cmst-window

The analytically sound, zero-preserving, interlace-preserving compact window.

🚀 The Problem

Signal processing engineers often need a "Flat-Top" window to preserve signal amplitude while filtering noise. Existing standard solutions force a dangerous compromise:

• Truncated Super-Gaussians: Create geometric singularities (infinite acceleration) at the boundaries, causing ringing in control loops.
• Planck-taper (LIGO Standard): Theoretically perfect flatness, but relies on piecewise "stitching" of functions. This creates impulsive "kicks" (discontinuities) in higher-order derivatives (Jerk/Snap) and makes hardware optimization difficult.

🔬 Theoretical Basis: CMST Theory

This window is an implementation of CMST (Cosh Moment Sturm Transform). Unlike standard windows which are often heuristic curve-fits, the CMST window is constructed as a Geometric Mollifier with three rigorous guarantees:

1. Analytically Sound ($C^\infty$): The function belongs to the Gevrey class of regularity ($s=2$), ensuring super-algebraic decay in the frequency domain. It is infinitely differentiable ($C^\infty$) with no discontinuities in any derivative. This eliminates the "spectral ringing" and mechanical jerk caused by piecewise functions like the Planck-taper or Tukey window.

2. Zero-Preserving : Derived from CMST theory, the kernel guarantees the preservation of realness in the signal chain. It does not introduce artificial complex roots (phantom oscillations) into the passband.

3. Interlace-Preserving Transform: For all derivatives, the window acts as a variation-diminishing operator. It preserves the root-interlacing structure of the underlying signal, ensuring that derivative noise is bounded and geometric topology is maintained even at the boundaries.

💡 The Formula (CMST)

We utilize a compensated log-concave mollifier that cancels low-order curvature to achieve Flatness:

$$ w(t) = \exp\left(1+t^p - \frac{1}{1-t^p}\right), $$

where p is even.

• Compensating Term $t^p$: Cancels the Gaussian curvature at the origin, extending the "Table-Top" flatness to order $2n$.
• Mollifier Term $(-1/(1-t^p))$ : Enforces strict compact support with essential singularities at the boundaries, ensuring all derivatives decay to zero smoothly.

🎛️ Tunable Flatness (p-Control)

Unlike traditional windows which are locked to a single profile (e.g., Hann, Blackman), the CMST window is a parametric family. The power parameter (p) allows you to tune the window's behavior to match your specific engineering constraint:

Mode A: The "Brick Wall" (p=6 or higher)
    Goal: Maximal Amplitude Accuracy.

    Behavior: The window remains effectively flat (>0.99) for over 70% of the duration, ensuring that signals are not attenuated in the center.

Mode B: The "Silencer" (p=2)
    Goal: Maximal Spectral Purity.

    Behavior: The window converges to an analytically smooth Gaussian-like profile. This sacrifices the "flat top" to achieve significantly faster side-lobe decay, often over 100dB improvement, diving into the noise floor deeper than standard piecewise functions like the Planck-taper.
    Generally I have found p=2 gives the best results.

📉 Tunable Flatness

📉 Spectral Leakage Comparison

Note on Precision Limits: The CMST response (Blue) was calculated using 100 digit precision arithmetic to demonstrate the asymptotic behavior beyond standard 64-bit machine limits (~ -320 dB). Polynomial Windows (e.g., Blackman-Harris, Planck): Their leakage floors are theoretical. Even with infinite precision, they would not drop much further and the graph wouldnt change much

📉 Performance Analysis:

Detection of a -100 dB weak signal (1.3 kHz) next to a strong carrier (1.0 kHz).

The standard Planck-taper (Red) buries the target in spectral leakage. The CMST window (Blue) resolves it clearly with >30 dB of headroom

Transform of a Sinc function

Lets remember that when I say these should be a box, we are using a log scale, visually they are all boxes.

Total Integrated Leakage (0.05 - 0.07 Hz):

Planck Taper: -20.86 dB

7-term BH: -16.21 dB

CMST (p=2): -28.44 dB

Improvement: 7.58 dB

Planet sim.

