Pulsed signal generation

The proposed channel model is additive white guassian noise. Train of pulses is generated by taking the LFM as an intra-pulse modulation example. This type of modulation is of great interest in today's radar applications due to its high resolution in range and increasing of maximum range capability. Sweep bandwidth is considered to be 500 kHz sampled at 5 MHz and the duty cycle is 10% (pulse width (PW) = 0.1 ms and PRI (pulse reppitition interval) = 1 ms). Figure 2 shows the pulse train and its noisy version. For the false alarm testing, pure noise is also generated as shown in Fig. 1.

Fig. 2.

Pulse-train (top), noisy with SNR −18 dB (bottom).

Wavelet denoising

WT is a new technique that maps from L ²(R) to L ²(R) and gives a time–frequency distribution of the input signal. It contributes in high frequency and time resolution that can localize the spectral components in precise time. Making an evolution in digital signal processing, WT is widely used in many applications such as multimedia compression. It decomposes the signal into what is called “approximations” and “details.” What makes WT different than WPD is that in WT only approximations are considered in sub-division, whereas in the latter one, both approximation and details are divided which results in a full tree of all parameters.

Equation (1) shows the WT of f(x) ∈ L ²(R) relative to (ψ(x)) wavelet and scaling function “ϕ (x)”, where j ₀ is an arbitrary starting scale and “c _j0(k)'s” are normally called the approximations or scaling coefficients, the “d _j(k)'s” are called the details or wavelet coefficients:

(1)$$f\,( x) = \mathop \sum \limits_k \;c_{\,j0}\,( k) \phi _{\,j0\comma k}\,( x) + \mathop \sum \limits_{\,j = j0}^\infty \mathop \sum \limits_k \;d_j\,( k) \psi _{\,j\comma k}\,( x) $$

In this paper, WT is used for denoising; three steps are included to make that: decompose, apply threshold, and signal regeneration. After WT decomposition hard threshold is applied, consisting of establishing the coefficients to zero whose absolute values are less than the threshold, otherwise, the coefficient values are not modified, as shown in equation (2) where c (n) represents the coefficients and “T” the threshold value:

(2)$$f_H = \left\{{\matrix{ {C( n ) \comma \;} \hfill & {{\rm for}\,\vert {c( n ) } \vert \gt T} \hfill \cr {0\comma \;} \hfill & {{\rm Otherwise}} \hfill \cr } } \right.\;\;$$

Threshold selection rules are derived by mathematical calculations that can provide a representative noise threshold. We propose the use of “Sqtwolog” method which was proposed by “Donoho and Johnstone” [Reference Joy, Peter and John5]. Threshold values are calculated by using the universal method (square root record) given by equations (3) and (4) where σ is the mean absolute deviation (MAD), N _j is the length of the noisy signal, and ω represents the wavelet coefficient to scale j:

(3)$$th_j = \sigma _j\sqrt {2\,{\rm log( }N_j) } $$

(4)$$\sigma _j = \displaystyle{{MAD_J} \over {0.6745}} = \displaystyle{{{\rm median( \vert }\omega {\rm \vert ) }} \over {0.6745}}$$

Another essential choice is the mother wavelet. Combining both threshold technique and mother wavelet was subjected to root mean square error (RMSE) testing, “Daubechies” mother wavelet shows the lowest RMSE with the original signal. Different mother wavelets show close results by this test; no specific condition is required; and hence no dependency between the signal denoising and the required denoising technique. “Daubechies” mother wavelet of order 8 is used with six levels decomposition and after that the threshold is applied and signal is regenerated. Figure 3 shows the signal before and after denoising at SNR −18 dB.

Fig. 3.

Pulse train (top), noisy with SNR −18 dB (middle), and denoised (bottom).

STFT and statistical threshold

These are used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time. In a short-time basis STFT creates the time–frequency grid that describes the spectral components in time basis. The signal is divided into shorter frames windowed and overlapped. The window size is an important parameter; it is preferable in our case to be small enough to get high-resolution spectral localization for both short and long pulses. STFT in discrete time is described in equation (5):

(5)$$X_l( k) = \mathop \sum \limits_{n ={-}( N/2) }^{N/2-1} w( n) x( n + lM) e^{{-}j2\pi kn/N}$$

where l is the frame number, M is the frame length, and w is the analysis window.

It can be seen from Fig. 4 that the denoised signal spectrogram contains the original spectral information. In our solution, we need to automatically calculate a threshold to be applied on the denoised signal on time–frequency basis. The threshold is adaptive; meaning that it is calculated each time the STFT is performed. Because this algorithm is a power-based algorithm, the intra-pulse modulation is an independent condition.

Fig. 4.

Spectrogram: pulse train (top), noisy with SNR −18 dB (middle), and denoised (bottom).

