Estimation of loudspeaker power response from impulse response

A simple and "fairly obvious" in-room speaker power response measurement without measuring off-axis responses with high spatial resolution.

July 1, 2024

Sound power response refers to the total sound radiated by a sound source in all direction. Sound power curve is important in closed spaces, especially when the listening position is outside the "room acoustical" near field or critical distance. On-axis room response of a loudspeaker is a combination of the sound power response (scaled according to room gain) and the on-axis response.

Speaker power response measurements (traditional methods)

Traditional measurement methods:

Spatial averaging of free field (anechoic) frequency responses taken from different angles (e.g. 10 deg resolution in the horizontal and vertical plane). This method is based on the definition of sound power response.
Measuring off-axis frequency response in a reverberant chamber or highly reflective room. This method works only in rooms with low and frequency independent sound absorption.

Some notes on the relationship between room gain, sound power and on-axis room response. Since the room gain in a typical listening room for a distance of 2-3 meter is between 8 dB and 12 dB, the most important region of the sound power is the "upper" 15 dB. Below 15 dB the sound power response has no contribution to the on-axis room response, only to the energy decay curve and some reverberation measurements (EDT, C₅₀, C₈₀).

Method of measurement

The impulse response of a loudspeaker in a room contains the direct sound, the early reflections and late reflections (reverb). By excluding the direct sound and some early reflections and removing the effect of room absorption, the sound power response can be measured.

There are three questions to be answered:

How does the spectrum of the impulse response of a loudspeaker in a room change as a function of time?
Why does the spectrum of the impulse response of a loudspeaker in a room change in a particular way?
How can we minimize the effect of room absorption (frequency dependent room absorption)?

The first question can be answered by analyzing the in-room impulse response of a full range speaker driver in 5 ms long segments (windows). A small "one-way" loudspeaker is more suitable for this measurement than a two-way loudspeaker with a dome tweeter.

Spectral decay as frequency response (1/3 octave smoothing)
Distance: 2 meter; room: 20 m², loudspeaker: Logitech X-120 with 3 inch driver (6 cm cone diameter)

We can see that the transition from the direct response (0ms-5ms) to the power response is very fast and not a gradual process. Between 5ms and 10 ms the response is almost the power response. The response between 10-15 ms is slightly different due to the strong reflection from the wall behind the head (on-axis), but a longer window would lower the difference to an insignificant level.

Why is the spectral change in the impulse response a fast event and not a gradual transition? After the direct wave, we have six first-order reflections from the walls and only one comes from on-axis (this is the reflection from the wall behind the head). And then we have 18 second-order reflections and only one comes from on-axis and the majority comes from 40-60 deg off-axis.

This also means that since the spectral change "from direct to late reflections" in the impulse response is very fast (about 5 ms), "early reflections response" and "early reflections DI" (used in spinorama) are questionable and just take up space. Since we can't hear the "early reflections response", why should we bother with it?

How can we take into account the effect of room absorption? The effect of room absorption can be minimized by comparing gated measurements with different window length and staying close to the start of the pulse.

Sound power can be estimated by measuring the response with a 100 ms long analysis window set 10ms after the direct sound.

The finalized method:

From a reasonable distance (minimum one meter, depends on size of the speaker) measure the impulse response.
For frequency analysis use different windows: 10 ms - 30 ms (a), 80 ms - 100 ms (b) and 10 ms - 100 ms (c).
If "10 ms - 100 ms" and "10 ms - 30ms" frequency responses run close above 1 kHz (after normalization), "10ms - 100 ms" gives a good estimation of the power response.
The difference between "10 ms - 30 ms" and "80 ms - 100 ms" responses shows the effect of sound absorption.

Measurements for the same speaker (1/3 octave smoothing)

The responses should be equalized with the microphone's diffuse field response and not with the free-field response.

Drawback of the method: it is difficult to interpret the measurement below ~300 Hz, because the response will be riddled with peaks (room modes) and notches (due to strong early reflections).

Csaba Horváth