# How to calculate time of arrival using room impulse response

I need to find the timestamp of arrival of a particular signal at a receiver location within a room. The known parameters are the room impulse response h for a fixed source and receiver location, the input signal x which is a linear sine chirp.

Currently the steps that i am doing are as follows:

1. Convolution of input signal x and the impulse response h to generate output y

2. Cross-correlation of y with x, on which i am doing peak picking to determine the arrival timestamp.

Is the mentioned procedure right or am i missing something?

• Approach you are taking seems correct to me. the cross correlation peak will tell you the delay introduced by the Room Impulse response, that is the time of arrival for you. Mar 15, 2018 at 11:27

If you do not have a very controlled room, there might be something attenuating the direct path and a reflection might then be the highest peak. In that case you should accept also a somewhat attenuated peak if it comes earlier than the strongest peak.

Coupling to structures is not that much of an issue. For example, when a sound wave reflects at the interface of wood and air, only a fraction of about 0.0007 (see Ref. 1) of the energy of the wave will be passed to the other medium. The wave will then spend a lot of time in the first medium until much of its energy is passed to the second medium, so you can imagine that the impulse response of this coupling is not very concentrated in time and doesn't disturb the peak detection much.

References:

The mathematical procedure asked, is right and you may be missing something.

Let's assume a room with concrete walls and floor.

The speed of sound in air is around 300m/s.

The speed of sound in concrete is around 3400m/s and concrete supports slower sheer waves are about 1900m/s and there are also Rayleigh waves.

In a big room and at frequencies where the coupling from air to concrete is negligible, you can usually expect that the direct acoustic path with be the shortest.

There are also cases where you get coupling (like a speaker sitting on the floor) and the concrete path (P-wave) will arrive first through the floor to your sensor supported on a metal stand. Let's say you suspend your sensor on a string attached the ceiling. Even if the sensor is in the exact same place, the impulse response will not be the same. In other words, your sensor setup is part of the impulse response. Now fill the room with people and the impulse response is changed again.

There is also the practical matter that an ideal impulse response has an infinite bandwidth so necessarily, you need to restrict the frequencies to some band, and that depends on your actual problem.

None of this says that the simple direct line-of-sight air path is wrong, but I wouldn't count on repeatable results. It's also easy to overthink the physics.