class="prettyprint"> /* Cookbook formulae for audio EQ biquad filter coefficients ---------------------------------------------------------------------------- by Robert Bristow-JohnsonAll filter transfer functions were derived from analog prototypes (that are shown below for each EQ filter type) and had been digitized using the Bilinear Transform. BLT frequency warping has been taken into account for both significant frequency relocation (this is the normal "prewarping" that is necessary when using the BLT) and for bandwidth readjustment (since the bandwidth is compressed when mapped from analog to digital using the BLT). First, given a biquad transfer function defined as: b0 + b1*z^-1 + b2*z^-2 H(z) = ------------------------ (Eq 1) a0 + a1*z^-1 + a2*z^-2 This shows 6 coefficients instead of 5 so, depending on your architechture, you will likely normalize a0 to be 1 and perhaps also b0 to 1 (and collect that into an overall gain coefficient). Then your transfer function would look like: (b0/a0) + (b1/a0)*z^-1 + (b2/a0)*z^-2 H(z) = --------------------------------------- (Eq 2) 1 + (a1/a0)*z^-1 + (a2/a0)*z^-2 or 1 + (b1/b0)*z^-1 + (b2/b0)*z^-2 H(z) = (b0/a0) * --------------------------------- (Eq 3) 1 + (a1/a0)*z^-1 + (a2/a0)*z^-2 The most straight forward implementation would be the "Direct Form 1" (Eq 2): y[n] = (b0/a0)*x[n] + (b1/a0)*x[n-1] + (b2/a0)*x[n-2] - (a1/a0)*y[n-1] - (a2/a0)*y[n-2] (Eq 4) This is probably both the best and the easiest method to implement in the 56K and other fixed-point or floating-point architechtures with a double wide accumulator. Begin with these user defined parameters: Fs (the sampling frequency) f0 ("wherever it's happenin', man." Center Frequency or Corner Frequency, or shelf midpoint frequency, depending on which filter type. The "significant frequency".) dBgain (used only for peaking and shelving filters) Q (the EE kind of definition, except for peakingEQ in which A*Q is the classic EE Q. That adjustment in definition was made so that a boost of N dB followed by a cut of N dB for identical Q and f0/Fs results in a precisely flat unity gain filter or "wire".) _or_ BW, the bandwidth in octaves (between -3 dB frequencies for BPF and notch or between midpoint (dBgain/2) gain frequencies for peaking EQ) _or_ S, a "shelf slope" parameter (for shelving EQ only). When S = 1, the shelf slope is as steep as it can be and remain monotonically increasing or decreasing gain with frequency. The shelf slope, in dB/octave, remains proportional to S for all other values for a fixed f0/Fs and dBgain. */ //Then compute a few intermediate variables: /* A = sqrt( 10^(dBgain/20) ) = 10^(dBgain/40) (for peaking and shelving EQ filters only) w0 = 2*pi*f0/Fs cos(w0) sin(w0) alpha = sin(w0)/(2*Q) (case: Q) = sin(w0)*sinh( ln(2)/2 * BW * w0/sin(w0) ) (case: BW) = sin(w0)/2 * sqrt( (A + 1/A)*(1/S - 1) + 2 ) (case: S) FYI: The relationship between bandwidth and Q is 1/Q = 2*sinh(ln(2)/2*BW*w0/sin(w0)) (digital filter w BLT) or 1/Q = 2*sinh(ln(2)/2*BW) (analog filter prototype) The relationship between shelf slope and Q is 1/Q = sqrt((A + 1/A)*(1/S - 1) + 2) 2*sqrt(A)*alpha = sin(w0) * sqrt( (A^2 + 1)*(1/S - 1) + 2*A ) is a handy intermediate variable for shelving EQ filters. Finally, compute