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plate_reverb.m
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plate_reverb.m
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% -------------------------------------------------------------------------
% Script for simulating a plate reverb.
%
% This plate reverb model uses modal synthesis approach and includes:
% 1. Second-order accurate and exact difference schemes.
% 2. Modal pruning.
% 3. Custom T60 approximation.
% 4. Saturation effect.
% 5. Reverb dry/wet and pre-delay controls.
% 6. Stereo output.
%
% There are three input forcing options for the algorithm:
% 1. "impulse" (dirac delta function).
% 2. "sweep" (sine sweeping technique to obtain impulse response).
% 3. "file" (to apply reverb to a recording).
%
% NB: there is no built-in resampling in a script so the sample rate of an
% input file should match the simulation sample rate. Otherwise the input
% will be changed to sweep.
%
% References:
% [1] Stefan Bilbao. Numerical Sound Synthesis: Finite Difference Schemes
% and Simulation in Musical Acoustics.
% https://doi.org/10.1002/9780470749012
% [2] M. Ducceschi and C. J. Webb. Plate Reverberation: Towards the
% Development of a Real-Time Plug-In for the Working Musician. Proc.
% ICA 2016.
% http://mdphys.org/PDF/icaPLA_2016.pdf
%
% Author: Victor Zheleznov
% Date: 20/03/2023
% -------------------------------------------------------------------------
clc; close all; clear all;
% sample rate
SR = 48e3; % [Hz]
% flags
USE_EXACT = true; % use exact update scheme [1]
USE_PRUNING = true; % apply modal pruning
% physical parameters
Lx = 2; % dimensions [m]
Ly = 1; %
H = 5e-4; % thickness [m]
T = 700; % tension [N/m]
rho = 7.87e3; % density [kg/m^3]
E = 200e9; % youngs modulus [N/m^2]
v = 0.29; % poisons ratio
% input/output
pos_in = [0.5, 0.5]; % normalised input point (0-1)
pos_out = [0.2, 0.6; 0.4, 0.2]; % normalised output points (0-1)
input = "sweep"; % impulse, sweep or file
file_path = "tabla_loop.wav";
sweep_dur = 0.5; % sweep duration [sec]
% pruning
tol_pr = 1e-6; % discard modes with modal shape function close to zero at input/output points
ncent = 0.1; % discard modes based on this cents distance
% reverb controls
rev_drywet = 0.3; % reverb dry/wet (0-1)
sat_drywet = 0.2; % saturation dry/wet (0-1)
sat_amp = 5; % saturation gain
Tpd = 50e-3; % pre-delay [sec]
% T60
T60_arr = [7, 7, 5, 8, 5, 4, 3, 2, 1 ]; % T60 values [sec]
f_arr = [63, 125, 250, 500, 1e3, 2e3, 4e3, 8e3, 16e3]; % corresponding frequency bands [Hz]
% check parameters
assert(SR > 0, "Sample rate must be positive!");
assert(Lx > 0 && Ly > 0, "Plate dimensions must be positive!");
assert(H > 0, "Plate thickness must be positive!");
assert(T > 0, "Tension must be positive!");
assert(rho > 0, "Density must be positive!");
assert(E > 0, "Youngs modulus must be positive!");
assert(min((pos_in > 0) & (pos_in < 1) & min((pos_out > 0) & (pos_out < 1))) == 1,...
"Normalised input/output points must be between 0 and 1!");
assert(min(T60_arr) > 0, "T60 times must be positive!");
assert(min(f_arr) > 0 & max(f_arr) < SR/2, "Frequency bands for T60 approximation should be within (0,SR/2)!");
assert(tol_pr > 0 & ncent > 0, "Modal pruning parameters must be positive!");
assert((rev_drywet >= 0 & rev_drywet <= 1) & (sat_drywet >= 0 & sat_drywet <= 1),...
