Spectral Subband Centroid Frequencies (SSCFs)

• SSCFs are short-time speech features that estimate the center of gravity of spectral energy within predefined subbands, providing a clear spectral shape descriptor.
• They are used to derive both static and dynamic trajectory features through polar coordinate and ratio-plane representations that capture transitions in speech.
• When integrated with MFCCs or used as stand-alone polar features, SSCFs enhance noise robustness and reduce inter-speaker variability in automatic speech recognition.

Searching arXiv for papers on SSCFs and related ASR usage to ground the article. arxiv.search({"query":"\"Spectral Subband Centroid Frequencies\" OR SSCF ASR polar coordinate centroid frequency", "max_results": 10, "sort_by": "submittedDate"}) Fetching the most relevant arXiv records for SSCFs and centroid-based speech features. Spectral Subband Centroid Frequencies (SSCFs) are short-time speech features that estimate the “centre of gravity” of the power spectrum within predefined subbands, thereby encoding where spectral mass is concentrated rather than only how much energy is present. In the cited literature, SSCFs are treated as frequency-centroid descriptors computed over fixed or Mel-spaced filter-bank bands, and are positioned as an alternative or complement to Mel-Frequency Cepstral Coefficients (MFCCs) for automatic speech recognition (ASR) and word recognition. The recent literature emphasizes three recurring roles for SSCFs: as static subband-centroid features, as the basis for trajectory descriptors in a two-dimensional SSCF plane, and as ingredients in ratio- or polar-coordinate parameterizations intended to capture dynamic structure while reducing discontinuities and inter-speaker variability (Leang et al., 2022, Berjon et al., 2022, Leang et al., 30 Jul 2025).

1. Formal definition and signal-theoretic interpretation

SSCFs are defined as subband-wise frequency centroids of a short-time spectrum. In one formulation, the speech spectrum $P(f)$ is divided into $M$ overlapping or non-overlapping subbands with lower and upper limits $l_m$ and $h_m$, and weighting function $w_m(f)$. The SSCF in band $m$ is then

$$\mathrm{SSCF}_m = \frac{ \int_{l_m}^{h_m} f \cdot w_m(f) \cdot P^\gamma(f)\, df }{ \int_{l_m}^{h_m} w_m(f) \cdot P^\gamma(f)\, df }.$$

In the studies considered here, $\gamma$ is a dynamic-range coefficient and is often set to $1$ (Leang et al., 2022). An equivalent time-indexed form uses the short-time power spectrum $P(t,f)$:

$M$0

A discrete version is also reported, replacing the integrals by sums over FFT bins (Leang et al., 30 Jul 2025).

The defining interpretation is consistent across the literature: SSCFs estimate the “centre of gravity” of spectral power in each subband, which makes them akin to formant tracking in spirit, but computed over fixed filter-bank bands rather than by explicit formant estimation (Leang et al., 2022). In the frequency-centroid formulation for non-native English word recognition, the same object is termed a Frequency-Centroid (FC) feature and explicitly identified as equivalent to SSCFs; there, each Mel subband yields a single real-valued centroid in hertz (Berjon et al., 2022).

This representation differs fundamentally from MFCCs. MFCCs summarize the spectral energy distribution after Mel filtering, logarithmic compression, and a DCT, whereas SSCFs preserve the frequency-weighted location of energy within each subband. The papers therefore frame SSCFs as encoding spectral shape and local spectral shift rather than only energy magnitude (Berjon et al., 2022).

2. Subband construction, extraction, and preprocessing

The cited works use filter-bank-defined subbands, but with two distinct operating regimes. In the non-native English word-recognition study, SSCFs are extracted with a 24-filter triangular Mel filterbank spanning from $M$1 Hz to the Nyquist frequency, with subbands defined by the overlapping triangular windows $M$2. The centroid in subband $M$3 is computed as

$M$4

with $M$5 filters and triangular windows anchored at Mel-warped frequencies $M$6 (Berjon et al., 2022).

In the ASR studies using SSCF trajectories, the frontend is more compact. One system uses $M$7 subbands, yielding SSCF$M$8 through SSCF$M$9, and constructs five adjacent-band planes from these values (Leang et al., 2022). A later Vietnamese system likewise uses six triangular Mel-spaced bandpass filters indexed $l_m$0, identical to the 6-filter MFCC bank employed in that work (Leang et al., 30 Jul 2025).

