0.4 The impact of digital tuning on the overall design of an fdm  (Page 4/7)

One implication of f _s being significantly larger than B _t is that many of the channels or filters in the transmux-based filter bank are not useful. To visualize this, consider [link] . [link] (a) shows the power transfer function of the tuner filter before its output is decimated to the rate f _s . The passband of the filter is B _t Hz wide, the transition band on each side of the passband is $\Delta f_t$ Hz wide, and the stopband extends from ${\displaystyle \frac{B_t}{2}+\Delta f_t}$ Hz to the Nyquist folding frequency $\frac{f_{in}}{2}$ . [link] (b) shows the power transfer fumction of the decimated filter. In this case, we assume that the transition band $\Delta f_t$ is slightly less than $f_s-B_t$ . With this choice, some energy passed by the tuner through the transition bands folds back into the output, but none falls in the passband. [link] (c) shows the channels of the transmux-based filter bank overlaying the tuner's power transfer function. The channels falling within the passband are clean, that is, the tuner's passband ripple and stopband rejection apply there, but the channels falling in the transition band are subject to several degradations (for example, gain slope and out-of-band signal aliasing) and are therefore not useful in most cases. Thus even though the transmultiplexer breaks the f _s Hz band at the output of the tuner into N channels, only C of them, where $C=\frac{B_t}{\Delta f}=\frac{N·B_t}{f_s}$ , are typically used for downstream processing.

Hardware examples of tuner/transmux tradeoffs

This company has built a number of digital transmultiplexers for various applications and all of them employ some form of digital tuner. The next three sections present a few of these designs with the intent of demonstrating how the overall system design decisions were made.

A single-card supergroup tuner/transmux

As a part of an IR&D program, the company developed an FDM supergroup transmultiplexer during 1985. Its basic requirements were to accept an FDM supergroup (that is, 60 voice grade channels spaced at regular intervals of 4 kHz over a band of 240 kHz) located at any of several possible spectral bands. These bands include 2-242 kHz, 12-252 kHz, 60-300 kHz, 312-552 kHz, and 564-804 kHz. Another key goal was excellent technical performance. To achieve this, the transmultiplexer portion was designed to use 16-bit arithmetic and key design parameters of $f_s=4$ kHz, $K=1$ , and $Q=16$ .

Since a supergroup only occupies 240 kHz, a convenient choice of f _s would be 256 kHz. This value exceeds 240 kHz and makes N equal 64, an integer power of two and four. This value proves not to be globally optimum, however, as we will see after examining the tuner's requirements.

The highest input frequency of interest to the tuner is 804 kHz. The sampling rate must therefore exceed this value by two or more. The actual rate chosen was 2.048 kHz. This was based on several considerations:

• It satisfies the Nyquist sampling theorem and includes some allowance for the imperfections of analog antialias filtering.
• It is a power-of-two integer multiple of $\Delta f=4$ kHz.
• It was the highest sampling rate attainable with financially acceptable 12-bit A/D converters of the era. Twelve-bit digitization was desired to maximize the unit's noise power ratio (NPR) and dynamic range.