"Dry/wet parameters should be in [0,1]!");
assert(Tpd >= 0, "Pre-delay time should be non-negative!");
assert(sweep_dur > 0, "Sweep duration should be positive!");
% derived parameters
k = 1/SR; % time step
K = sqrt(E*H^2/(12*rho*(1-v^2))); % stiffness
c = sqrt(T/(rho*H)); % wavespeed
Nf = floor(max(T60_arr)*SR); % simulation duration in samples
Npd = floor(Tpd*SR); % pre-delay duration in samples
% generate reverb input
if strcmp(input, "impulse") == 1
in = [1; zeros(Nf-1,1)];
elseif strcmp(input, "sweep") == 1
[in, in_f] = gen_sine_sweep(sweep_dur, max(T60_arr), SR, 20, SR/2, 0.95, 1e-2, false);
elseif strcmp(input, "file") == 1
[in, inSR] = audioread(file_path);
if size(in,2) == 2
in = mean(in,2);
end
if inSR == SR
in = [in; zeros(Nf,1)]; % add zeros for T60
Nf = length(in); % account for file size in simulation duration
else
warning("Input file has a different sample rate! Changed input to sweep");
input = "sweep";
[in, in_f] = gen_sine_sweep(sweep_dur, max(T60_arr), SR, 20, SR/2, 0.95, 1e-2, false);
end
else
error("Input should be set to impulse, sweep or file!");
end
% derive maximum frequency and wavenumber from stability condition
if USE_EXACT == true
wmax = pi*SR;
else
wmax = 2/k;
end
bmax = sqrt(-(c^2/(2*K^2)) + sqrt((c^2/(2*K^2))^2 + wmax^2/K^2));
Mx = floor(sqrt(bmax^2*Lx^2/pi^2 - (Lx/Ly)^2));
My = floor(sqrt(bmax^2*Ly^2/pi^2 - (Ly/Lx)^2));
% create modal index pairs
[mx,my] = meshgrid(1:Mx, 1:My);
mx = mx(:);
my = my(:);
% calculate wavenumbers
beta = sqrt((mx*pi/Lx).^2 + (my*pi/Ly).^2);
mx(beta > bmax) = [];
my(beta > bmax) = [];
beta(beta > bmax) = [];
% calculate modal frequencies
omega = sqrt(c^2*beta.^2 + K^2*beta.^4);
% calculate damping coefficients
sigma = calc_sigma(omega, T60_arr, f_arr, SR);
assert(min(sigma > 0), "Damping coefficients should be positive! Check specified T60 values...");
% calculate modal shape functions
Phi_in = Phi(pos_in(1), pos_in(2), mx, my, Lx, Ly);
Phi_outL = Phi(pos_out(1,1), pos_out(1,2), mx, my, Lx, Ly);
Phi_outR = Phi(pos_out(2,1), pos_out(2,2), mx, my, Lx, Ly);
% modes pruning
if USE_PRUNING == true
Mst = length(mx);
% find modal shape functions close to zero in input/output points
pr_idx_in = find(abs(Phi_in) < tol_pr);
pr_idx_outL = find(abs(Phi_outL) < tol_pr);
pr_idx_outR = find(abs(Phi_outR) < tol_pr);
pr_idx = union(pr_idx_in, intersect(pr_idx_outL, pr_idx_outR));
% apply pruning
mx(pr_idx) = [];
my(pr_idx) = [];
omega(pr_idx) = [];
sigma(pr_idx) = [];
Phi_in(pr_idx) = [];
Phi_outL(pr_idx) = [];
Phi_outR(pr_idx) = [];
Mpr1 = length(mx);
% pruning based on cents distance [2]
pr_idx = [];
freq = omega/(2*pi);
if ncent > 0
while min(freq) ~= Inf
[f_fix, idx_fix] = min(freq);
pr_idx_cur = find(1200*log2(freq/f_fix) < ncent);
freq(pr_idx_cur) = Inf;
if length(pr_idx_cur) > 1
pr_idx_cur(pr_idx_cur == idx_fix) = [];
pr_idx = [pr_idx; pr_idx_cur];
end
end
end
% apply pruning
mx(pr_idx) = [];
my(pr_idx) = [];
omega(pr_idx) = [];
sigma(pr_idx) = [];
Phi_in(pr_idx) = [];
Phi_outL(pr_idx) = [];
Phi_outR(pr_idx) = [];
Mpr2 = length(mx);
disp("Pruning: " + num2str(Mst) + " -> " + num2str(Mpr1) + " -> " + num2str(Mpr2) + " modes!");
end
% define output
out = zeros(Nf,2);
% allocate memory
M = length(mx);
p = zeros(M,1);
p1 = zeros(M,1);
p2 = zeros(M,1);
% precompute update matrices for loop
if USE_EXACT == true
