Transfer Functions And Impulse Responses.

-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.