- Details

Laboratory measurements of loudspeaker drivers are necessary for speaker system design. They involve theoretical background of humans engaged and measuring equipment. Thus many people believe it is a lost case for them.

However things are different. During the last years lab equipment is not as expensive as we think it is. With less than 500 USD one can get a quality measuring device that actually works as a computer peripheral; a PC 'addon' device. Surely if one intends to build a speaker system NOT just once, the price of this equipment will not discourage him or her.

In the past there has been a long discussion on the space requirements of an acoustics' lab environment, leading to the conclusion that quality measurements can only be carried out in a very large hall rather than in a conventional living room. The low frequency tail of acoustic measurements is jeopardized in typical 3m room height. Fortunately there are more than one alternatives to overcome this bad situation. In fact these alternatives have become mainstream practice even for industry engineers needing repetitive lab access and assessment of design progress.

**-What about theoretical background ? Is there any off-the-shelf alternative too ?**

No there isn't. Theoretical knowledge is a prerequisite that can not be exchanged with automatic setups of measuring devices. Measuring the gain response of a filter or the SPL response of a speaker system is not like taking pictures with a portable digital camera set to 'AUTO'. Fortunately there is a level of engineering math accessible by most high school graduates that can make things easy provided that one wishes to work for it.

Having spent years in the lab myself, taking electroacoustical measurements with different hardware (older Bruel & Kjaer gating system, MLSSA, PRAXIS) and for different reasons (speaker design, research on sound diffraction, audio amp power assessment, harmonic distortion), I will attempt to provide the required math substrate in a series of simplified articles and worked out examples.

I am confident that mixing several extensively illustrated examples with mathematical principles will successfully compensate for this site's visitors' lack of previous engineering background.

Material is divided in the following sections :

- Responses And Impedances As Complex Transfer Functions.
- Transfer Functions As Sine Sweeps in Reverberant Spaces.
- Transfer Functions And Impulse Responses.
- Impulse Responses As MLS Noise in Reverberant Space.
- Improving Low Frequency Accuracy in Reverberant Spaces

**-What do we want to measure in the lab ?**

We can't initiate the design process unless we have the following data stored in our computer hard disk:

- The SPL and phase responses of the loudspeakers drivers to be mounted on our final speaker system.
- The SPL and phase response of any bass reflex port or vent present on our enclosure baffle.
- The magnitude and phase of the (complex) impedances of all drivers.

During the design process we may also need to measure the crossover filters' (voltage) gain/phase response or the gain/phase response of an audio power or pre- amplifier.

- SPL responses are ratios of sound pressure level (detected by our lab microphone) to voltage signal (fed to the loudspeaker driver).
- Gain responses (of passive or other active sort of filters or of amplification units) are ratios of output to input voltage.
- Impedance of a component or a circuit or a loudspeaker driver is the ratio of voltage applied across its terminals to the current flowing into it. In lab measurements this current is usually monitored by the voltage drop across a small fixed (and known) resistor in series with the component of which the impedance we seek. So again impedance is measured by a voltage gain.

In all these cases the devices or components involved are assumed to be Linear and Time Invariant systems (LTI systems' theory plays a key role in Electronics and Engineering). In these ratios we take the quantity in the nominator to express the system's output and the one in the denominator to stand for the system's input or stimulus. LTI system theory calls these ratios 'Transfer Functions'.

Theory of LTI systems states that when these systems are stimulated by an oscillatory signal of frequency f their output is also an oscillation of frequency f. Output oscillation has a different amplitude and a phase difference with respect to input.

Let input be an oscillation x(t)= Xo * cos(ω*t), where Xo is its amplitude and ω the angular frequency 2*π*f.

Output will be of the form y(t)= Yo * cos(ω*t+φ), where Υo is the respective amplitude and φ the phase difference or delay between output and input oscillation. Yo and φ change with frequency.

Generally speaking LTI systems do two things: they change the amplitude and shift the phase of input oscillations.

The reason for writing these two expressions is only to prove that when we try to get their ratio and define the associated Transfer Functions we end up with a problem; the ratio of two oscillatory cosine functions gives nothing.

The only way to escape this problem and define useful transfer functions is to move this discussion to the field of complex numbers.

Now input oscillation becomes X(t) = Xo * e** ^{iωt}**, where constant e stands for the base of natural logarithms e=2.1718..

