The traditional TLM method provides output information in the time domain and it is often necessary to transform this data into the frequency domain. Fourier transform techniques have traditionally been used for this purpose. If an impulse excitation is used then the impulse response is obtained and this is generally taken as valid from d.c. up to a maximum frequency corresponding to a wavelength of ten nodes. The discrete Fourier transform (DFT) is often used in preference to the fast Fourier transform (FFT) since it allows greater flexibility in the way it is used and any additional calculation time is generally negligible when compared with the total simulation time.
With the DFT, it is possible to obtain any number of points in the frequency domain from any number of points in the time domain (although the information content in the frequency domain will always be less than that of the time domain data). By increasing the number of points in the frequency domain, a better evaluation of the signal is obtained although there is no increase in resolution. Flexibility is necessary so that the same technique can be used to obtain, for example, a single frequency transform over a plane (to obtain the field profile of a particular resonance), or, a transform at many closely spaced frequencies (in order to accurately predict the frequency of a resonance). The problem with the FFT is that it produces output uniformly distributed over a large frequency bandwidth extending from d.c. to half the sampling frequency; well beyond the accepted maximum working frequency of the mesh.
In general, models can be described as low, medium or high loss systems. For low loss systems, the spectral content of the truncated time series is required since the amplitude of any resonances will increase in proportion to the number of timesteps (tending to infinity in the limit). For high loss systems, in which the signal effectively decays to zero within the simulation time, the actual spectral content of the complete time series is obtained. Moderately damped systems present a problem since an excessive number of timesteps would be required for the signal to decay significantly. In this case, the spectral content of the truncated time series is obtained as an estimation of the spectral content of the complete time series.
A typical result is shown below, for a perfect 1m^3 cavity excited with an impulse. Both the time (a) and frequency (b) responses are shown.
To predict resonant frequencies, the frequency step must be smaller than the width of the peaks. For a truncated time series, the width is given by the sin(x)/x function. The 3dB point occurs at x=1.392, requiring that for the worst case, the frequency step should be selected from
If there are two closely spaced resonance peaks then the predicted frequencies will be shifted slightly.
The DFT is only one technique which can be used to transform data into the frequency domain. With Prony's method it has been possible to reduce the number of timesteps required by an order of magnitude [Eswarappa]. There are also a large number of digital filtering and modern spectral estimation techniques which can be employed. Although these techniques undoubtedly give good results, experience is often required to make best use of them, for example, in the choice of model order.
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