{1}{N}\mathop \sum \nolimits_{k = 1}^{N} \left| {g\left[ k \right]} \right|^{2} }}\) where \(g\left[ k \right]\), with \(k = 1, 2, \ldots ,N\), 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 \(\delta = 0.01\) 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 \(- 10 \cdot \log_{10} \left( {{\text{PAPR}}} \right)\) [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 2*BT* 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.

### 2.5 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 \(\sigma^{2}\) from the Gaussian one, the *negentropy* \(J\left( X \right)\) is used [60]:

$$J\left( X \right) = h_{G} \left( X \right) - h\left( X \right)$$

(5)

where \(h_{G} \left( X \right) = \frac{1}{2}\ln \left( {2\pi e\sigma^{2} } \right)\) is the *entropy* of a Gaussian random variable of variance \(\sigma^{2}\). It is well known that the Gaussian has the maximum entropy among all distributions with the same variance. Then, \(J\left( X \right) > 0\) 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, \(\sigma^{2} = 0.5\)) varying the PAPR, where \(h_{G} \left( X \right) = \ln \left( {\sqrt {\pi e} } \right) \cong 1.0724\), while \(h\left( X \right)\) is estimated using the histogram approximation of the probability density function. With the PAPR decreasing up to a value of \(4\sigma \cong 2.8\), 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.