Ferkans — Interactive Telecom Tutor

From Perfect to Practical CSIT

The precoding strategies of Section 16.5 assume that the transmitter has perfect channel knowledge. In practice, the receiver estimates the channel and feeds back a quantised representation using a finite number of bits $B$ . This section develops the theory of codebook-based precoding and Grassmannian quantisation, quantifying the capacity loss due to imperfect CSIT.

Definition:
Precoding Codebook

A precoding codebook $\mathcal{F} = \{\mathbf{f}_1, \mathbf{f}_2, \ldots, \mathbf{f}_{2^B}\}$ is a set of $2^B$ unit-norm precoding vectors in $\mathbb{C}^{N_t}$ , where $B$ is the number of feedback bits.

The receiver selects the codebook entry that maximises the effective channel gain:

$\hat{\mathbf{f}} = \arg\max_{\mathbf{f} \in \mathcal{F}} |\mathbf{h}^H \mathbf{f}|^2$

and feeds back the $B$ -bit index of $\hat{\mathbf{f}}$ to the transmitter. The transmitter then uses $\hat{\mathbf{f}}$ as its beamforming vector.

The codebook is known to both transmitter and receiver, so only the index (not the vector itself) needs to be communicated. The design of $\mathcal{F}$ determines how well the finite set covers the space of possible channel directions.

Definition:
Grassmannian Quantisation

Grassmannian quantisation designs the codebook $\mathcal{F}$ to maximise the minimum chordal distance between any two codewords:

$\mathcal{F}^* = \arg\max_{\mathcal{F}} \min_{i \neq j} d_c(\mathbf{f}_i, \mathbf{f}_j)$

where the chordal distance is:

$d_c(\mathbf{f}_i, \mathbf{f}_j) = \frac{1}{\sqrt{2}} \sqrt{1 - |\mathbf{f}_i^H \mathbf{f}_j|^2}$

This is a packing problem on the Grassmann manifold $\mathcal{G}(N_t, 1)$ — the space of all one-dimensional subspaces of $\mathbb{C}^{N_t}$ . Grassmannian codebooks provide the best worst-case quantisation of the channel direction.

Grassmannian codebooks are optimal in the sense of minimising the maximum quantisation error. However, finding exact Grassmannian packings is NP-hard in general. Practical alternatives include DFT codebooks, random vector quantisation (RVQ), and structured codebooks (e.g., the 5G NR Type I codebook).

Capacity Loss with Limited Feedback

Explore how the achievable rate degrades as the number of feedback bits $B$ decreases. Compare against perfect CSIT and no CSIT baselines.

Parameters

N_t

(transmit antennas)4

B

(feedback bits)4

Codebook type

Theorem: Rate Loss with Finite-Rate Feedback

For a MISO channel with $N_t$ transmit antennas and $B$ feedback bits using random vector quantisation (RVQ), the average rate loss compared to perfect CSIT beamforming is upper-bounded by:

$\Delta R \leq \log_2\!\left(1 + \frac{P}{\sigma^2} (N_t - 1) \cdot 2^{-B/(N_t - 1)}\right)$

At high SNR, this simplifies to:

$\Delta R \approx (N_t - 1)\left(1 - \frac{B}{N_t - 1}\right) \cdot \frac{\log_2 e}{1} \cdot 2^{-B/(N_t-1)} \cdot \frac{P}{\sigma^2}$

To maintain a constant rate gap of $\Delta R \leq c$ bits/s/Hz as SNR increases by 3 dB, the number of feedback bits must increase by $(N_t - 1)$ bits.

The quantisation error in the beamforming direction creates a residual "interference" proportional to $2^{-B/(N_t-1)}$ . More antennas increase the dimensionality of the space to quantise, requiring exponentially more bits. The scaling $B/(N_t - 1)$ reflects the $2(N_t - 1)$ real degrees of freedom of a unit-norm vector in $\mathbb{C}^{N_t}$ (after removing the phase ambiguity).

Proof

Quantisation error bound

With RVQ using $2^B$ i.i.d. isotropic random vectors, the expected quantisation error for a random channel direction $\tilde{\mathbf{h}} = \mathbf{h}/\|\mathbf{h}\|$ is:

$\mathbb{E}\!\left[1 - |\tilde{\mathbf{h}}^H\hat{\mathbf{f}}|^2\right] \leq (N_t - 1) \cdot 2^{-B/(N_t - 1)}$

This follows from the beta distribution of $|\tilde{\mathbf{h}}^H\mathbf{f}|^2$ for isotropic $\mathbf{f}$ and the order statistics of $2^B$ i.i.d. beta random variables.