The extraction pipeline reported for SSCF-derived polar features is explicit: 25 ms frames, 10 ms hop, Hamming window, pre-emphasis $l_m$1, and 3-frame moving average smoothing (Leang et al., 2022). The Vietnamese study reports 25 ms windows, 10 ms shift, 0.97 pre-emphasis, 512-point FFT, Hamming window, and liftering $l_m$2 for the accompanying MFCCs (Leang et al., 30 Jul 2025). The frequency-centroid study uses a similar but not identical frontend: 20 ms frame length, 10 ms frame shift, pre-emphasis $l_m$3, Hamming window, FFT-based power spectrum computation, and weighted sums over the filterbank (Berjon et al., 2022).

A high-level extraction procedure common to these reports consists of pre-emphasis, framing, Hamming windowing, FFT, power-spectrum estimation, filter-bank weighting, and then, for each band, a centroid numerator divided by the corresponding band energy denominator (Berjon et al., 2022). The additional computation relative to MFCC extraction is described as one extra weighted sum per filter, and the reported cost is negligible relative to FFT and moderate-sized CNN inference (Berjon et al., 2022).

3. Trajectory modeling: angles, polar coordinates, and ratio planes

A central line of work uses SSCFs not merely as framewise static features but as coordinates in low-dimensional acoustic trajectories. Earlier work described in the 2022 ASR paper observed that vowel-to-vowel transitions in the SSCF$l_m$4–SSCF$l_m$5 plane are quasi-straight, motivating the definition of a transition angle

$l_m$6

where $l_m$7 denotes the difference between the endpoint and start of a short sliding window on axis $l_m$8 (Leang et al., 2022). The stated difficulty is that $l_m$9 has a branch cut at $h_m$0, which produces abrupt steps in the measured angle despite smooth underlying trajectories.

To address this, the paper proposes converting adjacent SSCF pairs into polar coordinates. For a point on the $h_m$1–$h_m$2 plane at time $h_m$3,

$h_m$4

the polar coordinates are

$h_m$5

The radius measures distance from the origin in the selected SSCF plane, while the angle measures direction using a continuous two-argument arctangent (Leang et al., 2022). The paper’s motivation is explicitly twofold: to avoid discontinuities and to capture both direction and magnitude of the spectral trajectory.

The 2025 Vietnamese study adopts a related but not identical construction by first forming a ratio plane:

$h_m$6

These ratio features are then mapped to polar coordinates,

$h_m$7

with the interpretation that the radius captures the overall magnitude of spectral shift in the ratio plane and the angle captures the direction of transition (Leang et al., 30 Jul 2025). The same study also notes an alternative formulation in which a two-dimensional point is built from current and previous SSCF values in one band, followed by polar conversion, although the reproducible implementation details specify that the reported system computes polar ratios only on the $h_m$8 versus $h_m$9 plane (Leang et al., 30 Jul 2025).

The claim that these trajectory parameters are relatively speaker- and rate-independent appears in both the 2022 and 2025 ASR-oriented papers, though their empirical conclusions about gender independence differ. This suggests that the trajectory abstraction is attractive as a dynamical descriptor, but that its invariance properties depend materially on the exact embedding, normalization, and task (Leang et al., 2022, Leang et al., 30 Jul 2025).

4. Dynamic parameters, derivatives, and pseudo-$w_m(f)$0

Dynamic augmentation is a recurring theme in SSCF-based ASR. In the 2022 polar-coordinate study, first- and second-order finite-difference derivatives are defined in the usual way over a frame shift $w_m(f)$1, for example

$w_m(f)$2

with $w_m(f)$3 wrapped into $w_m(f)$4 if necessary, and similarly

$w_m(f)$5

Using six SSCF bands yields five adjacent-band planes; each plane contributes $w_m(f)$6, producing a 10-dimensional static polar representation and a 30-dimensional representation when $w_m(f)$7 and $w_m(f)$8 are included (Leang et al., 2022).