(Perhaps it would be a good idea to check your high school algebra for the properties of this exponential that relate it to cosines and sines : e** ^{i * ..}** = cos(..) + i sin(..) ).

Output becomes Y(t) = Yo * e** ^{iωt+iφ}**. The ratio of output to input is easily defined :

H = Y/X = Yo/Xo * ( e** ^{iωt+iφ}** / e

So a system's transfer function H(f) is now defined to be a complex function of frequency.

The Algebra of complex numbers states that a complex number can be represented in two ways :

- either in the usual form : real part +imaginary part *i , or
- in the exponential form : magnitude * e
.^{i*phase argumant}

Both forms relate to each other : a complex number's magnitude equals √(real^2+imag^2) and its phase argument is the angle having a tangent equal to imag/real.

Let us have some examples of all these absurd notions.

A filter is fed with a single frequency (100Hz) oscillatory signal of amplitude 2 Volts peak. Its output oscillation was found to have an amplitude of 1 Volt and a phase lead of 45 degrees.

Input is denoted by complex number X=2 * e** ^{i*2*π*100*t}**.

We write 45 degrees as π/4 radians and filter's output becomes Y=1 * e** ^{i*2*π*100*t+i*π/4}**.

Taking Y/X for the filter's transfer function at this frequency we get H=Y/X= 0.5 e** ^{i*π/4}**.

Magnitude of the transfer function gives the gain response of this filter device : 0.5 or -6dB

The phase argument of the transfer function gives the phase difference that this device establishes between its output and input oscillatory signals : π/4 or +45 degrees.

Of course a transfer function changes with frequency; so change its magnitude (gain response) and its phase argument (phase response).

Now let us have an example of impedance.

A loudspeaker driver is fed with a single frequency (400Hz) oscillatory signal of amplitude 3.5 Volts peak. Its current was found to be an oscillation with a peak value of 0.5 Amperes and a phase lag of -30 degrees with respect to voltage.

Input voltage is denoted by complex number V=3.5 * e** ^{i*2*π*400*t}**.

We write 30 degrees as π/6 radians and current becomes I=0.5 * e** ^{i*2*π*400*t-i*π/6}**.

Taking V/I for the drivers's impedance at this frequency we get Z=V/I= 7 e** ^{+i*π/6}**.

Impedance magnitude gives the driver's absolute impedance at this frequency 7 Ohms, which is rather typical for '8-Ohm' woofer drivers at 300-400 Hz.

The phase argument of impedance gives the phase difference that this device establishes between its current and its voltage at this frequency. In this case voltage leads the current (or current lags the voltage) by +30 degrees. This behaviour is said to be inductive. Actually most woofer drivers behave inductively above 300Hz.

Impedance changes with frequency; so change its magnitude (absolute impedance) and its phase argument (phase lead or lag between voltage and current).

**-How complex valued transfer functions relate to our lab measurement task ?**

The main idea behind lab measurements is to have the measuring equipment do the following steps:

- Establish an oscillatory stimulus (input) of fixed amplitude to the device under test (DUT),
- Measure the (oscillatory) output of the DUT in terms of amplitude and phase difference,
- Compute and store the complex value of the associated ratio (transfer function) at this frequency.
- Change frequency and repeat previous steps 1,2,3.

Some useful notes :

- The complex value of a transfer function is stored as a pair of numbers : either in the form [real part,imaginary part] or [magnitude,phase argument].
- When measuring impedance the DUT receives an oscillatory voltage across its terminals and current oscillation is measured in terms of amplitude and phase difference.
- Transfer functions are therefore stored as consecutive text lines; each line holds three numbers, one for the frequency value and two for the (pair of numbers of the) transfer function measured.

**-How does our measuring equipment realize these four steps ? Can it detect amplitudes and phase differences ?**

Measuring systems sweep a frequency range defined by their user, us. We say that they make a 'sine sweep'; sine stands for the sinusoidal form of their stimulus. In fact they create a special oscillatory (voltage) signal with a frequency changing in time. This signal is a cosine oscillation with frequency sweeping from a low frequency limit up to a high frequency limit. It is called 'chirp'. Figure at the right illustrayes how frequency changes with time in a chirp. However its voltage amplitude remains fixed as frequency increases (at least that is what the electronic circuitry of our measuring device is trying to do). |

While chirp stimulus is fed to the DUT our measuring device monitors the latter's output. The DUT's output is also of an oscillatory shape with increasing frequency. However its amplitude changes as stimulus frequency advances.

