# 0.4 The impact of digital tuning on the overall design of an fdm

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## Problem statement

So far we have presumed that the FDM input signal to the transmux has been magically provided and that it has been sampled at the proper rate. In fact, the signal available to the processor might not be in the desired form and signal processing may be required to convert it appropriately. As we shall see, the computation required for this can be significant in itself. As a result, these signal conditioning steps must be taken into account in the optimal design of the whole system. In this section, we focus on the use of digital tuners for this signal conditioning and examine the tradeoffs between the parameters of a tuner and the transmultiplexer that follows it.

## Total computational requirements

There are a few practical applications in which the input signal is complex-valued, sampled at the desired rate, and spectrally registered with the filters produced by the transmux-based filter bank. More typically, however, applications involve real-valued input signals, the signal is not aligned with the filters in the bank, or the signal of interest must be extracted from a wideband signal. It is common in these cases to use a digital tuner to select the portion of the spectral band in which the transmux will operate. This tuner will usually have a block diagram exactly like that seen in Figure 1 from "Derivation of the equations for a Basic FDM-TDM Transmux" . The incoming sampled signal is quadrature downconverted, filtered using an FIR linear phase filter, and then decimated We assume one-step decimation in this analysis. An important exception to this approach is described in Section 5.4.3. . The decimated tuner output is applied to the preprocessor portion of the transmultiplexer. For the analysis here we assume that the input is real-valued (from an A/D converter, for example), that the tuner input sampling rate is given by ${f}_{in}$ , that the pulse response duration of the tuner's filter is given by L t and that its decimation factor is M t . The spectral band over which the tuner offers rated passband performance and adjacent signal rejection is denoted by B t . The combined block diagram of the tuner and FDM-TDM transmultiplexer is shown in [link] , along with the key variables needed to determine the joint optimal design.

We obtain an equation for the total number of multiply-adds required by adding the transmux expression found in Equation 18 from "Derivation of the equations for a Basic FDM-TDM Transmux" with the computation requirements of the preceding tuner. This produces the following:

${G}_{\mathrm{total}}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}{G}_{\mathrm{tuner}}\phantom{\rule{4pt}{0ex}}+\phantom{\rule{4pt}{0ex}}{G}_{\mathrm{transmux}}$
$=2{f}_{in}\left\{1+\frac{{L}_{t}}{{M}_{t}}\right\}+\frac{2{f}_{s}N}{M}·\left\{Q+lo{g}_{2}N\right\}.$

By inspection we see that $\frac{fin}{{M}_{t}}={f}_{s}$ and that ${f}_{\mathrm{out}}=\frac{{f}_{s}}{M}=\frac{{f}_{in}}{{M}_{t}M}$ .

We observe that the bandwidth of the signal exiting the tuner, denoted B t , must be less than f s , the transmux input rate, in order to satisfy the Nyquist sampling theorem. Their ratio is a key element in the computational tradeoff between the tuner and the transmux. With B t fixed, an increase in f s increases the computation needed for the transmux while decreasing that needed for the tuner. We make this explicit by developing a formula for the tuner's pulse response duration L t . Again assuming one-step decimation and appealing to the design formulas discussed in [link] , L t is closely approximated by

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