The Vietnamese work instead combines 6 static MFCCs with their $w_m(f)$9 and $m$0, producing 18 dimensions, and then appends the ratio-plane $m$1 and $m$2, with optional inclusion of SSCF$m$3 as a pseudo-$m$4. This yields either 20 dimensions without SSCF$m$5 or 21 dimensions with SSCF$m$6 (Leang et al., 30 Jul 2025).

The treatment of SSCF$m$7 is notable. The 2025 study defines SSCF$m$8 as the centroid over the lowest-frequency band and uses it as a robust stand-in for the true fundamental frequency. To compensate gender-dependent pitch scaling, utterance-level mean-variance normalization is applied:

$m$9

where $$\mathrm{SSCF}_m = \frac{ \int_{l_m}^{h_m} f \cdot w_m(f) \cdot P^\gamma(f)\, df }{ \int_{l_m}^{h_m} w_m(f) \cdot P^\gamma(f)\, df }.$$0 and $$\mathrm{SSCF}_m = \frac{ \int_{l_m}^{h_m} f \cdot w_m(f) \cdot P^\gamma(f)\, df }{ \int_{l_m}^{h_m} w_m(f) \cdot P^\gamma(f)\, df }.$$1 are the utterance-level mean and standard deviation (Leang et al., 30 Jul 2025).

At the same time, the 2022 paper lists SSCF$$\mathrm{SSCF}_m = \frac{ \int_{l_m}^{h_m} f \cdot w_m(f) \cdot P^\gamma(f)\, df }{ \int_{l_m}^{h_m} w_m(f) \cdot P^\gamma(f)\, df }.$$2 as a limitation and open question: SSCF$$\mathrm{SSCF}_m = \frac{ \int_{l_m}^{h_m} f \cdot w_m(f) \cdot P^\gamma(f)\, df }{ \int_{l_m}^{h_m} w_m(f) \cdot P^\gamma(f)\, df }.$$3, described there as relative $$\mathrm{SSCF}_m = \frac{ \int_{l_m}^{h_m} f \cdot w_m(f) \cdot P^\gamma(f)\, df }{ \int_{l_m}^{h_m} w_m(f) \cdot P^\gamma(f)\, df }.$$4, moves very differently from higher SSCF bands, and mixing it into trajectories may be physically meaningless unless careful normalization or exclusion is used (Leang et al., 2022). These statements are not contradictory so much as method-dependent. One paper warns against unnormalized inclusion of SSCF$$\mathrm{SSCF}_m = \frac{ \int_{l_m}^{h_m} f \cdot w_m(f) \cdot P^\gamma(f)\, df }{ \int_{l_m}^{h_m} w_m(f) \cdot P^\gamma(f)\, df }.$$5 in generic trajectory planes; the other uses SSCF$$\mathrm{SSCF}_m = \frac{ \int_{l_m}^{h_m} f \cdot w_m(f) \cdot P^\gamma(f)\, df }{ \int_{l_m}^{h_m} w_m(f) \cdot P^\gamma(f)\, df }.$$6 as a separate pseudo-pitch channel with explicit normalization. A plausible implication is that SSCF$$\mathrm{SSCF}_m = \frac{ \int_{l_m}^{h_m} f \cdot w_m(f) \cdot P^\gamma(f)\, df }{ \int_{l_m}^{h_m} w_m(f) \cdot P^\gamma(f)\, df }.$$7 is useful when isolated and normalized, but problematic when indiscriminately fused into higher-band geometric trajectories.

5. Integration into recognition pipelines

SSCFs have been integrated into both hybrid DNN–HMM ASR systems and CNN-based closed-set word recognizers.

In the French ASR study, the acoustic model is a hybrid DNN–HMM implemented in Kaldi. The input uses a $$\mathrm{SSCF}_m = \frac{ \int_{l_m}^{h_m} f \cdot w_m(f) \cdot P^\gamma(f)\, df }{ \int_{l_m}^{h_m} w_m(f) \cdot P^\gamma(f)\, df }.$$8-frame context splice, producing a $$\mathrm{SSCF}_m = \frac{ \int_{l_m}^{h_m} f \cdot w_m(f) \cdot P^\gamma(f)\, df }{ \int_{l_m}^{h_m} w_m(f) \cdot P^\gamma(f)\, df }.$$9 input where $\gamma$0 is the feature dimension. The network contains 3 hidden layers of 512 units with tanh activation, followed by a fully connected softmax over approximately 2000 triphone states. Training proceeds through monophone and triphone GMM stages and then DNN training over 20 epochs with learning rate $\gamma$1. Decoding uses a WFST decoder with a 3-gram LM built by SRILM and a lexicon generated by Phonetisaurus (Leang et al., 2022).