the coefficients for whichever filter type you want: (The analog prototypes, H(s), are shown for each filter type for normalized frequency.) LPF: H(s) = 1 / (s^2 + s/Q + 1) b0 = (1 - cos(w0))/2 b1 = 1 - cos(w0) b2 = (1 - cos(w0))/2 a0 = 1 + alpha a1 = -2*cos(w0) a2 = 1 - alpha HPF: H(s) = s^2 / (s^2 + s/Q + 1) b0 = (1 + cos(w0))/2 b1 = -(1 + cos(w0)) b2 = (1 + cos(w0))/2 a0 = 1 + alpha a1 = -2*cos(w0) a2 = 1 - alpha BPF: H(s) = s / (s^2 + s/Q + 1) (constant skirt gain, peak gain = Q) b0 = sin(w0)/2 = Q*alpha b1 = 0 b2 = -sin(w0)/2 = -Q*alpha a0 = 1 + alpha a1 = -2*cos(w0) a2 = 1 - alpha BPF: H(s) = (s/Q) / (s^2 + s/Q + 1) (constant 0 dB peak gain) b0 = alpha b1 = 0 b2 = -alpha a0 = 1 + alpha a1 = -2*cos(w0) a2 = 1 - alpha notch: H(s) = (s^2 + 1) / (s^2 + s/Q + 1) b0 = 1 b1 = -2*cos(w0) b2 = 1 a0 = 1 + alpha a1 = -2*cos(w0) a2 = 1 - alpha APF: H(s) = (s^2 - s/Q + 1) / (s^2 + s/Q + 1) b0 = 1 - alpha b1 = -2*cos(w0) b2 = 1 + alpha a0 = 1 + alpha a1 = -2*cos(w0) a2 = 1 - alpha peakingEQ: H(s) = (s^2 + s*(A/Q) + 1) / (s^2 + s/(A*Q) + 1) b0 = 1 + alpha*A b1 = -2*cos(w0) b2 = 1 - alpha*A a0 = 1 + alpha/A a1 = -2*cos(w0) a2 = 1 - alpha/A lowShelf: H(s) = A * (s^2 + (sqrt(A)/Q)*s + A)/(A*s^2 + (sqrt(A)/Q)*s + 1) b0 = A*( (A+1) - (A-1)*cos(w0) + 2*sqrt(A)*alpha ) b1 = 2*A*( (A-1) - (A+1)*cos(w0) ) b2 = A*( (A+1) - (A-1)*cos(w0) - 2*sqrt(A)*alpha ) a0 = (A+1) + (A-1)*cos(w0) + 2*sqrt(A)*alpha a1 = -2*( (A-1) + (A+1)*cos(w0) ) a2 = (A+1) + (A-1)*cos(w0) - 2*sqrt(A)*alpha highShelf: H(s) = A * (A*s^2 + (sqrt(A)/Q)*s + 1)/(s^2 + (sqrt(A)/Q)*s + A) b0 = A*( (A+1) + (A-1)*cos(w0) + 2*sqrt(A)*alpha ) b1 = -2*A*( (A-1) + (A+1)*cos(w0) ) b2 = A*( (A+1) + (A-1)*cos(w0) - 2*sqrt(A)*alpha ) a0 = (A+1) - (A-1)*cos(w0) + 2*sqrt(A)*alpha a1 = 2*( (A-1) - (A+1)*cos(w0) ) a2 = (A+1) - (A-1)*cos(w0) - 2*sqrt(A)*alpha */ /* FYI: The bilinear transform (with compensation for frequency warping) substitutes: 1 1 - z^-1 (normalized) s <-- ----------- * ---------- tan(w0/2) 1 + z^-1 and makes use of these trig identities: sin(w0) 1 - cos(w0) tan(w0/2) = ------------- (tan(w0/2))^2 = ------------- 1 + cos(w0) 1 + cos(w0) resulting in these substitutions: 1 + cos(w0) 1 + 2*z^-1 + z^-2 1 <-- ------------- * ------------------- 1 + cos(w0) 1 + 2*z^-1 + z^-2 1 + cos(w0) 1 - z^-1 s <-- ------------- * ---------- sin(w0) 1 + z^-1 1 + cos(w0) 1 - z^-2 = ------------- * ------------------- sin(w0) 1 + 2*z^-1 + z^-2 1 + cos(w0) 1 - 2*z^-1 + z^-2 s^2 <-- ------------- * ------------------- 1 - cos(w0) 1 + 2*z^-1 + z^-2 The factor: 1 + cos(w0) ------------------- 1 + 2*z^-1 + z^-2 is common to all terms in both numerator and denominator, can be factored out, and thus be left out in the substitutions above resulting in: 1 + 2*z^-1 + z^-2 1 <-- ------------------- 1 + cos(w0) 1 - z^-2 s <-- ------------------- sin(w0) 1 - 2*z^-1 + z^-2 s^2 <-- ------------------- 1 - cos(w0) In addition, all terms, numerator and denominator, can be multiplied by a common (sin(w0))^2 factor, finally resulting in these substitutions: 1 <-- (1 + 2*z^-1 + z^-2) * (1 - cos(w0)) s <-- (1 - z^-2) * sin(w0) s^2 <-- (1 - 2*z^-1 + z^-2) * (1 + cos(w0)) 1 + s^2 <-- 2 * (1 - 2*cos(w0)*z^-1 + z^-2) The biquad coefficient formulae above come out after a little simplification. */