cond = (sigma <= omega);
idx = find(cond == 0);
B = 2*exp(-sigma*k).*cos(sqrt(omega.^2-sigma.^2)*k);
B(idx) = exp(-sigma(idx)*k).*(exp(sqrt(sigma(idx).^2-omega(idx).^2)*k) + exp(-sqrt(sigma(idx).^2-omega(idx).^2)*k));
C = -exp(-2*sigma*k);
else
B = (2-k^2*omega.^2)./(1+k*sigma);
C = (k*sigma-1)./(1+k*sigma);
end
J = (k^2*Phi_in/(rho*H))./(1+k*sigma);
% main loop
tic
in = in./max(abs(in));
in_sat = (1 - sat_drywet)*in + sat_drywet*tanh(sat_amp*in);
for n = 1:Nf
% update state
p = B.*p1 + C.*p2 + J*in_sat(n);
out(n,1) = Phi_outL.'*p;
out(n,2) = Phi_outR.'*p;
% shift state
p2 = p1;
p1 = p;
end
exc_time = toc;
disp("Audio length = " + sprintf("%.2f sec.", Nf/SR))
disp("Execution time = " + sprintf("%.2f sec.", exc_time))
% process output
if strcmp(input, "impulse") == 1
ir = diff(out);
ir = ir./max(abs(ir),[],'all');
fig_spec = myspec(ir(:,1), SR, 512, 0.75);
soundsc(ir, SR);
audiowrite('plate_reverb_ir.wav', ir, SR);
elseif strcmp(input, "sweep") == 1
ir = [conv(out(:,1), in_f), conv(out(:,2), in_f)];
ir(1:length(in_f),:) = [];
ir = diff(ir);
ir = ir./max(abs(ir),[],'all');
fig_spec = myspec(ir(:,1), SR, 512, 0.75);
clim = caxis;
caxis([clim(2)-120, clim(2)]);
soundsc(ir, SR);
audiowrite('plate_reverb_ir.wav', ir, SR);
elseif strcmp(input, "file") == 1
% normalise and differentiate
out_diff = diff(out);
out_diff = out_diff./max(abs(out_diff),[],'all');
in = in(1:length(out_diff));
in = [in, in];
% apply pre-delay
out_diff = [zeros(Npd,2); out_diff];
in = [in; zeros(Npd,2)];
% apply dry/wet
out_diff_drywet = (1 - rev_drywet)*in + rev_drywet*out_diff;
% listen and save
soundsc(out_diff_drywet, SR);
[~,file_name] = fileparts(file_path);
audiowrite(append(file_name, '_plate_reverb', '.wav'), out_diff_drywet, SR);
end
%% FUNCTIONS
% calculate modal functions
% input:
% x, y --- normalised 2D coordinates;
% mx, my --- modal index pairs;
% Lx, Ly --- dimensions.
% output:
% z --- modal functions evaluated at x,y.
function z = Phi(x, y, mx, my, Lx, Ly)
z = (2/sqrt(Lx*Ly))*sin(mx*pi*x).*sin(my*pi*y);
end
% calculate loss parameters
% input:
% omega --- modal angular frequencies [rad];
% T60_arr --- T60 values [sec];
% f_arr --- corresponding frequency bands [Hz];
% SR --- sample rate [Hz];
% output:
% sigma --- loss parameters vector.
function sigma = calc_sigma(omega, T60_arr, f_arr, SR)
% check input size
if size(T60_arr,1) ~= 1
T60_arr = T60_arr.';
end
if size(f_arr,1) ~= 1
f_arr = f_arr.';
end
if size(omega,2) ~= 1
omega = omega.';
end
% sort input
[f_arr, sort_idx] = sort(f_arr);
T60_arr = T60_arr(sort_idx);
% add boundary
f_arr = [1e-6, f_arr, SR/2];
T60_arr = [T60_arr(1), T60_arr, T60_arr(end)];
% calculate linear approximation (based on cents distance)
f = omega/(2*pi);
ineq = (f >= f_arr);
f = f.';
idx = sum(ineq,2);
sigma_arr = (6*log(10)./T60_arr).';
w = (log2(f./f_arr(idx)) ./ log2(f_arr(idx+1)./f_arr(idx))).';
sigma = sigma_arr(idx).*(1-w) + sigma_arr(idx+1).*w;
end
% generate logarithmic sine sweep
% input:
% t_dur --- sweep duration [sec];
% t_sil --- silence duration [sec];
% fs --- sample rate [Hz];
% f0 --- lowest sweep frequency [Hz];
% f1 --- highest sweep frequency [Hz];
% max_amp --- maximum amplitude of the sweep;
% t_fade --- fade in/out duration [sec];
% PLOT_SPECTRUM --- flag to plot spectrum of sweep and inverse filter.
