Liberty Instruments, Inc.
Maximum Length Sequence (MLS) based measurements with LAUD
by Bill Waslo
(portions of this text were previously published in "The IMP Goes MLS" in Speaker Builder Magazine, 6/1993)
There is a one-to-one relationship between an impulse response (the waveform with which a device responds to a sharp impulse at its input) and that device's frequency response (the magnitude and delay changes that the device imparts to sinewaves of different frequencies applied to its input). If you are able to determine the impulse response, you can transform this information into the frequency response. Similarly, if you can determine the frequency response you can derive from this the impulse response, (but you must have both magnitude and true phase to do such a transformation). The mathematical operations that do these conversions are the Fourier Transform and the Inverse Fourier Transform. A measurement of a system such as a loudspeaker can start from either point of to obtain these equivalent types of measurement data: it can apply sinewaves to find the frequency response, or in can apply impulsive stimuli to obtain the impulse response. Once either type data is obtained, transformation from one to the other is a simple matter of using computer postprocessing.
A narrow pulse is attractive as a measurement stimulus for several reasons. It is easy to generate using inexpensive circuitry. It contains a wide frequency spectrum allowing simultaneous measurement of most or all of a speaker's or amplifier's range. Both the phase and magnitude of a narrow pulse's spectrum are essentially uniform over a wide range of frequencies. In the time domain, echoes in a device's pulse response are easily identified and removed, to that mesurements equivalent to those from an anechoic chamber can be obtained
NOISE INTRUDES. A weakness of pulse testing, however, is the rather low energy content of the test signal . The pulse stimulus is very brief while the response data collection time must be many times longer in order to provide low frequency information. This allows a long opportunity for noise to intrude into the measurement.
The test signal energy cannot be arbitrarily increased by "turning up the volume" into the unit under test. The crest factor, defined as the ratio of the signal's peak power to its average power, is very high for a pulse stimulus. You may clip your amplifier or drive your speaker into nonlinear operation with many watts of peak power, while only applying microwatts of average power to do battle with the noise. Due to the nature of most environmental noise and the need for long response collection times, the problem is most severe at lower frequencies.
In the original IMP measurement system (and in Fincham and Berman's original direct measurement technique ), noise is dealt with by employing repetitive pulse stimuli and averaging. The stimuli must be spaced enough in time so that low frequency information can be obtained and device pulse response can decay sufficiently. For every doubling of the number of time responses averaged the signal-to-noise ratio, neglecting quantization and coherent noise, should theoretically increase by 3 dB . The tradeoff is measurement time versus increased noise immunity. Averaging two responses gets you 3 dB improvement. Four responses get you 6 dB and eight yield 9 dB.
The simple averaging scheme works quite well for most speaker measurement purposes and for those who can afford the extra time and have reasonably quiet surroundings. However, if you want to increase the noise rejection by, say, 30 dB, you'll need to average over one thousand responses. That might take longer than you care to wait.
There are other broadband stimuli also characterized by flat specta magnitudes, from which frequency response or impulse response data can be derived, but which possess friendlier crest factors than does the impulse. One is the frequency sweep, or its optimized version, the chirp. This stimulus is not as simply generated and getting phase information can be difficult unless the system being measured is already definitely known to be minimum phase. It also takes a rather long time to obtain a full response measurement, as each frequency is is, in effect, measured separately. Another broadband test signal is random white or pink noise, in which the magnitude spectrum is generally flat or gradually sloped, but the phase is random. Due to this random factor, two channels (input and output) must be measured simultaneously to accurately determine a system's full response (though magnitude-only information can be obtained using a single channel).
PRECISION NOISE. A very convenient stimulus is pseudo-random noise, which is an analog version of a digital signal known as a Pseudorandom Number (PN) pattern or Maximum Length Sequence (MLS). PN sequences are used extensively in spread-spectrum radio communications, data encryption, music synthesizers, and even computer games. In pseudo-random noise, the magnitude of the spectrum is basically flat, while the phase is scrambled -- but not really random. The spectrum is absolutely deterministic and repeatable, like that of the pulse, so only a single measurement channel is required.