With the planet dB at -80 dB and a noise floor of -90 dB. Slightly contrived, but the windows I am comparing against are good! Note that not only has Planck lost 4 planets, but it has also created non existant mountains (side lobes). We are looking at this in crazy detail, if we were back in the real world, the -80dB planets would not show up on your screen, your eyes couldnt see them. Those "mountains" are invisible as well. 80dB is the ratio between the Empire State Building and a golf ball.

📉 Resolution Law:

As part of this work we produce a resolution law for CMST(2) for the resolution of two signals in terms of bins, namely

$$m = \left\lceil \frac{(\ln R)^2}{\pi} \right\rceil$$

Where:

• $R$ is the linear ratio of amplitudes (e.g., for -100 dB, $R = 10^5$).
• $m$ is the distance required between the signals in bins.

Theortical law vs practical. (There is actually a small log term that gives the law a bit of a boost, by which I mean m bins are sufficient)

📉 GW150914 (The first Gravity wave detected)

To benchmark performance on real-world signals, the CMST window was applied to the raw 4kHz data from GW150914, the first direct detection of gravitational waves.

📉 Kepler 10

Running the window on the Kepler 10 data, a detection of a planet orbitting a sun, we get a frequency plot with clear harmonics. When we run a regression on these harmonics we get an R^2 of ~0.99997 and an estimate of the rotation period of 0.8386 days, which is within 0.1% of the NASA number, or roughly out by only 114 seconds.

📉 Data Preparation

Do not pre-filter your data.

The CMST algorithm operates well on raw time-series data. Applying standard signal processing filters before the transform will degrade performance since they have algerbraic side lobes which CMST wont be able to remove.

You can:

Fill Gaps: Use linear interpolation to ensure a rigid time grid. Center the Data: Subtract the mean to remove the DC offset. and then apply CMST.

Note: The CMST window provides sufficient dynamic range to isolate other signals from the signal of interest in the frequency domain. Pre-filtering is redundant and destructive.

🎛️ Performance Trade-offs: SNR vs. Spectral Decay

The CMST window family allows you to tune the shape parameter p to balance Coherent Gain (SNR) against the speed of Spectral Decay.

Where spectral decay is $$|\hat{\Psi}(\omega)| \propto e^{a \sqrt{\omega}}$$

Window Variant	Parameter (p)	SNR Loss (vs Rect)	Spectral Decay (a)
Rectangle	N/A	0.00 dB	N/A
Std. Bump	N/A	-1.30 dB	-1
CMST(2)	p=2	-0.96 dB	-1
CMST(4)	p=4	-0.50 dB	$$-\frac{1}{\sqrt{2}}$$
CMST(6)	p=6	-0.34 dB	$$-\frac{1}{\sqrt{3}}$$

Key Metrics:

• SNR Loss: Signal loss relative to a perfect Rectangular window. Higher is better (closer to 0 dB).
• Spectral Decay: The slope constant 'a' describing how fast the side-lobes vanish. More negative is better.

The numbers here show there is a balance between SNR and Side lobe decay. CMST(2) is a general workhorse but CMST(4) for instance got me down to within 2 seconds of NASA's orbit time for Kepler 10 where SNR is probably more important than spectral resolution.

The Std. Bump (often denoted as $\Psi$ in distribution theory) is the canonical example of a smooth, compactly supported function ($C_c^\infty$). It serves as the baseline ancestor for the CMST window family.

$$ \Psi(t) = \exp\left(\frac{1}{t^2 - 1} \right) $$

It has the best spectral decay, but CMST(2) is almost the same and has a better SNR.

Traditional windows (like Hann, Hamming, or Blackman) exhibit algebraic spectral decay. Because they decay much slower (following a polynomial curve rather than an exponential one), they do not possess a comparable decay constant $a$.

📉 The Math:

Behind all of this there is a CMST theory paper here CMST

📦 Installation

git clone https://github.com/aronp/CMST.git

cd CMST

pip install .