# 0.2 Derivation of the equations for a basic fdm-tdm transmux  (Page 2/10)

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In principle, the parameters f s (and hence T ), f 0 , L , and M can be chosen arbitrarily. In fact, significant simplications to the implementation of the tuner occur if they are carefully chosen. To do this we must first develop a general equation for the decimated tuner output $y\left(r\right)$ .

The undecimated filter output $\overline{y}\left(k\right)$ can be written as the convolutional sum of $\rho \left(k\right)$ and the filter pulse response $h\left(k\right)$ :

$\overline{y}\left(k\right)=\sum _{l=0}^{L-1}h\left(l\right)\rho \left(k-l\right).$

Substituting the expression for $\rho \left(k\right)$ yields

$\overline{y}\left(k\right)=\sum _{l=0}^{L-1}h\left(l\right)x\left(k\phantom{\rule{4pt}{0ex}}-l\right){e}^{-j2\pi {f}_{0}T\left(k-l\right)}.$

Separating the two terms in the exponential produces the next expression:

$\overline{y}\left(\mathrm{k}\right)={e}^{-j2\pi {f}_{0}Tk}·\sum _{l=0}^{L-1}h\left(l\right)x\left(k-l\right){e}^{j2\pi {f}_{0}Tl}.$

Decimation by the factor M is introduced by evaluating $\overline{y}\left(k\right)$ only at the values of k where $k=rM$ . We denote the decimated output as $y\left(r\right)$ , given by

$y\left(r\right)\equiv \overline{y}\left(k=rM\right)={e}^{-j2\pi {j}_{0}TrM}·\sum _{l=0}^{L-1}h\left(l\right)x\left(rM-l\right){e}^{j2\pi {f}_{0}Tl}$

## Choosing various system parameters to simplify the general equation for the tuner output

Equation 4 holds for arbitrary choice of L , M , f 0 , and f s . To obtain the equations for the basic FDM-TDM transmultiplexer, we must first simplify the general equation for the output of the digital tuner. We do this by making the three key assumptions:

1. We assume that the sampling rate f s and the tuning frequency f 0 are integer multiples of the same frequency step $\Delta f$ . In the case of FDM multichannel telephone systems for example, $\Delta f$ is typically 4 kHz. We define the integer parameters N and n with the expressions ${f}_{s}\equiv N·\Delta f$ and ${f}_{0}\equiv n·\Delta f$ .
2. We next assume that the pulse response duration L is an integer multiple of the factor N defined above. We define the positive integer parameter Q where $L\equiv Q·N$ . This is a nonrestrictive assumption since Q can be chosen large enough to make it true for any value of L . If $QN$ exceeds the minimum required value of L , then $h\left(k\right)$ can be made artificially longer by padding it with zero values. The factor Q turns out to be an important design parameter. The parameters Q and N are determined separately and the resulting value of L follows from their choice.
3. We also assume that the decimation factor M is chosen to be closely related to the parameter N . Typical values are $M=N$ and $M=\frac{N}{2}$

We can now examine the effects of these assumptions. First, the relationship between f s , f 0 , and $\Delta f$ allows $y\left(r\right)$ to be written as

${y}_{n}\left(r\right)={e}^{-j2\pi \frac{nrM}{N}}·\sum _{l=0}^{L-1}h\left(l\right)x\left(rM-l\right){e}^{j2\pi \frac{nl}{N}}.$

We subscript the decimated output $y\left(r\right)$ by the parameter n to indicate that it depends on the tuning frequency ${f}_{0}=n·\Delta f$ .

The second assumption, the definition of the parameter Q , permits the single sum to be split into a nested double sum. To do this, define the new integer indices q and p by the expressions

$l\equiv qN+p,\phantom{\rule{4pt}{0ex}}where\phantom{\rule{4pt}{0ex}}0\le q\le Q-1\phantom{\rule{4pt}{0ex}}and\phantom{\rule{4pt}{0ex}}0\le p\le N-1.$

Examination of [link] shows that the pulse response running index l has a unique value in the range from 0 to $L-1$ for each permissible value of p and q . This permits the single convolutional sum over the index l to be replaced (for reasons to be shown) with a double sum over the indices p and q . In particular,

$\begin{array}{ccc}{y}_{n}\left(r\right)\hfill & =\hfill & {e}^{-j2\pi \frac{nrM}{N}}·\sum _{l=0}^{L-1}h\left(l\right)x\left(rM-l\right){e}^{j2\pi \frac{nl}{N}}\hfill \\ & =\hfill & {e}^{-j2\pi \frac{nrM}{N}}·\sum _{p=0}^{N-1}\sum _{q=0}^{Q-1}h\left(qN+p\right)x\left(rM-qN-p\right){e}^{j2\pi \frac{n\left(qN+p\right)}{N}}\hfill \\ & =\hfill & {e}^{-j2\pi \frac{nrM}{N}}·\sum _{p=0}^{N-1}\sum _{q=0}^{Q-1}h\left(qN+p\right)x\left(rM-qN-p\right){e}^{j2\pi \frac{nqN}{N}}{e}^{j2\pi \frac{np}{N}}\hfill \\ & =\hfill & {e}^{-j2\pi \frac{nrM}{N}}·\sum _{p=0}^{N-1}{e}^{j2\pi \frac{np}{N}}\left[\sum _{q=0}^{Q-1}h\left(qN+p\right)x\left(rM-qN-p\right)\right].\hfill \end{array}$

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