---
title: Phase Tracking Partial Response DFE
url: https://www.emergentmind.com/topics/phase-tracking-partial-response-dfe
type: topic
---

# Phase Tracking Partial Response DFE

Phase Tracking Partial Response Decision-Feedback Equalizer (PT-PRDFE) is an advanced signal processing architecture that jointly addresses intersymbol interference (ISI) and carrier-phase noise in high-speed coherent communication systems. It integrates partial-response channel equalization, decision-directed phase tracking via digital phase-locked loop (PLL), and iterative turbo equalization with soft-input soft-output decoders. The PT-PRDFE demonstrates measurable performance benefits when applied to Faster-Than-Nyquist (FTN) 16QAM signaling over fiber links, notably achieving a 0.9 dB optical power margin improvement over multi-band probabilistically shaped 64QAM (MB-PCS-64QAM) when used in amplifier-less coherent links with turbo-FEC [2601.17803].

## 1. System Model and Partial-Response Targeting

The communication system under consideration employs FTN-16QAM transmission, with each in-phase (I) and quadrature (Q) channel filtered by a square-root-raised-cosine (SRRC) pulse shape whose roll-off (α=0.8) is matched to the FTN baud-rate compression. The received signal model post-coherent detection, chromatic dispersion compensation, timing recovery, and frequency offset estimation (FOE), is

$$
y[n] = \sum_{k=0}^{2} h[k] \, a[n-k] \, e^{j\phi[n]} + w[n]
$$

where:
- $a[n]$: transmitted 16QAM symbol sequence (FTN-compressed)
- $h[0]=h[1]=h[2]=1$, $h[k]=0$, $k\geq3$: 2nd-order partial-response impulse
- $\phi[n]$: cumulative phase noise plus residual carrier frequency offset
- $w[n]$: complex additive white Gaussian noise (AWGN), combining amplifier spontaneous emission (ASE) and receiver thermal noise

Amplifier-less short-reach links present unique equalization demands: the deliberate partial-response shaping ($H(z)=1+z^{-1}+z^{-2}$) forces controlled ISI, sharpening spectral efficiency at the cost of post-compensation complexity.

## 2. Joint Phase Tracking Algorithm

PT-PRDFE embeds carrier-phase extraction within the equalization loop. The block diagram is as follows:

- The input sample $y[n]$ passes through an adaptive feedforward filter ($c_0 \ldots c_M$).
- The filtered symbol is phase-rotated using the latest phase estimate $e^{-j\hat{\phi}[n]}$.
- Hard decision via a minimum-distance slicer yields $\bar{a}[n]$ (nearest 16QAM constellation point).
- Feedback filtering ($b_1 \ldots b_N$) subtracts estimated ISI, generating equalized output $y_{\mathrm{eq}}[n]$.

Phase error is sensed by

$$
\epsilon[n] = \Im \left\{ y_{\mathrm{eq}}[n] \bar{a}[n]^* \right\}
$$

The PLL uses a second-order digital loop filter:

$$
\hat{\phi}[n] = \hat{\phi}[n-1] + \alpha\epsilon[n] + \beta\epsilon[n-1]
$$

where $\alpha,\beta$ set bandwidth and damping. Phase correction is then applied:

$$
y_c[n] = y_{\mathrm{eq}}[n] e^{-j\hat{\phi}[n]}
$$

This approach leverages decision-directed tracking, assuming upstream FOE and timing recovery have resolved coarse synchronization.

## 3. Decision-Feedback Equalizer (DFE) Architecture

Equalization attacks channel ISI by combining feedforward filtering, feedback subtraction based on past decisions, and phase compensation. The canonical equations are:

- Feedforward:
  $$
  u[n] = \sum_{i=0}^{M} c_i[n] \, y[n-i]
  $$
- Hard decision:
  $$
  \bar{a}[n] = Q(u[n] e^{-j\hat{\phi}[n]})
  $$
- Feedback subtraction:
  $$
  v[n] = u[n] - \sum_{j=1}^{N} b_j[n] \, \bar{a}[n-j]
  $$
- Output:
  $$
  y_{\mathrm{eq}}[n] = v[n] e^{-j\hat{\phi}[n]}
  $$

