In principle, the parameters
f
_{s} (and hence
T ),
f
_{0} ,
L , and
M can be chosen arbitrarily. In fact, significant simplications to the implementation of the tuner occur if they are carefully chosen. To do this we must first develop a general equation for the decimated tuner output
$y\left(r\right)$ .
The undecimated filter output
$\overline{y}\left(k\right)$ can be written as the convolutional sum of
$\rho \left(k\right)$ and the filter pulse response
$h\left(k\right)$ :
Decimation by the factor
M is introduced by evaluating
$\overline{y}\left(k\right)$ only at the values of
k where
$k=rM$ . We denote the decimated output as
$y\left(r\right)$ , given by
Choosing various system parameters to simplify the general equation for the tuner output
Equation 4 holds for arbitrary choice of
L ,
M ,
f
_{0} , and
f
_{s} . To obtain the equations for the basic FDM-TDM transmultiplexer, we must first simplify the general equation for the output of the digital tuner. We do this by making the three key assumptions:
We assume that the sampling rate
f
_{s} and the tuning frequency
f
_{0} are integer multiples of the same
frequency step$\Delta f$ . In the case of FDM multichannel telephone systems for example,
$\Delta f$ is typically 4 kHz. We define the integer parameters
N and
n with the expressions
${f}_{s}\equiv N\xb7\Delta f$ and
${f}_{0}\equiv n\xb7\Delta f$ .
We next assume that the pulse response duration
L is an integer multiple of the factor
N defined above. We define the positive integer parameter
Q where
$L\equiv Q\xb7N$ . This is a nonrestrictive assumption since
Q can be chosen large enough to make it true for any value of
L . If
$QN$ exceeds the minimum required value of
L , then
$h\left(k\right)$ can be made artificially longer by padding it with zero values. The factor
Q turns out to be an important design parameter. The parameters
Q and
N are determined separately and the resulting value of
L follows from their choice.
We also assume that the decimation factor
M is chosen to be closely related to the parameter
N . Typical values are
$M=N$ and
$M=\frac{N}{2}$
We can now examine the effects of these assumptions. First, the relationship between
f
_{s} ,
f
_{0} , and
$\Delta f$ allows
$y\left(r\right)$ to be written as
We subscript the decimated output
$y\left(r\right)$ by the parameter
n to indicate that it depends on the tuning frequency
${f}_{0}=n\xb7\Delta f$ .
The second assumption, the definition of the parameter
Q , permits the single sum to be split into a nested double sum. To do this, define the new integer indices
q and
p by the expressions
Examination of
[link] shows that the pulse response running index
l has a unique value in the range from 0 to
$L-1$ for each permissible value of
p and
q . This permits the single convolutional sum over the index
l to be replaced (for reasons to be shown) with a double sum over the indices
p and
q . In particular,
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Source:
OpenStax, An introduction to the fdm-tdm digital transmultiplexer. OpenStax CNX. Nov 16, 2010 Download for free at http://cnx.org/content/col11165/1.2
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