# How can I obtain the impulse response using a dual channel FFT?

I want to obtain the impulse response of a space to calculate reverberation time using the Schroeder method. This space is characterized by the short reverberation time within 0.5 seconds due to the properly installed sound absorption materials.

Theoretically, I know that impulse response can be obtained using the inverse FFT of the frequency response function which is usually obtained from a dual channel FFT analyzer. This is theoretically very simple. I thought that there would be many factors that could cause errors experimentally, but there are not many references in the literature.

Anyway, I tried to measure the frequency response function (FRF) using a volume acceleration source (which can measure volume acceleration m3/s2) and a diffuse field microphone. These signals are measured by a Siemens LMS mobile scadas data acquisition system.

Test setting are given below:

• Sampling frequency: 40960 Hz
• Resolution: 0.3125 Hz (=Frame size is about 3.2 s), which is much longer than expected reverberation time.

Input signal for volume acceleration source:

1. Burst random (burst time 70% is set)
2. Exponential sine sweep (sweep time 70% is set.)

For both signals, a uniform window is applied and 15 averaging is performed to calculate the FRF.

As in attached Fig.1, it was confirmed that both FRFs measured with the two signals were very similar in magnitude and phase. Finally, the impulse response measured using the inverse FFT is shown in Fig.2. Of the total 3.2s of data, only the data corresponding to the initial 0.27s is indicated.

To summarize my question:

1. Is it reasonable to calculate the impulse response by the inverse FFT of the frequency response function whose unit corresponds to Pascal/volume acceleration? I have seen several papers that use an exponential sine sweep or maximum length sequences to obtain impulse response, but none of the papers mention units, so I am very confused. For example, is it okay to measure the FRF and find the impulse response by assuming the input signal as the voltage level of the signal transmitted to the power amp or speakers?

2. If there is something I have overlooked in the above method, please let me know. I'm not familiar with digital processing, so I'd appreciate it if you could explain.

3. If the method of obtaining impulse response using the inverse FFT is effective, is there any other reason to use an exponential sine sweep or maximum length sequences to obtain the impulse response?

Is it reasonable to calculate the impulse response by the inverse FFT of the frequency response function

Yes. The transfer function can be calculated as

$$H[k] = \frac{Y[k]}{X[k]}$$

The tricky part here is that the input signal $$X[k]$$ has to have sufficient energy at ALL frequencies, otherwise you run into "divide by zero" problems or "noise amplification" issues. Your impulse responses look fairly decent but there is clearly some low frequency noise visible.

whose unit corresponds to Pascal/volume acceleration?

That's a bit of an odd ball. "Volume Acceleration" seems to be a term coined mostly by Siemens for vibration measurements, it's very unusual for a room acoustics application. More common units would Pa/V, i.e. sound pressure relative to voltage at the speaker terminal. If you want to an acoustic reference you would typically us the volume velocity of the speaker which, I guess would be the integral over the volume acceleration.

I also have never seen a "volume acceleration" source, so I don't know exactly what this is and how it works. The tricky part of using an acoustic reference quantity is that it's often dependent on direction of radiation. Loudspeakers tend to radiate more to the front and less to the rear.

have seen several papers that use an exponential sine sweep or maximum length sequences to obtain impulse response,

MLS are not a great choice for measuring room impulse responses. Their spectrum is typically white while your acoustic noise spectrum is anywhere between pink and brown. That means you typically get great SNR and high frequencies but terrible SNR at low frequencies (where you need it the most). It was a fad in the 80s/90s but has mostly disappeared.

by assuming the input signal as the voltage level of the signal transmitted to the power amp or speakers

You can defined the transfer function in whatever way is best for your application. There is no absolute "right" or "wrong", it really depends what you are trying to do,

If there is something I have overlooked in the above method, please let me know. I'm not familiar with digital processing, so I'd appreciate it if you could explain.

Measuring good room impulses is tricky. Firstly you have to carefully look at the placement of source and microphone. Every room has an infinite number of impulse response, not just one. If you want to characterize the room, you either need to measure a lot of positions or be very smart about placement (distance to boundaries and diffractors, staying out of the dominant modes, polar patterns, etc).

Another difficult part is noise management. There is ALWAYS acoustical noise and it's particularly bad at low frequencies (HVAC, traffic, building vibrations, etc). Ideally you shape the spectrum of your excitation so it matches the noise spectrum and you end up with constant SNR over your frequency range of interest. It's also a good idea to carefully dial in the excitation level: too much results in non-linear distortions of the source and too little results in poor SNR. There is an optimal spot somewhere in between.

If the method of obtaining impulse response using the inverse FFT is effective, is there any other reason to use an exponential sine sweep or maximum length sequences to obtain the impulse response?

See comment above. Exponential sweeps are much better than MLS since they are at least pink and not white. On the downside, they tend to concentrate a lot of energy into a very small bandwidth at any given point in time, which has the tendency to create more non-linear distortion in the source. IMO the best choice is periodic pseudo random noise that's spectrally matched to the noise floor.

• Very very good answer. How would you generate the spectrally matched periodic pseudo random noise - by filtering the random noise? Jan 16 at 9:53