flaw

Embedded signal filtering, no-std and no-alloc compatible.

This library provides a simple method for initializing and updating single-input, single-output infinite-impulse-response filters using 32-bit floats, as well as tabulated filter coefficients for some common filters. Filters evaluate in 4N-1 floating-point operations for a filter of order N.

The name flaw is short for filter-law, but also refers to the fact that digital IIR filtering with small floating-point types is an inherently flawed approach, in that higher-order and lower-cutoff filters produce very small coefficients that result in floating-point roundoff error. This library mitigates that problem by providing filter coefficients for a tested domain of validity. The result is a limited, but useful, range of operation where these filters can achieve both accuracy and performance as well as be formulated and initialized in an embedded environment.

Capabilities

• IIR (f32-only for now)
  • General IIR filter using state-space canonical form
  • Interpolated low-pass filters w/ gain error correction
  • Baked coefficients for Butterworth filters of order 1-6
• SOS representation of IRR filters (generic number type)
  • General IRR filter using cascaded second order sections
  • Interpolated low-pass filters w/ gain error correction
  • Baked coefficients for Butterworth filters of order 2, 4 and 6 for f32 and f64
• FIR (generic number type)
  • General FIR filter
  • Lagrange polynomial fractional-delay filter construction

Example: Second-Order Butterworth Filter

// First, choose a cutoff frequency as a fraction of sampling frequency
let cutoff_ratio = 1e-3;

// Construct a filter, interpolating coefficients to that cutoff ratio.
// Initializes internal state to zero by default.
let mut filter = flaw::butter2(cutoff_ratio).unwrap();  // Errors if extrapolating

// Initialize the internal state of the filter
// to match the steady-state associated with some input value.
let initial_steady_measurement = 1.57;  // Some number
filter.set_steady_state(initial_steady_measurement);

// Update the filter with a new raw measurement
let measurement = 0.3145; // Some number
let estimate = filter.update(measurement);  // Latest state estimate

Coefficient Tables

Tabulated filters are tested to enforce

• <0.01% error in converged step response at the minimum cutoff frequency
• <1ppm error in converged step response at the maximum cutoff frequency
• <5% error to -3dB attenuation of a sine input at the cutoff frequency at the maximum cutoff ratio
  • This error appears to be mainly an issue of discretization in test cases, and could be reduced by using a better method for testing (fit a sine curve to the result or do gradient-descent on a cubic interpolator)

Each filter with tabulated coefficients has a minimum and maximum cutoff ratio. The minimum value is determined by floating-point error in convergence of a step response, while the maximum value is determined by the accuracy of attenuation at the cutoff frequency as the cutoff ratio approaches the Nyquist frequency.

Coefficients for a given filter are interpolated on these tables using a cubic Hermite method with the log10(cutoff_ratio) as the independent variable. Tabulated values are stored and interpolated as 64-bit floats, and only converted to 32-bit floats at the final stage of calculation.

After interpolation, the state-space measurement coefficient vector (C) is scaled to correct steady-state gain for interpolation error, targeting unity gain.

Filter coefficients are extracted from scipy's state-space representations, which are the result of a bilinear transform of the transfer function polynomials.

License

Licensed under either of

• Apache License, Version 2.0, (LICENSE-APACHE or http://www.apache.org/licenses/LICENSE-2.0)
• MIT license (LICENSE-MIT or http://opensource.org/licenses/MIT)

at your option.