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//! Second Order Sections (SOS) filters.
// Required for float conversions from 64 to 32 bit and log10 on no-std targets
#[cfg(not(feature = "std"))]
use num_traits::Float;
use core::ops::Neg;
use crunchy::unroll;
use num_traits::{FromPrimitive, MulAdd, Num, ToPrimitive};
pub mod tables;
pub use tables::butter2::butter2;
pub use tables::butter4::butter4;
pub use tables::butter6::butter6;
use crate::AlignedArray;
#[cfg(test)]
mod test_helpers;
/// Single-Input-Single-Output, cascaded Second Order Sections filter.
#[derive(Clone, Copy)]
pub struct SisoSosFilter<
const SECTIONS: usize,
T: Num + Copy + MulAdd<Output = T> + Neg<Output = T>,
> {
/// Latest output
y: T,
/// Internal state: two filter delays for each section
z: AlignedArray<[T; 2], SECTIONS>,
/// SOS coefficients ordered as [b0, b1, b2, a1, a2] per section.
/// These correspond to a filter transfer function of:
///
/// H(z) = (b0 + b1*z^-1 + b2*z^-2) / (1 + a1*z^-1 + a2*z^-2)
///
/// Note that some SOS implementations (e.g. SciPy) use six coefficients,
/// including a0, but here a0 is assumed to be unity, and is omitted to reduce memory usage.
sos: AlignedArray<[T; 5], SECTIONS>, // Using AlignedArray did not measurably change performance on an i7-8550U CPU, but might help on other platforms
}
impl<const SECTIONS: usize, T> SisoSosFilter<SECTIONS, T>
where
T: Num + Copy + MulAdd<Output = T> + Neg<Output = T> + FromPrimitive + ToPrimitive, // FromPrimitive needed for conversion from f64 in SOS coef lookup tables.
{
/// Evaluate the next estimated value based on the latest measurement
/// in 9N floating-point ops for a filter with N sections (order up to 2*N).
#[inline]
pub fn update(&mut self, u: T) -> T {
let mut input = u; // Input to each section
let mut output = T::zero(); // Output of each section
// `crunchy::unroll` requires a literal upper bound; use a conservative
// limit and let the assert catch unsupported configurations.
const {
assert!(
SECTIONS <= 8,
"SOS filters with SECTIONS > 8 are not supported"
);
}
// Unrolling this loop gives a ~5% speedup of the sos butter4 f64 benchmark
// on a i7-8550U CPU.
unroll! {
for s < 8 in 0..SECTIONS {
let b0 = self.sos[s][0];
let b1 = self.sos[s][1];
let b2 = self.sos[s][2];
let a1 = self.sos[s][3];
let a2 = self.sos[s][4];
#[cfg(not(feature = "fma"))]
{
// Direct Form II Transposed implementation
output = b0 * input + self.z[s][0];
// Update the filter delays, they will be used the next time this function is called
self.z[s][0] = b1 * input - a1 * output + self.z[s][1];
self.z[s][1] = b2 * input - a2 * output;
}
// The FMA implementation is ~40% faster, with target-cpu=x86-64-v3 on a i7-8550U CPU.
#[cfg(feature = "fma")]
{
// Direct Form II Transposed implementation
output = b0.mul_add(input, self.z[s][0]); // b0 * input + self.z[s][0]
// Update the filter delays, they will be used the next time this function is called
self.z[s][0] = b1.mul_add(input, a1.mul_add(-output, self.z[s][1])); // b1 * input - a1 * output + self.z[s][1]
self.z[s][1] = b2.mul_add(input, -a2 * output); // b2 * input - a2 * output
}
// Cascaded sections: output of this section is input to next
#[allow(unused_assignments)] // rustc warns because the assignment to input is unused on the last section
{
input = output;
}
}
}
// Overall output of the filter is the output of the last sections
self.y = output;
self.y
}
/// Reset internal state to zero.
pub fn reset(&mut self) {
self.y = T::zero();
self.z = AlignedArray([[T::zero(); 2]; SECTIONS]);
}
/// Set filter internal state to the steady value
/// achieved for input `u`. For filters with unity steady-state gain,
/// this will also produce an output reading of `u`.
