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Beyond Linear Equalization

Linear equalizers face a fundamental trade-off: suppressing ISI requires boosting frequencies where the channel is weak, which amplifies noise. The decision-feedback equalizer (DFE) breaks this trade-off by using past symbol decisions to subtract out the "trailing" ISI caused by previously detected symbols. Because the feedback filter operates on decisions (not the noisy received signal), it cancels ISI without amplifying noise --- provided the decisions are correct.

Definition:
Decision-Feedback Equalizer (DFE)

A decision-feedback equalizer consists of two filters:

Feedforward filter $\mathbf{w}_f$ with $N_f$ taps, operating on the received samples $\{y_k\}$ .
Feedback filter $\mathbf{w}_b$ with $N_b$ taps, operating on past symbol decisions $\{\hat{a}_{k-1}, \ldots, \hat{a}_{k-N_b}\}$ .

The equalizer output is

$\tilde{a}_k = \mathbf{w}_f^H \mathbf{y}_k - \mathbf{w}_b^H \hat{\mathbf{a}}_k$

where $\mathbf{y}_k = [y_k, \ldots, y_{k-N_f+1}]^T$ and $\hat{\mathbf{a}}_k = [\hat{a}_{k-1}, \ldots, \hat{a}_{k-N_b}]^T$ . The final decision is $\hat{a}_k = \text{dec}(\tilde{a}_k)$ (slicing/quantisation to the nearest constellation point).

The feedback filter subtracts the ISI from already-detected symbols without noise enhancement, since it operates on clean decisions rather than noisy observations.

Definition:
Feedforward Filter

The feedforward filter of a DFE is the linear filter with coefficients $\mathbf{w}_f = [w_{f,0}, w_{f,1}, \ldots, w_{f,N_f-1}]^T$ that processes the received signal. Unlike a pure linear equalizer, the feedforward filter in a DFE only needs to suppress precursor ISI (interference from future symbols $a_{k+1}, a_{k+2}, \ldots$ ) because the feedback filter handles postcursor ISI.

Definition:
Feedback Filter

The feedback filter of a DFE has coefficients $\mathbf{w}_b = [w_{b,1}, w_{b,2}, \ldots, w_{b,N_b}]^T$ and operates on past decisions $\hat{a}_{k-1}, \ldots, \hat{a}_{k-N_b}$ . Its role is to subtract the postcursor ISI --- the interference caused by symbols that have already been detected. When past decisions are correct ( $\hat{a}_{k-i} = a_{k-i}$ ), the feedback filter perfectly cancels postcursor ISI without introducing any noise.

Theorem: MMSE-DFE

The MMSE-DFE minimises $J = E[|a_k - \tilde{a}_k|^2]$ jointly over the feedforward filter $\mathbf{w}_f$ and feedback filter $\mathbf{w}_b$ . Assuming correct past decisions ( $\hat{a}_{k-i} = a_{k-i}$ ), the MMSE-DFE achieves

$J_{\text{MMSE-DFE}} = \exp\!\left[\frac{1}{2\pi} \int_{-\pi}^{\pi} \ln\!\left(\frac{N_0/E_s}{|H(e^{j\omega})|^2 + N_0/E_s}\right) d\omega\right]$

This is always less than or equal to $J_{\text{MMSE}}$ of the linear MMSE equalizer, with equality only when the channel is flat.

The feedforward filter is

$W_f(e^{j\omega}) = \frac{H^*(e^{j\omega})}{|H(e^{j\omega})|^2 + N_0/E_s} \cdot \frac{1}{F(e^{j\omega})}$

where $F(e^{j\omega})$ is a spectral factor, and the feedback filter coefficients are obtained from the causal part of the cascade response.

The DFE achieves a lower MSE than a linear equalizer because it uses decisions to subtract postcursor ISI without noise penalty. The improvement is greatest for channels with deep spectral nulls, where the linear equalizer suffers severe noise enhancement.

Proof

Decomposition into causal and anticausal parts

The key idea is to decompose the MMSE target response into a causal part (handled by the feedback filter) and an anticausal part (handled by the feedforward filter).

Perform the spectral factorisation of $|H(e^{j\omega})|^2 + N_0/E_s = |F(e^{j\omega})|^2$ where $F(z)$ is minimum-phase (all zeros inside the unit circle).

Minimum MSE

Applying the orthogonality principle to the DFE output and using the spectral factorisation, one obtains

$J_{\text{MMSE-DFE}} = \exp\!\left[\frac{1}{2\pi} \int_{-\pi}^{\pi} \ln\!\left(\frac{N_0/E_s}{|H|^2 + N_0/E_s}\right) d\omega\right]$

By Jensen's inequality applied to the concave $\ln$ function:

$J_{\text{MMSE-DFE}} \leq \frac{1}{2\pi} \int_{-\pi}^{\pi} \frac{N_0/E_s}{|H|^2 + N_0/E_s}\, d\omega = J_{\text{MMSE}}$

with equality iff $|H(e^{j\omega})|$ is constant. $\blacksquare$

DFE Demonstration

Compare the BER of a linear MMSE equalizer and an MMSE-DFE on a multipath channel. The DFE provides significant gains on channels with spectral nulls. Toggle error propagation to see its effect at lower SNR values.

