[Scipy-tickets] [SciPy] #1105: Logarithmic chirp frequency sweep incorrect

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Sat Feb 6 19:14:41 CST 2010

#1105: Logarithmic chirp frequency sweep incorrect
 Reporter:  johntryan                    |       Owner:  somebody
     Type:  defect                       |      Status:  new     
 Priority:  normal                       |   Milestone:  0.8.0   
Component:  scipy.signal                 |     Version:  0.7.0   
 Keywords:  chirp logarithmic frequency  |  

Comment(by warren):

 If we accept that 'logarithmic' should implement the formula from the
 web site (which seems to be the conclusion from ticket #547), then
 the main problem is that the implementation of the
 formula is more complicated than necessary.  The problem is not that
 a log of 0 is taken, but that if f1-f0 = 1, then beta=0 and the formula
 currently implemented divides by beta.

 The formula for the frequency, f(t), can be written

     f(t) = f0 + (f1 - f0)**(t/t1)

 (After eliminating beta and doing some algebra, that's what's left.)
 Then the phase angle is given by the integral of 2*pi*f(t):

     phase = 2*pi * (f0*t + ((f1-f0)**(t/t1) - 1) * t1 / log(f1 - f0))

 If the sweep is implemented with this formula, then the only requirement
 is that f1 - f0 > 0.

 However, I agree that the formula given on that web page is, if not
 "incorrect", a strange formula to use.  Note that, as pointed out on
 the web page, it gives f(0) = f0 + 1! So the frequency sweep goes from
 f0+1 to f1, which seems strange and unnecessary.

 One "fix" is the formula

    f(t) = f0 + (f1 - f0 + 1)**(t/t1) - 1

 for which f(0) = f0 and f(t1) = f1.  This formula results in

    phase = 2*pi * ((f0 - 1)*t + (f1 - f0 + 1)**(t/t1) * t1 / log(f1 - f0 +

 but this still does not seem like a natural choice.

 The formula you suggest, which I agree makes more sense, can be simplified

     f(t) = f0*(f1/f0)**(t/t1)

 However, your formula for the phase is not correct; to get the phase, you
 must integrate 2*pi*f(t), not multiply by 2*pi*t.  Integration gives

     phase = ((f1/f0)**(t/t1) - 1) * t1 / log(f1/f0)

 With this formula, the only constraint is that f0 and f1 are postive and
 have the same sign.

 I'm working on a revision of chirp() that includes this formula.  I'll
 it to the mailing list soon and get some feedback.

Ticket URL: <http://projects.scipy.org/scipy/ticket/1105#comment:1>
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