# Dispersion Filter Design

### From CCRMA Wiki

## Reference

"Robust, Efficient Design of Allpass Filtersfor Dispersive String Sound Synthesis", by Jonathan S. Abel, Vesa Välimäki, and Julius O. Smith III, IEEE SIGNAL PROCESSING LETTERS, VOL. 17, NO. 4, APRIL 2010

## Matlab for dispersion allpass filter design

function sos = adf(f0,B,df,beta) % adf.m % % Allpass dispersion filter design % % REFERENCE: % "Robust, Efficient Design of Allpass Filters for % Dispersive String Sound Synthesis," by % Jonathan S. Abel, Vesa Valimaki, and Julius O. Smith, % IEEE Signal Processing Letters, 2010. % % PARAMETERS: % f0 = fundamental frequency, Hz % B = inharmonicity coefficent, ratio (e.g., B = 0.0001) % df = design bandwidth, Hz % beta = smoothing factor (e.g., beta = 0.85) % % EXAMPLES: % sos = adf(32.702,0.0002,2100,0.85); % sos = adf(65.406,0.0001,2500,0.85); % sos = adf(130.81,0.00015,4300,0.85); % % The output array sos contains the allpass filter coefficients % as second-order sections. % % Created: Vesa Valimaki, Dec. 11, 2009 % (based on Jonathan Abel's Matlab code) % (ported to CCRMA Wiki by jos 2010-02-23)

%% initialization

% system variables fs = 44100; %% sampling rate, Hz nbins = 2048; %% number of frequency points

%% design dispersion filter

% period, samples; df delay, samples; integrated delay, radians tau0 = fs/f0; %% division needed pd0 = tau0/sqrt(1 + B*(df/f0).^2); %% division and sqrt needed mu0 = pd0/(1 + B*(df/f0)^2); phi0 = 2*pi*(df/fs)*pd0 - mu0*2*pi*df/fs;

% allpass order, biquads; desired phases, radians nap = floor(phi0/(2*pi)); phik = pi*[1:2:2*nap-1];

% Newton single iteration etan = [0:nap-1]/(1.2*nap) * df; %% division needed pdn = tau0./sqrt(1 + B*(etan/f0).^2); %% division and sqrt needed taun = pdn./(1 + B*(etan/f0).^2); phin = 2*pi*(etan/fs).*pdn; theta = fs/(2*pi) * (phik - phin + (2*pi/fs)*etan.*taun) ./ (taun - ... mu0); % division needed

% compute allpass pole locations delta = diff([-theta(1) theta])/2; cc = cos(theta * 2*pi/fs); eta = (1 - beta.*cos(delta * 2*pi/fs))./(1 - beta); %% division %needed alpha = sqrt(eta.^2 - 1) - eta; % sqrt needed

% form second-order allpass sections temp = [ones(nap,1) 2*alpha'.*cc' alpha'.^2]; sos = [fliplr(temp) temp];