*4.2. Impulse Response for the Defined Acoustic Channels*

We define three different urban channel models, which are linear combinations of sets of paths received at a sensor, each with an acoustic pressure and phase shift as described by Equation (9):

$$y(n) = \sum\_{i=1}^{N} x(n - \tau\_i) a\_i e^{j\tau\_i 2\pi w} \,\prime \tag{9}$$

where *y*(*n*) is the received signal at the sensor, *x*(*n*) is the transmitted signal through each path whose amplitude (i.e., *ai*) is computed as explained in Equation (5), N is the number of paths of each model,

*τ<sup>i</sup>* is the delay suffered by each path, and *ejτi*2∗*π<sup>w</sup>* is the phase delay of each path, which is proportional to the delay and frequency.

In order to study the effect of time-dispersive channels on acoustic signals, we propose three different scenarios. We first define a simple channel (model A) with just two paths, which can describe a wide street with a direct path and a reflection path on the ground. Models B and C describe two scenarios with more dense scatterers than model A: While model B depicts a scenario with a high delay spread, which may be distinctive of a wide street, model C represents a highly scattered scenario with half the delay spread of model B, which may be characteristic of a narrower street. Table 2 shows the length and delay of each path per model (where the propagation velocity was taken as 343 m/s).

**Table 2.** Length and delay of each path, *rms* delay spread, and coherence bandwidth of each channel model.


In Figure 7, we show a block diagram of the synthesis of the channel impulse response of a *N*-path time-dispersive channel, as described by Equation (9). The input signal *x*(*n*) is first delayed a number of samples proportional to the delay, *τi*, suffered by each path. It is then attenuated with an inverse proportion to the distance travelled and, finally, the phase is shifted proportionally to the distance travelled.

**Figure 7.** Block diagram for the N-path channel.

Figure 8 depicts a snapshot of the ideal impulse response of each of the three models used in this study. Each impulse response is normalised to the first path, which is the direct path and that with highest intensity at the receiver. The remaining paths are reflected paths showing negative amplitude, which denotes a value of phase between *π* and 2*π*. Model A shows two taps, whereas models B and C show four taps. The taps of model B spread in time, over a much longer period than the taps of model

A and C. Therefore, it is worth noting that model B will spread the energy of the propagated signal throughout a higher time-span than models A and C, which have closer paths.

**Figure 8.** Channel ideal impulse response of models A, B, and C.
