NAMDF: Robust Pitch Estimation

• NAMDF is a normalized correlation measure that computes the average magnitude difference between a frame and its lagged version to reduce amplitude variability.
• The process integrates adaptive sigmoid mapping and harmonic summation, enhancing pitch detection reliability in noisy and reverberant conditions.
• Temporal aggregation combined with Viterbi decoding enforces smooth pitch trajectories, significantly reducing errors like octave doubling and spurious detections.

The Normalized Average Magnitude Difference Function (NAMDF) is a correlation-based measure introduced for robust pitch estimation in speech, particularly under high-distortion conditions including additive noise and reverberation. NAMDF integrates time-domain difference computation with normalization and modern probabilistic mapping, supports aggregation across harmonics and frames, and incorporates temporal continuity via Viterbi decoding constraints to enhance resilience against noise-induced errors. Its design addresses the longstanding challenge of accurate pitch tracking in environments where conventional methods often suffer from octave doubling, halving, and spurious voicing errors.

1. Mathematical Definition and Computation of NAMDF

NAMDF is computed for each frame $f_i$ of an energy-normalized speech signal. For candidate lags $l$ spanning a range determined by plausible pitch periods, the function evaluates the average magnitude difference between the frame and its lagged counterpart, then normalizes by a scale-invariant measure to suppress amplitude effects:

$$\phi^{i}_{l} = \frac{\sum |f_i - f_{i+l}|}{\sqrt[4]{\|f_i\|_2^2 \|f_{i+l}\|_2^2}},\quad \forall\; l_{\min} \leq l \leq l_{\max}$$

The numerator quantifies the dissimilarity at each lag, while the denominator—given as the fourth root of the product of the squared $\ell_2$ norms—provides normalization robust to varying energy, reducing sensitivity to amplitude fluctuations across frames.

This produces, per frame, a vector $\Phi_i$ of NAMDF values for each candidate lag, forming the basis for subsequent probabilistic pitch analysis and aggregation.

2. Likelihood Mapping via Sigmoid Normalization

To transform raw NAMDF difference scores into interpretable likelihoods suitable for probabilistic state modeling, a robust sigmoid mapping is applied to each NAMDF vector within a frame. Centering and scaling are dynamically adapted to the distribution of NAMDF values in the frame using the 10th ($\eta_{0.10}$) and 90th ($\eta_{0.90}$) percentiles:

$$\phi^{i}_{l} = \frac{1}{1 + \exp\!\left(-\left( k\, \frac{\phi^{i}_{l} - \frac{\eta_{0.10} + \eta_{0.90}}{2}}{\eta_{0.90} - \eta_{0.10}}\right) \right)},\quad \forall\; l_{\min} \leq l \leq l_{\max}$$

Here, $k$ is a scaling factor controlling sigmoid sharpness. The use of adaptive centering and scaling ensures that local dynamic ranges are respected, so pitches salient in the present context are emphasized while the influence of periodic noise is suppressed. The outputs are likelihoods in [0,1], encoding confidence in the presence of periodicity at each lag.

3. Harmonic Summation and Temporal Aggregation

To further enhance robustness, NAMDF-based likelihoods are aggregated both harmonically and temporally.

Harmonic Summation: Given the harmonic nature of speech, likelihoods are reinforced by summing contributions from integer multiples of the putative fundamental period. For each lag $l$:

$$\phi^{i}_{l} = \phi^{i}_{l} + \sum_{h=2}^{H+1} w_n \max\left(\phi^{i}_{h\, l \pm r}\right),\quad \forall\; l_{\min} \leq l \leq \frac{l_{\max}}{H+1}$$

where $w_n$ are weights for each harmonic, $H$ is the highest harmonic considered, and $r$ accounts for possible harmonic misalignments. This process accentuates periodic structures in the signal, mitigating typical octave errors and suppressing incidental, non-harmonically aligned peaks.

Temporal Accumulation: To exploit the slow temporal variation of pitch, per-frame likelihoods are temporally smoothed by accumulation over a symmetric window:

$\phi^{i}_{l} = \sum_{k=-K}^{K} \phi^{i+k}_{l}$

with $K$ determining the window's half-width. This leverages inter-frame continuity, attenuating spurious frame-level fluctuations caused by transient noise.

4. Constrained Viterbi Decoding for Smooth Pitch Trajectories

For final voiced/unvoiced decisions and pitch trajectory recovery, a continuity-constrained Viterbi algorithm is deployed across the frame-wise likelihood grid. Transition between consecutive frames is restricted to adjacent pitch states, ensuring temporal smoothness and physical plausibility. The cost between states is formulated as:

$C^i_{t,t'} = \phi^{i+1}_{t'} - \phi^{i}_{t}$

This transition structure ensures that only realistic, continuous pitch trajectories are selected, strongly rejecting abrupt, noise-induced jumps or outlier estimates. The approach yields a sequence of pitch states representing the most plausible and smooth path, given the observed likelihoods and the model constraint.

5. Experimental Evaluation and Comparative Performance

NAMDF-based pitch estimation was benchmarked on the TUG and Keele speech corpora under diverse acoustic conditions, including additive noise (varying SNR) and room reverberation. Evaluation employed:

• Gross Pitch Error (GPE): Percentage of voiced frames where estimated pitch deviates by more than 5% from reference.
• Voicing Decision Error (VDE): Rate of incorrect voiced/unvoiced frame classification.

Key empirical findings:

Metric	Setting	Observed Performance
GPE	SNR = 10 dB or lower	Up to 15% reduction vs. baselines
GPE	Reverberant environments	Consistently lower values
VDE	SNR ≤ 10 dB	Consistently lower rates

When compared against time-domain and frequency-domain state-of-the-art methods (YIN, PEFAC, SHRP, SWIPE), the system leveraging NAMDF with harmonic and temporal aggregation plus continuity-constrained Viterbi decoding produced more robust pitch estimates, especially in nonstationary noise and reverberant field conditions. Harmonic and temporal aggregation provided major contributions by reducing spurious and octave errors. In scenarios such as car noise, the aggregated likelihood and trajectory constraint mitigated pitch-doubling or halving effects prevalent in rival algorithms.

6. Relation to and Advancement over Prior Approaches

NAMDF differs from classical Average Magnitude Difference Function (AMDF) and autocorrelation-based approaches by its combination of robust normalization, dynamic likelihood transformation, and aggregation strategies. The normalization via the fourth root of the energy product is particularly noteworthy for addressing amplitude variability—a challenge for vanilla AMDF calculations. The adaptive sigmoid mapping yields frame-localized confidence measures, enabling subsequent probabilistic modeling.

Harmonic summation extends prior methods focused on single-lag evidence by integrating harmonic structure explicitly; temporal accumulation leverages the quasi-stationarity of pitch over short intervals, analogous to strategies in dynamic Bayesian networks. The continuity constraint in Viterbi decoding enforces a smooth trajectory, preventing the selection of implausible pitch jumps that often arise in high-noise situations.

7. Applications and Impact

NAMDF and associated framework elements are suited for deployment in automatic speech recognition frontends, speaker diarization, and other applications requiring robust pitch tracking in adverse environments. The architecture is well matched to real-world scenarios—such as telephony, in-vehicle speech systems, and far-field microphones—where conventional approaches are susceptible to environmental distortions.

Experimental confirmation of substantial GPE and VDE reductions demonstrates that NAMDF-based pitch estimation provides more reliable speaker and prosody cues under realistic, high-noise conditions. This suggests its adoption can improve downstream tasks dependent on pitch, including speech synthesis alignment, voice conversion, and prosodic analysis in clinical or communications devices.