Rate with quantised beamforming

The achieved rate with beamforming vector $\hat{\mathbf{f}}$ is:

$R = \mathbb{E}\!\left[\log_2\!\left(1 + \frac{P}{\sigma^2} |\mathbf{h}^H\hat{\mathbf{f}}|^2\right)\right]$

Decompose: $|\mathbf{h}^H\hat{\mathbf{f}}|^2 = \|\mathbf{h}\|^2 \cdot |\tilde{\mathbf{h}}^H\hat{\mathbf{f}}|^2$ . The rate loss compared to MRT ( $|\tilde{\mathbf{h}}^H\hat{\mathbf{f}}|^2 = 1$ ) is:

$\Delta R = \mathbb{E}\!\left[\log_2\!\left(\frac{1 + \frac{P}{\sigma^2}\|\mathbf{h}\|^2} {1 + \frac{P}{\sigma^2}\|\mathbf{h}\|^2|\tilde{\mathbf{h}}^H\hat{\mathbf{f}}|^2}\right)\right]$

Bounding the rate loss

Using $\log(1+a) - \log(1+b) \leq \log(1 + a - b)$ for $a \geq b \geq 0$ :

$\Delta R \leq \log_2\!\left(1 + \frac{P}{\sigma^2}\|\mathbf{h}\|^2 (1 - |\tilde{\mathbf{h}}^H\hat{\mathbf{f}}|^2)\right)$

Taking expectations and applying Jensen's inequality:

$\Delta R \leq \log_2\!\left(1 + \frac{P}{\sigma^2}N_t \cdot (N_t-1) \cdot 2^{-B/(N_t-1)}\right)$

where we used $\mathbb{E}[\|\mathbf{h}\|^2] = N_t$ for i.i.d. Rayleigh fading. $\blacksquare$

,

Example: Required Feedback Bits for a 4-Antenna System

A MISO system has $N_t = 4$ transmit antennas operating at SNR = 20 dB. How many feedback bits $B$ are needed to keep the rate loss below 1 bit/s/Hz?

Solution

Apply the rate loss bound

We need:

$\Delta R \leq \log_2\!\left(1 + \frac{P}{\sigma^2}(N_t - 1) \cdot 2^{-B/(N_t-1)}\right) \leq 1$

This requires:

$\frac{P}{\sigma^2}(N_t - 1) \cdot 2^{-B/(N_t-1)} \leq 1$

Solve for B

$2^{-B/(N_t-1)} \leq \frac{\sigma^2}{P(N_t-1)} = \frac{1}{100 \cdot 3} = \frac{1}{300}KATEXPLACEHOLDER0END\frac{B}{N_t - 1} \geq \log_2(300) = 8.23KATEXPLACEHOLDER1ENDB \geq 3 \times 8.23 = 24.7$ $We need$ B \geq 25$ feedback bits.

Interpretation

At 20 dB SNR with 4 antennas, 25 bits are needed to maintain less than 1 bit/s/Hz rate loss. If SNR increases by 3 dB, we need $N_t - 1 = 3$ additional bits. This linear scaling of $B$ with SNR (in dB) is a fundamental characteristic of limited feedback systems. $\blacksquare$

Quick Check

For a MISO system with $N_t = 4$ , how should the number of feedback bits $B$ scale to maintain a fixed rate gap as SNR increases by 3 dB?

Increase $B$ by 1 bit

Increase $B$ by 3 bits

Increase $B$ by 4 bits

$B$ does not need to increase

Correction:

Increase

B

by 3 bits

The required scaling is $(N_t - 1) = 3$ bits per 3 dB increase in SNR. This ensures the quantisation "interference" scales at the same rate as the useful signal power.

Common Mistake: Neglecting Feedback Overhead

Mistake:

Increasing the number of feedback bits $B$ without considering the uplink overhead. In FDD systems, the feedback consumes uplink resources proportional to $B$ , which reduces the time/ bandwidth available for actual data transmission.

Correction:

There is an optimal $B$ that balances the downlink rate improvement from better precoding against the uplink rate cost of feedback. In massive MIMO ( $N_t \gg 1$ ), the required $B$ scales linearly with $N_t$ , making explicit feedback impractical. This motivates TDD reciprocity-based approaches (where the base station estimates the channel directly from uplink pilots) and implicit feedback mechanisms.

Why This Matters: Limited Feedback in Cellular Standards

LTE uses a codebook of $2^B$ precoding matrices ( $B = 4$ bits for rank-1) drawn from DFT-based designs, fed back via the PMI (Precoding Matrix Indicator) field. 5G NR introduces hierarchical Type I and Type II codebooks: Type I uses DFT beams with oversampling factors, while Type II provides linear-combination codebooks with higher accuracy at the cost of more feedback bits (up to 11 bits for wideband PMI). The move to massive MIMO in 5G makes TDD reciprocity increasingly attractive, avoiding explicit feedback altogether.

Precoding Codebook

A finite set of $2^B$ precoding vectors/matrices shared between transmitter and receiver, enabling quantised channel feedback via a $B$ -bit index.

Grassmannian Quantisation

The optimal codebook design criterion that maximises the minimum distance between codebook entries on the Grassmann manifold, providing the best worst-case quantisation of channel directions.

Limited Feedback

A MIMO system design paradigm where the receiver quantises the channel information to $B$ bits and feeds the index back to the transmitter, enabling approximate precoding with finite overhead.

Limited Feedback

From Perfect to Practical CSIT

Definition: Precoding Codebook

Definition: Grassmannian Quantisation

Capacity Loss with Limited Feedback

Parameters

Theorem: Rate Loss with Finite-Rate Feedback

Quantisation error bound

Rate with quantised beamforming

Bounding the rate loss

Example: Required Feedback Bits for a 4-Antenna System

Apply the rate loss bound

Solve for B

Interpretation

Quick Check

Common Mistake: Neglecting Feedback Overhead

Why This Matters: Limited Feedback in Cellular Standards

Precoding Codebook

Grassmannian Quantisation

Limited Feedback

Definition:
Precoding Codebook

Definition:
Grassmannian Quantisation