Sanjoy Nath's Qhenomenology presents a radical departure from conventional signal processing, proposing
a novel, time-domain, and topology-centric framework for wave analysis. This abstract outlines its core
tenets, suggests mathematical formalizations, and compares it to related fields like Topological Data
Analysis, signal grammars, and symbolic dynamics.
Research Abstract: Qhenomenology: A Topological-Combinatorial Framework
for Waveform Analysis
Abstract: Conventional digital signal processing, heavily reliant on Fourier analysis, often focuses on
spectral content, potentially obscuring time-domain morphological features critical for certain information
extraction tasks. This paper introduces Qhenomenology, a novel axiomatic framework for waveform
analysis that fundamentally redefines signal interpretation. Diverging from sinusoidal decomposition,
Qhenomenology treats waveforms as ordered sequences of topologically classified Axis-Aligned Bounding
Box (AABB) objects: "Crest AABBs," "Trough AABBs," and "Silence AABBs."
The core innovation lies in: (1) a precise methodology for normalizing signals via a median-centered
baseline, yielding "crisp zero-crossing points"; (2) the extraction and intrinsic topological numbering of
individual AABBs based on scale-invariant properties (e.g., local extrema counts, monotonic segment
lengths, and percentile ranks of boundary amplitudes), encapsulated in a "Sensitiveness Number"; (3) a
novel classification and topological numbering system for zero-crossing "junctions" (e.g., Crest-to-Trough,
Trough-to-Silence), incorporating neighborhood rank information; and (4) the identification and topological
categorization of "Container AABBs" representing complete wave cycles or meaningful segments, derived
from specific combinatorial patterns of constituent AABBs and junctions. Qhenomenology asserts that
information critical for perception (e.g., musical tonality, percussions) resides in these topological-geometric
arrangements rather than exact amplitude values or harmonic superpositions. This framework transforms
time-series analysis into a "stringology" or "grammar parsing" problem, enabling pattern matching and
algebraic reasoning on sequences of topologically numbered symbols. This approach promises to uncover
distinct waveform features essential for applications where conventional methods fall short, without
recourse to Fourier transforms or statistical machine learning for core classification.
Mathematical Formalizations for Peer-Reviewed Publication
To formalize Sanjoy Nath's Qhenomenology for peer-reviewed publication, a precise mathematical
language is essential. Here are proposed formalizations:
1. Signal Representation and Baseline Normalization
Let be a discrete-time signal of length , where is the amplitude at sample index .
Axiom (Median Baseline): The fundamental zero-amplitude reference line is established by the global
median of the signal.
Definition 1.1 (Median-Centered Signal):
Let . The normalized signal is defined as:
2. AABB Object Definition and Properties
Definition 2.1 (Amplitude Type Function): Given a silence threshold :
S = {s[n]}n=0
N−1 N s[n] n
s med = median(S) S′ = {s [n]}
′
n=0
N−1
s′[n] = s[n] − s med
δ > 0
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Definition 2.2 (AABB Object): An AABB object is a contiguous segment of the normalized signal
defined by indices such that:
1. All samples for have the same (ignoring boundary transitions
for definition of the AABB itself).
2. It is maximally extended, meaning (if exists) and (if exists) have a different
than samples within .
3. Each is assigned its .
Definition 2.3 (AABB Properties for ):
Let be an AABB with samples.
Width: (in samples). Can be converted to microseconds:
.
Area Under Curve: .
Monotonicity Counts:
where is the indicator function (1 if condition is true, 0 otherwise).
Local Extrema Counts: , (number of local minima/maxima within ).
Absolute Amplitude Percentile Ranks:
Let be the set of absolute amplitudes within . Let be this set
sorted in ascending order.
.
.
(The percentile_rank(value, sorted_list) function returns the proportion of values in
sorted_list less than or equal to value ).
3. Topological Sensitiveness Number for AABBs
Definition 3.1 (AABB Topological Sensitiveness Number, ):
This metric quantifies the topological "shape" of an AABB, designed to be largely scale-invariant and to
"forget" exact amplitude details.
