Channel file format specifications

The uwa-channels-compatible .mat files must be saved with the following flags:

• -v7.3 to support large variables and the HDF5 file format.
• -nocompression to speed up loading.

Required fields

Each .mat file must include the following variables:

h_hat

• Type: Multi-dimensional complex tensor.
• Dimensions: [delay, receiver, time]
• Units:
  • Delay axis: sampled at params.fs_delay [Hz], denoted $f_s = 1/T_s$
  • Time axis: sampled at params.fs_time [Hz]
  • Amplitude: complex baseband impulse response (unitless)
• Description: The estimated time-varying channel impulse response (TVIR) as a function of delay $\tau$ and time $t$. When delay tracking is used, this is the drift-free response $\hat{\underline{h}}(\tau, t)$; otherwise it is the drifting response $\hat{h}(\tau, t)$. For a given receiver and time index $n$, the slice along the delay axis is the channel vector $\hat{\underline{\mathbf{h}}}[n]$, or $\hat{\mathbf{h}}[n]$ in the drifting case.

params

A structure with the following scalar fields:

version

• Type: Numeric scalar.
• Description: Dataset format version number (currently 1.0).

Phase/delay tracking fields (optional)

phi_hat, theta_hat, and f_resamp may coexist; the replay engine applies them according to the following precedence:

• If none of the three is present, no Doppler correction is applied.
• phi_hat takes precedence over theta_hat: if phi_hat is present, it is used and theta_hat is ignored; if only theta_hat is present, it is used instead.
• f_resamp (see Optional fields) is applied independently of the two phase/delay fields, whether or not either of them is present.

phi_hat (delay tracking)

• Type: Numeric matrix, size [receiver, time]

• Units: Radians

• Sampling rate: params.fs_delay

• Description: The phase estimate $\hat\varphi(nT_s)$, a time-varying phase that encodes both phase rotation and delay drift. In this mode, h_hat holds the drift-free response $\hat{\underline{h}}(\tau, t)$. Let $\Delta\tau$ denote the delay drift common to all propagation paths. It is related to the phase by

  $$\hat\varphi(nT_s) = -2\pi f_c\, \Delta\tau(nT_s)$$

  Unpacking first reinserts the phase, multiplying the drift-free response by $e^{j\hat\varphi(nT_s)}$ to obtain the signal $\bar y_{\text{r}}(nT_s)$. It then reinserts the delay drift by evaluating that signal at shifted instants,

  $$\bar v_{\text{r}}(nT_s) = \mathcal{I}\left[\bar y_{\text{r}}\left(nT_s + \frac{\hat\varphi(nT_s)}{2\pi f_c}\right)\right]$$

  where $\mathcal{I}[\cdot]$ denotes interpolation, implemented here as spline interpolation.

theta_hat (phase tracking only)

• Type: Numeric matrix, size [receiver, time]

• Units: Radians

• Sampling rate: params.fs_delay

• Description: The phase estimate $\hat\theta(nT_s)$, a time-varying phase correction. In this mode, h_hat contains the drifting impulse response $\hat{h}(\tau, t)$ (delay drift is embedded in the taps). Only the phase is tracked separately. The baseband received signal is modeled as:

  $$v(t) = \sum_n d(n)\, h(t - nT, t)\, e^{j\theta(t)} + w(t)$$

  where $d(n)$ are the transmitted data symbols, $h(\tau, t)$ is the time-varying impulse response at delay $\tau$ and time $t$ with drifting taps, $T$ is the symbol interval, $\theta(t)$ is the channel phase, and $w(t)$ is the additive complex baseband noise.

Duration constraint

The time dimension of theta_hat or phi_hat and the third dimension of h_hat must span the same duration. The check is written below for theta_hat; the same applies to phi_hat:

size(theta_hat, 2) / params.fs_delay == size(h_hat, 3) / params.fs_time

Optional fields

f_resamp

• Type: Scalar (double precision)
• Units: Unitless resampling factor
• Description: A time-invariant resampling factor applied to the output signal. This is typically the inverse of a resampling operation applied to remove the nominal Doppler frequency offset before channel estimation. It is applied after the time-varying convolution.

meta

The meta structure is optional but strongly encouraged. The following fields are recognized by the toolbox:

Users are free to add additional fields to meta to capture experiment-specific metadata.

Noise file format

Each noise .mat file contains the following fields:

The noise generation function noisegen uses the mixing equation:

Here $\hat{n}_i$ is the synthetically generated noise on hydrophone $i$, with the hydrophone indices running as $i, j = 0, \ldots, M-1$. The sampling interval is $T_s = 1/f_s$, the sampling rate $f_s$ being stored in the Fs field, and $L$ is the maximum discrete time lag with non-negligible covariance, so that $\beta_{ij}(kT_s)$ has $L+1$ taps, $k = 0, \ldots, L$. The innovations $\eta_j(nT_s) \sim \mathcal{S}_\alpha(0, 1/\sqrt{2})$ are i.i.d. symmetric $\alpha$-stable variates with zero location and scale $1/\sqrt{2}$; this scale ensures that the distribution reduces to the standard Gaussian $\mathcal{N}(0, 1)$ when $\alpha = 2$.