What does settling time in Fractional Octave analysis mean?
Octave spectrum are created from special filters (band-pass) which are designed using equations from IEC 61260-1:2014. These filters can be used to evaluate frequency content of signals.
Fractional Octave Spectrum are commonly used in Audio and Acoustic analysis. Most commonly used are 1/3rd and 1/12th Octave Spectrum. 1/3rd Octave filters between 1000 Hz to 10,000 Hz are shown below. After designing these filters, the signal or the data can be passed through those filters.
As an example, let us say that the signal has the following frequencies and amplitudes:
· 50 Hz sine wave at an amplitude of 10
· 100 Hz sine wave at an amplitude of 5
· 1000 Hz sine wave at an amplitude of 3
· 5000 Hz sine wave at an amplitude of 1
The filtered signal might look like the graphs below. The output is usually amplitude modulated. This is due to the reactive component, which is the filter, which requires a short period of time to output the steady state values. The period it takes for the transient waveform to settle within a given percentage of steady state is known as settling time.
Why do we have settling time?
Filters are defined in the Laplace plane using Zeros-Poles-Gain. In the s plane, the transfer function (TF) for each pair of poles can be defined through an expression. The normalized transfer function of the filter will be a product of the pole pair expressions.
By applying Laplace Transform to the filter TF and the input signal, the frequency domain output can be obtained.
Output(s) = Input(s) * H(s)
where H(s) is the Filter's Transfer Function
After transforming the output to the time domain, the solution can be separated into two parts: a steady state solution and a transient solution. The transient portion of the solution is an exponentially decaying sine wave which drives towards zero shortly after a transient event. This gives rise to amplitude modulation and hence the settling time.
The settling time is typically higher for lower frequencies due to its small bandwidth and low cut-off frequency.
A practical approach would be to skip the settling data and use the steady state portion.
--
4 年You can also make a very simple estimate of filter settling time by considering the BT (bandwidth time) product First Select an averaging criteria, like BT=10....then, using a 1/3 octave filter centered at 1000 Hz, since we know B=230 Hz, we compute that T must be 1/23 seconds, or about 43 milliseconds. This is a reasonable estimate for steady state response time of a 1/3 rd octane band filter at 1kHz
Acoustics Engineer
4 年This is great. Now I’m inspired to start writing few articles on LinkedIn myself. Keep up the good work.
Key Accounts Telecom Manager + Audio Engineer at Hottinger, Bruel and Kjaer
4 年Thanks Trinoy! In the example above the signal gets stable after ~0.2s. Can you accurately estimate the filter settling time using the IIR coefficients?