The Vietnamese gender-independent ASR study reports essentially the same DNN–HMM backbone: Kaldi nnet2 recipe, 9-frame context, 3 hidden layers of 512 tanh units, 20 epochs, and learning rate $\gamma$2 (Leang et al., 30 Jul 2025). In that system, SSCF-derived parameters are concatenated with MFCC-based features rather than replacing them outright.

By contrast, the non-native English word-recognition study uses a two-stage CNN. Each utterance is transformed into a time–filterbank matrix for MFCCs and another for FC/SSCFs; the matrices are zero-padded or interpolated to $\gamma$3. A single feature type yields a $\gamma$4 single-channel input, while fused MFCC+FC input forms a $\gamma$5 tensor. The CNN contains a first convolutional layer with 32 filters and ReLU activation, a second convolutional layer with 64 filters and ReLU activation, and a fully connected softmax output with 5 units corresponding to the five target words (Berjon et al., 2022).

The following table summarizes the principal configurations reported in the cited literature.

Study	SSCF role	Model
(Leang et al., 2022)	Adjacent-band polar coordinates, with optional $\gamma$6	Hybrid DNN–HMM (Kaldi)
(Leang et al., 30 Jul 2025)	Ratio-plane polar parameters plus optional SSCF$\gamma$7	DNN–HMM backbone, Kaldi nnet2
(Berjon et al., 2022)	24-band SSCF/FC maps fused with MFCC maps	Two-stage CNN

Across these systems, normalization is applied either per speaker or per utterance, following standard Kaldi-style mean and variance normalization in the ASR papers, while the CNN study reports mean normalization and does not report additional whitening or cepstral mean variance normalization beyond zero-padding (Leang et al., 2022, Leang et al., 30 Jul 2025, Berjon et al., 2022).

6. Empirical performance and comparative findings

The three papers support distinct but related empirical conclusions.

For French read-speech ASR on BRAF100, the 2022 study reports the following DNN–HMM word error rates (WER). In the normal train/test split, 13-MFCC+$\gamma$8 achieves $\gamma$9, while raw angle features perform extremely poorly at $1$0. Raw polar coordinates reduce WER to $1$1, and Polar+$1$2 further reduces it to $1$3. The paper states that Polar+$1$4 significantly outperforms raw polar and angle features, with $1$5 via paired $1$6-test over multiple test sentences (Leang et al., 2022).

In cross-gender recognition on the same corpus, 13-MFCC+$1$7 yields $1$8 WER in the cross-male condition and $1$9 in the cross-female condition, whereas Polar+$P(t,f)$0 yields $P(t,f)$1 and $P(t,f)$2, respectively. The conclusion drawn in that study is that, although polar coordinates are superior to transition angles for defining acoustic trajectories, they are not much more gender-independent than conventional MFCCs (Leang et al., 2022).

The non-native English word-recognition study reaches a different conclusion about SSCFs’ role: SSCFs alone are not competitive in quiet conditions, but fusion with MFCCs consistently improves performance. Clean-speech accuracy for MFCC alone is $P(t,f)$3, $P(t,f)$4, and $P(t,f)$5 for Arabic, French, and Spanish accents, respectively; FC alone yields $P(t,f)$6, $P(t,f)$7, and $P(t,f)$8; MFCC+FC yields $P(t,f)$9, $M$00, and $M$01. Under NOISEX-92 additive noise, MFCC+FC outperforms MFCC alone by 3–8 percentage points at every SNR level and for each of White, Babble, and HF-channel noise. At $M$02 dB White noise, the reported improvement is from approximately $M$03 to approximately $M$04 (Berjon et al., 2022).

