0.8 Appendix b  (Page 4/5)

In the section Derivation of the equations for a Basic FDM-TDM Transmux , we defined the basic FDM-TDM transmultiplexer as the special case of general transmultiplexer in which we set the decimation factor M to equal the number of channels N . This implies directly that $K=1$ . This assumption leads to the corresponding basic transmux equations for the offset-bin case:

and $IDF{T}_o\{.\}$ indicates the offset-bin inverse DFT.

While not immediately obvious, it can be shown that the offset-bin DFT can be computed with an FFT-like algorithm. A listing of one is shown in the following code. It results from a simple modification (that is, the initialization of U ) in the FFT routine appearing on page 367 of [link] .

When an offset-bin transmux is performed, it is common not to premultiply by ${(-1)}^{\mathrm{r}}$ as shown in [link] . This has the effect of frequency-converting the output signal by ${\displaystyle \frac{f_{out}}{2}=\frac{f_s}{2M}}$ . When $M=N$ (hence $K=1$ ), the spectral effect of this is as shown in [link] (a) and (b). Instead of producing a complex signal centered at DC, not premultiplying by ${(-1)}^{\mathrm{r}}$ centers the signal at ${\displaystyle \frac{f_{out}}{2}}$ In addition to obviating the need for a multiplication, this has the effect of moving the signal away from DC. This tends to improve signal quality since many finite-word length effects arising from hardware implementations (for example, truncation) produce spurious signals at DC.

Operation with real-valued inputs

There are practical applications of FDM-TDM transmultiplexers in which the designer wants to extract all channels from a real-valued input. Such a signal can be applied directly to an FFT-based transmux of the variety described in the section Derivation of the equations for a Basic FDM-TDM Transmux but, since such a transmux is designed for use with complex-valued data, it might appear that unneeded computation is being performed. That is in fact the case. This section shows how the real-valued nature of the input can be exploited to reduce the required computation by slightly less than a factor of two.

Assume for this discussion that the input signal $x\left(k\right)$ is real-valued and sampled at a rate of $f_s=\mathit{2}N\Delta f$ , where $\Delta f$ , as before, is the frequency spacing between channels, and N is the maximum number of unique channels. The Nyquist theorem requires that f _s be twice $N\Delta f$ since the input is real-valued. Half of the $2N$ channels present in the real-valued input are sideband reversed images of the other N channels. Thus we work to find an expression for those N unique channels. Using the basic equation for the FDM-TDM transmux (see equation 14 from Derivation of the equations for a Basic FDM-TDM Transmux ), the n -th output is given by

where $v(r,p)$ is given by

Since both the input $x\left(k\right)$ and the pulse response $h\left(k\right)$ are real-valued, so is $v(r,p)$ . Thus the DFT in [link] is taken over real-valued data. We now pursue a two-step approach to exploiting the reality of the data.