The MLS additionally has the property that its autocorrelation function yields an impulse signal and the cross-correlation function of a system's response to an MLS with the MLS itself is the system's impulse response. The meaning of that intimidating sounding statement might perhaps be clarified by an example:
A maximum length sequence of length seven, modified so that digital zeroes are represented as negative ones, is the series
After the last value, the sequence repeats starting back at the beginning. If a copy of this sequence is lined up beneath another, then corresponding values are multiplied and all the products are summed up, the result is the value 7.
Because of the periodic nature of an MLS, a circular shift of digits to the left corresponds to a time delay (any digits that run off the left end can be pasted back onto the right). If the same multiply-and-sum operation (a form of correlation) done in the previous paragraph is done instead with a time-shifted MLS as the lower sequence, the result will be the value -1. For example, here's a shift to the left of 5 places:
If you plot the autocorrelation result versus the time shift you'll see a large peak at zero shift (and at multiples of 7) and a small negative value when the shift is any other value.
This plot is quite similar to a periodic impulse signal. If the MLS we used had a length of 4095, the peak value would be 4095, it would repeat only after 4095 points and the "baseline" value would still be only -1.
CROSS-CORRELATION. Suppose now, that an N-point MLS were converted into an analog signal, say by making each value a sample into a D/A converter, and then fed repeatedly through a system to be tested. Then suppose you take the resulting response from the system, digitize N points of it, and do the multiply-and-sum-up operation of this data stream versus that of the original MLS stream for every circular shift (this is called a cross-correlation operation). A plot of the result versus time shift would be the same as the system's time impulse response -- or very nearly so, assuming that the system's impulse response decayed sufficiently within the time period of the N samples.
This is the idea behind Maximum Length Sequence testing. Several cycles of the N-length sequence are fed into the unit being tested, and the unit's response is subjected to some intensive number crunching. The result is the same impulse response you would get from an IMP type stimulus without the extra math, but with an important difference. The benefit of adding this more complex stimulus and all of the computational overhead is that the impulse response you get has the noise immunity of averaging the responses to N actual pulses. Additionally, any unintentional transient noise picked up during the measurement is spread evenly by the cross-correlation operation across the time interval and appears as more benign low background noise.
This is achieved all within the time period of a single data collection and two N-length sequences, allowing for one "warm-up" run. When N is equal to 4095 or more, the improvement in noise immunity and the reduction in measurement time is considerable.
The technique was used as far back as 1979, with a clever technique called the Fast Hadamard Transform (FHT) being applied to greatly speed up the correlation arithmetic around 1982 . MLS stimulus testing was more recently publicized in a paper , by D. Rife and J. Vanderkooy and has been made commercially available in the DRA Laboratories MLSSA system, which allows use of sequence lengths from 4095 points to 65535 points.
When using MLS to measure a device such as a speaker, an additional transformation step is now added to obtain the Impulse response from the device's response to the MLS stimulus. But this transformation can be done very quickly. In LAUD and IMP, the FHT is performed in a high resolution lossless integer format, preserving dynamic range and speeding the process even further -- the time taken to perform the FHT is nearly insignificant, particularly when compared to the time that would otherwise be spent in averaging impulse responses.
With MLS type measurements, it is possible to take repeated frequency response measurements of a loudspeaker, with update rates of once every two seconds or less. Such capability enables speaker developers to tune "on the fly", watching the results change as the crossover is adjusted rather than only after long, tedious, response measurements made using swept sine or pink noise/RTA methods.
1. Berman, J. M. and L. R. Fincham, "The Application of Digital Techniques to the Measurement of Loudspeakers", J. Audio Eng. Soc., vol. 25, 6/77, p. 370.
2. Rife, Douglas D. and John Vanderkooy, "Transfer-Function Measurement with Maximum-Length Sequences", J. Audio Eng. Soc., vol. 37, 6/89, p. 419.
3. Fincham, L. R., "Refinements in the Impulse Testing of Loudspeakers", J. Audio Eng. Soc., vol. 33, 3/85, p. 133.
4. Borish, Jeffrey and James B. Angell, "An Efficient Algorithm for Measuring the Impulse Response Using Pseudorandom Noise", J. Audio Eng. Soc., vol. 31 7/83, p. 478