0.4 The impact of digital tuning on the overall design of an fdm  (Page 3/7)

The value of f _s which leads to the minimum amount of computation is a complicated and nonlinear function of virtually all of the design parameters. While an exact closed form equation for this minimum point is not attainable, it is possible to develop a useful approximation. We now proceed to do that.

We have made various assumptions about f _s along the way, the most important being that it is an integer multiple (and usually a power-of-two multiple) of the filter bank's channel separation $\Delta f$ . For this analysis, however, we temporarily release that constraint and treat it as a continuous variable. To find its optimal value we can then evaluate the first derivative of G _total with respect to f _s and then find the value of f _s which makes the first derivative equal to zero. We first find that the derivative is given by

Setting the derivative to zero leads to an implicit, nonlinear expression. While it can be solved numerically, a practically valid assumption allows a closed form solution. We first define the variable γ , given

With this definition we can write the equation determining the optimum point as

For convenience, we also define the factor ρ , a function of the tuner bandwidth reduction ratio, by $\rho =\frac{f_{in}}{B_t}$ . Using this definition, [link] can be compactly, but deceptively, written as

This expression is deceptive since it proves to be implicit. The term γ depends on f _s , keeping [link] from being easily solved exactly. However, the equation proves to be useful anyway. Examination of the definition of γ shows that it depends on the logarithm of f _s and, in fact, is often quite insensitive to the actual choice of f _s . Once a general range of f _s has been determined, a nominal value of γ can in turn be found and plugged into [link] to find a value of f _s very close to the unconstrained optimum.

We can use the hypothetical supergroup tuner/transmux to demonstrate this procedure. Suppose we guess the optimum value of f _s to be 480 kHz, twice the required tuner bandwidth B _t of 240 kHz. Plugging this into the expression for γ yields 10.4 and using that in [link] indicates that the optimum value for f _s should be about 625 kHz. [link] shows the curve to be quite flat in the vicinity of the optimum point, allowing the actual value of f _s to be chosen consistently with some of the constraints so far ignored in this analysis. In particular, we desire f _s to be a power of two or four times the channel spacing of 4 kHz in this case. Thus a reasonable choice for f _s in this case is 512 kHz.

We can observe some general trends affecting the optimal choice of f _s . It grows higher as the tuner input sampling rate $f_{in}$ does, reflecting the associated growth in tuner computation. It tends to decrease with growth in Q , K , and N , all of which imply more computation in the transmultiplexer. We note also that this formula depends strongly on the assumption of one-step decimation in the tuner. If a multistage tuner is used, the balance will be different. A rule of thumb can be developed by using [link] . Over a broad range of practical examples,the optimal ratio between f _s and B _t attains values between 1.3 and 2.3 for one-stage decimation. When this ratio (that is, $1+\sqrt{\frac{\rho }{\gamma }}$ ) exceeds 2.5 or so, the tuner computation overwhelms that of the transmux and alternative designs for the tuner should be examined. Multistage decimation is only one possible alternative. [link]