**3. Radar Signal Processing**

On the basis of the used configuration we have four available receiving lines. To perform our analysis we need just one of them, so we can sum the complex samples coming from the analog-to-digital converters in order to improve the signal-to-noise ratio. We thus obtain a vector of complex samples which can be reorganized in the form of matrix, as shown in Figure 4. Along the rows of the matrix, also called *fast time*, we have samples from single chirps, while on the columns, or *slow time*, we samples from different chirps.

**Figure 4.** Slow Time and Fast Time Matrix.

From this raw matrix we can extract two types of map to classify different types of movement. The first one contains information about distance and speed of the subject and it is obtained by applying a Fourier transform to columns and then to rows; this map is defined as *Range-Doppler Map*. Each element of the map is a complex number and it is built considering the total acquisition, as shown in Figure 5. Besides distance and speed, this allows to extract also the micro-Doppler components characterizing the movement.

**Figure 5.** Range-Doppler Map data processing.

Since our subjects are moving during the acquisitions, we can extract the spectrogram from each Range bin and hence characterize their micro-Dopplers along the entire activity.

During each acquisition all the objects are stationary with the exception of the subject under exam, therefore the only significant micro-Doppler components are related to her/him. The presence of stationary objects does not influence either the Range-Doppler map, but only the zero Doppler. As described in Reference [19], from this map is possible to identify the kind of movement carried out by the subject.

The second type of map can be extracted through spectrograms and is denoted as *Doppler-Time Map*. A spectrogram is the most common time-frequency representation [41], and it is derived from the Short Time Fourier Transform (STFT) according to the following equation

$$STFT\_x[k,n] = \sum\_{r=-\infty}^{+\infty} x[r]w[n-r]e^{-j2\pi rk/N}, \; k = 0, 1, \dots, N-1,\tag{2}$$

where *n* represents a discrete index of time, *k* is a discrete index of frequency and *<sup>w</sup>*[·] is a window function. The Short Time Fourier Transform (STFT) can in fact be considered as the Fourier transform of a signal multiplied by a window sliding over time. A trade-off between resolution in time and in frequency must be found, and overlapping frequencies can help in this sense [42].

Starting from the range matrix, the second matrix in Figure 5, and applying the Short Time Fourier Transform (STFT) along the rows, we obtain the Doppler-Time map. This function uses windows of 512 samples, with an overlap of 98% and an Hann window is applied. Figure 6 depicts this process.

**Figure 6.** Doppler-Time Spectrograms.

By using both the mentioned maps it is possible to classify different kinds of movements, as will be explained in the next section.
