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OTFS Channel Estimation And Data Detection Designs With Superimposed Pilots

Himanshu B. Mishra, Prem Singh, Abhishek K. Prasad, Rohit Budhiraja

arXiv:2010.15066v1eess.SP

TL;DR

The paper addresses OTFS channel estimation and detection without dedicated pilot resources, while retaining low-complexity processing for sparse delay-Doppler channels. It proposes non-iterative and iterative superimposed-pilot designs, optimizes data–pilot power allocation, and reports higher spectral efficiency than existing OTFS designs with negligible BER degradation for SP-I.

  • Problem

    Existing OTFS channel estimation and detection designs incur pilot overhead, while extending superimposed pilots is difficult for rapidly varying channels and frame-varying delay-Doppler gains.

  • Method

    The paper superimposes pilots on delay-Doppler data, proposes SP-NI and SP-I designs, uses sparsity-exploiting message passing, and optimizes data–pilot power allocation through an SINR lower bound.

  • Results

    The proposed designs achieve higher spectral efficiency than existing OTFS designs, with SP-I showing significantly higher SE and negligible BER degradation.

  • Takeaways & Limitations

    Superimposed pilots provide an OTFS channel-estimation and detection framework that avoids the spectral-efficiency loss associated with dedicated pilot resources.

Abstract

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This work proposes a superimposed pilot (SP)-based channel estimation and data detection framework for orthogonal time-frequency space (OTFS) scheme, wherein low-powered pilots are superimposed on to data symbols in the delay-Doppler domain. We propose two channel estimation and data detection designs for SP-OTFS systems which, unlike the existing OTFS designs, do not designate any slots for pilots, which improves their spectral efficiency (SE). The first SP design estimates channel by treating data as interference, which degrades its performance at high signal to noise ratio. The second SP design alleviates this problem by iterating between channel estimation and data detection. Both these designs detect data using message passing algorithm which exploits OTFS channel sparsity, and consequently has low computational complexity. We also derive a lower bound on the signal-to-interference-plus-noise ratio of the proposed designs, and maximize it by optimally allocating power between data and pilot symbols. We numerically validate the derived analytical results, and show that the proposed designs have superior SE than the two state-of-the-art OTFS channel estimation and data detection designs.

I. INTRODUCTION

OTFS transforms doubly selective channels into an almost time-invariant, sparse delay-Doppler representation, enabling reduced pilot overhead and lower-complexity processing. The paper extends superimposed pilots to OTFS with two designs that avoid dedicated pilot slots or frames while targeting higher spectral efficiency.

  • Motivation: OTFS converts doubly selective channels into an almost time-invariant delay-Doppler channel whose symbols experience nearly constant gain.This representation can reduce pilot overhead for rapidly time-varying channels.
  • Motivation: Delay-Doppler channel sparsity can reduce the complexity of channel estimation and data detection, motivating sparsity-exploiting receivers.The channel matrix otherwise has size MN ×MN, increasing OTFS system complexity.
  • Existing designs: Existing OTFS estimators either use an entire pilot frame or insert zeros around embedded pilots, reducing spectral efficiency and potentially suffering channel aging at high Doppler spread.Time-frequency estimators are also computationally complex because the channel is non-sparse there and must be transformed into the delay-Doppler domain.
  • Existing designs: Prior superimposed-training methods assume nearly time-invariant channels over many frames, limiting their performance for rapidly time-varying channels.OTFS extension is nontrivial because the time-frequency channel changes rapidly and delay-Doppler channel gain varies across frames.
  • Proposed framework: The proposed SP-OTFS framework superimposes pilots on data without inserted zeros, dedicated delay-Doppler pilot slots, or a dedicated pilot frame.The authors state that this should provide significantly higher spectral efficiency than and, with minor BER degradation.
  • Proposed designs: SP-NI treats data as interference during channel estimation, whereas SP-I iterates between channel estimation and data detection to mitigate that interference.Both designs exploit delay-Doppler sparsity for message-passing detection; SP-NI performance degrades at high SNR, while SP-I has better BER and SE.
  • Power allocation: The proposed designs derive and optimize a lower bound on SINR through power allocation between data and pilot symbols.The paper reports that optimal power allocation minimizes BER and maximizes SE for both designs.
  • Results: At SE 3 bps/Hz, SP-I requires approximately 15 dB and 5 dB lower SNR than designs and, respectively.The reported comparison is a numerical result for the proposed SP-I design.

II. OTFS SYSTEM MODEL WITH SUPERIMPOSED PILOT

The SP-OTFS transmitter adds pilot and data symbols arithmetically in the delay-Doppler domain, then maps the combined symbols through OTFS time-frequency and time-domain processing.

