fft - Basic question about chirp signal

Suppose I generate chirp signal with the following python code:

 t = np.linspace(0, 10, 512)
 f0 = 2
 f1 = 5
 fs = 2*f1

 w = chirp(t, f0=f0, f1=f1, t1=10, method='linear')
 result  = rfft(w)
 phase = np.zeros(257)
 mag = np.zeros(257)
 for binnum in range(len(result)):
   mag[binnum] = abs(result[binnum])
   phase[binnum] = np.arctan2(result[binnum].imag, result[binnum].real)
 plt.plot(fs * np.linspace(0, 0.5, 257),np.unwrap(phase))
 plt.show()
 plt.plot(fs * np.linspace(0, 0.5, 257),mag)

 plt.show()  

And the phase plot is:

Magnitude plot is:

But it is very different from what is described here.

Well, looking at the phase graph I can see parabola from 0 to 0.8. The magnitude is definitly not a constant.

What I'm doing wrong or misunderstood?

Thanks in advance.

Answer

I dont understand the definition of the chirp in the link you provided (perhaps chirp in frequency domain?), but maybe the following explanation could help.

In general, a linear chirp with a varying frequency given by $$f(t) = \alpha + \beta t$$ is defined by $$ x_1(t) = e^{2\pi j (\alpha t + 0.5\beta t^2)} $$

In that case, the phase of $x_1(t)$, the signal in time-domain, is indeed parabolic and the magnitude is constant and equals to 1.
Now, I am not familiar with python, but the chirp function in matlab produces a real signal rather than complex one. That is $$ x_2(t) = \sin(2\pi (\alpha t + 0.5\beta t^2)) $$ In that case, you will get different results. However, you can use the Hilbert transform (I guess this is in python) which, given $x_1(t)$ will produce $x_2(t)$ using fft.