-Can we measure impulse responses directly or such a measurement has no advantage over sine sweeps ?
In several cases measuring the impulse response of an LTI system or device proves to be advantageous. Historically things followed a different route: The first acoustical measurements were based on sine sweeps without the ability to compute impulse responses via FFT (Fourier Transform). Measuring equipment was completely 'analog' without a PC barebone to compute the time domain IR and allow for echo suppresion. Chirp bursts were released by analog electronics and special filters were responsible for the detection of (amplitude and phase of) microphone signal within specific time intervals. These time windows allowed for the reduction of reflection signals. This (time) gating technique led speaker industry up to 1989.
In 1989 Douglas Rife introduced a new type of signal stimulus suitable for acoustical measurements : the maximum length sequence (MLS).
|It was a noisy signal consisting of spikes within specified (voltage) limits. If one could apply a special mathematical operation called crosscorrelation between output and input signals for a device (stimulated by MLS noise) the device's impulse response would be generated with minimal errors (minimal electric noise or nonlinear distortion). Crosscorrelation also proved to be invulnerable to unwanted sound noise in the laboratory during measurement.|
Apparently 1989 was the right moment of time since PC industry had grown enough to supply hardware able to produce MLS signal, record two channels (DUT's input/output), compute complicated mathematical operations and visualize time domain plots. MLSSA analyzer was born by that time. In fact it was the first time an engineer could observe time domain IRs along with their reverberant tails. The idea of echo detection and suppression developed quickly. Contemporary measuring equipment jump from one domain to the other quite easily since mathematical operations are a piece of cake for modern CPUs. This fact allows for the use of both sine sweeps and MLS stimulation. The choice is up to the equipment's user. MLS stimulation is even used at plain electric measurements of crossover filters or driver impedance where no reflections are present.
-Can we elaborate on echo removal applied on an impulse response ?
Let us give a complete example of a real world measurement of real world's loudspeakers.
We used a PRAXIS measurement (PC based) system with a calibrated microphone. We applied an MLS stimulus to a FOCAL woofer and a SEAS metal dome tweeter. For each measurement we used MLS sequences 128ksamples each (131072 samples). Usually signal points or samples are a power of two so that all computations involving FFT and inverse FFT algoritmhs are fast. These algorithms are radix-2 ie. can only be applied to number of points that is a power of two : 4-points,8-points,16-points,..512-points,..8192-points,..65536-points, etc.
Sampling frequency of our hardware was 48kHz ie. 48000 samples per second. As a result our 131072 samples' MLS sequence corresponded to a time period of 2.731 seconds approx. A long MLS stimulus is often needed so that very late reflections that arrive at microphone position have decayed almost completely before the sequence ends. In this way no significant reflections raised by one MLS sequence enter the time span of the next one, corrupting all computations and thence the resulting impulse response or transfer function plot.
An estimate of sound wave energy decay of a space is the RT60: the time needed for sounde energy to decay by a value of 60dB. An MLS stimulus sequence should last longer than the RT60 value of a measurement environment.
In order to suppress any electric or other sort of measurement noise we adjusted PRAXIS to repeat the MLS stimulus 16 times and take the average. This sort of averaging technique is very popular in lab practice. Our lab room was rather small in dimensions : 5.5m x 7.0m x 2.7m.
MLS stimulus amplitude is usually produced by a DtoA hardware quite unable to furnish adequate current and power to drive our loudspeaker drivers. A power amplifier is connected in between. The amplifier's gain is usually adjusted so as to provide a peak voltage of a few volts across the DUT terminals when the MLS stimulus is active. If this peak value is too high it can either burn our tweeter or generally raise distortion products.
Figure below illustrates the very first 30msec part of our long tweeter impulse response. The first two reflections are indicated by markers 2 and 3. Vertical axis is given in pressure mPascal units. Impulse response starts again at 3msec approx.
By theory tweeter impulse responses contain oscillations that decay almost completely within a few milliseconds. So what we see in this plot is actually a signal full of reflections some of them being dispersed in time, some of them being concentrated. A quantity that can visualize the instantaneous energy of an impulse response is the Energy Time Curve-ETC. Next figure gives a very good picture of this decay.Reflections are located in time instants at which energy stops decaying and rises again as time advances. Markers 2,3 and 4 describe the first three reflections of significant amplitude.
Now let us get the SPL response via an FFT:
This plot is computed by an FFT algorithm applied to the overall 2.7sec impulse response of our tweeter. It is highly corrupted by the vast number of reflections.
