The Art Of Noise or
high performance audio measurements with noise reference

© 2010-2018 Marcel Müller

Abstract:
When characterizing transfer functions of audio equipment, measurements with noise reference lacks support for characterizing non-linearity. The method shown here overcomes this restriction by choosing an appropriate reference signal combined with an appropriate analysis method. This enables you to measure the double logarithmic, complex frequency response, the harmonics and even two distinct channels with one measurement.
The method is suitable for measurements of the speaker or room response as well as for network analysis of analog filters or crossovers.

The pitfalls of using a noise reference

SNR

The signal to noise ratio in audio measurements is usually a compromise. On the one hand you need many samples for your FFT to get reasonable frequency resolution at low frequencies. On the other hand this resolution is far to high at high frequencies. In effect the SNR degrades with the frequency resolution, because each frequency channel gets less energy.

Non-linearity

Speakers are non-linear devices. Only at low amplitudes the response is approximately linear. Usually one wants to know about the non-linearities, at least for the 2nd and 3rd harmonic. This cannot be done with ordinary noise references because the harmonics are masked by the intensity in the reference signal.

Getting the optimum result

Synchronization

First of all we must get rid of any window functions, if we had not already done that. The solution is quite simple. All we have to do is to make the reference signal periodic. If its cycle time matches the length of the FFT in the analysis there is simply no need for a window function, because the reference spectrum is discrete in the same way as the frequency channels of the FFT.

Logarithmic frequency axis

In Bode Diagrams the frequency scale is usually logarithmic. Choosing equidistant points in the frequency is not appropriate for getting smooth graphs with a logarithmic frequency axis.

There are two ways out of this.

1. We could choose frequencies in approximately logarithmic distances rounded to the closest frequency in the FFT result. If we fix the frequencies before the measurement the reference signal could be modeled in a way that the energy is only spread to this certain frequencies, and the analysis could discard any other frequencies. This does significantly improve the SNR of the remaining points.

2. The energy distribution could be adjusted e.g. by choosing pink noise. However, this degrades the SNR at high frequency end. To compensate for that, we could join the response of adjacent frequencies to get bins of an approximately constant f_max/f_min. But we have to be careful. If we simply average the complex coefficients the amplitude decreases, because of phase noise. So we should do the averaging in polar space (amplitude and phase). This is less sensitive to systematic errors. But still we have to be careful, because of the cyclic nature of the phase. The simple average will badly fail if the phase jumps over it's limits. We have to unwrap the phase before the averaging.

Large delay

The phase unwrap to determine the group delay might still be not that easy if the signal response has a large delay. In this case the phase unwrap of a noisy signal could fail. We can overcome this if we compensate for large delays before we unwrap the phase. To get a reasonable value for the delay if it is not a known constant of the setup, a full spectrum cross correlation with the reference signal is helpful. The average delay for the cross correlation result c_i is then:

$\mathit{delay}=\frac{N}{2\mathrm{\pi }{f}_{\mathit{samp}}}·\arg \sum _{i=1}^N{c_i}^2·{\mathrm{e}}^{2\mathrm{\pi }\mathbf{i }i/N}$

Harmonic distortion

To get information about harmonic distortions it is essential that the reference signal does not contain any intensity at the harmonic frequencies. This could be assured by using only frequencies that are prime numbers multiplied by some base frequency. Now all multiples of the analysis frequencies have no initial intensity. So any intensity found at these frequencies have to be harmonics. But we still have to take care, since the harmonics also must be unique in their base frequency.

Furthermore the definition of harmonics only apply to sinusoidal wave forms. In general the result is intermodulation. This generates sums and differences of the frequencies in the source spectrum. It is almost impossible to avoid all of them. But a few things can be done: i.e. if you choose only odd frequencies all sums and differences of two frequencies appear at even frequencies. Unfortunately this is only sufficient for 2^nd order intermodulation.