Estimating the Impulse Response of the Room Using Sweep Signal Microphone Recorded Signal (Input & Output of a Convolution)

I played this signal A (a 20Hz to 20000Hz sinusoidal sweep in 10 seconds) with a studio monitor speaker in a big church, and I recorded the result B with good microphones.

The result is very reverb-ish, that's exactly what I wanted to catch.

Now a software (such as Deconvolver but non open-source) can build an impulse response from A + B, that can be later used in a convolution reverb.

It works well. But I would like to learn how to do this myself via DSP / programming.

How can I use a sweep (signal A) + recorded output (signal B) to get an impulse reponse?

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Edit: In other words, if a is the original sweep, b the recorded output, and h the impulse response, how to get h from

$$a * h = b$$

Is this formulation correct? is the solution $h=a^{-1} * b$, where $a^{-1}$ is the inverse of $a$ for the convolution? How to compute a convolution-inverse of a discrete signal?

Answer

this is the two-channel FFT method of spectrum analyzer:

$$y[n] = h[n] \ \circledast \ x[n]$$

just make sure that the length of the FFT $N$ is at least as large as the length of sound $x[n]$ plus the expected length of the impulse response $h[n]$. the length of sound $y[n]$ is also as long as the FFT. you can round $N$ up to the nearest power of two. just zero-pad everything to that length and then

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

is functionally true.

you might sometimes have to worry about division by zero, but if your driving signal $x[n]$ is sufficiently broad-banded (which a linear sweep or a maximum-length sequence is), then you don't have to worry too much about division by zero.

if you know $H[k]$, then you know $h[n]$.