# 0.2 Derivation of the equations for a basic fdm-tdm transmux  (Page 3/10)

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The first portion of the exponential term in the sum vanishes since its argument is always an integer multiple of $2\pi$ . Moving the terms of the summation in the last step is possible since the remaining term of the exponential does not depend on the running index q . It is useful to give a short name to the terms in brackets in the last equation. Noting that it is a function of the decimated time index r and the running index p , we define the variable $v\left(r,p\right)$ by the expression

$v\left(r,p\right)\equiv \sum _{q=0}^{Q-1}h\left(qN+p\right)·x\left(rM-qN-p\right).$

Notice that $v\left(r,p\right)$ is a function of the input data $x\left(k\right)$ , the filter pulse response $h\left(l\right)$ , and the system parameters Q , M , and N , but it is not a function of the selected conversion frequency f 0 , represented in the equation for ${y}_{n}\left(r\right)$ by the integer n .

Substituting $v\left(r,p\right)$ into the equation for the decimated output ${y}_{n}\left(r\right)$ of the tuner tuned to frequency ${f}_{0}=n·\Delta f$ yields

${y}_{n}\left(r\right)={e}^{-j2\pi \frac{nrM}{N}}·\sum _{p=0}^{N-1}{e}^{j2\pi \frac{np}{N}}v\left(r,p\right)$

Notice that the frequency dependency of the tuner shows up only in the exponential terms.

Before discussing this result in detail it remains to examine the effects of the third assumption. To do this, define the decimation factor M by the expression $M\equiv \frac{N}{K}$ , where $K=1$ , 2, or 4. Look first at the exponential terms preceding the sum. It can now be written as

$\begin{array}{ccc}{e}^{-j2\pi \frac{nrM}{N}}\hfill & =\hfill & {e}^{-j2\pi \frac{nr}{K}}\hfill \\ & =\hfill & {\left[{e}^{-j\frac{2\pi }{K}}\right]}^{nr}\hfill \\ & =\hfill & {\left[-{j}^{\frac{4}{K}}\right]}^{nr}.\hfill \end{array}$

With K defined this way, the most general expression for ${y}_{n}\left(r\right)$ is

${y}_{n}\left(r\right)={\left[-{j}^{\frac{4}{K}}\right]}^{nr}·\sum _{p=0}^{N-1}{e}^{j2\pi \frac{np}{N}}v\left(r,p\right),where$
$v\left(r,p\right)=\sum _{q=0}^{Q-1}h\left(qN+p\right)·x\left(\frac{rN}{K}-qN-p\right).$

It can be verified that for $K=1$ , 2, or 4, the factor multiplying the sum is at most a negation or a swapping from imaginary to real or vice versa. Thus no actual multiplication is needed. By far the cleanest case is the one in which the other system parameters (for example, N , Q , and $h\left(k\right)$ ) are selected so that the decimation factor $M$ exactly equals $N$ , or equivalently that $K=1$ . In this case, the exponential preceding the sum collapses to unity, yielding what will be termed in this technical note as the basic FDM-TDM transmux equation Because of the $K=1$ assumption, this equation is the simplest of all those seen to this point and will be referred to as the basic equation . Many applications require M to be chosen differently however (see Section 4 for example) and in these cases equation 12 should be used. :

${y}_{n}\left(r\right)=\sum _{p=0}^{N-1}{e}^{j2\pi \frac{np}{N}}v\left(r,p\right),\phantom{\rule{4pt}{0ex}}where$
$v\left(r,p\right)=\sum _{q=0}^{Q-1}h\left(qN+p\right)·x\left(\left(r-q\right)N-p\right).$

## Interpretation of the basic tuner equation in terms of the discrete fourier transform

Examination of [link] shows that each sample of the tuner output, when tuned to frequency ${f}_{0}=n\Delta f$ , is the N-point inverse discrete Fourier transform (DFT) of the preprocessed data $\left\{v\left(r,p\right)\right\}$ , evaluated at frequency index n . The signal flow described by the equation is shown in [link] . The sampled input data $x\left(k\right)$ passes into a digital tapped delay line of length $QN$ at the sampling rate f s . Every M-th sample, the complete contents of the delay line, all $QN$ samples, are used to compute $\left\{v\left(r,p\right)\right\}$ . Thus the $v\left(r,p\right)$ are computed at the decimated rate $\frac{{f}_{s}}{M}$ . Each of the N elements of $v\left(r,p\right)$ is computed by weighting Q of the delayed input samples by the appropriate coefficient from the pulse response vector $h\left(k\right)$ and summing them together. Notice that at each decimated sampling interval all of the delayed data and all of the pulse response coefficients are used to compute the $v\left(r,p\right)$ . Notice also that since $QN$ is usually much greater than M , each input sample is used in the production of the $\left\{v\left(r,p\right)\right\}$ over several consecutive values of the decimated sampling index r .

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