Given expressions such as those shown in Equation 5 and Equation 7 it is possible to accurately estimate the total amount of multiplyadd computation needed for a tuner/transmux processor. It is also possible to perform tradeoffs between the various parameters in order to optimize the resulting design. While this can in principle be done with any of the design parameters, we demonstrate in this section the computational implications of varying the parameter f_{s}, the input sampling rate to the transmultiplexer. In practice, this usually turns out to be one of the designer's most important parameter choices.
Figure 2 shows the computational requirements for a hypothetical transmultiplexer. In this case, the input sampling rate finfin is assumed to be 6.4 MHz. The tuner must select an FDM telephone supergroup from the input signal and demultiplex all 60 voice channels in the supergroup. The tuner's bandwidth B_{t} must therefore be greater than or equal to 240 kHz and f_{s} must exceed that. For the telephone demultiplexing application, the channel spacing ΔfΔf is usually 4 kHz and the oversampling factorK is typically chosen to be unity. Figure 2 shows five curves, one for each segment of the computation and one for the composite. The number of multiplyadds required by the input mixer is constant, since the input sampling rate finfin is fixed. The computation required by the tuner's filter falls as f_{s} rises from 240 kHz and tends toward the input Nyquist frequency of 3.2 MHz. The cause of this can be ascertained by examining Equation 4. As f_{s} decreases toward B_{t}, the transition band decreases, L_{t} increases hyperbolically, and the amount of computation needed for the tuner's filter grows without bound.
The next two curves describe the effect of f_{s} on the two components of the transmultiplexer. For a given value of Q, the computation required by the preprocessor is strictly proportional to f_{s}. The FFT's computation rises slightly faster than proportionally since the number of FFT bins grows as f_{s} does. The sum of these constituent curves represents the total amount of multiplyadd computation needed. Note that it has a broad minimum. It rises precipitously as f_{s} decreases toward B_{t} and more slowly as f_{s} increases toward its other limit fin2fin2.
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 poweroftwo multiple) of the filter bank's channel separation ΔfΔf. 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
d
G
total
d
f
s
=

2
α
t
f
i
n
,
B
t
(
f
s

B
t
)
2
+
2
K
(
Q
+
1
)
+
2
K
·
l
o
g
2
f
s
Δ
f
.
d
G
total
d
f
s
=

2
α
t
f
i
n
,
B
t
(
f
s

B
t
)
2
+
2
K
(
Q
+
1
)
+
2
K
·
l
o
g
2
f
s
Δ
f
.
(8)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
γ
=
K
{
(
Q
+
1
)
+
{
l
o
g
2
[
f
s
Δ
f
]
}
}
α
t
.
γ
=
K
{
(
Q
+
1
)
+
{
l
o
g
2
[
f
s
Δ
f
]
}
}
α
t
.
(9)With this definition we can write the equation determining the optimum point as
f
i
n
B
t
(
f
s

B
t
)
2
=
γ
.
f
i
n
B
t
(
f
s

B
t
)
2
=
γ
.
(10)For convenience, we also define the factor ρ, a function of the tuner bandwidth reduction ratio, by ρ=finBtρ=finBt. Using this definition, Equation 10 can be compactly, but deceptively, written as
(
f
s
)
optimum
=
B
t
(
1
+
ρ
γ
)
.
(
f
s
)
optimum
=
B
t
(
1
+
ρ
γ
)
.
(11)This expression is deceptive since it proves to be implicit. The term γ depends on f_{s}, keeping Equation 11 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 Equation 11 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 Equation 11 indicates that the optimum value for f_{s} should be about 625 kHz. Figure 2 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 finfin 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 onestep decimation in the tuner. If a multistage tuner is used, the balance will be different. A rule of thumb can be developed by using Equation 11. 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 onestage decimation. When this ratio (that is, 1+ργ1+ργ) 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. [1]
One implication of f_{s} being significantly larger than B_{t} is that many of the channels or filters in the transmuxbased filter bank are not useful. To visualize this, consider Figure 3. Figure 3(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 ΔftΔft Hz wide, and the stopband extends from Bt2+ΔftBt2+Δft Hz to the Nyquist folding frequency fin2fin2. Figure 3(b) shows the power transfer fumction of the decimated filter.
In this case, we assume that the transition band ΔftΔft is slightly less than fsBtfsBt. With this choice, some energy passed by the tuner through the transition bands folds back into the output, but none falls in the passband. Figure 3(c) shows the channels of the transmuxbased 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 outofband 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=BtΔf=N·BtfsC=BtΔf=N·Btfs, are typically used for downstream processing.