Suppose that our design objective is to build a digital processor capable of demodulating all of the FSK canals found in the R.35 signal shown in Figure 1 from "An Introduction to the FDM-TDM Digital Transmultiplexer: Introduction". Suppose further that we choose to build the demodulator for each FSK signal along the lines of the one shown in Figure 1(a). Both are used in practice. This type of FSK demodulator uses two fllters: one centered at the mark frequency fmkfmk and another the space frequency fspfsp. The powers or amplitudes of the two filter outputs are compared to determine whether the signal instantaneously falls mostly in the vicinity of the mark or is closer to the space. The bit synchronizer logic monitors the transitions between mark and space (and vice versa), using the information to determine the right instants to sample the thresholded difference waveform and produce binary decisions.
Figure 1 shows the block diagram of the demodulating process when extended to handle all 24 FSK canals in an R.35 signal. Initially it appears to only be 24 parallel demodulators. On closer inspection however, it may be recognized that the center frequencies of all the filters differ by integer multiples of a single frequency increment ΔfΔf. This suggests the use of a digital filter bank to compute all ofthe required bandpass filters. We now proceed to see how the system design for this filter bank is done.
The system design of the transmultiplexer/filter bank is specified by a small set of parameters. We determine these parameters as follows:
- ΔfΔf: Inspection of the frequency allocations for the R.35 signal shows that all possible mark and space frequencies are separated by integer multiples of 60 Hz. Thus it is natural to set Δf=60Δf=60 Hz.
- fs and N: With ΔfΔf determined, the choice of N, the DFT dimension, and the choice of the input sampling rate fs, are locked together. We bound N from below, by noting that at least 48 filters are needed, two for each FSK canal. In principle, the value of N can be chosen to be any number higher than 48. If the use of the FFT is contemplated then N is usually chosen to the first power of 2 or 4 higher than the minimum value. Assuming the use of either a radix-2 or radix-4 FFT, plus the use of complex-valued input data, the prudent value of N is 64. This immediately leads to a complex-valued input sampling rate of N·Δf=3840N·Δf=3840 Hz. If the input were real-valued instead, then the chosen sampling rate would be twice that, or 7680 Hz.
- L and Q: Wth N determined, we find that L and Q are locked together and that they are a function of the exact filter design used to select the pulse response (or, equivalently, the window function) used to determine the shape of the bandpass fllters. The issues to be considered in the design of the pulse response are discussed in "An Introduction to the FDM-TDM Digital Transmultiplexer: Appendix A". Without reiterating them here, we observe that following those rules yields a minimum pulse response duration L of about 174. For this application we extend the filter pulse duration to 192, allowing Q to equal exactly 3.
- M: With the input sampling rate set, the decimation factor M determines the output sampling rate at each of the filter outputs. Thus foutfout, the output sampling rate, equals 3840M3840M Hz. The required output rate depends on the types of signals present and the types of processing to be done to them. In the case of demodulating asynchronous FSK signals, experience has shown that the output sampling rate needs to exceed the highest FSK baud rate expected by a factor of four or more. The highest baud rate allowed by the CCITT for an R.35 canal is 75 Hz. Thus the output sampling rate foutfout must exceed 300 Hz. By choosing M=12M=12, we obtain an output sampling rate of fout=320fout=320 Hz.
A demodulator using these parameters is shown in Figure 2. When developed in 1985 it was termed the filter bank card and was used to simultaneously demodulate 24 voice grade channels, each containing 24 R.35 FSK canals. Thus 24·24·2=115224·24·2=1152 fllters were needed to isolate the mark and space energy of all FSK canals. When demodulating R.35 signals, each filter produces outputs at a rate of 320 per second. The input sampled waveforms are tuned and filtered by a preceding tuner to spectrally align the filter bank's bins with the mark and space frequencies of the R.35 signal.
Arrows in Figure 2 point to the key computational elements on the filter bank card. One multiplier-accumulator device handled the window-and-fold, or preprocessing, function for all 24 voice channel inputs. The second computed the needed FFTs. When processing R.35 signals in all 24 channels, 320·24=76800320·24=76800 FFTs of dimension N=64N=64 were performed per second. The third arrow points to a floating-point processor used to measure the instantaneous power at the bandpass filter outputs and threshold their differences to produce binary decisions.
In passing, we note that this design exploits the fact that the FSK demodulator only requires knowledge of the magnitude of each filter output. Recalling the general transmux equation given in Equation 9 from "Derivation of the equations for a Basic FDM-TDM Transmux" and substituting the appropriate values of N,QN,Q, and M produces the equation:
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(2)This same filter bank card is also capable of demodulating FSK signals conforming to the R.37 and R.38A ITU-T recommendations. The R.37 signal, for example, uses 12 canals instead of 24, and each of them operates at twice the rate and with twice the mark-space frequency separation of the R.35 signal. Repeating the system design just performed yields the following:
- Δf=120Δf=120 Hz
- N=32N=32 and fs=3840fs=3840 Hz
- L=96L=96 and Q=3Q=3
- fout=640fout=640 Hz
Note that Q and fs remain the same as for R.35, and that ΔfΔf and foutfout double because of the increased mark-space separation and allowable baud rates, while L and N are halved. The impact of processing this additional signal can be assessed by using the formula for the number of multiply-adds required, specifically
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(3)It can be verified that the number of multiply-adds needed to perform the window-and-fold function for the filter bank is exactly the same as is needed for the R.35 signal, since the ratio NMNM holds constant. Moreover, the amount of computation needed for the radix-2 FFT is 5656 of that needed for R.35, the ratio between log 232 log 232 and log 264 log 264. Thus a filter bank with the computational horsepower to handle R.35 can also handle R.37. The R.38A standard represents another factor of two in frequency separations and allowable baud rates. Once again it can be verified that the window-and-fold computation is the same and the FFT computation is smaller yet. Thus a properly designed filter bank processor capable of handling the R.35 standard can also handle R.37 and R.38A as well. A subtle difference is that the input signal must be tuned slightly differently for the three different standards.