pub fn set_steady_state(&mut self, u: T) -> Result<(), &'static str> {
let mut input = u;
let mut overall_ss_gain = 1.0;
for s in 0..SECTIONS {
let b0 = self.sos[s][0];
let b1 = self.sos[s][1];
let b2 = self.sos[s][2];
let a1 = self.sos[s][3];
let a2 = self.sos[s][4];
// Calculate the steady-state output of this section
let input_f64 = input.to_f64().ok_or("Conversion to f64 failed")?;
let section_f64: [f64; 5] = [
b0.to_f64().ok_or("Conversion to f64 failed")?,
b1.to_f64().ok_or("Conversion to f64 failed")?,
b2.to_f64().ok_or("Conversion to f64 failed")?,
a1.to_f64().ok_or("Conversion to f64 failed")?,
a2.to_f64().ok_or("Conversion to f64 failed")?,
];
let ss_gain = steady_state_gain_sos(§ion_f64);
overall_ss_gain *= ss_gain; // accumulate the gain of each section
let output: T = T::from_f64(input_f64 * ss_gain).ok_or("Conversion from f64 failed")?;
// Set the internal states based on the steady state input and output of this section
self.z[s][1] = b2 * input - a2 * output;
self.z[s][0] = b1 * input - a1 * output + self.z[s][1];
// Cascaded sections: output of this section is input to next
input = output;
}
if (overall_ss_gain - 1.0).abs() < 1e-6 {
// Try updating the filter and verify that the output matches the input
debug_assert!(
// use debug_assert so that release builds can be panic-free
(self.update(u) - u)
.to_f64()
.ok_or("Conversion to f64 failed")?
.abs()
< 1e-6,
);
}
Ok(())
}
pub fn new(sos: &[[T; 5]]) -> Self {
let mut sos_ = [[T::zero(); 5]; SECTIONS];
sos_.copy_from_slice(sos);
Self {
y: T::zero(),
z: AlignedArray([[T::zero(); 2]; SECTIONS]),
sos: AlignedArray(sos_),
}
}
/// Build a new low-pass with coefficients interpolated on baked tables.
pub fn new_interpolated(
cutoff_ratio: f64,
log10_cutoff_ratio_grid: &[f64],
sos_tables: [[&[f64]; 5]; SECTIONS],
) -> Result<Self, &'static str> {
let log10_cutoff_ratio = cutoff_ratio.log10();
// Check table bounds
let mut extrapolated = [false; 1];
interpn::multicubic::rectilinear::check_bounds(
&[log10_cutoff_ratio_grid],
&[&[log10_cutoff_ratio]],
1e-6,
&mut extrapolated,
)?;
if extrapolated[0] {
return Err("Selected cutoff ratio is outside the grid");
}
let mut sos = [[T::zero(); 5]; SECTIONS];
for sec in 0..SECTIONS {
for coeff in 0..5 {
let val_f64: f64 = interpn::MulticubicRectilinear::<'_, _, 1>::new(
&[log10_cutoff_ratio_grid],
sos_tables[sec][coeff],
true,
)?
.interp_one([log10_cutoff_ratio])?;
sos[sec][coeff] = T::from_f64(val_f64).ok_or("Conversion from f64 failed")?;
}
}
// Correct the DC gain of the filter to 1.
// First, calculate the DC gain we'd get with the coefficients as is after interpolation.
let mut dc_gain: f64 = 1.0;
for section in sos.iter() {
let b0 = section[0].to_f64().ok_or("Conversion to f64 failed")?;
let b1 = section[1].to_f64().ok_or("Conversion to f64 failed")?;
let b2 = section[2].to_f64().ok_or("Conversion to f64 failed")?;
let a1 = section[3].to_f64().ok_or("Conversion to f64 failed")?;
let a2 = section[4].to_f64().ok_or("Conversion to f64 failed")?;
let sec_dc_gain = (b0 + b1 + b2) / (1.0 + a1 + a2);
dc_gain *= sec_dc_gain;
}
// Now scale the numerator coefficients to get unity gain at DC.
