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The Case for Interference Cancellation

MRT ignores interference. At the other extreme, we can design the precoding matrix to completely eliminate multi-user interference. This is the zero-forcing (ZF) strategy: choose $\mathbf{v}_{k}$ so that $\mathbf{h}_j^H \mathbf{v}_{k} = 0$ for all $j \neq k$ . The price is noise amplification, but when the SNR is high enough, removing interference is worth the cost.

Definition:
Zero-Forcing Precoding

The ZF precoding matrix is the normalised pseudo-inverse of the channel:

$\mathbf{W}^{\text{ZF}} = \mathbf{H}^{H} (\mathbf{H} \mathbf{H}^{H})^{-1} \mathbf{D}_{\text{ZF}}$

where $\mathbf{D}_{\text{ZF}}$ is a diagonal normalisation matrix ensuring unit-norm columns: $[\mathbf{D}_{\text{ZF}}]_{kk} = 1/\|[\mathbf{H}^{H}(\mathbf{H}\mathbf{H}^{H})^{-1}]_{:,k}\|$ .

Equivalently, the unnormalised ZF precoding vector for user $k$ is the $k$ -th column of $\mathbf{H}^{H} (\mathbf{H} \mathbf{H}^{H})^{-1}$ , which is the $k$ -th row of the right pseudo-inverse of $\mathbf{H}$ .

ZF exists whenever $\mathbf{H}$ has full row rank, which requires $N_t \geq K$ .

The constraint $\mathbf{h}_j^H \mathbf{v}_{k} = 0$ for $j \neq k$ means each user's precoder lies in the null space of all other users' channels. This is possible only when $N_t > K$ , leaving $N_t - K + 1$ degrees of freedom for the precoder direction.

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Theorem: ZF Eliminates Multi-User Interference

With ZF precoding, the received signal at user $k$ simplifies to

$y_k = \sqrt{p_k}\, s_k + w_k$

i.e., an interference-free AWGN channel. The achievable rate is

$R_k^{\text{ZF}} = \log_2\!\left(1 + \frac{p_k}{\sigma^2}\right).$

ZF converts the multi-user MIMO channel into $K$ parallel single-user channels. The cost is that the effective channel gain for each user is reduced because the precoder must avoid the other users' channel directions.

Proof

Verify interference cancellation

By construction, $\mathbf{H} \mathbf{W}^{\text{ZF}} = \mathbf{D}_{\text{ZF}}^{-1}$ (before normalisation). After normalisation, $\mathbf{h}_j^H \mathbf{v}_{k} = 0$ for $j \neq k$ and $\mathbf{h}_k^H \mathbf{v}_{k} = 1/\|[\mathbf{H}^{H}(\mathbf{H}\mathbf{H}^{H})^{-1}]_{:,k}\|$ .

Write received signal

$y_k = \mathbf{h}_k^H \sum_{j=1}^{K} \sqrt{p_j}\,\mathbf{v}_{j} s_j + w_k = \sqrt{p_k}\,(\mathbf{h}_k^H \mathbf{v}_{k}) s_k + w_k.$ $With appropriate power normalisation, this is a scalar AWGN channel with SNR$ = p_k |\mathbf{h}k^H \mathbf{v}{k}|^2 / \sigma^2 $.$ \blacksquare$

Theorem: ZF Noise Amplification (Power Penalty)

With ZF precoding and equal power allocation, the effective SNR at user $k$ is

$\text{SNR}_{k}^{\text{ZF}} = \frac{P_t/K}{[(\mathbf{H}\mathbf{H}^{H})^{-1}]_{kk}\, \sigma^2}.$

For i.i.d. Rayleigh fading, $[(\mathbf{H}\mathbf{H}^{H})^{-1}]_{kk}$ is an inverse chi-squared random variable, and

$\mathbb{E}\left[\frac{1}{[(\mathbf{H}\mathbf{H}^{H})^{-1}]_{kk}}\right] = N_t - K.$

The ZF precoder thus suffers a power penalty of factor $N_t - K$ compared to the single-user MRT gain of $N_t$ : it "wastes" $K$ degrees of freedom on interference nulling.

ZF forces the precoder into the $(N_t - K + 1)$ -dimensional null space of the other users' channels. With fewer degrees of freedom available, the precoder cannot align as well with the intended user's channel, resulting in a reduced effective channel gain. The penalty vanishes when $N_t \gg K$ .

Proof

Express effective gain

The unnormalised ZF vector for user $k$ is the $k$ -th column of $\mathbf{H}^{H}(\mathbf{H}\mathbf{H}^{H})^{-1}$ . Its squared norm is $[(\mathbf{H}\mathbf{H}^{H})^{-1}]_{kk}$ , which is the inverse of the effective channel gain after ZF.

Use Wishart inverse moments

For $\mathbf{H}$ with i.i.d. $\mathcal{CN}(0,1)$ entries, $\mathbf{H}\mathbf{H}^{H} \sim \mathcal{W}_{K}(N_t, \mathbf{I})$ . The diagonal entries of its inverse satisfy

$\mathbb{E}[(\mathbf{H}\mathbf{H}^{H})^{-1}_{kk}] = \frac{1}{N_t - K}$

which exists only when $N_t > K$ .

