0.9 Appendix c  (Page 2/3)

Substituting the decomposition of k as $rM+p$ yields

Note that ${\overline{x}}_n\left(k\right)$ has the sifting property, that is, it is non-zero only when $p-v=0$ , because of its zero-filling. Using this, we can write $\rho \left(k\right)\equiv \rho (r,p)$ as

Note the close relationship of this expression to the ones developed for $v(r,p)$ in previous sections. It is a weighted combination of the input data and, so far, does not depend on the frequency to which the signal will be upconverted.

Now we produce the multiplexer output by upconverting each interpolated input, indexed by n , to its desired center frequency ω _n and then summing them. This sum is given by

where N is the number of components to be multiplexed.

If we substitute the expression of ${\rho }_n\left(k\right)={\rho }_n(r,p)$ into [link] , decompose k in the exponential's argument into r and p , and reverse the order of summation, we obtain a general expression for a digital frequency-division multiplexer:

This equation assumes that all of the N constituent input signals are sampled at the same rate and that the same lowpass interpolating filter is used for each. The upconversion frequencies (the $\left\{{\omega }_n\right\}$ ) are arbitrary, however.

Suppose now that we choose the upconversion frequencies to be regularly spaced in the spectrum between $-\frac{f_s}{2}$ and $\frac{f_s}{2}$ . Mathematically, we do this by assuming that ω _n is given by

We also define K by the familiar ratio $\frac{N}{M}=K$ . With these assumptions, the expression for $y\left(k\right)=y(r,p)$ further reduces to

the general form of the DFT-based TDM-to-FDM transmultiplexer.

An important special case of the general equation is the one in which the interpolation factor M is chosen to equal the potential number of upconversion carriers N . In this case, $K=1$ . For this case to be practical, the bandwidth of the input processes $\left\{x_n\left(r\right)\right\}$ must all be less than ${\displaystyle \frac{f_s}{N}}$ Hz and the pulse response $h\left(k\right)$ must be properly designed. When it is true, [link] reduces to

The sum inside the braces can be recognized as the N-point inverse discrete Fourier transform of all N inputs $x_n\left(r\right)$ at time r . To make this clear, we define $D_p\left(t\right)$ by the expression

for integer time index t . With this definition, the equation for the basic TDM-to-FDM transmultiplexer becomes

Thus each sample of the FDM output $y\left(k\right)$ is a weighted combination of the current and $\overline{Q}-1$ past DFTs of the N channel inputs.

A block diagram of the processor implied by [link] is shown in [link] . At each input sampling instant r , all N inputs to the transmultiplexer are Fourier transformed and the resulting N-point DFT stored in a buffer. The transmultiplexer output for each interpolated time instant $k=rN+p$ is computed with a dot product of the $\overline{Q}$ points of the pulse response $h(qN+p)$ , for $0\le q\le \overline{Q}-1$ , and the stored DFT points $D_p(r-q)$ , for q over the same range. Thus $2\overline{Q}$ real multiplies are needed for each output, assuming that $h\left(k\right)$ is real-valued, and therefore $2\overline{Q}{f}_s$ multiply-adds/sec are needed for this weighting operation.