### Transfer Functions As Sine Sweeps in Reverberant Spaces.

-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:

1. Establish an oscillatory stimulus (input) of fixed amplitude to the device under test (DUT),
2. Measure the (oscillatory) output of the DUT in terms of amplitude and phase difference,
3. Compute and store the complex value of the associated ratio (transfer function) at this frequency.
4. 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:

1. A chirp voltage stimulus of fixed amplitude and predefined frequency range is sent to the DUT for a given period of time.
2. The DUT's input and output are digitized for the same time interval. This involves two-channel Analog-to-Digital conversion.
3. 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.
4. 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.

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