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    <p>The trick in this sort of problems is to understand the relation between <a href="http://en.wikipedia.org/wiki/Frequency_modulation" rel="nofollow noreferrer">frequency modulation</a> and <a href="http://en.wikipedia.org/wiki/Phase_modulation" rel="nofollow noreferrer">phase modulation</a>, these two are closely related. A sine with a constant frequency <code>f</code> and amplitude <code>A</code> can be described as (formulas, not python code):</p> <pre><code>x(t) = A sin(2 * pi * f * t) </code></pre> <p>but a different way to write this is by first defining a phase <code>phi</code> as a function of time:</p> <pre><code>phi(t) = 2 * pi * f * t x(t) = A sin(phi(t)) </code></pre> <p>The thing to note here is that frequency <code>f</code> is the derivative of the phase, divided by 2*pi: <code>f = d/dt(phi(t)) / (2*pi)</code>.</p> <p>For a signal which has a frequency that is varying in time, you can similarly define an <strong>instantaneous frequency</strong> <code>f_inst</code>:</p> <pre><code>x(t) = A sin(phi(t)) f_inst = d/dt(phi(t)) / (2*pi) </code></pre> <p>What you want to do is the opposite of this, you have a given instantaneous frequency (your logarithmic sweep), which you need to convert into a phase. Since the opposite of derivation is integration, you can calculate the appropriate phase like this (still formulas):</p> <pre><code>phi(t) = 2 * pi * Integral_0_to_t {f_inst(t) dt} x(t) = A sin(phi(t)) </code></pre> <p>What you are doing here is some sort of phase modulation of a signal (with zero frequency) to obtain the required instantaneous frequency. This is pretty easy to do in numpy:</p> <pre><code>from pylab import * n = 1000 # number of points f1, f2 = 10, 30 # frequency sweep range in Hertz t = linspace(0,1,1000) dt = t[1] - t[0] # needed for integration # define desired logarithmic frequency sweep f_inst = logspace(log10(f1), log10(f2), n) phi = 2 * pi * cumsum(f_inst) * dt # integrate to get phase # make plot plot(t, sin(phi)) xlabel('Time (s)') ylim([-1.2, 1.2]) grid() show() </code></pre> <p>Resulting image:</p> <p><img src="https://i.stack.imgur.com/Ezwxh.png" alt="Logarithmic frequency sweep"></p> <p>But (as also noted in the <a href="https://stackoverflow.com/a/11199832/2647279">dupe</a> mentioned by Dave), you probably don't want a logarithmic sweep, but an exponential one. Your ear has a logarithmic perception of frequency, so a smooth/linear musical scale (think the keys on a piano) are therefore spaced exponentially. This can be achieved by simply redefining your instantaneous frequency <code>f_inst(t) = f1 * exp(k * t)</code>, where <code>k</code> is chosen such that <code>f_inst(t2) = f2</code>.</p> <p>If you want to use <a href="http://en.wikipedia.org/wiki/Amplitude_modulation" rel="nofollow noreferrer">amplitude modulation</a> at the same time, you can simply change <code>A</code> to a time dependent <code>A(t)</code> in the formulas.</p>
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