Now let us study the impulse response part that includes the first reflection (3 to 19 msec approx.). We apply a a multiplicative window function to our impulse response samples. A window is a series of plain decimal numbers in the range 1.00 to 0.00 which are multiplied with the samples of our impulse response. Figure below illustrates some popular window functions used in Signal Processing. The part of a window series with values near 1.00 leaves the corresponding IR samples unaffected while the part with numbers close to 0.00 minimizes their importance. A window is a weighting function. In acoustical measurement we use Hamming, Hann, Blackman Harris or the conventional Rectangular window (not shown here). The rectangular window keeps a 1.00 value through its time length and falls abruptly to zero outside it. Hamming window function decrease slowly from 1.00 to 0.00 while Blackman Harris drops quickly.
When we want to apply an FFT isolating a part of an impulse response sequence we first apply a window function to that part of the IR sequence. In this way the time interval where the useful part of th DUT' impulse response lies, is multiplied by window values close to 1.00 (thus left unaffected) while the time interval just before the instant at which the unwanted part begins is multiplied by window values close to 0.00 (thus fading out. All the rest of the IR sequence where the ubwanted reflections occur is multiplied by zero.
Let us apply a rectangular window to the section 3-19msec that includes one reflection only.
Our SPL plot is getting better though the reflection included in our analysis still does some serious damage. Often a smoother (than the abrupt rectangular window) window function like the Blackman Harris is used. Figure below was computed with such a window function having an improvement to some extent.
Finally we decide to restrict our analysis to the anechoic part of the tweeter's impulse response 3 to 7.7msec approximately :
We apply this time the Blackman Harris window (for better results) and let the FFT produce the drivers anechoic SPL response :
Our driver has a 90-91dB SPL sensitivity (dB SPL per 2.83Vrms) in the frequency range 1.5 to 5 kHz where it is expected to co-operate with a midrange or a small woofer driver. Note that measurement does not exceed half the sampling frequency point (24kHz).
-What about a woofer driver measurement ? Is it practically the same ?
No it is not. Let us have an example of a 10-inch driver by FOCAL. Again we used a 16-averaging scheme of an MLS sequence 2.731sec long (48kHz sampling frequency, 128ksamples).
Woofer impulse responses contain low frequency oscillations in their 'tail' so one should be very careful to use length IR sequences capable of containing at least two complete cycles of the lowest frequency present in the speaker system. Generally speaking the lowest characteristic frequencies present in a woofer+vent impulse response is approximately the vent resonance frequency. Let it be 20Hz in our example so that its period is 1/20 = 50msec. Allowing for two such complete oscillations to be present in IR measurement we end up with a 100msec requirement for our IR measurement. Surely our 2.7 sec IR length exceeds this limit by far. Figure below illustrates an 80msec segment of our woofer+vent impulse response measured at a distance of 1m away on tweeter axis. Low frequency oscillations of our woofer+vent impulse response's 'tail' along with reflections compose this measurement.
The ETC (Energy Time Curve) diagram of the first 20msec in the figure below reveals that the first reflection arrives just after 7.3 msec, roughly 4.3 msec after IR onset at 3.00msec. This floor+ceiling reflection comes too early allowing for a trully anechoic segment of only 4.3msec. Unfortunately this is too short to let us measure our woofer+vent behaviour in very low frequencies.
Applying our previous consideration of the critical 2-cycle oscillation time span we induce that an anechoic IR segment of length dT defines a critical frequency fc=2/dt under which measurement should be discarded, disregarded, ignored as erroneous.
Figure below zooms to the first 10msec part of our example IR. A marker indicates the end of the useful anechoic segment to be utilized for further processing.
In this case the time span of our anechoic segment, 7.30 - 3.00 = 4.30msec defines a critical frequency of fc=2/4.3 kHz = 465 Hz.
Applying an FFT with a Blackman Harris window produces the following SPL response of our woofer+vent example. Although PRAXIS stops the SPL measurement at 233 Hz we must keep in mind that our measurement is safe above 450-465 Hz. A marker indicates that our woofer has a sensitivity value just above 91dB on the enclosure used. Resonances above 1kHz are due to the stiff kevlar cone material used by FOCAL. Our crossover network should make sure that these frequencies be strongly attenuated at final design.
-What about phase responses ?
Phase responses are delivered by the same FFT algorithm that gives an SPL response. We chose not to present them here and to pay them special attention in our design examples.
Now let us summarize: