Filter Methods

The Mathematics of Market Signal Processing

📐 Technical Reference

Like master craftsmen who understand not just how to use their tools but why they work, this guide unveils the mathematical foundations underlying our curated collection of digital signal processing filters. Here, we bridge the gap between academic DSP theory and practical trading applications.

Note

This reference assumes familiarity with basic calculus and signal processing concepts. For practical implementation, see our FiltersToolkit and individual indicator documentation.

Foundation Concepts

The Nature of Financial Time Series

Financial markets generate time series data with unique characteristics that challenge traditional DSP approaches:

Non-Stationarity

Unlike audio signals, market data exhibits regime changes where statistical properties shift dramatically. Filters must adapt or maintain robustness across these transitions.

Heavy Tails

Price movements follow distributions with fatter tails than Gaussian, requiring filters that handle extreme events gracefully without distortion.

Multi-Scale Dynamics

Markets operate on multiple timescales simultaneously — from microsecond arbitrage to decade-long cycles. Effective filters must isolate relevant scales.

Frequency Domain Perspective

Fourier Transform

The Discrete Fourier Transform (DFT) decomposes price series into frequency components:

X(k) = Σ(n=0 to N-1) x(n) × e^(-j2πkn/N)

Where:

• x(n) = price at time n
• X(k) = frequency component k
• N = number of samples

This reveals the cyclical patterns hidden in price data.

Transfer Functions

Filters are characterised by their transfer function H(z):

H(z) = Y(z)/X(z) = (b₀ + b₁z⁻¹ + ... + bₘz⁻ᵐ)/(1 + a₁z⁻¹ + ... + aₙz⁻ⁿ)

The poles and zeros of H(z) determine:

• Frequency response
• Phase characteristics
• Stability conditions

Critical Frequencies

Key frequency relationships in trading:

• Nyquist Frequency: fₙ = 0.5 (highest representable)
• Normalised Frequency: f = 1/period
• Angular Frequency: ω = 2πf
• Critical Period: Where filter response drops to -3dB

Low-Pass Filters (Smoothers)

Low-pass filters attenuate high-frequency noise while preserving trend information. In trading, these create smoothed price representations for trend following.

The general form of an Infinite Impulse Response (IIR) low-pass filter:

y[n] = Σ(i=0 to M) bᵢ×x[n-i] - Σ(j=1 to N) aⱼ×y[n-j]

Where:

• bᵢ = feedforward coefficients
• aⱼ = feedback coefficients
• M, N = filter orders

Implementation Gallery

SuperSmoother

2-Pole Butterworth Design

Critical frequency: ωc = √2 × π / period
Pole location: a₁ = e^(-ωc)

Transfer function:
H(z) = K × (1 + 2z⁻¹ + z⁻²)/(1 - 2a₁cos(ωc)z⁻¹ + a₁²z⁻²)

Where K ensures unity gain at DC

Characteristics:

• Maximally flat passband
• -12dB/octave rolloff
• Minimal phase distortion

Ultimate Smoother

Zero-Lag Through Subtraction

Concept: LPF = APF - HPF

2-Pole Implementation:
ω = √2 × π / period
a₁ = e^(-ω)

HPF coefficients:
c₀ = (1 + 2a₁cos(ω) - a₁²) / 4
c₁ = 2a₁cos(ω)
c₂ = -a₁²

Final output subtracts HPF from input

Unique Properties:

• Zero phase delay in passband
• Frequency cancellation principle
• Counterintuitive parameter behaviour

BiQuad LPF

Parametric Second-Order Section

Normalised frequency: fc = 0.5 / period
Angular frequency: ω = 2π × fc

Q-factor influence:
α = sin(ω)/(2Q)

Coefficients:
b₀ = b₂ = (1 - cos(ω))/2 × 1/(1+α)
b₁ = (1 - cos(ω)) × 1/(1+α)
a₁ = -2cos(ω) × 1/(1+α)
a₂ = (1 - α) × 1/(1+α)

Q-Factor Effects:

