Check for IIR Bad Coefficients

checkPoles.hs

import Data.Complex (Complex((:+)))

data IIRParams = IIRParams
    { b0 :: Float
    , b1 :: Float
    , b2 :: Float
    , a0 :: Float
    , a1 :: Float
    , a2 :: Float
    }
    deriving (Show)

-- Example of an unstable filter (IIR BAD Coefficients)
unstableFilter :: IIRParams
unstableFilter = IIRParams
    { b0 = 0.5
    , b1 = 0.5
    , b2 = 0.5
    , a0 = 1.0
    , a1 = -1.8  -- These coefficients
    , a2 = 0.9   -- produce poles outside 'circle'
    }

-- Calculate poles of IIR filters
calculatePoles :: IIRParams -> (Complex Float, Complex Float)
calculatePoles (IIRParams _ _ _ a0 a1 a2) =
    let discriminant = a1 * a1 - 4 * a0 * a2
        twoA = 2 * a0
        sqrtDisc = sqrt discriminant
        realPart = (-a1) / twoA
        imagPart = sqrtDisc / twoA
        root1 = realPart :+ imagPart
        root2 = realPart :+ (-imagPart)
    in (root1, root2)


-- Magnitude of complex numbers
magnitude :: Complex Float -> Float
magnitude (r :+ i) = sqrt (r * r + i * i)

-- Check IIR filter is stable
isStable :: IIRParams -> Bool
isStable params =
    let (pole1, pole2) = calculatePoles params
    in magnitude pole1 < 1 && magnitude pole2 < 1

main :: IO ()
main = do
    let params = unstableFilter
    print params
    print $ calculatePoles params
    print $ isStable params  -- should be False

The theory comes from here: Audio EQ Cookbook

One needs to calculate two things:

The calculatePoles finds the roots of the characteristic equation; the equation that represents the poles of the filter. The characteristic equation is the filter transfer function denominators setting them equal to zero, then multiplying this by z^2

$a_0 + a_1 z^{ -1} + a_2 z^{-2} = 0$ then multiply this by $z^2$ →

$$ a_0 z^2 + a_1 z + a_2 = 0 $$

The isStable function ensures both poles have magnitudes less than 1.