  • Transmit representation: The delay-Doppler symbol is the arithmetic sum of data and pilot symbols, x[l, k] = xd[l, k] + xp[l, k].The corresponding data, pilot, and superimposed matrices are arranged as Xd, Xp, and X.
  • OTFS modulation: The combined delay-Doppler symbols are transformed into the time-frequency domain using the inverse symplectic finite Fourier transform.The time-frequency symbol matrix is represented as XTF = FMXFH.
  • Frame structure: The time-frequency frame spans duration NT and bandwidth M∆f, with T∆f = 1.M and N index the delay and Doppler dimensions, respectively.
  • Time-domain generation: The transformed symbols are pulse-shaped and converted into a continuous-time transmit signal using the Heisenberg transform.The resulting signal is sampled at fs = M∆f = M/T and represented by the matrix S and vector s = vec(S).
  • Guard interval: A cyclic prefix of length lmax samples is appended to mitigate inter-frame interference in the time domain.Here lmax corresponds to the tap associated with the maximum delay τmax.

B. Channel impulse response

The OTFS channel is modeled sparsely in the delay-Doppler domain using a small number of propagation clusters, while receive processing maps the signal back into that domain.

  • Channel model: The delay-Doppler channel contains Q clusters or propagation paths, each with a complex gain, delay tap, and Doppler tap.The impulse response spans delays up to τmax and Doppler shifts within [−νmax, νmax].
  • Channel assumptions: The model assumes integer-valued delay taps and neglects fractional Doppler effects, although the proposed designs can be extended to fractional Doppler.The maximum delay is represented by τmax = (lmax)T/M.
  • Received signal: After cyclic-prefix removal, the received signal is modeled as r = Hs + w with independent zero-mean complex Gaussian noise.The received vector is subsequently processed in the OTFS receiver chain.
  • Receive processing: The receiver applies Wigner transformation and SFFT processing to obtain time-frequency and delay-Doppler receive representations.The receive vector is y = (FN ⊗ Grx)r.
  • Effective channel: The effective OTFS channel incorporates inter-symbol and inter-carrier interference and differs substantially from the OFDM channel model.This difference changes the signal processing required for superimposed-pilot transmission.

III. SUPERIMPOSED PILOT -BASED OTFS CHANNEL ESTIMATOR AND DATA DETECTOR

The proposed SP-NI design estimates the delay-Doppler channel while treating data as interference, then uses sparse message passing for detection; its detector achieves BER close to MAP with lower complexity.

  • Channel estimation in SP-NI design: The SP-NI design computes an MMSE delay-Doppler channel estimate by treating data symbols as interference.Its interference statistics are characterized to support subsequent message-passing detection.
  • Message passing-aided data detection in the SP-NI design: The detector uses a reduced-complexity message passing algorithm that exploits delay-Doppler channel sparsity.Each observation or variable node connects to only Q neighboring nodes rather than all MN nodes.
  • Message passing-aided data detection in the SP-NI design: The resulting BER is very close to that of the maximum-a-posterior probability detector.The message calculations account for channel-estimation error, noise, and partially canceled pilot interference.
  • Channel estimation in SP-NI design: The channel estimator exploits sparsity by inverting a Q × Q matrix, where Q ≪ MN is the number of delay-Doppler taps.The resulting estimate includes mutual interference between superimposed data and pilot symbols in its covariance model.
  • Message passing-aided data detection in the SP-NI design: The message-passing procedure iteratively updates means, probability mass functions, and data decisions using a damping factor to control convergence.The effective channel estimate is updated from the estimated channel vector during the procedure.

B. Proposed superimposed pilot-iterative (SP-I) design

The SP-I design addresses SP-NI’s high-SNR degradation by iterating channel estimation and message-passing data detection, using initial SP-NI data estimates for data-aided refinement.

  • B. Proposed superimposed pilot-iterative (SP-I) design: SP-NI performance degrades at high SNR because its channel estimator treats data as interference.This limitation motivates the iterative design.
  • B. Proposed superimposed pilot-iterative (SP-I) design: SP-I iterates between channel estimation and message-passing-aided data detection.The design begins with an initial data estimate produced by SP-NI.
  • B. Proposed superimposed pilot-iterative (SP-I) design: SP-I uses the initial SP-NI data estimate together with the superimposed pilot vector for data-aided channel estimation.The refined estimate is then used within the iterative framework.

1) Channel estimation in SP-I design:

The SP-I design iterates between data-aided channel estimation and message-passing data detection, updating channel and effective-channel estimates across iterations.

  • The estimation-error and noise-plus-interference statistics depend on the current data estimate because the channel estimate itself depends on detected data.
  • The receiver calculates message-passing statistics from the received vector, channel estimate, and noise-plus-estimation-error statistics.
  • The iterative procedure repeats channel estimation and data detection until its stopping criterion is met.
  • SP-I initializes with data detected by SP-NI and uses those estimates for iterative data-aided channel estimation.
  • Each iteration computes a data-aided MMSE channel estimate and feeds the resulting effective-channel estimate to the message-passing receiver.

