*4.3. PN Sequence Channel Estimation*

The goal of this section is to present a method to estimate the channel impulse response and frequency response of a time-dispersive and static channel (no movement in either transmitter or receiver is expected). In order to characterise the effects of this type of channel on acoustic signals, we use a PN sequence of type M, which has good cyclic cross-correlation properties [37]. The sequence is sampled at *Fs* = 44.10 kHz, with 1023 chips length and four samples per chip. Then, the chip period is *Tc* = 4/*Fs* and, hence, the detection bandwidth equals 1/*Tc* and the delay resolution equals *Tc*. The PN sequence is also low-pass filtered with a Finite Impulse Response (FIR) root raised cosine filter (RRCOSFIR) with a roll-off factor equal to 0.9. The aim of this low-pass filtering is to limit the bandwidth of the PN sequence to 10 kHz, which is the most common band for audible signals. In Figure 9, we show the time domain (upper figure) and frequency domain (lower figure) of the PN sequence (in blue), as well as the same PN sequence after filtering (in red). It is worth noting that the time response displays a pseduo-random nature and that the bandwidth is limited after being filtered.

The analysis of the channel by means of a PN sequence is a key factor to characterise the time dispersion and, hence, to evaluate whether the signal arrived with any multi-path component at the receiver. First, the received signal *x*[*n*] is correlated with the original PN sequence *S*. The correlation function is computed as explained in Equation (1).

From the channel impulse response (see Equation (2)), we can define the root mean square (*rms*) delay spread (*τrms*) as the standard deviation value of the delay reflections of the transmitted signal, weighted proportionally to the energy in the reflection waves. Table 2 shows the *rms* delay spread

expected for each of the three channel models studied in this paper. Figure 10 depicts the channel impulse response of each of three models, computed by means of a PN sequence as explained in Equations (1) and (2). Model A has two paths and models B and C have four paths. Despite having different numbers of paths, models A and C showed a similar delay spread (see Table 2), as the paths were close together in channel C.

**Figure 9.** PN sequence time response and frequency response. The sequence without filtering is shown in blue, and after filtering (with a root raised cosine) is shown in red.

Once the channel impulse response is computed, we can derive the channel frequency transfer function as in Equation (10). A key parameter of the frequency characterisation of the channel is the coherence bandwidth. The coherence bandwidth is a statistical measure of the range of frequencies over which the channel can be considered flat (i.e., the bandwidth for which the auto co-variance of the signal amplitude at two extreme frequencies reduces from 1 to 0.5). The coherence bandwidth is inversely proportional to the *rms* delay spread of the channel (i.e., *Bc* ≈ 1/*τ*[*rms*]) [20]; but there is no precise relationship between both. However, a widely accepted definition considers the bandwidth over which the frequency correlation is above 0.9; or a more relaxed one considers a frequency correlation of just 0.5:

$$B\_{\mathbb{C}}(0.9) = \frac{1}{50\pi\_{rms}}, B\_{\mathbb{C}}(0.5) = \frac{1}{5\pi\_{rms}}.\tag{10}$$

In Table 2, we show the coherence bandwidth of each channel model, with frequency correlation above 0.5. In Figure 11, we compare the frequency response of the PN sequence with the frequency response of each channel model computed with the same PN sequence, following Equations (1), (2), and (4). In Figure 11, we do not show the coherence bandwidth (for the sake of practicality), but we underline the bandwidth between consecutive fadings of the channel, which is proportional to the coherence bandwidth for each of the studied models. It is clear that, the higher the *rms* delay spread of

the channel, the narrower the coherence bandwidth and, hence, the higher the probability of acoustic signal distortion.

**Figure 10.** Channel impulse responses of models A, B, and C obtained using a PN sequence.

**Figure 11.** Channel frequency responses of models A, B, and C by means of a PN sequence. An approximate frequency range of the channel coherence bandwidth is shown for each channel model.
