0.8 Appendix b

Production of real-valued outputs

It is frequently the case that the best system-wide choice is the use of a transmultiplexer producing complex-valued outputs even though some of the system outputs need to be real-valued. An example of this is when a transmultiplexer is used to supply all transmuxed signals to an analyzer of some sort, plus a few selected signals for a downstream processor or transmission system. If the analysis processing is best done with complex-valued data (and it often is), then the best system-level choice is often to use a complex-output transmux and then to convert the relatively few system outputs to a real-valued representation. This appendix describes how this can be done.

Producing a real-valued signal from a complex-valued one is as simple as taking its real part, but for this to be valid, the complex-valued signal must be sampled frequently enough. In actual practice, this condition is not usually met and it is necessary to increase the complex-valued signal's sampling rate by a factor of two before extracting the real part. Another complication is that the user may desire to choose the spectral orientation of the resulting real-valued signal.

[link] shows the spectral implications of the steps usually taken to interpolate the complex-valued signal, specify its sideband orientation, and produce a real-valued representation.

[link] (a) shows the assumed spectrum of the complex-valued input signal $z\left(r\right)$ . The signal $z\left(r\right)$ is assumed to be the output of a complex-valued FDM-TDM transmultiplexer. It is sampled at the rate f and is spectrally oriented so that an increasing input frequency results in a higher output frequency. We'll term this upper sideband . Note that the bandwidth of $z\left(r\right)$ is less than f .

We now upconvert $z\left(r\right)$ by $\frac{f}{2}$ Hz by multiplying it by ${(-1)}^r$ . This is shown in [link] (b). The signal is now centered around $\frac{f}{2}$ instead of 0 Hz. The sampling rate is doubled by zero-filling $\overline{z}\left(r\right)$ , that is, $\overline{z}\left(k\right)$ is created by setting $\overline{z}(k=2r)=z\left(r\right){(-l)}^r$ for even values of k and $\overline{z}\left(k\right)=0$ for odd values of k . The spectral effects of this zero-filling are shown in [link] (c). The sampling rate is now $2f$ and two upper sideband images of the original signal are present, one centered at $\frac{f}{2}$ and the other at $-\frac{f}{2}$ .

The next step is to bandpass filter the desired image. Two filter transfer functions are shown in [link] (d). We focus first on the one drawn with the solid line for now. When the zero-filled input signal $\overline{Z}\left(k\right)$ is convolved with the filter having this response, the higher of the two images is preserved and the lower one suppressed. This is shown in [link] (e).

Obviously, the use of a filter with transmission characteristics shown in the dashed lines in [link] (d) would have the effect of selecting the other image, the one shown in dashed lines in [link] (e).

As a result of the filtering, only one image remains and which image it is depends on the bandpass filter chosen. The last step is the production of the real-valued output by simply taking the real part of the complex-valued signal that appears at the filter's output. Since real-valued signals must have spectral symmetry about the origin, this extraction of the real part has the effect of producing an another image of the input, only this one has the opposite sideband orientation. If the upper image is selected by the bandpass filter, then spectrum of the output signal is as shown in [link] (f). We term this as upper sideband since the positive frequency image has the same orientation as the original complex-valued input signal. Choosing the lower image with the bandpass filter has the effect of producing the spectral relationship shown in [link] (g), the so-called lower sideband or inverted case .