IV. OPTIMAL POWER ALLOCATION BETWEEN DATA AND PILOT SYMBOLS

The paper derives an SINR lower bound for superimposed-pilot detection and optimizes the data–pilot power split, linking the allocation to spectral efficiency and BER.

  • The proposed designs superimpose pilots on data symbols under a total power constraint.
  • The resulting power allocation maximizes SINR, which the paper relates to maximizing spectral efficiency and minimizing BER.
  • The SINR lower bound incorporates channel-estimation error and noise-plus-interference effects in the effective received signal.
  • The optimal pilot power is obtained by differentiating the effective-SINR lower bound and solving the resulting stationarity equation.
  • Pilot power is selected independently of the instantaneous channel because pilots are used for channel estimation.

V. COMPUTATIONAL COMPLEXITY ANALYSIS

The complexity analysis counts arithmetic operations for the proposed SP-NI and SP-I schemes and expresses their costs in terms of frame dimensions, sparsity, and iterations.

  • The analysis counts multiplication/division and addition/subtraction operations for the proposed schemes.
  • SP-NI requires MN + 3Q^2 + O(Q^3) + O(N_I MNQS) operations.
  • The SP-I scheme has an iteration-dependent complexity, with its total cost governed by the number of SP-I iterations.
  • When Q ≪ MN, the SP-NI complexity is O(MN) + O(N_I MNQS).

O(MN) + O(NIMNQS)

The simulations evaluate complexity, power allocation, channel-estimation MSE, and BER for the proposed SP-NI and SP-I designs under specified OTFS settings.

  • Simulation setup: The simulations use M,N ∈ {16,32}, a 5-tap delay-Doppler channel, rectangular pulses, and BPSK signaling.
  • Simulation setup: The proposed designs allocate normalized total frame power across superimposed data and pilot symbols, while EP reserves slots for pilots.
  • Power allocation: The optimal pilot and data powers depend on SNR, tap count Q, channel statistics, and delay-Doppler grid dimensions.
  • Power allocation: For Q = 5 and M = N = 16, the average optimal pilot and data powers are approximately 0.3 and 0.7, respectively.
  • Power allocation: Allocating 30% of total power to pilots and 70% to data maximizes SE and minimizes BER for the proposed SP-aided designs.
  • Estimator and BER results: At the optimal pilot power, both proposed schemes achieve minimum BER even though that allocation does not minimize channel-estimator MSE.
  • Estimator and BER results: SP-I MSE depends on data-detection accuracy, while its theoretical MSE assumes perfect data availability and demonstrates the design’s accuracy.

B. BER of the proposed SP-NI and SP-I designs

The iterative SP-I design consistently achieves lower BER than SP-NI, while optimal pilot–data power allocation minimizes BER and maximizes spectral efficiency. Both proposed designs avoid the pilot overhead of conventional schemes, yielding higher SE than CPA and, in many settings, EP.

  • Power allocation: Optimal power allocation gives the proposed designs minimum BER, overlapping with the BER obtained using analytically calculated channel-estimation error values.
  • BER comparison: SP-I has much lower BER than SP-NI across all SNR values, with the gap widening at high SNR.SP-NI suffers mutual interference between data and pilots because it treats data as interference during channel estimation.
  • Message-passing convergence: For 0 < ∆≤0.8, BER remains constant because message passing converges within NI = 20 iterations; for ∆>0.8, BER degrades.
  • SE comparison: Optimal power allocation maximizes SE, and SP-I has significantly higher SE than SP-NI because of its lower channel-estimation MSE.

VII. CONCLUSIONS

The paper proposes non-iterative and iterative superimposed-pilot OTFS designs that estimate channels and detect data without dedicated pilot slots. It combines sparsity-exploiting message passing with SINR-based power allocation, and reports higher SE for SP-I with negligible BER degradation.

  • The proposed SP-NI and SP-I designs superimpose pilots on data symbols and avoid the SE loss associated with dedicated pilot slots.
  • Both designs use computationally efficient message passing for data detection by exploiting OTFS channel sparsity in the delay-Doppler domain.
  • The derived SINR lower bound supports optimal pilot–data power allocation, which minimizes BER and maximizes SE.
  • SP-I achieves significantly higher SE than existing state-of-the-art designs with negligible BER degradation.
  • The study does not consider data-dependent SP schemes or mean-removal-based techniques, which are identified as future work.

APPENDIX A

The appendix derives analytical expressions for channel-estimation error and SINR by simplifying covariance and expectation terms under the model’s statistical assumptions. It also relates the derivations to the EP benchmark.

  • The derivation evaluates covariance terms for the effective noise and interference using statistical independence of data symbols, channel parameters, and noise samples.
  • A positive-definite matrix trace result is used to obtain a lower bound for the SP-NI channel-estimation MSE.
  • The proposed and EP SINR expressions are formed by simplifying numerator and denominator expectations before substitution into the final formulas.
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