Trellis-Based Noise Modulation with Soft-Decision Viterbi Detection

I. INTRODUCTION

• Most noise-modulation schemes detect symbols independently, while trellis sequence detection can exploit temporal dependencies between transmitted noise-power symbols.The proposed soft receiver jointly uses noise-energy statistics and trellis constraints in a likelihood-based framework.Source: §I · p. 1
• The main extension maps each noise-power level jointly from the current input symbol and trellis state, creating sequence constraints for Viterbi detection.The binary rate-one filtering configuration is illustrative and does not provide coding gain.Source: §I · p. 1

II. SYSTEM MODEL

• The system represents each symbol with T independent Gaussian noise samples whose power depends on the symbol and, in a trellis, the current state.The receiver observes samples with additive receiver noise and uses their energy to estimate the transmitted power level.Source: §II · p. 2
• Increasing the number of samples per symbol makes the normalized energy a more reliable estimate of received variance.The system model identifies this property as important for noise-power detection.Source: §II · p. 2
• Valid trellis transitions constrain symbol sequences, but the resulting detection gain depends on whether the mapping has redundancy or nontrivial state-dependent constraints.The receiver can quantize energy before Viterbi decoding or incorporate unquantized energy directly into its likelihood-based branch metric.Source: §II · p. 2

III. BINARY FILTERING-BASED NOISE MODULATION: AN ILLUSTRATIVE TRELLIS REPRESENTATION

• The binary MODEM filters the input before noise modulation, introducing finite memory and a trellis representation for sequence recovery.The configuration is used to compare hard and soft energy-based detection.Source: §III · p. 2
• The filter output depends on previous input bits, and each state-input transition determines an output bit and next state.The state comprises M binary elements, producing 2^M trellis states.Source: §III · p. 3
• Each transmitted bit is represented by T Gaussian noise samples, with the filtered bit selecting the high or low power level.The special trellis shown uses M = 2 and h = [1, 1, 1], and its four states support Viterbi recovery.Source: §III · p. 3, Figure 2 · p. 3
• The rate-one causal filter is invertible for a known initial state, so it adds no redundancy and is not expected to provide conventional coding gain.The binary case illustrates filtering-based sequence recovery; the state-dependent N-ary scheme is the main trellis extension.Source: §III · p. 3
• The soft receiver places received energy directly in the Viterbi branch metric, whereas the hard receiver quantizes energy into a binary decision first.Direct use of energy avoids the information loss caused by the intermediate hard decision.Source: §III · p. 3

A. Separate Noise Demodulator and Viterbi Sequence Detector (Hard Receiver)

• The hard receiver quantizes received energy to a binary power-level decision before supplying it to the Viterbi detector.This discards information about each energy observation's reliability before sequence detection.Source: A. Separate Noise Demodulator and… · pp. 3–4
• The receiver's default threshold is the optimum ML threshold obtained by equating the conditional energy likelihoods; a midpoint-mean threshold is also evaluated.The midpoint threshold assesses sensitivity to threshold selection.Source: A. Separate Noise Demodulator and… · p. 4
• For each valid trellis transition, the branch metric is the Hamming distance between the detected bit and that transition's output bit.Survivor paths are traced back with depth D, which determines decoding delay.Source: A. Separate Noise Demodulator and… · p. 4

B. Joint soft Viterbi detector-demodulator (soft receiver)

• The soft receiver uses the received energy's negative log-likelihood as the branch metric for joint Viterbi detection and demodulation.The energy observation is incorporated directly rather than first being converted to a hard decision.Source: B. Joint soft Viterbi… · p. 4
• The branch metric is derived from the received-energy distribution, which the text identifies as a scaled chi-square distribution.The passage connects the chi-square distribution to the Gamma distribution when calculating the metric.Source: B. Joint soft Viterbi… · p. 4
• The receiver calculates state metrics in the classic manner and applies the same approach to the joint Viterbi detector-demodulator.Source: B. Joint soft Viterbi… · p. 4

IV. PROPOSED N-ARY TRELLIS-BASED NOISE MODULATION AND DEMODULATION

• The N-ary extension assigns transmitted noise power jointly from the current trellis state and input symbol, introducing sequence constraints for trellis detection.Source: §IV · p. 5
• When Q = N M+1, each state-input transition can map to a distinct power level; for smaller Q, transitions may share levels.The mapping preserves the state-dependent trellis structure.Source: §IV · p. 5
• Bhattacharyya-distance design distributes power levels by statistical separability of received noise distributions rather than absolute power differences.A larger distance indicates greater statistical separability; the design can impose a prescribed or approximately equal distance between adjacent levels.Source: §IV · pp. 5–6
• The soft receiver uses continuous received energy in the branch metric, preserving reliability information that quantization loses in the hard receiver.The Viterbi algorithm minimizes accumulated branch metrics to estimate the most likely state and symbol sequence.Source: §IV · p. 6
• Modulation order N, memory order M, and the number Q of distinct power levels are separate design parameters, with Q ≤ N M+1.The number of states is N^M and the number of transitions is N^(M+1).Source: §IV · p. 6
• The simulations use M = 2 unless specified otherwise and compare N = 2, 4, and 8; the Bhattacharyya-based design is used for subsequent evaluations.For the initial power-level comparison, the Bhattacharyya-based design is selected for later BER evaluations.Source: §IV · p. 6