AmpType(x) =
⎩ ⎨
⎧Crest
Trough
Silence
if x > δ
if x < −δ
if − δ ≤ x ≤ δ
Ak S′
[n , n ] k
start
k e
nd
s′[n] n ≤ k
start n ≤ n k
end AmpType(s′[n])
s [n ′ −
k s
tart 1] s [n ′ +
k e
nd 1]
AmpType A k
A k Type(A k) ∈ {Crest, Trough, Silence}
Ak
A k = {s [n]}
′
n=n k
start
n k
end W k = n − k
end n + k
start 1
W k = n − k
end n + k
start 1 W k =
μs
W k/SampleRate × 106
Area(A k) = Σ s [n] n=n k
start
n k
end ′
M (A I k) = 1 Σn=n +1 k
start
nk end
s′[n]>s′[n−1]
M (A D k) = 1 Σn=n +1 k
start
n k
end
s′[n]<s′[n−1]
1 condition
L (A min k) L (A max k) A k
S (A abs k) = {∣s [n]∣}
′
n=n k
start
n k
end
A k S (A abs )
sorted k
PR (A L k) = percentile_rank(∣s [n ]∣, S (A ′ ))
k s
tart
abs
sorted
k
PR (A R k) = percentile_rank(∣s [n ]∣, S (A ′ ))
k e
nd
abs
sorted
k
T (A S k)
T (A S k) = ⌊ ⋅
Σ ∣s [j]∣ + ϵ j=n k
start
n k
end ′
Area(A k) 105⌋ (Normalized Area Term)
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where is a small positive constant to prevent division by zero, and floor operations discretize the
contributions. (Note: The TotalArea_ofThisAABB in the original text is interpreted as sum of absolute
samples for normalization, making the area term a ratio reflecting general amplitude distribution, insensitive
to scale).
Definition 3.2 (AABB Global Rank): AABB objects are ranked globally based on their values.
.
Definition 3.3 (AABB Scale Factors): For each topological class (AABBs with the same ), identify
the widest AABB, .
4. Zero-Crossing Junction Classification
Definition 4.1 (Zero-Crossing Junction): A zero-crossing junction occurs at sample index where
(or equivalent, between AABB and ).
Definition 4.2 (Junction Type, ): Categorized based on the types of adjacent AABBs:
Definition 4.3 (Junction Topological Number, ): A unique, scale-invariant identifier for a junction
based on its type and the topological properties of its neighboring AABBs.
This hash captures the local topological context of the crossing.
5. Container AABB (Cycle) Definition and Classification
Definition 5.1 (Container AABB, ): A Container AABB is a sequence of contiguous AABB objects
that represents a "complete cycle" or a perceptually meaningful segment of the
+⌊ ⋅
W k
MI (Ak ) 104⌋ (Normalized Monotonic Increase Term)
+⌊ ⋅
W k
M (A D k) 103⌋ (Normalized Monotonic Decrease Term)
+PR (A L k) ⋅ 102 (Leftmost Sample Percentile Rank Term)
+PR (A R k) ⋅ 101 (Rightmost Sample Percentile Rank Term)
+ (Normalized Width Term)
103
W k
ϵ ⌊⋅⌋
T (A S k)
Rank(A k) = position of A when all A are sorted by T (A k j S j )
Rank(A k)
A widest
SF (A X k) = W /W k widest
SF (A Y k) = (MaxAmp(A k) − MinAmp(A ))/(MaxAmp(A k widest) − MinAmp(A widest))
J m m
AmpType(s′[m − 1]) =AmpType(s′[m]) A k A k+1
T ype(J m)
T ype(J m) ∈ {CT, TC, TT, CC, SS, ST, TS, SC, CS, Undefined}
T (J J m)
T (J J m) = Hash(Type(J ), PR (A ), PR (A ), Type(A ), Type(A m R k L k+1 k k+1))
C p C p
{A , A , … , A i i+1 j}
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waveform. In this framework, such a container is typically delimited by specific topological junction types
(e.g., a cycle starting with a TC junction and ending just before the next TC junction of a specific type).
Definition 5.2 (Container AABB Topological Sensitiveness Number, ): A composite metric reflecting
the topological "essence" of the entire container, derived from its aggregated internal properties.
This function needs to be explicitly defined, but it would combine the scale-invariant properties of its
contained AABBs and junctions.
Definition 5.3 (Container AABB Global Rank): Containers are ranked globally based on their values.
Definition 5.4 (Container AABB Scale Factors): Similar to AABBs, but for containers. Identify the widest
container within the same topological class (same ).
(where
MaxAmp/MinAmp of container means global max/min of samples within it).
6. Waveform as a Formal Language
Proposition 6.1 (Symbolic Waveform Representation): The timeline of a wave can be represented as a
formal string of symbols , where is the alphabet of topologically numbered AABB objects and zerocrossing
junctions.
Example: .
A waveform might be represented as:
Proposition 6.2 (Grammar and Pattern Matching): Specific wave "patterns" (e.g., musical phrases, distinct
vibrations) can be recognized and categorized by defining a formal grammar over . Regular
expressions or more complex parsing algorithms can be applied to identify occurrences of these patterns,
allowing for "compilability checking" of wave structures.
Comparison to Specific TDA, Signal Grammar, or Symbolic Dynamics Models
Sanjoy Nath's Qhenomenology is unique due to its foundational axioms and explicit rejection of Fourier.
