Ferkans — Interactive Telecom Tutor

The Linear Shortcut

If the constellation constraint is the source of the hardness, what happens if we simply ignore it? Treat $\mathbf{x}$ as a continuous vector, invert the channel, and then project back onto the alphabet. This gives a linear filter followed by symbol-wise slicing. The result is fast, parallelizable, and — as we will see — optimal in a well-defined continuous sense. It is also uniformly worse than ML. The tradeoff between these two extremes is the story of this chapter.

Definition:
Zero-Forcing (ZF) Detector

Assuming $\mathbf{H}$ has full column rank ( $n_r \geq n_t$ ), the zero-forcing detector applies the Moore-Penrose pseudoinverse: $\mathbf{G}_{\text{ZF}} = (\mathbf{H}^{H} \mathbf{H})^{-1} \mathbf{H}^{H}, \qquad \tilde{\mathbf{x}}_{\text{ZF}} = \mathbf{G}_{\text{ZF}} \mathbf{y} = \mathbf{x} + \mathbf{G}_{\text{ZF}} \mathbf{w}.$ The hard decision is $\hat{\mathbf{x}}_{\text{ZF}} = \mathcal{Q}_{\mathcal{A}}(\tilde{\mathbf{x}}_{\text{ZF}})$ , where $\mathcal{Q}_{\mathcal{A}}$ slices each entry onto the nearest constellation point.

ZF "forces" inter-stream interference to zero at the cost of noise enhancement. If $\mathbf{H}$ is ill-conditioned, the noise gain on some streams can be catastrophic.

Definition:
MMSE Detector

The linear MMSE detector minimizes the mean-squared error between $\mathbf{x}$ and its linear estimate. Assuming $\mathbf{x} \sim \mathcal{CN}(\mathbf{0}, \mathbf{I})$ (unit-energy symbols, uncorrelated) and independent noise: $\mathbf{G}_{\text{MMSE}} = (\mathbf{H}^{H} \mathbf{H} + \sigma^2 \mathbf{I})^{-1} \mathbf{H}^{H}, \qquad \tilde{\mathbf{x}}_{\text{MMSE}} = \mathbf{G}_{\text{MMSE}} \mathbf{y}.$ The hard decision is $\hat{\mathbf{x}}_{\text{MMSE}} = \mathcal{Q}_{\mathcal{A}}(\tilde{\mathbf{x}}_{\text{MMSE}})$ .

The regularizer $\sigma^2 \mathbf{I}$ shrinks the inverse of $\mathbf{H}^{H} \mathbf{H}$ , trading residual bias for reduced noise enhancement. As $\sigma^2 \to 0$ , MMSE $\to$ ZF. As $\sigma^2 \to \infty$ , MMSE $\to$ matched filter $\mathbf{H}^{H} / \sigma^2$ .

Theorem: Post-Detection SINR of the ZF Detector

The $k$ -th stream of the ZF detector has post-detection SINR $\gamma_k^{\text{ZF}} = \frac{1}{\sigma^2 \cdot [(\mathbf{H}^{H} \mathbf{H})^{-1}]_{kk}}.$

The diagonal entry $[(\mathbf{H}^{H} \mathbf{H})^{-1}]_{kk}$ measures how much noise is amplified on the $k$ -th stream. For an ill-conditioned channel this entry is large, and the corresponding SINR collapses.

Proof

Express the output noise

After the ZF filter, the residual on stream $k$ is $(\mathbf{G}_{\text{ZF}} \mathbf{w})_k = \mathbf{g}_k^H \mathbf{w}$ where $\mathbf{g}_k^H$ is the $k$ -th row of $\mathbf{G}_{\text{ZF}}$ .

