Advice on seperation of impulse responses for harmonic distortion orders

Question

I am working based on Angelo Farinas paper: Simultaneous measurement of impulse response and distortion with a swept-sine technique.

The purpose of the work is part of a loudspeaker measurement and analysis system.

At present my "measurement" data is generated via a non-linear transducer model, so simulated to include typical electroacoustic transducer non-linearities. The simulation is time-domain based and outputs displacement, velocity and acceleration data.

The process as it stands is:

1. Excite the transducer model with a exponential sweep defined by:

$ x(t)=\sin\left(\frac{\omega_1T}{\ln(\frac{\omega_2}{\omega_1})}(e^{\frac{t\ln(\omega_2)}{\omega_1}}-1)\right) $

2. Create an inverse filter

(which gives a delayed Dirac delta function when convolved with the input function $x(t)\circledast f(t)\rightarrow\delta(t)$):

$ f(t) = \frac{x_\text{inv}(t)}{e^{\frac{t\cdot \ln(\frac{\omega_2}{\omega_1})}{T}}} $

3. Convolve the transducer model output signal $y(t)$ with the inverse filter to retreive the systems impulse response $h(t)$:

$ h(t)=y(t)\circledast f(t) $

$h(t)$" />

4. Isolate and plot the FFT of each impulse in the system impulse response (right to left) in order of harmonics:

My main questions are, am I on the right track with this? Are there alternative methods I have missed? What improvements I can make?

Answer 1

You are doing in the right way. There are some additional points to improve the result.

1. In this JAES paper the authors present a new type of swept-sine named synchronized swept-sine, which outperforms the exponential swept-sine in Farinas's paper when the phase response of harmonics is important for you.

2. It can be seen that the measured impulse responses have significant "pre-ringing" artifact in your figure. This is maybe caused by the frequency range of your swept-sine. You should ensure the frequency of swept-sine cover the whole frequency band which goes from nearly 0 and up to Nyquist, otherwise the spectrum of swept-sine looks like a rectangular window and its time-domain impulse response will be a sinc function rather than a delta function.

3. However, we can't set $\omega_1=0$, and in some cases low frequency can cause pre-ringing as well. In this case you may use frequency-domain deconvolution with proper regularization.

In this AES convention paper Farinas concludes some problems of this method and also the solutions to solve them. You may have a look at it.