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Zero-Forcing (ZF) Receiver Summary: Uplink vs Downlink (2-User MIMO)

This summary compares Zero-Forcing (ZF) techniques for both uplink and downlink in a multi-user MIMO scenario with:

  • 2 users with single antennas
  • 1 Access Point (AP) with 2 antennas

Common Goal

Eliminate inter-user interference by linearly separating data streams using CSI.

System Model

  • Users transmit:
    $$ \mathbf{y} = \mathbf{H} \mathbf{x} + \mathbf{n} $$
  • \( \mathbf{y} \in \mathbb{C}^{2 \times 1} \): signal at AP
  • \( \mathbf{H} \in \mathbb{C}^{2 \times 2} \): uplink channel matrix
  • \( \mathbf{x} \in \mathbb{C}^{2 \times 1} \): user symbols
  • \( \mathbf{n} \sim \mathcal{CN}(0, \sigma^2 \mathbf{I}) \): AWGN

ZF Detection

AP applies:

\[ \mathbf{W}_{ZF} = (\mathbf{H}^H \mathbf{H})^{-1} \mathbf{H}^H \]

Recovered:

\[ \hat{\mathbf{x}} = \mathbf{W}_{ZF} \mathbf{y} = \mathbf{x} + (\mathbf{H}^H \mathbf{H})^{-1} \mathbf{H}^H \mathbf{n} \]

Interference-free
Noise amplified

System Model

  • AP transmits:
    $$ \mathbf{s} = \mathbf{W} \mathbf{x} $$
  • Each user receives:
    $$ y_k = \mathbf{h}_k \mathbf{s} + n_k $$ where \( \mathbf{h}_k \) is the row vector for user \( k \)

ZF Precoding

Design:

\[ \mathbf{W}_{ZF} = \mathbf{H}^H (\mathbf{H} \mathbf{H}^H)^{-1} \]

Then:

\[ \mathbf{H} \mathbf{W}_{ZF} = \mathbf{I} \Rightarrow y_k = x_k + n_k \]

Interference-free
Requires full CSI at AP (CSIT)

2-User Example

Let:

\[ \mathbf{H} = \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix} \]

Then:

  • Uplink ZF Receiver:
    $$ \mathbf{W}_{ZF}^{\text{UL}} = (\mathbf{H}^H \mathbf{H})^{-1} \mathbf{H}^H = \frac{1}{2} \begin{bmatrix} 1 & 1 \ 1 & -1 \end{bmatrix} $$

  • Downlink ZF Precoding:
    $$ \mathbf{W}_{ZF}^{\text{DL}} = \mathbf{H}^H (\mathbf{H} \mathbf{H}^H)^{-1} = \frac{1}{2} \begin{bmatrix} 1 & 1 \ 1 & -1 \end{bmatrix} $$

Same matrix! But different interpretation.

Summary Table

Feature Uplink ZF Downlink ZF
Interference removed? Yes Yes
Noise amplified? Yes Yes (post equalization)
CSI required at AP? (Receive side) (Transmit side)
CSIT needed? No Yes
Complexity at AP Linear filter (RX) Precoding (TX)

Conclusion

  • ZF uplink: separates signals at the receiver (no CSIT needed).
  • ZF downlink: cancels interference before transmission (needs CSIT).

_Last updated: June 06, 2025