0.7 Appendix a

The section Derivation of the equations for a Basic FDM-TDM Transmux showed the FDM-TDM transmultiplexer can be viewed as an efficient implementation of a bank of digital tuners, and that the data-weighting function $h\left(k\right)$ is just the pulse response of the FIR lowpass filter used in these equivalent tuners. We therefore approach the design of $h\left(k\right)$ by designing the proper tuner pulse response.

The perfect filter pulse response would pass the signal of interest with no gain or phase distortion, would completely suppress all other FDM channels, and would require little computation. These are not all simultaneously achievable, of course, and the design of the actual filter is a compromise between these issues. It is further complicated by the fact that software packages are not generally available to perform some of the types of optimization needed to design these filters. We proceed first by examining how an optimal equal-ripple linear phase FIR filter performs.

Use of optimal, linear phase, equal-ripple design techniques

The filter design problem at hand can be understood by examining [link] . The perfect filter, shown in [link] (a), passes the channel of interest with unity gain and zero phase shift across its bandwidth of B Hz, centered at DC, and completely attenuates energy at all other frequencies between $\frac{-f_s}{2}$ and $\frac{f_s}{2}$ .

In fact, it is not necessary to suppress all out-of-band energy to protect the signal of interest. The principal reason for this filtering is to suppress the out-of-band components that alias into the band of interest when the output of the tuner (that is, transmultiplexer) is decimated by the factor M . These bands are shown in [link] (b) for the general case in which $M\ne N$ , while [link] (c) shows the important special case of the basic FDM-TDM transmux in which $N=M$ . In the latter case, the FDM channels not of interest alias directly onto the signal of interest while, in the former, the channels not of interest may be spread around the band more.

As alluded to earlier, practical FIR filters of finite duration cannot pass the signal interest perfectly and suppress all other energy completely. The response shown in [link] (d) is the generalized response of a good practical approximation, the response provided by an optimal FIR linear-phase, equal-ripple filter of the sort designed by the Parks-McClellan software package. These filters provide flat differential group delay and allow the designer to optimally trade between passband ripple, stopband suppression, and transition band as a function of the filter order L . A description of this general filter design methodology can be found in [link] .

Superimposing [link] (c) with the optimal response shown in Figure 1 of [link] to produce [link] shows that we must specify the channel bandwidth B , the transition bandwidth $\delta f$ , the input sampling rate f _s , the degree of passband ripple tolerable, denoted PBR and the minimum tolerable stopband attenuation in dB, denoted SBR . A multi-term empirical formula can be found in [link] which determines the filter L quite accurately for a given set of design parameters. Reference [link] simplifies the Rabiner and Gold formula considerably to produce the design equation