### Non-Linear Amplitude Modulation (NLAM) of Loudspeakers

### What is it about?

The membrane of a loudspeaker travels at low frequencies considerable distances back and forth. In the following example the distance change from the receiver is assumed to be ±7 mm from the rest position.

The human eardrum, or the membrane of the condenser microphone used for the recording travel negligible distances of about 1 µm @ 25 Hz and 100 dB SPL.

This factor of approximately 10000 between the travel amplitudes of the membranes (and the sourrounding air molecules) between recording and play back leads to a Non-Linear Amplitude Modulation (NLAM) of the Sound Pressure.

Thought experiment:

One can imagine a high note being played back by the loudspeaker. Simultaneously a low note with considerable membrane travel amplitude is being played back. The location at which the high note is generated travels with the membrane position back and forth. Because of the inverse proportional law between sound pressure and distance, the high note is louder when the membrane comes closer to the receiver (ear, microphone) and vice versa.

This principle is a general one, described by the formula below and is valid for any note or frequency, even, when only one frequency is played back.

### Detailed consideration

The inverse proportional law tells us that the Sound Pressure (SP) is inverse proportional to the distance between sound source and receiver. The distance D at the rest position is modified by the time depending travel amplitude s(t) of the membrane.

The following time domain figure shows in **blue** the theoretical SP generated by the membrane and in **red** the resulting SP at the receiver considering the inverse proportional law for a 25 Hz frequency with 7 mm membrane travel amplitude. There is a difference between both SPLs shown in **green** (10 times magnified) at a receiver distance of 10 cm.

X axis: Time in Seconds, Y axis: SP in arbitrary units:

Obviously this difference is always negative at the maxima of the waveforms and zero at the zero crossings of the waveforms. This time depending difference represents the Non-Linear Amplitude Modulation.

From the above figure in the time domain, it is visible that the resulting main difference frequency is twice the fundamental frequency.

The Fourier Transformation of the waveform at the receiver shows that there are further harmonic frequencies generated with smaller amplitudes.

X axis: Frequency in Hz, Y axis: SPL in dB:

As can be seen there is a DC component, which is not further discussed.

The second harmonics is 29 dB below the fundamental frequency and the third harmonics is about 58 dB below.

### Consequences for the near field measurement of loudspeakers

We see that the moving loudspeaker membrane creates, because of the inverse proportional law, a non-linearity for the SPL at the receiver, which results in the generation of harmonic frequency components and intermodulation in case more frequencies are involved.

Microphone distances in the 10 cm range are usual for near field measurement of loudspeakers. As the membrane travel amplitudes are usually large at low frequencies the NLAM sets a limit for the measurement of harmonic distortions. The NLAM generates mainly second harmonics, significantly limiting the dynamic range for the measurement.

Repeating the above calculations for several frequencies shows a drop of the NLAM generated second harmonics by 12 dB per frequency octave, which corresponds exactly to the drop of the membrane amplitude as function of the frequency at constant SPL.

X axis: Frequency in Hz, Y axis: SPL in dB below fundamental frequency amplitude:

General rules:

- The NLAM is approximately proportional to the quotient of membrane travel amplitude and the distance but the exact dependency is non-linear. See figure below for a travel amplitude of 7 mm

- The NLAM is independent from the frequency.

- The NLAM decreases with growing distance

See figure below for a travel amplitude of 7 mm

- As can be seen the NLAM describing factor approaches quickly one with increasing distance

X-axis: Distance D in Meters, Y axis: NLAM describing factor 20 log (D/(D-7 mm-1)):

### Consequences for listening

The significant relationship by a factor of approximately 10000 of the membrane travel amplitudes between recording and play back leads to a non-linearity generating inter alias harmonic distortions. At deep notes and high SPL these distortions can easily dominate the distortions generated from a good loudspeaker at relatively small listening distances. This may e.g. be the case for studios.

For consumer loudspeakers, usually only High-End systems, especially with Motion Feed Back (MFB) generate low enough harmonic distortions such that the NLAM distortions become dominant.

As in a studio, one must not go below a minimum listening distance to achieve an undistorted sound experience.

### Elimination of the NLAM

As the generation of the NLAM is a physical effect, it can be compensated, however exactly only at a given listening distance. The elimination is based on an amplitude modulator in a Digital Audio Processor (DAP) using the membrane position as function of the time as control signal. Such it is possible to compensate (eliminate) the NLAM for a given distance. This requires, however, the exact knowledge of the membrane position at any point in time.

**AudioChiemgau** offers all electronic modules for building High-End motion feed back loudspeakers and a Digital Audio Processor which can eliminate both, the Doppler effect of the moving loudspeaker membrane as well as the generated NLAM.

*All measurments are done with our ModeCompensator *