// Apply the required scaling in even parts to each section.
let correction = T::from_f64(dc_gain.powf(-1.0 / SECTIONS as f64))
.ok_or("Conversion from f64 failed")?;
for section in sos.iter_mut() {
section[0] = section[0] * correction;
section[1] = section[1] * correction;
section[2] = section[2] * correction;
}
// Verify the DC gain after scaling.
dc_gain = 1.0;
for section in sos.iter() {
let b0 = section[0].to_f64().ok_or("Conversion to f64 failed")?;
let b1 = section[1].to_f64().ok_or("Conversion to f64 failed")?;
let b2 = section[2].to_f64().ok_or("Conversion to f64 failed")?;
let a1 = section[3].to_f64().ok_or("Conversion to f64 failed")?;
let a2 = section[4].to_f64().ok_or("Conversion to f64 failed")?;
let sec_dc_gain = (b0 + b1 + b2) / (1.0 + a1 + a2);
dc_gain *= sec_dc_gain;
}
if (dc_gain - 1.0).abs() > 1e-6 {
return Err("DC gain correction failed");
}
Ok(Self::new(&sos))
}
}
/// Calculate the steady-state gain of a single Second Order Section.
pub fn steady_state_gain_sos(section: &[f64; 5]) -> f64 {
let b0 = section[0];
let b1 = section[1];
let b2 = section[2];
let a1 = section[3];
let a2 = section[4];
// The formula for steady state gain is the z-transform evaluated at z = 1.
// Derivation: the steady state gain is the final value of the step response.
// Let H(z) be the z-transform of the second order section.
// The z-transform of the step response is:
// Y(z) = (1 / (1 - z^-1)) * H(z)
// The final value theorem for z-transforms states:
// lim_{n->inf} y[n] = lim_{z->1} (z - 1) * Y(z)
// Applying this to the step response gives:
// lim_{n->inf} y[n] = lim_{z->1} (z - 1) / (1 - z^-1) * H(z)
// The limit of the first term is 1, so we have:
// lim_{n->inf} y[n] = H(1)
// assuming that H(z) is stable so the limit exists.
// See https://en.wikipedia.org/wiki/Z-transform#Properties
(b0 + b1 + b2) / (1.0 + a1 + a2)
}
#[cfg(feature = "std")]
#[cfg(test)]
mod test {
use super::SisoSosFilter;
use super::test_helpers::simulate_gain_sinewave;
#[test]
fn test_sos_butter_gain() {
// Coefficients for a 4th order lowpass Butterworth filter with fc/fs = 0.05.
// Coefficients computed with scipy.signal.butter.
let mut filter = SisoSosFilter::<2, f64>::new(&[
[
4.16599204e-04,
8.33198409e-04,
4.16599204e-04,
-1.47967422e+00,
5.55821543e-01,
],
[
1.00000000e+00,
2.00000000e+00,
1.00000000e+00,
-1.70096433e+00,
7.88499740e-01,
],
]);
// Measure the gain of the filter on sine waves of different frequencies.
// Compare the measured gains to the expected gains. The expected gains were
// calculated with scipy.signal.freqz_sos in scripts/sos_test.py.
// f is frequency normalized to the sample frequency.
for (f, gain_expected) in [
(0.010, 1.000), // gain should be 1 well below the cutoff frequency
(0.050, 1.0 / f64::sqrt(2.0)), // gain should be ~1/sqrt(2) at the cutoff frequency
(0.100, 5.643e-2), // gain should be << 1 well above the cutoff frequency
] {
let gain = simulate_gain_sinewave(&mut filter, f, 1024);
let err = (gain - gain_expected).abs();
assert!(
err < 0.01,
"f = {f}: gain = {gain}, expected {gain_expected}, err = {err}"
);
}
}
#[test]
fn test_set_steady_state() {
// Coefficients for a 4th order lowpass Butterworth filter with fc/fs = 0.05.
// Coefficients computed with scipy.signal.butter.
let mut filter = SisoSosFilter::<2, f64>::new(&[
[
4.16599204e-04,
8.33198409e-04,
4.16599204e-04,
-1.47967422e+00,
5.55821543e-01,
],
[
1.00000000e+00,
2.00000000e+00,
1.00000000e+00,
-1.70096433e+00,
7.88499740e-01,
],
]);
for u in [-1.1, 0.0, 0.5, 1.0, 5.59823] {
filter.set_steady_state(u).unwrap();
let y = filter.update(u);
let err = (y - u).abs();
assert!(err < 1e-6, "u = {u}: y = {y}, expected {u}, err = {err}");
}
}
}