Adaptive tap updating employs the decision-directed least-mean-square (LMS) procedure:

- Error signal:
  $$
  e_{\mathrm{DFE}}[n] = y_{\mathrm{eq}}[n] - \bar{a}[n]
  $$
- Feedforward update:
  $$
  c_i[n+1] = c_i[n] - \mu_c \, e_{\mathrm{DFE}}[n] \, y[n-i]^*
  $$
- Feedback update:
  $$
  b_j[n+1] = b_j[n] + \mu_b \, e_{\mathrm{DFE}}[n] \, \bar{a}[n-j]^*
  $$

This LMS-driven adaptation rapidly converges given pre-equalization by pilot-tone FOE.

## 4. Outer Turbo Equalization Loop

To further suppress residual nonlinearities and error propagation, PT-PRDFE interfaces to an iterative turbo equalizer. The sequence is:

- PT-PRDFE processed samples are whitened by a post-filter $h_{\mathrm{pF}}$.
- Symbol log-likelihood ratios (LLRs) are forwarded to a BCJR sequence detector, exploiting channel memory.
- The BCJR output passes to a Forward Error Correction (FEC) decoder (soft-decision).
- The decoder’s extrinsic information recycles as a priori LLRs for the next BCJR iteration.

At bit index $n$,

$$
L_{\mathrm{app}}(n) = L_c(n) + L_a^E(n), \qquad L_e(n) = L_{\mathrm{app}}(n) - L_a^E(n)
$$

where $L_c(n)$ is the channel reliability, $L_a^E(n)$ is extrinsic input, and $L_e(n)$ is extrinsic output to FEC. This iterative turbo equalization loop typically yields steep “waterfall” BER performance as observed in the referenced work.

## 5. Performance Quantification and Comparative Results

Measured BER versus received optical power margin is the primary metric. For 16QAM under partial-response equalization (disregarding phase noise), the AWGN benchmark is:

$$
BER \approx \frac{3}{2(M-1)} \, \operatorname{erfc} \left(\sqrt{0.1\,\gamma\,/\, [1 + \text{ISI power}]} \right)
$$

where $\gamma$ is received SNR and ISI power is post-equalization energy in $h[1],h[2]$.

Power margin advantage is summarized in the following excerpt [2601.17803]:

| Format               | BER = 2e–2 crossing (dB PM) |
|----------------------|----------------------------|
| MB-PCS-64QAM         | 8.3 dB                     |
| FTN-16QAM w/ turbo   | 7.4 dB (≈0.9 dB gain)      |

At a hard-decision FEC threshold of $2\times 10^{-2}$ (7% overhead), FTN-16QAM with PT-PRDFE/turbo equalization achieves a $0.9\,\mathrm{dB}$ optical power margin improvement compared to MB-PCS-64QAM.

## 6. Implementation, Complexity, and Assumptions

Complexity per symbol comprises:
- $(M+N+1)$ complex multiplies (feedforward + feedback taps)
- One complex multiply plus argument or imaginary extraction for phase-error sensing
- Two real multiplies and add for PLL loop filter

The BCJR turbo equalizer’s state complexity grows exponentially with channel memory but remains manageable for 16QAM under memory truncation.

Key system-level assumptions:
- Perfect timing recovery and FOE upstream of PT-PRDFE
- Fast convergence of LMS adaptation after pilot-tone FOE
- Residual phase noise tracked with second-order digital PLL, analysis assumes small-phase error regime
- Turbo equalizer approximates post-filtered partial-response channel as full-response for BCJR trellis simplification

This approach yields practical implementation feasibility in amplifier-less coherent links, enabling improved spectral efficiency and reduced optical power requirements [2601.17803].

## 7. Broader Context and Implications

The integration of PT-PRDFE with turbo equalization provides a comprehensive methodology for countering ISI and phase noise in FTN transmission systems, notably where amplifier-less architectures restrict link budget. The documented 0.9 dB improvement in power margin for FTN-16QAM reflects the effectiveness of this collaborative equalization strategy. *A plausible implication is* that such architectures may facilitate next-generation short-reach coherent systems without recourse to power-hungry optical amplification or excessively high-order modulation under spectral constraints.

Source: https://www.emergentmind.com/topics/phase-tracking-partial-response-dfe