# 0.2 Derivation of the equations for a basic fdm-tdm transmux  (Page 7/10)

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${W}_{m}\left(\omega \right)=\sum _{p=0}^{N-1}{\overline{w}}_{m}\left(p\right)·{e}^{-j\omega pT},\phantom{\rule{4pt}{0ex}}\frac{-\pi }{T}\le \omega \le \frac{\pi }{T}.$

Suppose that we first choose the simple case with uniform weighting, that is, $w\left(p\right)=1$ for $0\le p\le N-1$ and, therefore, ${\overline{w}}_{m}\left(p\right)={e}^{j2\pi \frac{mp}{N}}$ . In this case, ${W}_{m}\left(w\right)$ is given by

${W}_{m}\left(\omega \right)={e}^{-j\frac{\pi m}{N}}·{e}^{-j\frac{\left(N-1\right)\omega T}{2}}·\frac{sin\frac{N\omega T}{2}}{sin\left(\frac{\pi m}{N}-\frac{\omega T}{2}\right)}$

The magnitude of this transfer function is plotted in [link] . From this plot we can conclude that the pulse response ${\overline{w}}_{m}\left(p\right)$ has what might be generally considered to be the frequency response of a bandpass filter. The filter is centered on bin m and its bandwidth is nominally $\frac{{f}_{s}}{N}$ . While it might be characterized as a bandpass filter, we also note that the passband is quite rounded and the stopband rejection is relatively poor. The first sidelobes are only 13 dB lower than the peak of the passband response.

We've now shown that the path from the input to the m-th bin can be described as a finite impulse response (FIR) filtering operation and that the transfer function of that filtering operation has a fairly sloppy bandpass characteristic, at least when the data weighting is uniform. What happens for other values of m then? The answer is "the same thing." For each value of m between 0 and $N-1$ , the pulse response ${\overline{w}}_{m}\left(p\right)$ is computed, leading to the transfer function ${W}_{m}\left(\omega \right)$ . An overlay of these bandpass responses is shown in the lower portion of [link] . From this we can conclude that the block diagram shown in [link] describes a single-input, N-output bank of filters. The filter center frequencies are spaced uniformly in increments of $\frac{{f}_{s}}{N}$ Hz. All N outputs are sampled in time as frequently as the input. When the weighting function $w\left(p\right)$ is uniform, then the bandpass filters have the form of [link] , shown in [link] . Transfer Functions of the Paths from the Data Input x ( k ) to the DFT Outputs X m ( k )

## The implications of attaining the desired bandpass characteristic

We've just shown that the DFT of delayed versions of the input sequence $x\left(k\right)$ has the general properties of a bank of regularly-spaced bandpass filters. Two considerations leave us short of our goals. The first is that the shape of the transfer function for each bandpass filter is not good enough for most applications and must be improved. The second is that of reducing the amount of computation required. We address the first one in this section, and temporarily defer the computation issue.

[link] shows two transfer functions. The one pointed to from the left is exactly the same as that shown in [link] and defined in [link] . The one pointed to from the right is representative of the type needed for demultiplexing FDM multichannel telephone signals. It offers essentially flat response for most of the passband, has very sharp transition bands, and suppresses all energy outside of the transition bands by 55 dB or more. While other applications may require different transfer functions, as a rule they will be much more stringent than unweighted transfer function shown in [link] . Comparison of the Unweighted Transfer Function W m ( ω ) and a typical Desired Characteristic

How then do we attain different transfer function characteristics? In fact, we use some of the remaining degrees of freedom, the weighting function $w\left(p\right)$ . By allowing $w\left(p\right)$ to be non-uniform we can now alter the shape of the transfer function of each bandpass filter. By using well-known FIR filter design techniques (see An Introduction to the FDM-TDM Digital Transmultiplexer: Appendix A ) it is possible to attain virtually any shape. It is not, however, possible to always attain the desired shape and the desired bandwidth while keeping the duration of the pulse response constant. In fact, as discussed in An Introduction to the FDM-TDM Digital Transmultiplexer: Appendix A , for a constant bandwidth, the pulse response duration must grow as the transition bandwidth is forced to be smaller and as the stopband suppression in increased. The chain of events described in [link] then unfolds.

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I think
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