Zero-Forcing and Minimum Mean-Square Error Equalization

ZF can cause severe noise enhancement. If a channel component is deeply faded (very weak), its inverse will have very high gain, amplifying noise on that path by an equally large factor. If noise already dominates on weak paths, the amplification can make the overall error rate worse. This is why MMSE is preferred when noise is non-negligible; it limits amplification by keeping interference if it reduces noise even more.

Common methods include: (1) Pilot-based: known training symbols are transmitted; the receiver computes the least-squares estimate of H. (2) Blind equalization: algorithms like CMA (Constant Modulus Algorithm) estimate H from the received signal using only the known modulation properties (constant amplitude, phase), without pilots. Blind methods save bandwidth but converge slowly. Practical systems often use pilot preambles followed by blind tracking.

DFE is a non-linear extension: once a symbol is detected (sliced), its contribution to future intersymbol interference is subtracted before equalization of the next symbol. This prevents error propagation that grows exponentially in deep channels. DFE is preferred for long channels (>20 taps) or very high SNR; however, it suffers error propagation if early symbols are misdetected.

In time-varying channels, a fixed equalizer becomes outdated. Adaptive algorithms continuously adjust equalizer weights based on error feedback. LMS uses gradient descent (slow, stable). RLS uses recursive least-squares (fast convergence, higher cost). Both track channel changes; LMS is simpler and more robust to model errors, RLS is faster but sensitive to noise. The choice depends on coherence time of the channel.