Essential condition earned from the pulsed radar signals is the duty cycle, usually below 5%. In the time domain, this contributes in the ideal case in most probable zeros and least probable values for the signal; therefore in a noisy environment, pure noise exists in all the PRI period and the signal plus noise exist just in the smaller PW time. In the time–frequency domain this gives an important theoretical result, noise spreads all over the band with low power and in all the time bins considered the centers of the frames, but the targeted signal waveform is condensed in fewer time bins corresponds to the transmitting period. The same view can be seen in the frequency band where in most cases the signal in the small time of the frame occupies small band in the entire band [ − f _s/2 f _s/2]. Different scenarios of intra-pulse modulations (LFM with different chirp coefficients, quadrature phase shift keying, and rectangular pulse) and using several duty cycle values implemented in pulsed radar can have the same result, which is an expected result, because calculating the threshold depends on the redundancy of noise in the PSD (spectrogram).

Statistically, the histogram validates this taking the example of 5% duty cycle, 32 samples window size, and 32 frequency bins in the band. The signal is generated at different low SNR values down to −18 dB, and then the histogram is calculated for the STFT matrix obtained (we take the PSD). Monte-Carlo simulations at different SNRs and duty cycles were performed to validate that. An example at the lowest SNR is shown in Fig. 5.

Fig. 5.

Histogram of the time-basis PSD showing the statistical aspect of the pulsed radar signal.

Applying threshold and localizing pulses were performed prior to ideal known pulse locations comparison. The results lead to an automatically validated threshold that was used.

Apply threshold and localize

The PSD matrix is subjected to the threshold “T,” forcing smaller than “T” values to 0 and higher to 1. Then the resulted plot will be a rectangular waveform representing a decision of signal at a set of time bins and noise at others. Each stream of ones corresponds to pulse with the corresponding width. However, some noise overcomes the “T” value to result in a pulse with random PW value and gives a false detection. Simulation of long signal with many pulses results in low probability of false detection. Figure 6 shows an example of two pulses case.

Fig. 6.

Applying threshold and localizing pulses in the PSD matrix.

Pattern analysis

Recognition and identification of pulsed radar parameters aims to blindly estimate first the radar class and second calculate its parameters (PW, pulse repetition frequency (PRF), bandwidth, etc.). Classes such as jittered, dwell and switch, staggered and others, each has special PW and PRF mode of variation, or it could be simply constant values. It is a wide range domain of R&D to solve these objectives accurately but in less complicated solutions. In the literature, algorithms exist based on neural network [Reference Yun and Jing-chao6], radar signature database analysis [Reference Matuszewski7], energy cumulate STFT [Reference Wang, Huang, Zhou, Tian, Yao and Gao8], and many others.

In this paper, pattern analysis is used to enhance the localization and decrease the false decisions. In this perspective we are going to implement all the cases of radar parameter classes. However, for simplicity and to prove the concept we are going to show the results of the considered example. The generated signal is considered to be dwell and switch mode, the values of PW and PRF are fixed for a train of pulses and switch to other values for another train.

The algorithm is built in a way that all the values of PW before pattern analysis (noise and pulses; Fig. 6) are taken each alone (consider no knowledge about the parameters of the generated signal PW and PRF). For each, the values of the time period between the equal PWs are calculated. Clearly, the glitches resulted from noise is randomly spread between the real pulses, therefore the time periods between these glitches are random and hence discarded. This procedure is repeated for all the possible values output from the rectangular plot of Fig. 6. The result outcome is a pure rectangular shape forming the time envelope of the pulses localized in the signal. Figure 7 shows an envelope example before and after the pattern analysis where three glitches are removed. Figure 8 shows the final result of the whole system suggested with pattern analysis at SNR −18 dB.

Fig. 7.

False detection removal by pattern analysis.

Fig. 8.

Original pulse train (black), noisy at SNR −18 dB (blue), localization after denoising and pattern analysis (red).

False alarm testing by pure noise signal

False alarm in radar may happen if decision is taken for target exists in the case of purely noise received signal. Now suppose that the input was pure noise. Referring to the flow chart, all the steps are executed normally; the signal is denoised, the PSD curves are extracted, a threshold is chosen, and localization is done. Figure 9 shows the result after the final step; pattern analysis. The results show the importance of the pattern analysis block. Taking advantage of the fact that it is impossible for a stochastic signal (pure noise) to produce random pulses with PW and PRF that coincides with any radar system pattern; the pattern analysis block will eliminate all the pulses that disagree with the rules. In fact, denoising and thresholding eliminates most of the noise glitches (the narrow one) that looks like pulses and efficiently decrease the false detection, but, pattern analysis improves strongly that by discarding the possible wider glitches that still exist as described.

Fig. 9.

Testing the case of pure noise.