0.8 Appendix b  (Page 3/5)

Suppose that we similarly define $\overline{h}(2u+1)$ by the expression $\overline{h}(2u+1)=h(2u+1){(-1)}^u$ , and then ${\overline{H}}_e$ and ${\overline{H}}_0$ as in [link] (fourth line). If we do this, we find that the expression for $y\left(k\right)$ becomes simpler yet:

A block diagram of the processing needed to implement these equations appears in [link] .

Offset bin operation

The analysis presented to this point assumes that the tuning frequencies are integer multiples of some fundamental step size $\Delta f$ . This implies that the $0-\underline{\mathrm{t}\mathrm{h}}$ bin or channel is centered at 0 Hz. While this is true in some applications, there are others in which the bin or channel centers are offset in frequency by $\frac{\Delta f}{2}$ An example is shown in [link] . For this example, we suppose that an FDM group of twelve channels is digitally tuned and filtered, that is, it is quadrature downconverted so as to center the group at 0 Hz. ASICs such as those discussed in the section The Impact of Digital Tuning on the Overall design of an FDM-TDM Transmux can perform this function. [link] shows the group centered at DC, which places channels 1-6 below DC and channels 7-12 above. The channels are still separated by 4 kHz but their center frequencies are offset from DC by 2 kHz.

There are several solutions to this problem, the most obvious being to off-tune the tuner by 2 kHz. As this appendix will show, however, the FDM-TDM transmultiplexer equations can be easily modified to introduce the desired offsets.

To produce the desired set of equations, we have to repeat some of the formulation developed in Section 3. Frequency steps of $\Delta f$ are still employed. The fundamental difference is that each tuner frequency is not an integer multiple of $\Delta f$ but rather is a half-integer multiple, for example, ${\displaystyle \omega =2\pi (n+\frac{1}{2})\Delta f}$ , where n is an integer. The effects of this substitution can be seen by joining the analysis in the section Derivation of the equations for a Basic FDM-TDM Transmux at [link] from Derivation of the equations for a Basic FDM-TDM Transmux. Substituting this new expression for the tuning frequency yields

As before, we subscript the decimate output $y\left(r\right)$ by the parameter n but in this case it indicates that the tuning frequency is given by ${\displaystyle f_0=(n+\frac{1}{2})·\Delta f}$ .

As before, we define the integer indices q and p by the expressions

yielding

Suppose we now define the variable $\overline{v}(r,p)$ by the expression

and the pulse response $\overline{h}(qN+p)$ by the expression

Substituting $\overline{v}(r,p)$ into the equation for the decimated output $y_n\left(r\right)$ of the tuner tuned to frequency $f_0=(n+\frac{1}{2})·\Delta f$ yields