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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 ( r ) .

Figure two is a five-part diagram of graphs with descriptions. Each graph plots f_0 on the horizontal axis, only displaying the first and second quadrants, and the horizontal axis ranges in value from -f_s/2 to f_s/2. The first graph, titled (a) Original FDM Spectrum, and described as the associated sequence x(k), is a graph of six congruent right triangles with bases on the horizontal axis. Three right triangles in the second quadrant face the center of the graph with their right angles on the left. Three right triangles in the first quadrant face the center of the graph with their right angles on the right. The center triangle in the first quadrant is shaded black. The second graph, titled (b) Spectrum after Downconversion of Desired Channel, and described as the associated sequence ρ(k), is a graph of six congruent right triangles with bases on the horizontal axis, but unlike (a), they are scattered in a less symmetric pattern. Far on the left is the first right triangle, with its right angle on the left. Just before, just after, and on the vertical axis are the next three triangles, all facing to the right with their right angles on the right side. The center triangle in this series is shaded black. A final two triangles further to the right in the first quadrant face away from the vertical axis with their right angles on the left. The third graph, titled (c) Channel Filter response, and described as the associated sequence h(k), contains a short wavering graph with a flat peak centered on the vertical axis. The fourth graph, titled (d) Resulting Filtered Output, and described  as the associated sequence y-bar(k), contains one small black right triangle centered at the origin with its right angle on the right side. The fifth graph, titled (e) Output after Decimation by Factor of M, and described as the associated sequence y(r), contains a large shaded right triangle with a base approximately twice as long as its height, with its base centered in the graph at the vertical axis.
Spectral Description of Each Step in the Digital Tuning of a Single Channel

The undecimated filter output y ¯ ( k ) can be written as the convolutional sum of ρ ( k ) and the filter pulse response h ( k ) :

y ¯ ( k ) = l = 0 L - 1 h ( l ) ρ ( k - l ) .

Substituting the expression for ρ ( k ) yields

y ¯ ( k ) = l = 0 L - 1 h ( l ) x ( k - l ) e - j 2 π f 0 T ( k - l ) .

Separating the two terms in the exponential produces the next expression:

y ¯ ( k ) = e - j 2 π f 0 T k · l = 0 L - 1 h ( l ) x ( k - l ) e j 2 π f 0 T l .

Decimation by the factor M is introduced by evaluating y ¯ ( k ) only at the values of k where k = r M . We denote the decimated output as y ( r ) , given by

y ( r ) y ¯ ( k = r M ) = e - j 2 π j 0 T r M · l = 0 L - 1 h ( l ) x ( r M - l ) e j 2 π f 0 T l

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:

  1. We assume that the sampling rate f s and the tuning frequency f 0 are integer multiples of the same frequency step Δ f . In the case of FDM multichannel telephone systems for example, Δ f is typically 4 kHz. We define the integer parameters N and n with the expressions f s N · Δ f and f 0 n · Δ f .
  2. 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 Q · N . This is a nonrestrictive assumption since Q can be chosen large enough to make it true for any value of L . If Q N exceeds the minimum required value of L , then h ( k ) 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.
  3. 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 = N 2

We can now examine the effects of these assumptions. First, the relationship between f s , f 0 , and Δ f allows y ( r ) to be written as

y n ( r ) = e - j 2 π n r M N · l = 0 L - 1 h ( l ) x ( r M - l ) e j 2 π n l N .

We subscript the decimated output y ( r ) by the parameter n to indicate that it depends on the tuning frequency f 0 = n · Δ 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

l q N + p , w h e r e 0 q Q - 1 a n d 0 p N - 1 .

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,

y n ( r ) = e - j 2 π n r M N · l = 0 L - 1 h ( l ) x ( r M - l ) e j 2 π n l N = e - j 2 π n r M N · p = 0 N - 1 q = 0 Q - 1 h ( q N + p ) x ( r M - q N - p ) e j 2 π n ( q N + p ) N = e - j 2 π n r M N · p = 0 N - 1 q = 0 Q - 1 h ( q N + p ) x ( r M - q N - p ) e j 2 π n q N N e j 2 π n p N = e - j 2 π n r M N · p = 0 N - 1 e j 2 π n p N [ q = 0 Q - 1 h ( q N + p ) x ( r M - q N - p ) ] .

Questions & Answers

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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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