A DUT is stimulated with a linearly swept chirp voltage of constant 1 Volt amplitude shown at the figure on the right. | |

Figure on the right illustrates our DUT's output signal. A few moments after startup output drops below 1 Volt and then gradually rises to a maximum of 2 Volts. This means that at low frequencies gain response drops below 1.0 (or 0dB) then rises to a value of 2 (+6dB) at mid-frequencies. At the right-side (ie. at higher frequencies) of this figure amplitude drops to 0.5 Volts approximately. This gives a high frequency gain of 0.5 (-6dB). |

Clearly a chirp is the best way to unfold a DUT's transfer function (either voltage gain or impedance). However measuring devices can not directly detect and store changing amplitudes and phase differences hidden in the DUT's output signal.

What our measuring instrument actually does is to digitize both input and output signal and process them in order to reveal their complex transfer function. This involves a complicated mathematical operation called 'Fourier Transformation'.

Let us describe the actual steps taking place in our measuring system:

- A chirp voltage stimulus of fixed amplitude and predefined frequency range is sent to the DUT for a given period of time.
- The DUT's input and output are digitized for the same time interval. This involves two-channel Analog-to-Digital conversion.
- When chirp stimulus stops both digitized signals are Fourier Transformed producing two complex series of numbers. These complex numbers represent magnitude and phase argument of input and output signal for various frequencies.
- These series are then divided producing the transfer function. So at the end of this division we get a series of (complex) numbers each one describing the magnitude and phase argument of the DUT's transfer function at one frequency.

At this point we must pay attention to some details involved in signal acquisition and processing:

Frequency range of the chirp stimulus must match the frequency range we need to cover in our measurements for this DUT.

Amplitude of the chirp signal must not be very low (to allow for the noise to corrupt the corresponding output signal) or too high to drive the DUT into distortion.

Our hardware and its operating system must support true two-channel signal acquisition. A-to-D conversion should also be of increased accuracy : 24 or 32 bits.

Sampling frequency is of major importance. Its half value will define the highest frequency value at which our measurements will stop. A sampling frequency of 22 kHz will give us poor measurements up to only 10-11 kHz. In Hi-Fi speaker design we need sampling frequencies not less than 48kHz.

Fourier Transform should be (adjusted to be) carried out on a very large number of frequency points. This will increase the accuracy of the SPL or Gain or Impedance graphs at low frequencies.

**-Are we ready to face the challenge of lab measurements ?**

Transfer functions involving filters, amplifiers and generally voltage ratios can be successfully treated by sine sweeps. The same holds for impedance measurements.

However when it comes to wave phenomena as in the case of SPL measurements we must not forget that our lab microphone does not receive only our loudspeaker DUT's waves. Unfortunately there are hundreds of strong reflections (by walls and other reflecting surfaces) arriving at the microphone along with the primary wave emission of our DUT. These reflections add to the primary sound pressure wave thus corrupting it. As a result our measuring device actually computes the overall transfer function of our DUT combined to the Reverberant Space of our laboratory room.

Although industry engineers have addressed this problem in the past by using very large halls as measuring chambers, further action could be taken in order to control and overcome this problem in a relatively small lab room.

**-Are Transfer Functions (ie. gain/SPL and phase responses) the only tools to handle systems and devices in Electroacoustics ?**

From one point of view, yes, they are unique for they analyze the behaviour of a system under oscillatory excitation. We say that transfer functions describe systems in the frequency domain which is highly perceptible by people.

However reflections occuring in a typical room environment (usually called reverberation sound field) change the transfer function that a sound emitting device originally features. If such a sound source is placed in a vast space without enclosing surfaces our lab microphone will receive no reflected sound waves. Such a situtaion is described as Free-Field or Anechoic. |

**-Why should we be interested in measuring loudspeaker transfer functions in anechoic environment ? After all loudspeaker will eventually operate in reverberant space.**

A speaker system is expected to operate in a reverberant room that is completely unknown prior to its design. Its placement in such a room also plays a key role to the determination of reflection timing and strength. So our design should change according to both room properties, speaker placement and listener's position.

