TDD Uplink Training Protocol

The coherence interval $\tau_c$ (in complex samples) is the number of time-frequency samples over which the channel is approximately constant. It is determined by the coherence time $T_c$ and coherence bandwidth $B_c$:

$$\tau_c \approx T_c \cdot B_c$$

For a user moving at $v$ m/s at carrier frequency $f_c$, the Doppler spread is $f_D = v f_c / c$, giving $T_c \approx 1/(2f_D)$. A 3 km/h pedestrian at 3.5 GHz gives $T_c \approx 30$ ms and $B_c \approx 200$ kHz, so $\tau_c \approx 6000$ samples — a generous interval. At vehicular speeds (100 km/h), $\tau_c$ drops to roughly $180$ samples.

In TDD massive MIMO, each coherence interval is divided into three phases:

1. Uplink pilot phase ($\tau_p$ samples): All $K$ users simultaneously transmit known pilot sequences $\boldsymbol{\phi}_k \in \mathbb{C}^{\tau_p}$, $k = 1, \ldots, K$. The base station uses the received signal to estimate all $K$ channels.

2. Uplink data phase ($\tau_u$ samples): Users transmit data using the estimated uplink channels for receive combining.

3. Downlink data phase ($\tau_d$ samples): The base station transmits to all users using the estimated channels for precoding (exploiting TDD reciprocity).

Resource constraint: $\tau_p + \tau_u + \tau_d \leq \tau_c$.

The effective spectral efficiency accounts for the pilot overhead factor $(\tau_c - \tau_p)/\tau_c$, which is near 1 for large $\tau_c$ but significant for high-mobility scenarios.

The pilot matrix $\boldsymbol{\Phi} \in \mathbb{C}^{K \times \tau_p}$ stacks the $K$ pilot sequences as rows:

$$\boldsymbol{\Phi} = \begin{bmatrix} \boldsymbol{\phi}_1^T \\ \vdots \\ \boldsymbol{\phi}_K^T \end{bmatrix}$$

The sequences are semi-orthogonal when $\boldsymbol{\Phi}\boldsymbol{\Phi}^H = \tau_p \mathbf{I}_{K}$, i.e., $\boldsymbol{\phi}_k^H \boldsymbol{\phi}_j = \tau_p \delta_{kj}$. This requires $\tau_p \geq K$.

Common choices:

• Columns of $\sqrt{\tau_p} \mathbf{I}_{\tau_p}$ (standard basis, length $\tau_p = K$)
• Rows of a $\tau_p \times \tau_p$ DFT matrix scaled by $\sqrt{\tau_p}$
• Zadoff-Chu sequences (used in 5G NR for their good autocorrelation properties)

During the uplink pilot phase, the received signal matrix $\mathbf{Y}_p \in \mathbb{C}^{N_t \times \tau_p}$ at the base station is:

$$\mathbf{Y}_p = \sqrt{p_u} \sum_{k=1}^{K} \mathbf{H}_{k} \boldsymbol{\phi}_k^T + \mathbf{N}_p$$

where:

• $\mathbf{H}_{k} \in \mathbb{C}^{N_t}$ is the channel vector of user $k$
• $p_u > 0$ is the uplink pilot transmit power (same for all users; power control in Ch. 5)
• $\boldsymbol{\phi}_k \in \mathbb{C}^{\tau_p}$ is user $k$'s pilot sequence
• $\mathbf{N}_p \in \mathbb{C}^{N_t \times \tau_p}$ is additive noise with i.i.d. $\mathcal{CN}(0, \sigma^2)$ entries

In compact matrix form: $\mathbf{Y}_p = \sqrt{p_u}\, \mathbf{H}_{K} \boldsymbol{\Phi} + \mathbf{N}_p$, where $\mathbf{H}_{K} = [\mathbf{H}_{1}, \ldots, \mathbf{H}_{K}] \in \mathbb{C}^{N_t \times K}$.