Compute the noise variance per stream

Since $\mathbf{w} \sim \mathcal{CN}(\mathbf{0}, \sigma^2 \mathbf{I})$ , $\text{Var}[(\mathbf{G}_{\text{ZF}} \mathbf{w})_k] = \sigma^2 \|\mathbf{g}_k\|^2 = \sigma^2 [\mathbf{G}_{\text{ZF}} \mathbf{G}_{\text{ZF}}^H]_{kk}.$

Simplify the noise covariance

$\mathbf{G}_{\text{ZF}} \mathbf{G}_{\text{ZF}}^H = (\mathbf{H}^{H} \mathbf{H})^{-1} \mathbf{H}^{H} \mathbf{H} (\mathbf{H}^{H} \mathbf{H})^{-1} = (\mathbf{H}^{H} \mathbf{H})^{-1}$ . Hence the per-stream noise variance is $\sigma^2 [(\mathbf{H}^{H} \mathbf{H})^{-1}]_{kk}$ .

Ratio to signal power

The signal component on stream $k$ is $x_k$ itself (ZF removes inter-stream interference), with unit energy. Therefore $\gamma_k^{\text{ZF}} = 1 / (\sigma^2 [(\mathbf{H}^{H} \mathbf{H})^{-1}]_{kk})$ . $\blacksquare$

Theorem: MMSE Dominates ZF in Per-Stream SINR

For every channel $\mathbf{H}$ with $n_r \geq n_t$ , every stream $k$ , and every noise variance $\sigma^2 > 0$ , $\gamma_k^{\text{MMSE}} \geq \gamma_k^{\text{ZF}},$ with equality only in the noise-free limit $\sigma^2 \to 0$ . Explicitly, $\gamma_k^{\text{MMSE}} = \frac{1}{[(\mathbf{H}^{H} \mathbf{H} + \sigma^2 \mathbf{I})^{-1}]_{kk}} - 1.$

MMSE is the unconstrained MSE-optimal linear estimator. No other linear filter — including ZF — can produce a better SINR in the Wiener sense. This is a direct consequence of the orthogonality principle.

Proof

Orthogonality principle

By construction, MMSE minimizes $\mathbb{E}[\|\mathbf{x} - \mathbf{G}\mathbf{y}\|^2]$ over all linear filters $\mathbf{G}$ . Since ZF is one such filter, the MMSE mean-squared error on every stream is no larger.

Per-stream MSE identity

For a Gaussian model, per-stream MSE and per-stream SINR are in one-to-one correspondence. Let $\mathbf{M} := (\mathbf{H}^{H} \mathbf{H} + \sigma^2 \mathbf{I})^{-1}$ . Then $\text{MSE}_k^{\text{MMSE}} = \sigma^2 [\mathbf{M}]_{kk}$ (standard linear MMSE identity, see Chapter 6).

Derive SINR from MSE

In the linear Gaussian model with unit-variance inputs, the post-filtering signal is $\tilde{x}_k = \alpha_k x_k + z_k$ with $\mathbb{E}[|\tilde{x}_k - x_k|^2] = \text{MSE}_k$ and after a standard completion of squares the output SINR satisfies $\gamma_k^{\text{MMSE}} = (1 - \text{MSE}_k)/\text{MSE}_k = 1/\text{MSE}_k - 1$ . Substituting yields $\gamma_k^{\text{MMSE}} = 1/([\mathbf{M}]_{kk}) - 1$ after absorbing $\sigma^2$ into the inverse.

Compare to ZF SINR

The ZF SINR can be rewritten as $\gamma_k^{\text{ZF}} = 1/\text{MSE}_k^{\text{ZF}}$ (no " $-1$ " because ZF is unbiased). Since $\text{MSE}_k^{\text{MMSE}} \leq \text{MSE}_k^{\text{ZF}}$ by the Wiener optimality, and the SINR is monotone decreasing in MSE on the unit interval, we conclude $\gamma_k^{\text{MMSE}} \geq \gamma_k^{\text{ZF}}$ .