• Q < 0.5: Overdamped (very smooth)
• Q = 0.707: Critically damped
• Q > 1: Resonant peak at cutoff

Adaptive Filters

MESA Adaptive Moving Average (MAMA)

MAMA adjusts its smoothing based on the dominant market cycle:

1. Hilbert Transform extracts phase:
   HT(x) = 0.0962×x + 0.5769×x[2] - 0.5769×x[4] - 0.0962×x[6]

2. Homodyne Discriminator finds period:
   Phase difference → Instantaneous period

3. Adaptive alpha:
   α = FastLimit / DeltaPhase (constrained by SlowLimit)

4. Dual-speed smoothing:
   MAMA = α×price + (1-α)×MAMA[1]
   FAMA = α/2×MAMA + (1-α/2)×FAMA[1]

The result adapts to market conditions, tightening during trends and loosening during cycles.

High-Pass Filters (Detrenders)

High-pass filters remove trend components to isolate short-term fluctuations, creating oscillators for mean reversion strategies.

Butterworth High-Pass

1-Pole Design

Transfer Function:
H(z) = K × (1 - z⁻¹)/(1 - αz⁻¹)

Where:
ω = π / period
α = e^(-ω)
K = (1 - α)/2

Implementation:
y[n] = K×(x[n] - x[n-1]) + α×y[n-1]

Properties:

• 6dB/octave rolloff
• 45° phase lead at cutoff
• Minimal lag

2-Pole Design

Enhanced attenuation:
ω = √2 × π / period
a₁ = e^(-ω)

Coefficients:
c₀ = (1 + 2a₁cos(ω) - a₁²)/4
c₁ = 2a₁cos(ω)
c₂ = -a₁²

Output:
y[n] = c₀×(x[n] - 2×x[n-1] + x[n-2]) + c₁×y[n-1] + c₂×y[n-2]

Advantages:

• 12dB/octave rolloff
• Zero mean output
• Better trend removal

Band-Pass Filters (Cycle Isolators)

Band-pass filters extract specific frequency ranges, ideal for identifying market cycles.

Mathematical Foundation

A band-pass filter combines high-pass and low-pass characteristics:

Centre frequency: f₀ = 1/period
Bandwidth: BW (in frequency units)
Quality factor: Q = f₀/BW

Transfer function peak at f₀
Attenuation outside passband

Implementation Variants

BiQuad BPF

Centre frequency: fc = 0.5/period
ω = 2π × fc

Bandwidth to Q conversion:
Q = 1/(2×sinh(ln(2)/2 × BW × ω/sin(ω)))

Filter coefficients:
α = sin(ω)×sinh(ln(2)/2 × Q × ω/sin(ω))

b₀ = sin(ω)/2 × 1/(1+α)
b₁ = 0
b₂ = -b₀
a₁ = -2cos(ω) × 1/(1+α)
a₂ = (1-α) × 1/(1+α)

Ehlers BPF

Combined approach with AGC:

1. Highpass preprocessing
2. Bandpass filtering:
   BP = 0.5×(1-α)×(HP - HP[2]) + β×(1+α)×BP[1] - α×BP[2]

3. Automatic Gain Control:
   Peak = 0.991×Peak[1]
   if |BP| > Peak: Peak = |BP|
   Normalised = BP/Peak

Results in consistent amplitude output.

Roofing Filter

Cascaded design for ultra-clean extraction:

Stage 1: 2-pole Butterworth HPF (remove trend)
Stage 2: 2-pole SuperSmoother (remove noise)

Result: Pristine cycle component
Phase distortion: Minimal
Lag: Balanced across frequency range

Predictive Filters

Voss Predictor

Mathematical Basis The Voss filter achieves negative group delay through anticipatory coupling:

Filter order: N = 3 × prediction_horizon

Output computation:
y[n] = 0.5×(3+N)×x[n] - Σ(i=0 to N-1)((i+1)/N × y[n-N+i])

Key insight: Weighted feedback creates phase lead
Limitation: Input must be band-limited
Practical range: 1-3 bars prediction

Critical Understanding: Prediction is only valid for band-limited signals. Raw price cannot be meaningfully predicted; only filtered components.