Interpret the penalty

The average SNR is proportional to $N_t - K$ instead of $N_t$ . ZF "pays" $K$ degrees of freedom to null the interference, leaving $N_t - K$ for useful beamforming gain. $\blacksquare$

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Example: ZF vs MRT for 8 Antennas and 4 Users

A BS with $N_t = 8$ serves $K = 4$ users with i.i.d. Rayleigh channels. Compare the average per-user SNR (in dB) of MRT and ZF for $P_t/\sigma^2 = 10$ dB. Assume equal power allocation.

Solution

MRT average signal and interference

With MRT, the average useful signal power is $\mathbb{E}[\|\mathbf{h}_k\|^2] = N_t = 8$ . The average interference from each other user is $\mathbb{E}[|\mathbf{h}_k^H \mathbf{h}_j|^2/\|\mathbf{h}_j\|^2] = 1$ . So

$\overline{\text{SINR}}_k^{\text{MRT}} \approx \frac{(P_t/4) \cdot 8}{(P_t/4) \cdot 3 + \sigma^2} = \frac{2.5 \times 8}{2.5 \times 3 + 1} = \frac{20}{8.5} \approx 2.35 = 3.7 \text{ dB}.$

ZF average SNR

With ZF, all interference is zero. The average effective gain is $N_t - K = 4$ :

$\overline{\text{SNR}}_k^{\text{ZF}} = \frac{(P_t/4) \cdot 4}{\sigma^2} = \frac{2.5 \times 4}{1} = 10 = 10 \text{ dB}.$

Comparison

ZF achieves $10$ dB vs MRT's $3.7$ dB per-user SINR, despite losing $K = 4$ degrees of freedom. The interference penalty of MRT (denominator $= 8.5$ instead of $1$ ) dominates the ZF power penalty. This is the regime $K/N_t = 0.5$ where ZF dominates.

SINR Comparison — MRT vs ZF vs Users

Compare the average per-user SINR of MRT and ZF as the number of users $K$ increases from 1 to $N_t - 1$ . Observe how MRT saturates due to interference while ZF degrades due to the shrinking null space.

Parameters

N_t

64

Number of transmit antennas

P_t/\sigma^2

(dB)10

Transmit SNR in dB

Common Mistake: ZF Fails Near Rank Deficiency

Mistake:

Using ZF when $K$ is close to $N_t$ , expecting good performance because "all interference is cancelled."

Correction:

When $K \to N_t$ , the matrix $\mathbf{H}\mathbf{H}^{H}$ becomes ill-conditioned. The inverse amplifies noise dramatically: $[(\mathbf{H}\mathbf{H}^{H})^{-1}]_{kk} \to \infty$ in expectation as $N_t - K \to 0$ . In practice, if $K > 0.5 N_t$ , RZF/MMSE precoding should be used instead of ZF.

Common Mistake: ZF Requires Perfect CSI

Mistake:

Applying ZF precoding with estimated channels $\hat{\mathbf{H}}$ without accounting for the estimation error.

Correction:

ZF with imperfect CSI does not achieve zero interference. The residual interference is proportional to the estimation error variance. With imperfect CSI, one should either (a) use RZF/MMSE precoding, which inherently accounts for error, or (b) use the UatF bound from Chapter 4 with ZF based on $\hat{\mathbf{H}}$ , acknowledging the residual interference term.

Quick Check

A BS with $N_t = 32$ antennas serves $K = 8$ users with ZF precoding. How many degrees of freedom does each user's precoding vector have for signal enhancement?

$32$

$25$

$24$

$8$

Correction:

25

The precoder must lie in the null space of the other $K - 1 = 7$ users' channels. This removes $7$ dimensions, leaving $N_t - K + 1 = 25$ degrees of freedom.

Why This Matters: ZF Precoding in 5G NR

In 5G NR, ZF-based precoding is the workhorse for MU-MIMO in both sub-6 GHz (FR1) and mmWave (FR2) deployments. The gNB computes the ZF precoder from SRS-based channel estimates in TDD mode, or from Type II CSI feedback in FDD mode. The standard supports up to 12 simultaneously scheduled MU-MIMO layers, and practical implementations typically use RZF (the regularised variant from Section 6.3) for robustness.

Zero-Forcing (ZF) Precoding

A linear precoding strategy that designs each user's precoding vector to lie in the null space of all other users' channels, completely eliminating multi-user interference at the cost of noise amplification.

Key Takeaway

ZF precoding eliminates all multi-user interference by projecting each user's precoder into the null space of the other channels. The cost is a power penalty of $N_t - K$ degrees of freedom. ZF outperforms MRT in the high-SNR, moderate-loading regime, but becomes ill-conditioned when $K \to N_t$ .

Zero-Forcing Precoding

The Case for Interference Cancellation

Definition: Zero-Forcing Precoding

Theorem: ZF Eliminates Multi-User Interference

Verify interference cancellation

Write received signal

Theorem: ZF Noise Amplification (Power Penalty)

Express effective gain

Use Wishart inverse moments

Interpret the penalty

Example: ZF vs MRT for 8 Antennas and 4 Users

MRT average signal and interference

ZF average SNR

Comparison

SINR Comparison — MRT vs ZF vs Users

Parameters

Common Mistake: ZF Fails Near Rank Deficiency

Common Mistake: ZF Requires Perfect CSI

Quick Check

Why This Matters: ZF Precoding in 5G NR

Zero-Forcing (ZF) Precoding

Key Takeaway

Definition:
Zero-Forcing Precoding