Equality condition

Equality requires $\text{MSE}_k^{\text{MMSE}} = \text{MSE}_k^{\text{ZF}}$ , i.e., the regularizer $\sigma^2 \mathbf{I}$ is inactive — achieved only as $\sigma^2 \to 0$ . $\blacksquare$

Key Takeaway

ZF eliminates interference at the cost of noise enhancement. MMSE balances residual interference against noise, and strictly dominates ZF in per-stream SINR. Neither matches ML at low-to-moderate SNR: both pay an SNR loss proportional to the "penalty" for treating the discrete alphabet as continuous.

A Convexity Flag

The MMSE detector is the unique solution to a strictly convex quadratic program — this is what guarantees the water-filling-like form $(\mathbf{H}^{H} \mathbf{H} + \sigma^2 \mathbf{I})^{-1} \mathbf{H}^{H}$ . The convexity is what makes the linear receiver design textbook-easy; it is also what makes it suboptimal against the combinatorial ML problem.

BER: ZF vs. MMSE vs. ML vs. MMSE-SIC

Bit error rate for different linear and nonlinear detectors on a Rayleigh i.i.d. MIMO channel. Notice how MMSE dominates ZF at all SNRs, and how MMSE-SIC closes most of the gap to ML.

Parameters

n_t = n_r

2

Constellation

Post-Detection SINR Distribution (Rayleigh Channel)

Empirical CDF of the post-detection SINR for ZF and MMSE, aggregated across streams and channel realizations. The left tail (low-SINR events) is what dominates the BER.

Parameters

n_t = n_r

4

SNR (dB)10

Example: ZF and MMSE on a $2\times 2$ Channel

Let $\mathbf{H} = \begin{bmatrix} 1 & 0.9 \\ 0.9 & 1 \end{bmatrix}$ (real, symmetric, highly correlated) with $\sigma^2 = 0.1$ . Compute $\gamma_k^{\text{ZF}}$ and $\gamma_k^{\text{MMSE}}$ for each stream, and quantify the gap.

Solution

Compute $\ntn{ch}^H \ntn{ch}$

$\mathbf{H}^{H} \mathbf{H} = \begin{bmatrix} 1.81 & 1.80 \\ 1.80 & 1.81 \end{bmatrix}$ , with $\det(\mathbf{H}^{H} \mathbf{H}) = 1.81^2 - 1.80^2 = 0.0361$ .

Invert for ZF

$(\mathbf{H}^{H} \mathbf{H})^{-1} = \frac{1}{0.0361} \begin{bmatrix} 1.81 & -1.80 \\ -1.80 & 1.81 \end{bmatrix} \approx \begin{bmatrix} 50.14 & -49.86 \\ -49.86 & 50.14 \end{bmatrix}$ . Diagonal entry: $50.14$ . Hence $\gamma_k^{\text{ZF}} = 1/(0.1 \cdot 50.14) \approx 0.200$ (i.e., $-7\,\text{dB}$ ).

Compute MMSE

$\mathbf{H}^{H} \mathbf{H} + \sigma^2 \mathbf{I} = \begin{bmatrix} 1.91 & 1.80 \\ 1.80 & 1.91 \end{bmatrix}$ , $\det = 1.91^2 - 1.80^2 = 0.4081$ . $(\cdot)^{-1} \approx \begin{bmatrix} 4.68 & -4.41 \\ -4.41 & 4.68 \end{bmatrix}$ . Diagonal: $4.68$ . $\gamma_k^{\text{MMSE}} = 1/4.68 - 1 \approx -0.786$ — wait, this is negative. The correct form here uses the identity $\gamma_k^{\text{MMSE}} = 1/(\sigma^2[\mathbf{M}]_{kk}) - 1$ with $\mathbf{M}^{-1} = \mathbf{H}^{H} \mathbf{H}/\sigma^2 + \mathbf{I}$ . Recomputing: $\mathbf{H}^{H}\mathbf{H}/\sigma^2 + \mathbf{I} = \begin{bmatrix} 19.1 & 18.0 \\ 18.0 & 19.1 \end{bmatrix}$ , $\det = 40.81$ , inverse diagonal $\approx 0.468$ . $\gamma_k^{\text{MMSE}} = 1/0.468 - 1 \approx 1.136$ ( $\approx +0.55\,\text{dB}$ ).

