0.5 General design principles

There is a very large number of considerations that affect the selection of the best method of frequency-division demultiplexing signals in a particular application. As a result, it is virtually impossible to provide a simple cookbook methodology that always produces the best design. Even so, it is useful to systematically describe the design issues and choices evaluated so far in this technical note. Such a description, condensed into a design flowchart, is discussed in this section. Comparison of it with the design examples provided in the section "Example: Using an FDM-TDM Transmux to Demodulate R.35 Telegraphy Signals" and the section "The Impact of Digital Tuning on the Overall design of an FDM-TDM Transmux" shows excellent agreement. But while it is intended to be helpful, it must be used with care since relatively small differences in the application-dependent assumptions can influence the resulting choices quite considerably.

The decision flowchart presented in [link] assumes that the generalized demultiplexer has the block diagram shown in [link] . The system accepts $N_{in}$ digitized FDM signals, all sampled at $f_{in}$ Hz. These are made available to N _t digital tuners. All of these tuners are of the same design, employ the same decimation factor M _p and produce output samples at the same rate of f _s Hz. The tuner outputs are transmultiplexed, sending their output channel samples to a bus, which up to N _u user processes have access to. Since each transmux is fed by a tuner, there are N _t transmuxes, each parameterized by $Q,N$ , and M .

On the one hand, this architecture is not perfectly general, since parameters such as filter bandwidths are assumed to be identical, but it is representative of a very complex transmultiplexer-based system. On the other hand, it can be simplified considerably, by allowing $N_{in}$ or N _t to be unity for instance, and still be reasonably described by the flowchart.

The flowchart is shown in [link] . While perhaps self-explanatory, some commentary is provided for the faint-hearted.

• The first step is to determine whether an FDM-TDM transmultiplexer is really needed for the application. Generalizing wildly, a transmultiplexer is the right choice if three conditions are met:
  1. It is desired to simultaneously demultiplex a reasonably large (for example, 10 or more) number of contiguous channels from an FDM signal
  2. They are regularly spaced in frequency
  3. The same filter can be used for all of them without harm to the signals
  If these conditions aren't met, then altemative schemes, such as separate tuners for the desired channels, should be considered.
• Once it is determined that a transmultiplexer is needed, the next question is whether some form of digital tuner is needed to precede it. As a rule, no tuner is needed if:
  1. It is desired to demultiplex all of the channels seen in the full bandwidth of the input
  2. The input signal is sampled at a suitable rate
  If resampling is needed, or if only a subband of the input signal's bandwidth is to be dechannelized, then a tuner is called for. Usually the use of a digital tuner leads to the use of a transmultiplexer that accepts complex-valued data while the absence of a tuner implies the use of a transmux that accepts real-valued data.
• The last major question is whether the outputs of the transmultiplexer should be real- or complex-valued. This usually depends completely on the processes using the transmultiplexer outputs. In some cases, such as commercial telephony (see the example in Appendix C), the outputs are desired to be in real-valued form so that they can be switched or formatted for TDM/PCM transmission. In other applications, however, particularly those that involve signal processing (for example, spectrum analysis), the use of complex-valued outputs is desired.
• With these fundamental system-level questions answered, the preliminary design of the transmultiplexer itself can begin. Based on the channel spacing, the desired filter frequency response, and the nature of the follow-on processing, such parameters as $\Delta f,B,Q$ , and $f_{out}$ can be determined by using the rules presented in the section "Derivation of the equations for a Basic FDM-TDM Transmux" .
• If no tuners are needed, then the design of the transmux can be completed by determining $M,N,L$ , and the pulse response $h\left(k\right)$ . If tuners are needed, then the tradeoffs between the tuner and transmultiplexer design must be performed in order to know enough to finish the design of the transmux itself. The first step in this tradeoff is to determine the number of tuners N _t and their bandwidths B _t . The second step, given B _t , is the tradeoff identified in Section 5, which leads to the choice of the transmux input sampling rate f _s , and hence L _t and M _t .

Of these two steps, the first is often the more difficult since the optimization may be based on non-mathematical considerations. An example of this is the case in which a large number of contiguous FDM channels need to be demultiplexed from an even larger input band. Should there be a few tuners of large bandwidth or more with narrower bandwidth? A purely mathematical optimization using an objective function such as the number of multiply-adds will conclude that the former is better, while a user might prefer the selectivity (for example, cherry picking ) afforded by a multitude of narrower tuners.