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    <p>Your example</p> <pre><code>eq1=0.2*cos(2pi*t*3+phase1)+vertical offset1 eq2=0.7*cos(2pi*t*9+phase2)+vertical offset2 eq3=0.8*cos(2pi*t*5+phase3)+vertical offset3 eq_total=eq1+eq2+eq3 </code></pre> <p>where the maximum amplitude should be less than 0.8, has infinitely many solutions. Unless you have some additional objective you'd like to achieve, I suggest that you modify the problem such that you find the combination of phase angles that has a maximum amplitude of <em>exactly</em> 0.8 (or 0.79, such that you're guaranteed to be below). </p> <p>Furthermore only two out of three phase angles are independent; if you increase all by, say, <code>pi/3</code>, the solution still holds. Thus, you have only two unknowns in <code>eq_total</code>.</p> <p>You can solve the nonlinear optimization problem using e.g. <a href="http://www.mathworks.com/help/techdoc/ref/fminsearch.html" rel="nofollow noreferrer">FMINSEARCH</a>. You formulate the problem such that <code>max(abs(eq_total(phase1,phase2)))</code> should equal 0.79.</p> <p>Thus:</p> <pre><code>%# define the vector t, verticalOffset here %# objectiveFunction is (eq_total-0.79)^2, so the phase shifts 1 and 2 that %# satisfy this (approximately) should guarantee that signal never exceeds 0.8 objectiveFunction = @(phase)(max(abs(0.2*cos(2*pi*t+phase(1))+0.7*cos(2*pi*t*9+phase(2))+0.8*cos(2*pi*t*5)+verticalOffset)) - 0.79)^2; %# search for optimal phase shift, starting at no shift solution = fminsearch(objectiveFunction,[0;0]); </code></pre> <p><strong>EDIT</strong></p> <blockquote> <p>Unfortunately when I Try this code and plot the results the maximum amplitude is not 0.79 it's over 1. Am I doing something wrong? see code below t=linspace(0,1,8000); verticalOffset=0; objectiveFunction = @(phase)(max(abs(0.2*cos(2*pi*t+phase(1))+0.7*cos(2*pi*t*9+phase(2))+0.8*cos(2*p‌​i*t*5)+verticalOffset)) - 0.79)^2; s1 = fminsearch(objectiveFunction,[0;0]) eqt=0.2*cos(2*pi*t+s1(1))+0.7*cos(2*pi*t*9+s1(2))+0.8*cos(2*pi*t*5)+verticalOffs‌​et; plot(eqt)</p> </blockquote> <p><code>fminsearch</code> will find a minimum of the objective function. Whether this solution satisfies all your conditions is something you have to test. In this case, the solution given by <code>fminsearch</code> with the starting value <code>[0;0]</code> gives a maximum of ~1.3, which is obviously not good enough. However, when you plot the maximum for a range of phase angles from 0 to 2pi, you'll see that `fminsearch didn't get stuck in a bad local minimum. Rather, there is no good solution at all (z-axis is the maximum).</p> <p><img src="https://i.stack.imgur.com/rrPM2.png" alt="enter image description here"></p>
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    1. POSolving multiple phase angles for multiple equations
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    1. COUnfortunately when I Try this code and plot the results the maximum amplitude is not 0.79 it's over 1. Am I doing something wrong? see code below t=linspace(0,1,8000); verticalOffset=0; objectiveFunction = @(phase)(max(abs(0.2*cos(2*pi*t+phase(1))+0.7*cos(2*pi*t*9+phase(2))+0.8*cos(2*pi*t*5)+verticalOffset)) - 0.79)^2; s1 = fminsearch(objectiveFunction,[0;0]) eqt=0.2*cos(2*pi*t+s1(1))+0.7*cos(2*pi*t*9+s1(2))+0.8*cos(2*pi*t*5)+verticalOffset; plot(eqt)
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    2. CO@RickT: All you're doing wrong is to assume that the best solution is good enough. See my edit.
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