Zero-forcing precoding and generalized inverses

Ami Wiesel*, Yonina C. Eldar, Shlomo Shamai

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

459 Scopus citations


We consider the problem of linear zero-forcing precoding design and discuss its relation to the theory of generalized inverses in linear algebra. Special attention is given to a specific generalized inverse known as the pseudo-inverse. We begin with the standard design under the assumption of a total power constraint and prove that precoders based on the pseudo-inverse are optimal among the generalized inverses in this setting. Then, we proceed to examine individual per-antenna power constraints. In this case, the pseudo-inverse is not necessarily the optimal inverse. In fact, finding the optimal matrix is nontrivial and depends on the specific performance measure. We address two common criteria, fairness and throughput, and show that the optimal generalized inverses may be found using standard convex optimization methods. We demonstrate the improved performance offered by our approach using computer simulations.

Original languageAmerican English
Pages (from-to)4409-4418
Number of pages10
JournalIEEE Transactions on Signal Processing
Issue number9
StatePublished - 2008
Externally publishedYes

Bibliographical note

Funding Information:
Manuscript received February 20, 2007; revised January 24, 2008. Published August 13, 2008 (projected). The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Timothy N. Davidson. This work was supported by the EU 6/7th framework program, via the NEWCOM/NEWCOM++ network of excellence, by the Israel Science Foundation, and by the Glasberg-Klein Research Fund.


  • Beamforming
  • Generalized inverses
  • Per-antenna constraints
  • Semidefinite relaxation
  • Zero-forcing precoding


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