Senior Product Manager Mike Gough continues to discuss the importance of measuring frequency response and the accuracy of the results.

Impulse response

Whereas the concept of the swept sine wave resides conceptually in the frequency domain, the impulse response is regarded as being in the time domain. Of course, the two are equivalent – you can switch between the two domains using a bit of fairly advanced maths called the Fourier Transform – but using impulse response measurements made engineers think in a different way, both about measurements themselves and also how they relate to the listening experience.

An ideal or perfect impulse cannot exist in real life. It’s a bit like the frictionless or weightless objects that you meet in maths classes at school. The perfect impulse has infinite amplitude, lasts for zero time and has a flat amplitude and phase response from DC to infinity.

In practice, you have to use a pulse that has some duration. If you look at a rectangular pulse, its response is flat to a certain frequency and then goes into a series of dips. For instance, here we see the frequency response of a 0.5ms wide rectangular pulse. Its first minimum is at 2kHz and it’s around 4dB down at 1kHz. This is obviously no good as a signal for speakers that you want to measure to 20kHz and beyond, but divide the pulse width by 100 to 5µs and the first minimum moves up by a factor of 100 to 200kHz. The response is then only 0.15dB down at 20 kHz.

The trouble with such narrow pulses is that, even with a large peak  voltage, they have very little energy and the resulting output tends to have a poor signal to noise ratio. To overcome this, signal averaging is used. The signal is repeated, the measurements are added together and the result divided by the number of measurements. Every time the number of repetitions is doubled, the noise, being random, drops by 3dB. In addition, you find that the noise spectrum is weighted towards low frequencies, so it is beneficial apply boost to the bass of the signal so that the signal to noise ratio is more or less the same at all frequencies, then apply the inverse of that boost to the measured output.

It’s quite an eye opener to look back at the sort of computers that were being used to make such measurements back in the 1970s. This picture shows our Digital PDP11. Other computers were available, but all were huge, required air conditioning to keep them cool, yet had less memory capacity and computing power than today’s personal computers.

It shouldn’t matter which method you use to measure frequency response; the results should be the same, but there may be instances when they appear not to be. A classic case involved the measurement of the bass response of a system that used a wool-like material to absorb resonances inside the cabinet. The swept sine method gave a more extended bass with a slightly lower level. It turned out that there was enough energy in the sine wave signal to make the wool move, but the impulse did not have enough energy to overcome friction. The moving wool added to the effective mass of the driver’s moving parts, which explained the difference. Once the wool was constrained so that it couldn’t move, the results were the same.

Alternative ways of measuring impulse response

You don’t have to use an actual impulse signal to calculate the impulse response and today it is more usual to use signals with greater energy. One of the milestone developments was the Minimum Length Sequence (MLS). The brainchild of John Vanderkooy and Doug Rife, this involves the use of a pseudo-random noise signal. Gaussian noise contains all frequencies, but is truly random. The MLS signal also contains all relevant frequencies and sounds like noise, but it repeats the same sequence over and over so is completely known and calculable. The speaker’s response also sounds like noise, but you can correlate the output with the input and derive the impulse response. When DOS was the norm for PCs, Doug Rife made a commercial version of this method called MLSSA (pronounced Melissa), which used a hardware card slotted into the PC. It was never developed for modern Windows platforms and other software solutions tend now to be used. At B&W, we use WinMLS. This is based on the work of Vanderkooy and Rife, but uses a rapid sine sweep as the stimulus. It has the added advantage that the harmonic distortion can be derived from the result, something that is not possible with the low-energy impulse stimulus.

 Dealing with impulse response measurements

In a previous blog looking at anechoic chambers, I mentioned that you can make impulse measurements in a live room. Well, this is the sort of thing that you get

This is actually a simulation of a 3-way speaker which I have created to show general principles. The origin, or time zero, represents the time that the signal was applied to the speaker. The response of the speaker arrives some time later, being the time taken for the sound to travel the 2 metres to the microphone. You can see a smaller impulse arriving even later and this represents the first reflection off the walls of the live room. If we now do our mathematical trickery and apply the Fourier Transform to this result, we get the following frequency response – amplitude and phase:

You can see the effects of the reflection in the amplitude response, similar to the effects shown in my previous Anechoic Chamber article, but the phase response is a total mess and difficult to make any sense of at all. Actually, the practice of only plotting phase response between +/-180º limits hides the fact that the phase can have values outside these boundaries. But if it goes over the limits, we simply wrap it round. If it drops off the bottom, the plot shows it to jump up to the top again. So, for example, -200º would plot as (360º – 200º) or +160º. This phase plot, therefore, adds lots of wrapping to the general jaggedness apparent in the amplitude response.

The first thing we can do is get rid of the 2 metres worth of time delay by simply shifting the initial impulse to the origin.

 When we look at the frequency response, we can see that the amplitude response stays the same, but we can begin to see some improvement in the phase response as most of the wrapping is taken out.

There’s still some wrapping, but we’ll see why later.

Now we need to get rid of the reflection and here we run into a little problem. If we expand our impulse response, we can see that the reflection arrives before the response of the speaker itself has settled down.

If we simply chop off the reflection by putting all the values after it arrives to zero, you can see a distinct step in the plot:

This step is going to leave some raggedness on the response, so let’s be a little bit clever and truncate with a tapering function that gently approaches zero.

When we transform this to the frequency response, we get the black trace on this next graph, which you can compare to the red trace of the true response:

Our measured response is not bad down to 100Hz, it’s within 1dB, but it’s pretty bad at representing the true bass performance. In this case, it’s fairly obvious that the black trace is not accurate, but sometimes you can get traces that look believable but that are wrong. The error is simply due to the fact that we have not captured the complete impulse response of the speaker; it has been truncated.

There are techniques to improve the accuracy at low frequencies, but the general principle applies that, the lower in frequency you want accuracy, the larger the room you need. In our example, the reflection came some 11ms after the main impulse. With a measuring distance of 2 metres, this equates to a room approximately 8 metres in each dimension. That’s large by any standard and an anechoic chamber, although it cannot totally eliminate reflections, can give equivalent accuracy in a much smaller space, whatever the method you use to make your measurements.

The phase response is interesting. There is a downward slope as the frequency increases. This indicates time delay. An upward slope would indicate something arriving before it was created – not natural. The steeper the slope, the greater the delay and much of what we see in the trace is due to the crossover, with the rest down to the roll-off at each end of the system response.

Once captured, the impulse response can be manipulated and used for other design purposes. We’ll look into some of these techniques in the next article.

Mike Gough, Senior Product Manager