# 0.2 Derivation of the equations for a basic fdm-tdm transmux  (Page 6/10)

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The computational efficiency of the transmultiplexer can therefore be traced to two key items:

1. Separation of the tuning computation into two segments, one of which (the $\left\{v\left(r,p\right)\right\}$ ) need be computed only once
2. The use of the FFT algorithm to compute the inverse DFT

The first accrues from strategic choices of the sampling and tuning frequencies, while the second depends on N being chosen to be a highly composite integer.

## The transmux as a dft-based filter bank

We have just developed an FDM-TDM transmultiplexer by first writing the equations for a single, decimated digital tuner. The equations for a bank of tuners come from then assuming that (1) they all use the same filter pulse response and (2) their center frequencies are all integer multiples of some basic frequency step. In this section, we develop an alternate view, which happens to yield the same equations. It produces a different set of insights, however, making its presentation worthwhile.

## Using the dft as a filter bank

Instead of building a bank of tuners and then constraining their tuning frequencies to be regularly spaced, suppose we start with a structure known to provide equally-spaced spectral measurements and then manipulate it to obtain the desired performance.

Consider the structure shown in [link] . The sampled input signal $x\left(k\right)$ enters a tapped delay line of length N . At every sampling instant, all N current and delayed samples are weighted by constant coefficients $w\left(i\right)$ (where $w\left(i\right)$ scales $x\left(k-i\right)$ , for i between 0 and $N-1$ ), and then applied to an inverse discrete Fourier transform Whether or not it is implemented with an FFT is irrelevant at this point. Also, we happen to use the inverse DFT to produce a result consistent with that found in the proceeding subsction, but the forward DFT could also be used. . The complete N-point DFT is computed for every value of k and produces N outputs. The output sample stream from the m-th bin of the DFT is denoted as ${X}_{m}\left(k\right)$ .

Since DFTs are often associated with spectrum analysis, it may seem counterintuitive to consider the output bins as time samples. It is strictly legal from an analytical point of view, however, since the DFT is merely an N-input, N-output, memoryless, linear transformation. Even so, the relationship of this scheme and digital spectrum analysis will be commented upon later. We continue by first examining the path from the input to a specific output bin, the m-th one, say. For every input sample $x\left(k\right)$ there is an output sample ${X}_{m}\left(k\right)$ . By inspection we can write an equation relating the input and chosen output:

${X}_{m}\left(k\right)=\sum _{p=0}^{N-1}x\left(k-p\right)w\left(p\right){e}^{j2\pi \frac{mp}{N}}$

the m-th bin of an N-point DFT of the weighted, delayed data. We can look at this equation another way by defining ${\overline{w}}_{m}\left(p\right)$ by the expression

${\overline{w}}_{m}\left(p\right)\equiv w\left(p\right)·{e}^{j2\pi \frac{np}{N}}$

and observing that [link] can be written as

${X}_{m}\left(k\right)=\sum _{p=0}^{N-1}x\left(k-p\right)·{\overline{w}}_{m}\left(p\right).$

From this equation it is clear ${X}_{m}\left(k\right)$ is the output of the FIR digital filter that has $x\left(k\right)$ as its input and ${\overline{w}}_{m}\left(p\right)$ as its pulse response. Since the pulse response does not depend on the time index k , the filtering is linear and shift-invariant. For such a filter we can compute its transfer function, using the expression

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