However, it shares conceptual commonalities with several advanced analytical paradigms:
1. Comparison to Topological Data Analysis (TDA)
Similarities:
Focus on Shape/Structure: Both TDA and Qhenomenology prioritize the underlying shape and
structure of data over precise numerical values. Both seek "invariant properties."
Abstraction and Feature Extraction: Both aim to extract meaningful, low-dimensional features that
describe complex high-dimensional data.
Differences:
Fundamental Primitives: TDA typically constructs abstract topological spaces (e.g., simplicial
complexes, Vietoris-Rips complexes) from point clouds or metric spaces. Features are then
derived using algebraic topology (e.g., Betti numbers from persistent homology, which count
"holes" of different dimensions). Qhenomenology, conversely, uses concrete, geometric AABBs
directly on the time series as its fundamental building blocks. Its "topology" is defined by specific
arithmetic combinations of derived AABB properties rather than abstract homological invariants.
Metric Definition: TDA relies on metric spaces to define proximity and build complexes.
Qhenomenology's "Sensitiveness Number" is a custom, composite metric for AABBs themselves,
TS (Cp )
T (C S p) = AggregatedFunction(Count(A k ∈ C ), Count(J p m ∈ C ), Sum(L (A )), Sum(L (A p min k max k)),…)
T (C S p)
Cwidest Rank(C p)
SF (C X p) = Width(C )/Width(C p widest)
SF (C Y p) = (MaxAmp(C p) − MinAmp(C ))/(MaxAmp(C p widest) − MinAmp(C widest))
Σ∗ Σ
Σ = {(A , T (A)), (J , T T ype S T ype J (J))}
(A , T (A ))(J , T (J ))(A , T (A ))(J , T (J Crest S 1 CT J 1 T rough S 2 TC J 2))…
G Σ∗
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not directly for a point cloud representation of the wave.
Zero-Crossing Emphasis: While TDA can reveal features related to "loops" or "cycles" in data, it
does not have an inherent mechanism for "zero-crossing point classification" with specific
semantic labels like CT/TC based on signal sign changes as a primary topological feature.
2. Comparison to Signal Grammar and Formal Languages
Similarities:
Symbolic Representation: Qhenomenology's transformation of the continuous waveform into a
"strict queue of symbols" (AABBs and classified junctions) aligns perfectly with the core idea of
symbolic representation in signal processing and formal languages.
Pattern Recognition: The use of "regular expressions" and "grammar parsing" to identify patterns
in these symbolic sequences is a direct application of formal language theory.
Compositionality: Both allow complex signals/messages to be understood as compositions of
simpler, meaningful units according to defined rules.
Differences:
Symbol Granularity and Richness: In typical signal grammars (e.g., for speech, music), symbols
are often derived from simpler features like amplitude thresholds, frequency bands, or short-time
spectral characteristics. Qhenomenology's "symbols" (AABBs and Junctions) are inherently richer,
carrying internal topological "sensitiveness numbers" and other complex morphological properties
that are scale-invariant. This provides a much deeper semantic content for each symbol than a
simple categorical label.
Axiomatic Basis: The "compilability as existence" axiom and the derivation of "dictionary order"
from class dependencies are unique meta-level philosophical constructs not found in standard
signal grammar frameworks.
3. Comparison to Symbolic Dynamics Models
Similarities:
Discretization and Sequence Analysis: Both symbolic dynamics and Qhenomenology convert
continuous-time dynamics (or signals) into discrete sequences of symbols for analysis.
Focus on Patterns and Complexity: Both fields aim to understand the underlying patterns,
structure, and complexity within these symbolic sequences.
Differences:
State Space Partitioning: In symbolic dynamics, the continuous state space of a dynamical system
is partitioned into a finite number of regions, and each time the system enters a region, a
corresponding symbol is emitted. Qhenomenology's "partitioning" is explicitly based on the
segments of the waveform determined by zero-crossings and silence, creating the AABB objects
directly from the time domain signal.
Symbol Definition: While symbolic dynamics symbols are typically simple labels (e.g., '0' or '1'),
Qhenomenology's AABB symbols encapsulate rich, multi-dimensional topological information (the
sensitiveness number and other derived properties).
Reconstruction and Meaning: Symbolic dynamics focuses on properties of the symbolic sequence
that reflect the original dynamics. Qhenomenology explicitly claims that altering amplitude details
while preserving CG geometry can generate "same spectral behaviors," suggesting a unique
perspective on information and reconstruction.
In conclusion, while Qhenomenology shares a family resemblance with time-domain morphological
analysis, symbolic representation, and topological thinking in data analysis, its specific axiomatic
foundation, the definition of AABB primitives with their unique topological metrics, and the ambitious claim
of replacing Fourier analysis make it a distinct and novel theoretical framework. Further research would be
needed to rigorously validate its claims and explore its full potential across various signal domains.
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