What is also most interesting is that human brain does not perceive reflections through their overall contribution to sound wave field. Instead our brain and ears identify them separately. For this reason engineering practice prefers to design a speaker system with specific targets on its anechoic behaviour.

In other words if this speaker reaches a specific anechoic transfer function, its behaviour in reverberant space will be highly appreciated by human listeners for various (not all) listening positions and for a wide range (not all again) of rooms. Speaker design must therefore be based manily on anechoic measurements. Echo removal can only be done in time domain where each reflected wave arrives at our lab microphone at a discrete time instant. Before discussing this issue we have to develop the equivalent of transfer functions in time domain.

**-What sort of characteristic function of every LTI system (like loudspeaker drivers or even amplifiers) can be used in time domain ? How does it relate input and output signals ?**

This new tool is called Impulse Response (IR) of the system. It is a function of time and describes how the device or system responds to an abrupt pulse-like input stimulus.

Figure on the right illustrates the whole idea. Input is a very narrow pulse stimulus of a very small duration, suddenly occuring at zero time instant. Such a stimulus is usually called dirac signal δ(t). The device or system responds to such an input by creating an output signal according to its internal components. This output is defined as the Impulse Response h(t) of the system and integrates all the structural properties of the system without exception. |

In this figure the system is a highpass filter. Its impulse response is shown as measured in a digitized form (bullets at a specific sampling time interval) while a continuous line has been used for viewing purposes.

Now let us study the impulse response of a complete speaker system measured at a distance of 1m ie with our lab microphone placed on tweeter axis in front of the speaker and at a distance of 1m. Speaker used was one of old Quart models measured with and old MLSSA measurement system back at Christmas of 1991. Speaker was excited with an electric voltage of dirac-shape at time instant t=0. Microphone recorded the sound pressure wave producing some mVolts of electric signal as can be easily observed on the vertical axis. These mVolts correspond to certain mPascals of pressure not shown here. |

Input stimulus excited instantly the crossover filters and the latter responses excited the two loudspeaker drivers. Although the voice coils of these drivers responded instantly to the electric stimulus by vibrating the cone, the produced sound wave took some time to travel the distance to the microphone. At a speed of almost 330m/sec sound needs about 3milliseconds to travel this distance. That is why the impulse response of all sound sources measured through microphones at a distance of 1m, always features a silence period of 3msec before the outburst.

Experienced engineers will also notice the following hidden information in this IR measurement :

The IR's short-time tail in the time interval 3.1-3.6 msecs suffers from a small dense oscillation giving this black shade area. This is a very high frequency resonance usually called ringing and almost always created by metal dome tweeters around or above 20000 Hz ! Sometimes it is perceived as treble harshness.

IR's tail fades out as time advances. Loudspeaker cones and domes eventually stop vibrating. However our microphone seems to record a newly arrived wavefront at an instant of t=8.6-8.7 msec. It is a sound wave that arrives at our microphone position about 5.6msec after primary speaker wave. An engineer understands that it is a reflection that has travelled an additional distance of 5.6msec X 331m/sec = 185-186 cm. In most cases this is due to a reflecting surface placed some 186/2 =93 cm above or below speaker enclosure. Yes you guessed right, it is the reflection due to the floor.

If we extend time axis in an IR measurement we will have the chance to detect a lot of reflections along with their time and space stamps. Figure on the right gives us a rough estimate of how the IR of a sound source in a reverberant space incorporates reflections.
(figure drawn from a very interesting website of Art Ludwig) In this IR graph microphone seems to record a sound flight delay of 9msec ie a distance between microphone and acoustic source of about 3m. |

**-IR's seem quite interesting but what how can we use them to solve our initial problem of echo removal ?**

When an impulse response is acquired and stored all we have to do is to study it for reflections and simply replace the values of the respective 'bad' time intervals with zeros. After all the long term tail of an impulse response fades out to zero. We will come back to this trailing zero impregnation later on.

