Assuming that your Exponential Sweep Sine was generated using the formula:
$$x(t)=\sin\left(\frac{2\pi f_1 T}{R}\left(e^{\frac{t R}{T}} -1\right) \right)$$
where:
$f_1, f_2$ - Initial and final frequency of the sweep
$T$ - Duration of the sweep
$R = \ln\left(\frac{f_2}{f_1} \right)$ - Sweep rate
Then the inverse filter is calculated by scaling the amplitude of time reversed $x(t)$ by:
$k = e^{\frac{tR}{T}}$
Which will result in an exponentially decaying sweep:
$f(t) = \frac{x_{inv}(t)}{k}$
Example in Python:
#!/usr/bin/env python
from __future__ import division
import numpy as np
import scipy.signal as sig
import matplotlib.pyplot as plt
def dbfft(x, fs, win=None):
N = len(x) # Length of input sequence
if win is None:
win = np.ones(x.shape)
if len(x) != len(win):
raise ValueError('Signal and window must be of the same length')
x = x * win
# Calculate real FFT and frequency vector
sp = np.fft.rfft(x)
freq = np.arange((N / 2) + 1) / (float(N) / fs)
# Scale the magnitude of FFT by window and factor of 2,
# because we are using half of FFT spectrum.
s_mag = np.abs(sp) * 2 / np.sum(win)
# Convert to dBFS
ref = s_mag.max()
s_dbfs = 20 * np.log10(s_mag/ref)
return freq, s_dbfs
if __name__ == "__main__":
# Sweep Parameters
f1 = 10
f2 = 100
T = 3
fs = 1000
t = np.arange(0, T*fs)/fs
R = np.log(f2/f1)
# ESS generation
x = np.sin((2*np.pi*f1*T/R)*(np.exp(t*R/T)-1))
# Inverse filter
k = np.exp(t*R/T)
f = x[::-1]/k
# Impulse response
ir = sig.fftconvolve(x, f, mode='same')
# Get spectra of all signals
freq, Xdb = dbfft(x, fs)
freq, Fdb = dbfft(f, fs)
freq, IRdb = dbfft(ir, fs)
plt.figure()
plt.subplot(3,1,1)
plt.grid()
plt.plot(t, x)
plt.title('ESS')
plt.subplot(3,1,2)
plt.grid()
plt.plot(t, f)
plt.title('Inverse filter')
plt.subplot(3,1,3)
plt.grid()
plt.plot(t, ir)
plt.title('Impulse response')
plt.figure()
plt.grid()
plt.semilogx(freq, Xdb, label='ESS')
plt.semilogx(freq, Fdb, label='Inverse filter')
plt.semilogx(freq, IRdb, label='IR')
plt.title('Spectrum')
plt.xlabel('Frequency [Hz]')
plt.ylabel('Amplitude [dBFS]')
plt.legend()
plt.show()
And output:
For Robert, here is the spectrum plot for case without amplitude modulation of the inverse filter:
Relevant literature:
Q. Meng - Impulse Response Measurement with Sine Sweeps and Amplitude
Modulation
Schemes
A. Novak - Nonlinear System Identification Using Exponential
Swept-Sine
Signal
K. Vetter - ExpoChirpToolbox - a Pure Data implementation of ESS
impulse response
measurement