V. SIMULATION RESULTS

• The simulation study compares the conventional hard-demodulation Viterbi receiver (RX1) with the proposed joint soft Viterbi receiver (RX2).Unless stated otherwise, simulations use M = 2 and Nbits = 10,000, with Bhattacharyya-based power levels.Source: §V · p. 6
• The study investigates how power-level selection, traceback depth, Eb/N0, and symbol duration affect BER.Source: §V · p. 6

A. Power and Eb/N0 Definitions

• The instantaneous transmit power during a symbol interval equals the variance of its transmitted noise samples.Source: A. Power and Eb/N0 Definitions · p. 6
• The stated average transmitted noise power is P̄tx = 1.5.Source: A. Power and Eb/N0 Definitions · p. 7
• BER is reported as a function of Eb/N0, with noise variance adjusted to maintain the same Eb/N0 across modulation orders.The normalization accounts for the different numbers of information bits conveyed by each N-ary symbol.Source: A. Power and Eb/N0 Definitions · p. 7

• For the soft Viterbi receiver, performance approaches full traceback as depth increases, with no further improvement beyond D = 5M.D = 5M is adopted as a trade-off between decoding performance and traceback delay.Source: B. Binary Noise Modulation · p. 7
• As Eb/N0 decreases, BER increases; the soft receiver performs best, while midpoint- and ML-threshold hard receivers perform similarly.Source: B. Binary Noise Modulation · p. 7, Figure 7 · p. 8
• Power-level configuration affects BER, and the Bhattacharyya-based design is selected as the baseline for subsequent simulations.The comparison evaluates designs under identical conditions at Eb/N0 = 10 dB.Source: B. Binary Noise Modulation · p. 7, Table II · p. 8
• The soft Viterbi receiver performs best; hard quantization, rather than the choice of midpoint or ML threshold, accounts for the main performance loss.It uses continuous-valued energy observations, preserving reliability information discarded by hard detection.Source: B. Binary Noise Modulation · pp. 7–8, Figure 7 · p. 8
• The binary rate-one filtering operation is invertible and introduces no coding redundancy, so its soft-detection gain is not a conventional coding gain.Source: B. Binary Noise Modulation · p. 8
• Increasing T improves performance because more noise samples provide a more reliable noise-power estimate; the improvement is most pronounced for soft detection.The symbol-by-symbol ML detector performs worse because it processes observations independently, without exploiting the filtering relationship across symbols.Source: B. Binary Noise Modulation · p. 8, Figure 8 · p. 8

• The soft-decision Viterbi receiver outperforms the hard-decision receiver for all considered modulation orders.It incorporates received noise energy directly into the likelihood-based branch metric instead of quantizing the observed energy.Source: C. N-ary Noise Modulation · pp. 8–9, Figure 9 · p. 9
• Higher modulation orders degrade BER as more densely packed noise-power levels become harder to distinguish, while trellis states and transitions increase in number.The scheme trades more information bits per symbol against trellis complexity and reduced power-level separation; soft detection mitigates part of the degradation.Source: C. N-ary Noise Modulation · p. 9
• Increasing T improves detection for both receivers by providing more observations to estimate noise power, at the cost of longer symbols and more receiver processing.The soft receiver consistently outperforms the hard receiver across the considered modulation orders and symbol durations.Source: Figure 10 · p. 9, C. N-ary Noise Modulation · p. 9

• The evaluation uses block fading, with the channel coefficient constant across a symbol’s T samples and independently varying between symbol intervals.Source: D. Performance under Fading Channels · p. 9
• When CSI is available, the instantaneous channel gain is incorporated into the soft branch metric; without CSI, the receiver uses the AWGN-based metric without adapting to fading.Source: D. Performance under Fading Channels · p. 9
• With perfect CSI, the receiver significantly outperforms its no-CSI configuration across the Eb/N0 range; AWGN performs best overall, and Rician generally outperforms Rayleigh.Ignoring fading causes substantial degradation, particularly in Rayleigh channels.Source: Figure 11 · p. 10, D. Performance under Fading Channels · p. 10

VI. CONCLUSION

• The N-ary scheme jointly maps trellis state and input symbol to noise power, introducing transition constraints exploitable by Viterbi detection.This is the paper's main extension beyond the binary illustrative configuration.Source: §VI · p. 10
• The soft receiver incorporates received energy directly into a likelihood-based branch metric, avoiding information loss from intermediate hard decisions.The binary rate-one invertible filter provides an illustrative comparison, not conventional coding redundancy.Source: §VI · p. 10
• The soft Viterbi receiver consistently outperforms hard-decision and symbol-by-symbol receivers in the reported simulations.The receiver was also evaluated over Rayleigh and Rician fading channels, where incorporating channel state information into the metric substantially improves performance.Source: §VI · p. 10
• Observation length, modulation order, and noise-power-level configuration significantly affect detection performance.Bhattacharyya-distance-based design provides a systematic way to select statistically distinguishable noise-power levels.Source: §VI · p. 10
• The results support trellis-based soft-decision sequence detection as a framework for improving reliability in binary and higher-order noise modulation.The conclusion characterizes this as potential rather than a universal guarantee.Source: §VI · p. 10