**-Eventhough a corrected IR seems useless if we can't use it to get the familiar frequency domain transfer functions. **

Fortunately we can apply Fourier Transformation to an impulse response and get our complex-values transfer function along with its amplitude and phase responses. In general time and frequency domains are related through Fourier Tranformation. Such a mathematical operation is usually done by our measurement system software without any detectable delay or effort. Figure on the left illustrates this relationship. FFT stands for the computer version of the Fourier transform, the Fast Fourier Transform algorithm. |

Now let us summarize:

**Devices and systems can also be analyzed in time domain through their characteristic impulse responses (IR).****The IRs of electroacoustical systems easily reveal the reflections of a reverberant space.****A reverberant IR can be corrected (echo removal) by replacing its corrupted parts with trailing zeros.****An anechoic IR acquired through such a correction scheme, can lead to an anechoic version of the system's transfer function via plain Fourier transformation.**

**-So how do we use sine sweep measurements for speaker design ? **

- We use sine sweeps to acquire the magnitude and phase of loudspeaker impedances. These measurement involve only electrical voltage signals without corrupting reflections.
- We also use sine sweeps to get the complex transfer functions of all sound emitting sources (loudspeakers and bass reflex ports), ie all SPL and phase responses inherent to the speaker system under design. These measurements are corrupted by the reflective surfaces of the reverberant laboratory space. We apply an inverse Fourier transformation to them in order to get the respective time domain impulse responses.
- Then we follow some steps to achieve echo removal (better regarded as echo suppression) to these IRs and finally go back to frequency domain by one more Fourier transform.
- The overall process, ie. this interdomain journey, depends highly on lab space geometry and can not be automated. We must be able to get through it without loss of focus.

**-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:

**MLS sequences can be used to measure a DUT's transfer function or impedance. However they are mainly used for SPL measurement in reverberant spaces. Sine sweeps are preferred for ordinary electrical and impedance measurements.****The length of the required MLS sequence, the averaging scheme and the IR acquisition length are carefully selected prior to measurement according to certain rules based on DUT and lab room properties.****Reflections can be easily located on the Energy Time Curve (ETC) produced by the impulse response.****The anechoic IR segment confines the useful frequency range of the resulting SPL response.****The anechoic IR segment is treated by window functions to minimize FFT errors due to corrupting reflections.**

Without having a reliable SPL and phase response in the 20Hz-400Hz range no serious design can even be attempted for the crossover network filters. In theory all we have to do is to realize our measurements in a ..concert hall. Its large dimensions will 'move' all reflections away to time instants of 60msec or more. Anechoic segment will increase to 60msec or more providing a critical frequency of less than 32Hz !

Moving away reflections in time is also accompanied by increased attenuation of their amplitude. We must not forget that soundwave amplitude decreases rapidly with distance travelled away from the sound source. In open space (what we call spherical space or 4-pi space) amplitude is proportional to 1/r** ^{2}** (r is the distance travelled) while in a typical room this law is modified to 1/r approximately.

But we can't have a concert hall available every time we want to measure a speaker component or a complete system.

**-What about moving the microphone closer to the DUT ?**

Let us study such a case. The following figure describes a typical room with a height of 2.8m. We place the loudspeaker to be measured (mounted on its enclosure of course) in half this height at 1.4m. in this way floor and ceiling reflection will arrive at the microphone position at the same time which is the best scenario for our anechoic segment. We place our microphone at a distance of 1m for conventional reasons (let us not argue that choice for the moment).

As stated above our anechoic segment is about 6msec and our lowest reliable frequency is about 330Hz. Now let us move the microphone closer to our DUT driver. Let us place it just 20cm away from the driver's dustcap.

What we get is only an 80Hz improvement of the lowest reliable frequency as explained in the figure above. In addition primary sound gets louder. What we mean is that this setup increases the SPL of the direct sound wave while the floor-ceiling reflection maintains its previous SPL. In the IR plot the reflection will seem weaker than the direct response at 0.6msec. Weaker reflections corrupt the FFT results much less. However the problem remains.

**-So what can we do to improve drastically low frequency measurement accuracy ?**

The idea is not to broaden our anechoic segment in such a way as to drive our measurement down to 10Hz. The idea that has survived such arguments and technical discussions is to get another measurement, a dedicated one, that records the huge intensity of the direct sound wave emitted in the vicinity of the dustcap. These near-field measurements include reflections but at a level which is unimportant when compared to the SPL of the driver's sound wave. What is interesting is that these measurements have restrictions at high frequencies. So what we actually do is to use the curve of a near field measurement by splicing it to the curve acquired by a conventional measurement at 1m. Excellent descriptions of such a procedure are given throughout the internet by audio enthousiasts, amateurs or engineers. More on this in our design examples. |
(picture drawn from MH-audio) |