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Piecewise Linear Sine Function
Approximate the sine function in the first quadrant using a piecewise linear continuous function with two segments. Find the slope change point that minimizes the squared error difference between the piecewise linear function and its Fourier series fundamental sine component b1. The minimum is approximately:
a.) 0.014 b.) 0.018 c.) 0.023 d.) 0.027 e.) 0.030 Code:
samples = 100; X = 0:1/samples:1; Y = sin(X*pi/2); xc = 0.6; % initial x value for slope change yc = 0.8; % inital y value for slope change % fundamental sine coefficient from Fourier series of PWL function b1 = 8/(pi^2*(xc*(xc1)))*(xc*sin(xc*pi/2)yc*sin(xc*pi/2)+xc*ycxc); seg1 = yc/xc*X; % first linear segment samples change = ceil(xc*samples); % index of slope change seg1(change+1:samples+1) = 0; % remove seg1 above change point seg2 = (1yc)/(1xc)*X+ycxc*(1yc)/(1xc); % samples above change point seg2(1:change) = 0; % remove seg2 up to change point Ypwl = seg1+seg2; % combine pieces error = (Y*b1Ypwl).^2; thd = sqrt(sum(error))/sqrt(sum((Y*b1).^2)); % plot(X,Y*b1,'0',X,Ypwl,'1',X,error*100); axis([0 1 0 1], "square"); grid on; title('Piecewise Linear Sine Approximation'); legend('sin(x)','pwl(x)', '% error','Location', 'Northwest'); xlabel('X');ylabel('Y'); text(xc0.04, yc+0.03, 'xc, yc'); text(1.01, b1, 'b1'); text(0.5,0.5, cstrcat('THD = ', num2str(thd))); text(0.03, ycxc*(1yc)/(1xc),'a'); 
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