Parameters

Channel tap

h_1

0.7

Channel tap

h_2

-0.3

Feedforward taps

N_f

7

Feedback taps

N_b

3

Include error propagation

Example: DFE for a Two-Tap Channel

Consider the channel $h = [1,\; 0.8]$ with BPSK and $E_s/N_0 = 15$ dB. Design an MMSE-DFE with $N_f = 3$ feedforward taps and $N_b = 1$ feedback tap. Compare the MSE with the linear MMSE equalizer.

Solution

Channel model

The received signal is $y_k = a_k + 0.8\, a_{k-1} + \eta_k$ with $\sigma_\eta^2 = E_s / 10^{1.5} \approx 0.0316\, E_s$ .

The postcursor ISI is entirely due to $h[1] = 0.8$ , which can be cancelled by a single feedback tap.

Ideal DFE operation

Assuming correct past decisions, the DFE subtracts $w_{b,1}\, \hat{a}_{k-1}$ from the feedforward output.

With $w_{b,1} = 0.8$ , the feedback filter perfectly cancels the postcursor ISI, leaving only the feedforward filter to handle the remaining noise.

The effective channel seen by the feedforward filter is reduced to a single tap, and the feedforward filter becomes approximately a matched filter.

MSE comparison

For the linear MMSE equalizer on this channel, the MSE is relatively large because the channel $1 + 0.8 z^{-1}$ has a spectral null near $\omega = \pi$ .

The MMSE-DFE achieves significantly lower MSE because the feedback filter removes the deep spectral null without noise amplification.

Numerically: $J_{\text{MMSE}} \approx 0.15\, E_s$ vs. $J_{\text{DFE}} \approx 0.04\, E_s$ --- an improvement of about 5.7 dB. $\blacksquare$

Decision-Feedback Equalizer Signal Flow

Animate the signal flow through a DFE: the received signal passes through the feedforward filter, past decisions are fed back to cancel postcursor ISI, and the result is sliced to make a new decision.

The feedback filter operates on hard decisions (noise-free), cancelling ISI without amplifying noise — the key advantage of the DFE over linear equalizers.

DFE Block Diagram — The feedforward filter processes received samples; the feedback filter subtracts postcursor ISI using past hard decisions.

Quick Check

In a DFE, what is the primary advantage of the feedback filter over simply using a longer feedforward filter?

The feedback filter has lower computational complexity

The feedback filter cancels postcursor ISI without amplifying noise

The feedback filter eliminates all ISI including precursor ISI

The feedback filter makes the equalizer unconditionally stable

Correction:

The feedback filter cancels postcursor ISI without amplifying noise

Correct. Because the feedback filter operates on symbol decisions (which are noise-free when correct), it subtracts ISI without injecting any additional noise.

Common Mistake: Error Propagation in the DFE

Mistake:

Assuming that the MMSE-DFE always achieves its theoretical MSE in practice. The theoretical analysis assumes correct past decisions ( $\hat{a}_{k-i} = a_{k-i}$ ).

Correction:

When a symbol decision is wrong, the feedback filter subtracts the wrong ISI contribution, causing a larger error in the next symbol. This error propagation can cause bursts of errors. At low-to-moderate SNR, error propagation significantly degrades DFE performance below the "genie-aided" bound (which assumes perfect feedback). Mitigations include:

Using stronger error-correction coding
Tomlinson-Harashima precoding (moves DFE to the transmitter)
Reduced-state sequence estimation

Common Mistake: Incorrect DFE Filter Ordering

Mistake:

Applying the feedback filter before the feedforward filter, or feeding the equalizer output (before slicing) into the feedback filter instead of the hard decisions.

Correction:

The DFE processes in this order: (1) feedforward filter on received samples, (2) subtract feedback filter output on past hard decisions, (3) make a new hard decision. Feeding soft (unquantised) values into the feedback filter creates an unstable IIR structure that diverges.

Decision-Feedback Equalizer (DFE)

A nonlinear equalizer consisting of a feedforward filter on received samples and a feedback filter on past symbol decisions, which cancels postcursor ISI without noise enhancement.

Error Propagation

A phenomenon in decision-feedback equalizers where an incorrect symbol decision causes the feedback filter to subtract wrong ISI values, leading to a burst of subsequent errors.

Decision-Feedback Equalizers

Beyond Linear Equalization

Definition: Decision-Feedback Equalizer (DFE)

Definition: Feedforward Filter

Definition: Feedback Filter

Theorem: MMSE-DFE

Decomposition into causal and anticausal parts

Minimum MSE

DFE Demonstration

Parameters

Example: DFE for a Two-Tap Channel

Channel model

Ideal DFE operation

MSE comparison

Decision-Feedback Equalizer Signal Flow

DFE Block Diagram

Quick Check

Common Mistake: Error Propagation in the DFE

Common Mistake: Incorrect DFE Filter Ordering

Decision-Feedback Equalizer (DFE)

Error Propagation

Definition:
Decision-Feedback Equalizer (DFE)

Definition:
Feedforward Filter

Definition:
Feedback Filter