% output:
% x --- sweep signal;
% f --- inverse filter.
function [x,f] = gen_sine_sweep(t_dur, t_sil, fs, f0, f1, max_amp, t_fade, PLOT_SPECTRUM)
t = (0:1/fs:t_dur).';
R = log(f1/f0); % sweep rate
sweep_arg = 2*pi*(f0*t_dur/R)*(exp(t*R/t_dur)-1);
x = max_amp*sin(sweep_arg);
len_x = length(x);
% generate fade-in/out
len_fade = t_fade*fs;
fade = 0.5*(1 - cos(2*pi.*(0:len_fade-1)./(2*len_fade))).';
fade = [fade; ones(len_x-2*len_fade, 1); flipud(fade)];
x = x.*fade;
% generate inverse filter
f = flipud(x)./exp(t*R/t_dur);
% add silence
Nsil = floor(fs*t_sil);
x = [x; zeros(Nsil,1)];
% plot spectrum
if PLOT_SPECTRUM
NFFT = 2^(ceil(log(len_x)/log(2))); % next power of 2
NFFT_2 = NFFT/2 + 1;
X = fft(x, NFFT); X = X(1:NFFT_2);
F = fft(f, NFFT); F = F(1:NFFT_2);
fig_fft = figure;
freq = (0:fs/NFFT:fs/2).';
hold on;
plot(freq, 20*log10(abs(X)), 'r');
plot(freq, 20*log10(abs(F)), 'b');
xlabel('Frequency [Hz]', 'interpreter', 'latex');
ylabel('Magnitude [dB]', 'interpreter', 'latex');
title('Spectogram for the sine sweep', 'interpreter', 'latex');
leg = legend({'Sine sweep', 'Inverse filter'}, 'interpreter', 'latex');
set(leg,'location','southeast')
myspec(x, fs, 512, 0.95);
end
end
% create a spectogram plot of an input signal
% input:
% x - mono input signal;
% Fs - sampling frequency [Hz];
% N - frame length;
% O - overlap factor (between 0 and 1).
function fig_spec = myspec(x, Fs, N, O)
% find hop size
HA = round(N - O*N);
% generate window
win = 0.5*(1 - cos(2*pi.*(0:N-1)./N)).';
% calculate number of frames
L = length(x);
NF = ceil(L/HA);
x = [x; zeros((NF-1)*HA+N-L,1)];
% STFT size
NFFT = 2^(ceil(log(N)/log(2))); % next power of 2
NFFT_2 = NFFT / 2 + 1;
% calculate STFT
STFT = zeros(NFFT_2, NF);
for m = 0:NF-1
x_frame = win.*x((1:N).'+m*HA);
X = fft(x_frame, NFFT);
STFT(:,m+1) = X(1:NFFT_2);
end
% plot spectogram
fig_spec = figure;
t = ((0:NF-1).*HA/Fs).';
freq = (0:Fs/NFFT:Fs/2).';
STFT_dB = 20*log10(abs(STFT));
max_dB = max(max(STFT_dB));
imagesc(t, freq, STFT_dB, 'CDataMapping', 'scaled');
c = colorbar;
c.Label.String = 'dB';
colormap hot
caxis([max_dB-60, max_dB]);
xlim([0 t(end)]);
ylim([0 freq(end)]);
ax_spec = fig_spec.CurrentAxes;
set(ax_spec, 'YDir', 'normal');
set(ax_spec, 'YTick', 0:1000:Fs/2);
set(ax_spec, 'YTickLabel', 0:1000:Fs/2);
xlabel('Time [s]', 'interpreter', 'latex');
ylabel('Frequency [Hz]', 'interpreter', 'latex');
title_str = sprintf("Spectogram with frame length = $%d$ ms and overlap factor = $%d$\\%%", floor((N/Fs)*1e3), O*1e2);
title(title_str, 'interpreter', 'latex');
end