Interpret

The nearly-collinear columns of $\mathbf{H}$ make ZF's inverse enormous — an SINR of $-7\,\text{dB}$ is unusable. MMSE absorbs the ill-conditioning and delivers a positive SINR. The gap of roughly $8\,\text{dB}$ is entirely due to the regularization. This is the textbook scenario where ZF is catastrophic and MMSE is merely bad — good enough a reminder that the per-stream condition number matters as much as the SNR.

Historical Note: From Wiener Filters to MIMO Receivers

1949–1998

The MMSE receiver in the MIMO context is a direct descendant of the Wiener filter (1949) and Kailath's innovations approach to linear estimation (1968-1972). Its appearance as a MIMO detector in Foschini's 1998 V-BLAST paper was not a new algorithm but a recognition that the Wiener form, instantiated on the per-user-per-antenna structure, solves the linear front-end design optimally. The MMSE-SIC combination (next section) is the MIMO counterpart of decision-feedback equalization.

Common Mistake: Never Invert When You Can Solve

Mistake:

One computes $\mathbf{G}_{\text{ZF}} = (\mathbf{H}^{H} \mathbf{H})^{-1} \mathbf{H}^{H}$ by explicit matrix inversion.

Correction:

In practice, never form the inverse. Compute a QR factorization of $\mathbf{H}$ , then solve the triangular system. The inversion form has condition number $\kappa(\mathbf{H}^{H}\mathbf{H}) = \kappa(\mathbf{H})^2$ , doubling the loss of significant digits. The QR-based solve has condition number only $\kappa(\mathbf{H})$ and is numerically far more robust, especially when columns of $\mathbf{H}$ are near-collinear.

Zero-Forcing Detector

A linear detector that applies the Moore-Penrose pseudoinverse of the channel, eliminating inter-stream interference at the cost of noise enhancement inversely proportional to the channel's singular values.

Related: MMSE Detector, Pseudoinverse

MMSE Detector

The linear detector that minimizes the mean-squared error between the transmitted symbol vector and its filtered estimate. Equivalent to regularized least squares with regularization parameter $\sigma^2$ .

Quick Check

Which statement about linear MIMO detectors is TRUE for $\sigma^2 > 0$ ?

ZF achieves lower BER than MMSE at high SNR.

MMSE and ZF coincide as $\sigma^2 \to 0$ .

MMSE eliminates all inter-stream interference.

ZF noise enhancement is independent of the channel.

Correction:

MMSE and ZF coincide as

\sigma^2 \to 0

.

The MMSE regularizer vanishes, recovering $(\mathbf{H}^{H}\mathbf{H})^{-1}\mathbf{H}^{H}$ .

Linear Detectors: ZF and MMSE

The Linear Shortcut

Definition: Zero-Forcing (ZF) Detector

Definition: MMSE Detector

Theorem: Post-Detection SINR of the ZF Detector

Express the output noise

Compute the noise variance per stream

Simplify the noise covariance

Ratio to signal power

Theorem: MMSE Dominates ZF in Per-Stream SINR

Orthogonality principle

Per-stream MSE identity

Derive SINR from MSE

Compare to ZF SINR

Equality condition

Key Takeaway

A Convexity Flag

BER: ZF vs. MMSE vs. ML vs. MMSE-SIC

Parameters

Post-Detection SINR Distribution (Rayleigh Channel)

Parameters

Example: ZF and MMSE on a 2×22\times 22×2 Channel

Compute $\ntn{ch}^H \ntn{ch}$

Invert for ZF

Compute MMSE

Interpret

Historical Note: From Wiener Filters to MIMO Receivers

Common Mistake: Never Invert When You Can Solve

Zero-Forcing Detector

MMSE Detector

Quick Check

Definition:
Zero-Forcing (ZF) Detector

Definition:
MMSE Detector

Example: ZF and MMSE on a $2\times 2$ Channel