The Vietnamese study reports that adding only the SSCF-derived polar-ratio parameters and SSCF$M$05 to a 6-MFCC+$M$06 baseline reduces WER from $M$07 to $M$08 when SSCF$M$09 is mean-variance normalized, and reduces the male/female WER gap from $M$10 to $M$11. The paper explicitly states that the proposed parameters exhibit greater gender independence than the baseline MFCCs and match or exceed the 13-MFCC+$M$12 baseline, which reports $M$13 all-speaker WER and a $M$14 male/female gap (Leang et al., 30 Jul 2025).

These findings can be organized succinctly as follows.

Setting	Main comparison	Reported outcome
French BRAF100 ASR (Leang et al., 2022)	Angle vs Polar vs Polar+$M$15	Polar features improve strongly over angles; derivatives further improve WER
French cross-gender ASR (Leang et al., 2022)	Polar+$M$16 vs 13-MFCC+$M$17	No significant relative advantage in cross-gender robustness
Non-native English word recognition (Berjon et al., 2022)	MFCC vs FC vs MFCC+FC	SSCFs alone weak; fusion improves clean and noisy accuracy
Vietnamese ASR (Leang et al., 30 Jul 2025)	6-MFCC baseline vs +Polar-ratio vs +Polar-ratio+SSCF$M$18	Lower WER and smaller male/female gap with SSCF-derived additions

A common empirical pattern is therefore that SSCFs are most effective when used as complementary or structured descriptors rather than as stand-alone replacements for MFCCs.

7. Limitations, interpretive issues, and research directions

The available literature presents SSCFs as technically promising but methodologically sensitive. Several limitations are stated explicitly in the 2022 polar-coordinate study. First, SSCF$M$19 behaves very differently from higher SSCF bands, so mixing it into the same trajectory representation may be physically meaningless without careful normalization or exclusion. Second, beyond the first SSCF planes, trajectories are not quasi-linear, suggesting that the straight-line intuition motivating angle or polar parameterization may weaken for higher bands. Third, $M$20 and $M$21 are described as crude approximations of dynamic gestures, implying that more sophisticated temporal representations may be preferable (Leang et al., 2022).

The same paper identifies future directions including per-speaker or per-gender normalization of SSCFs, especially for the $M$22-related band; end-to-end embeddings of SSCF trajectories via RNNs or CNNs on two-dimensional “SSCF maps”; and unsupervised spatial normalization of polar trajectories to reduce inter-speaker variability (Leang et al., 2022). The Vietnamese work can be read as partially instantiating one of these directions by applying utterance-level normalization to SSCF$M$23 and using ratio-based geometry to reduce spectral variation (Leang et al., 30 Jul 2025).

The differing conclusions about gender independence deserve careful interpretation. One paper concludes that SSCF-derived polar coordinates are not inherently more speaker- or gender-independent than MFCCs under the BRAF100 cross-gender setup (Leang et al., 2022), whereas another reports greater gender independence than baseline MFCCs on Vietnamese speech when using polar-ratio parameters and normalized SSCF$M$24 (Leang et al., 30 Jul 2025). The concrete experimental conditions differ in corpus, language, feature composition, and normalization. This suggests that gender independence is not an intrinsic property of SSCFs alone, but depends on how SSCFs are parameterized and normalized within the larger frontend.

Another recurring interpretive issue is the relationship between static SSCFs and dynamic SSCF-derived geometry. The frequency-centroid study supports SSCFs primarily as complementary static features, especially under additive noise, and explains their utility by their encoding of the shape and shift of spectral subbands (Berjon et al., 2022). The ASR studies instead emphasize geometric trajectories in SSCF planes, with polar coordinates intended to characterize direction and magnitude of movement while avoiding angle discontinuities (Leang et al., 2022, Leang et al., 30 Jul 2025). A plausible implication is that SSCFs support two distinct methodological families: centroid augmentation of conventional spectral frontends, and low-dimensional dynamical modeling of articulatory-acoustic transitions.

Taken together, the literature characterizes SSCFs as a compact spectral representation with multiple operational forms: first-moment descriptors of Mel or fixed subbands, geometric coordinates in SSCF planes, and normalized low-frequency proxies for $M$25. Their practical value lies less in displacing MFCCs universally than in exposing spectral-center information, transition direction, and trajectory magnitude that MFCC-only pipelines do not directly encode (Berjon et al., 2022, Leang et al., 2022, Leang et al., 30 Jul 2025).