Exponential Chirp

Question

I'm trying to replicate in python the exponential chirp (here) but I cannot understand well the t and T in the formulas; here is what I wrote, but it is surely wrong, please could you help me?

timeLength = 6                        # seconds
fs = 44100
t = np.arange(0, timeLength, 1.0/fs)


# EXPCHIRP -----------------------------------------

t1 = np.arange(1, timeLength+1, 1.0/fs)
A = np.e**-t                                 
f0 = 2000.0                                     # Hz   
f1 = 10000.0
k = (f1/f0)*(1/t1) 
phi0 = np.pi / 2                                # phase, radiants.                                                 # sampling rate
s0 = A * np.sin(phi0 + 2*np.pi*f0*((k-1)/np.log(k)))      

UPDATE: Thank you, I made also another solution but it is an equivalent way to produce this; I like also your method, thank you for the explanation!!

timeLength = 6.0                        # seconds
fs = 44100
t = np.arange(0, timeLength, 1.0/fs)
ww = get_window("hamming", t.size)


# Real Sinusoid: EXPCHIRP 0 -----------------------------------------

A = np.e**-t                                    # Or np.arange(.8, .0, -.8/t.size)
f0 = 200.0                                     # Hz   
f1 = 1000.0
k = np.power((f1/f0),1/timeLength) 
phi0 = np.pi / 2                                # phase, radiants.                                                 # sampling rate
s0 = A * np.sin(phi0 + 2*np.pi*f0*((np.power(k,t)-1)/np.log(k)))       # time function, sinusoid.

Answer 1

The Python implementation looks almost exactly like the definitions in Wikipedia. One will need to define functions in order to make it work, like this:

# necessary imports:
import numpy as np

# start by defining a function that returns a sine wave with time-dependent frequency
# f_func is a function f(t, f0, k)
def chirp(t, f_func, f0, k):
    return np.sin(2*np.pi*f_func(t, f0, k)*t)

# for a linearly changing frequency, use this for f_func:
def f_linear_chirp(t, f0, k):
    return f0 + k*t

# for exponentially changing frequency, use:
def f_exp_chirp(t, f0, k):
    return f0*k**t

# for a visually useful example, take:
duration = 1 # seconds
fs = 2**13 # sampling rate
t = np.arange(0, timeLength, 1.0/fs)

f0 = 2 # Hz
f1 = 20 # HZ

# the coefficients are then given by
k_test_lin = (f1-f0)/timeLength
k_test_exp = (f1/f0)**(1/timeLength)

print(f'k_lin = {k_test_lin}, k_exp = {k_test_exp}')

# the time series for exsample signals can be created by:
linear_chirp_example = [chirp(x, f_linear_chirp, f0, k_test_lin) for x in t]
exp_chirp_example = [chirp(x, f_exp_chirp, f0, k_test_exp) for x in t]

# to look at the results:
import matplotlib.pyplot as plt
plt.plot(t, linear_chirp_example)
plt.show()
plt.plot(t, exp_chirp_example)
plt.show()

*NOTE that the instantaneous frequency at any given t is df(t)/dt!