 # New procedure for the testing of exponential sweep signals with the aid of Hilbert transform

06.06.2013

Acoustic signals are analyzed with the aid of sound level meters, using so-called filter banks. These must be investigated with respect to their real-time capability. For this purpose, test signals are used which must in turn fulfill specific conditions. A new procedure for the testing of test signals for the real-time testing of filters can help to avoid incorrect test results.

Problem

For the spectral analysis of acoustic signals, for example on sound level meters, filter banks with constant relative bandwidth are used. These are, typically, octave and third octave filters. To check the real-time capability of these filters, the new standard IEC 61260 recommends a sweeping sinusoidal sound signal with an exponential frequency increase as a test signal. This signal subsequently provides the same energy to all individual filters of the filter bank, and, therefore, all filters should show the same level value. This is the case if no input data are lost during signal processing (real-time requirement). A prerequisite of such a test is that the test signal used meets the following requirements:

1. Constant level during the total sweep,
2. Exponential frequency increase in the sweeping sinusoidal sound signal.

For simultaneous testing of the two conditions, a new procedure based on Hilbert transform has been recommended and realized.

Hilbert transform

This mathematical procedure is based on the Fast Fourier transform and provides a complex time function for a signal, the real part of which is the input signal. Together with the calculated imaginary part of the Hilbert transform, an envelope and, thus, a time-related amplitude signal can be generated.

If the signal is – at any time – characterised by only one frequency, the phase between the real part and the imaginary part can be utilized to calculate an instantaneous frequency which indicates, at any time of the analysis, the momentary frequency of the signal. In the case of an exponential sweep signal, the logarithmic representation of the frequency with respect to time furnishes a straight line (Figure 1). If a test signal deviates from the ideal form, the filter test furnishes an inaccurate result. Figure 1: Logarithmic representation of the instantaneous frequency with respect to time in the case of a mathematically exact exponential sweep signal.

Example

Using the example of a commercial signal synthesizer it can be shown that a correct filter cannot be classified as real-time capable when an erroneous test signal is used. In the case of this synthesizer, the exponential sweep signal is approximated by sections which increase linearly and piecewise. Proof of this can be furnished when the Hilbert transform is used (cf. Figure 2). Figure 2: Linear representation of the instantaneous frequency with respect to time of a linearized exponential sweep signal.

The use of such an erroneous test signal inevitably leads to an incorrect assessment of a real-time capable filter bank (cf. Figure 3). The deviations from the expected result, i.e. constant level L in all third octave bands, are considerably higher than the resolution of the display of the measured values of 0.1 dB. Figure 3: Incorrect result of a filter test by linearized exponential sweep signal.

### Contact person:

Ingolf Bork, Dept 1.6, WG 1.63, e-mail: ingolf.bork@ptb.de 