Filter Quality Metrics

Objective Measures

Group Delay

Group Delay Formula:

τg(ω) = -d(∠H(e^jω))/dω

Measures filter delay at each frequency. Zero-lag filters achieve τg ≈ 0 in passband.

Smoothness Ratio

Variance Reduction:

SR = σ²(input)/σ²(output)

Higher ratios indicate better noise suppression.

Frequency Response

Magnitude Response:

|H(e^jω)| across frequency spectrum

Ideal: Unity in passband, zero in stopband, sharp transition.

Implementation Optimisations

Computational Efficiency

Coefficient Caching

# Recalculate only on parameter change
if period != period[1]:
    omega = calculate_omega(period)
    update_coefficients(omega)

# Use cached coefficients for filtering
output = apply_filter(input, cached_coefficients)

State Management

# Efficient circular buffer for filter states
class FilterState:
    def __init__(self, order):
        self.buffer = [0] * order
        self.index = 0

    def update(self, value):
        self.buffer[self.index] = value
        self.index = (self.index + 1) % len(self.buffer)

Numerical Stability

# Prevent coefficient overflow
def stabilise_poles(poles):
    return [p * 0.999 if abs(p) >= 1 else p for p in poles]

# Handle startup conditions
def initialise_filter(source):
    return source if bar_index < warmup_period else filtered_value

Advanced Applications

Cascade architectures, or serial processing chains, allow complex signal processing by chaining multiple filters together:

1. Noise → Trend → Cycle:

   • Stage 1: SuperSmoother (remove noise)
   • Stage 2: High-pass (remove trend)
   • Result: Clean cycle component

2. Adaptive Cascade:

   • Stage 1: MAMA (adaptive smoothing)
   • Stage 2: Ultimate Smoother (zero-lag refinement)
   • Result: Responsive yet stable signal

3. Predictive Chain:

   • Stage 1: Band-pass (isolate cycle)
   • Stage 2: Voss Predictive (forecast cycle)
   • Result: Anticipatory signals

Multi-Resolution Analysis

Wavelet Decomposition

While not implemented in FiltersToolkit, wavelet concepts inspire multi-scale approaches:

Price = Trend + Cycles + Noise

Trend: Ultimate Smoother (long period)
Cycles: Multiple band-pass filters
Noise: Residual after filtering

Empirical Mode Decomposition

Adaptive decomposition philosophy:

1. Identify local extrema
2. Fit envelope
3. Extract mean as intrinsic mode
4. Repeat on residual

Similar results achievable with adaptive filter banks.

Practical Considerations

Market-Specific Adaptations

Cryptocurrency

• 24/7 trading requires robust startup handling
• High volatility benefits from adaptive approaches
• Consider shorter periods due to rapid evolution

Forex

• Lower noise allows aggressive filtering
• Session-based adjustments recommended
• Longer periods capture macro trends

Equities

• Opening gaps require special handling
• Intraday vs daily filtering differs significantly
• Volume-dependent variants often superior

Open Source Commitment

This mathematical reference is part of our commitment to sharing institutional-grade DSP knowledge with the trading community. By documenting these foundations, we enable traders to:

• Understand the mathematical principles behind effective filtering
• Implement custom variations for specific needs
• Contribute improvements to the community
• Build upon proven methodologies

Related Open Source Resources:

• FiltersToolkit Library - Implementation source code
• Filters Toolkit Indicator - Interactive experimentation
• Ultimate Smoother - Deep dive into zero-lag filtering

Next Steps

• Practical Implementation: Explore our FiltersToolkit for hands-on experimentation
• Specific Filters: Deep dive into the Ultimate Smoother or other implementations
• Custom Development: Use these mathematical foundations to create your own filters
• Community Discussion: Share insights in our Discord channel

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“Mathematics is the language in which the universe speaks. In financial markets, filters are our translators—converting the cacophony of price action into comprehensible signals. Master the mathematics, master the message.” 📊