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I am currently working on my Bachelor-Thesis about Real-Time convolution and Impulse Response Measurements. After reading different papers about the (exponential)sine sweep method I didn't find an answer on how to calculate the inverse filter for deconvolving the impulse response.

In the papers I read, it is described as the Time Reversal Mirror and after that some kind of scaling.

Could someone explain, how to calculate the inverse filter for a given sine sweep? If you could add an example, or an algorithm it would be appreciated

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    $\begingroup$ there is a terminology issue here. a "sine sweep" is a signal, not a filter (unless you mean that the sine sweep is an impulse response of a filter). filters have an inverse filter, but not signals. $\endgroup$ – robert bristow-johnson Jun 13 '17 at 16:10
  • $\begingroup$ so the papers I read most notably A. Farina's "Simultaneous Measurement of Impulse Response and Distortion with a Swept-Sine Technique" he is talking about creating an inverse filter so that the input signal x(t) convolved with said inverse filter f(t) results in a delayed dirac's delta function. So that the Output signal y(t) convolved with f(t) results in the impulse response h(t) $\endgroup$ – A. Hofmann Jun 13 '17 at 16:38
  • $\begingroup$ i might call that a form of a "matched filter". it is a delayed inverse filter of a filter having a sine sweep as its impulse response. $\endgroup$ – robert bristow-johnson Jun 13 '17 at 17:10
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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: enter image description here

enter image description here

For Robert, here is the spectrum plot for case without amplitude modulation of the inverse filter:

enter image description here


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

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  • $\begingroup$ where does $f_2$ figure into the equations? i might imagine that there is a relationship between $f_1$, $f_2$, $T$, and $R$, but what is it? $\endgroup$ – robert bristow-johnson Jun 13 '17 at 17:12
  • $\begingroup$ Should be T. Fixed now $\endgroup$ – jojek Jun 13 '17 at 17:15
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    $\begingroup$ @robertbristow-johnson: Short answer being: otherwise you will get a skewed frequency response after convolving the recording with unscaled time-reversed sweep. $\endgroup$ – jojek Jun 13 '17 at 17:49
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    $\begingroup$ it's skewed in a deterministic manner. you know exactly how it is skewed. you can compensate for the skewing in the frequency response. but the point of driving an acoustic system with a broadbanded test signal is to get a good representation of frequency response and to have as high S/N ratio as possible without clipping or going significantly non-linear. this is why low crest factor is salient. $\endgroup$ – robert bristow-johnson Jun 13 '17 at 18:15
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    $\begingroup$ well, you have the diamond, but i think that this is all technically relevant. $\endgroup$ – robert bristow-johnson Jun 13 '17 at 18:16
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@jojek answer is very good.

Long history short, the inverse ESS should be the ESS time reversed with amplitude scaling so after convolving with the ESS the frequency is constant in the [f1 f2] range. In other words the closest to a diraq impulse.

The matlab code would be (approximately):

invSweep(n) = sweep(N-1-n) .* (f2/f1)^(-n/(N-1))

invSweep(n) = flip(sweep) .* (f2/f1)^(-n/(N-1))

N being the duration in samples of the ESS (sweep)

Perhaps there's some error in the style N instead of N-1.

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