We can take the “time windowed” data and transform it into the frequency domain using FFT mathematics.
This transformation yields a result that shows how much energy is present at each frequency, as shown in Figure 3.
You can see the pronounced roll-off of low frequency energy. You can also notice the lack of LF resolution in this figure.
The lack of resolution at LF is offset by a excess of HF resolution.
This uneven resolution between LF and HF energy is the result of the FFT mathematics used to transform the data from the time domain to the frequency domain.
Standard FFTs yield data that is distributed linearly in frequency (one data point every X Hertz).
Unfortunately, humans perceive frequency logarithmically.
Figure 3: The frequency response of the direct sound portion of an impulse response of a 1250 seat multi-purpose hall. The response was calculated using a 512 point FFT (which equals a 512/48000 or ~11 msec). As you can see the frequency response shows a pronounced LF roll-off.
This lack of LF resolution in Figure 3 is a direct result of the use of a short time window in our transformation from the time domain to the frequency domain.
It is interesting to note that this plot does not correlate with what we hear.
Simply listening to the full range loudspeaker system we were measuring made it clear that the system was reproducing LF energy down to at least 100 Hz!
I would suggest that a primary goal of an effective measurement system should be to provide results that correlate well with what we hear.
So the lack of correlation between what we have heard and what we measured suggests a modification to our approach.
As an alternate approach to trying to find a measurement that correlates with what we hear, we can try using a longer time window to “see” the LF response with better resolution.
A longer time window of approximately 250 msec is shown in Figure 4.
Figure 4: The impulse response of a 1250 seat multipurpose hall. The vertical lines suggest a time window that INCLUDES most of the effects of the room. The time window shown is approximately 0.25 seconds.
To transform this longer “slice” of the impulse response into the frequency domain, we will use an 8k FFT which represents 8k/48000 seconds, or 0.171 seconds.
Notice again that this time window includes both the direct sound and the response of the room.
In Figure 5 the low frequency information is seen in adequate resolution, however the high frequency results look confusing. The plot shows data that has 5 Hz resolution (i.e. one data point every 5 Hz).
While this resolution provides excellent LF resolution (between 31 Hz and 62.5 Hz there are 15 data points.
However at HF we have excessive resolution - between 4 kHz and 8 kHz there are approximately 800 data points.
Simply stated, the longer time window provides good LF resolution, but excessive HF resolution.