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The computational efficiency of the transmultiplexer can therefore be traced to two key items:

  1. Separation of the tuning computation into two segments, one of which (the { v ( r , p ) } ) need be computed only once
  2. The use of the FFT algorithm to compute the inverse DFT

The first accrues from strategic choices of the sampling and tuning frequencies, while the second depends on N being chosen to be a highly composite integer.

Figure four is a traditional cartesian graph in the first quadrant, with horizontal axis labeled Channels to be Tuned Simultaneously C, and vertical axis labeled Mega Multiply Adds. The values on the horizontal axis range from 0 to 40 in increments of 10, and the vertical axis ranges from 0 to 150 in increments of 50. There is one horizontal line labeled FFT-based Transmux with a vertical value of approximately 25. At (10, 25) and approximately (32, 25) there are small squares on the line. Beginning near the origin and increasing with a shallow, constant slope is a line labeled DFT-Based Transmux, with two hash-marks near the end of the graph. Beginning at the origin and increasing with a strong positive constant slope is a third line, labeled Conventional One-step Digital Tuning. There is a label in the middle of the figure that reads: System Parameters. f_s = 512 kHz, N = 128, Q = 16, ∆f = 4 kHz, M = N.
The Number of Multiply-Adds Needed to Compute C Tuner Outputs for a Particular Set of System Parameters
Figure 5 is a flow chart. Moving from left to right, the flow chart begins with the caption Analog FDM Signal. An arrow points to the right at a large rectangle containing the label A/D. Below this rectangle is an arrow pointing up, labeled f_s = N ⋅ ∆f. A large arrow points to the right from the A/D rectangle at a second rectangle that contains the caption Preprocessor. Below this rectangle are three bullet points that read, block points, weight, computer v(r, p). A large arrow again points to the right at a third rectangle labeled N Point Inverse DFT. To the right of this are a series of small horizontal lines, an arrow pointing to the top-left corner of the figure, and a large arrow pointing to the right. This series of lines and arrows is labeled Commutator.
The Basic FDM-to-TDM Digital Transmultiplexer

The transmux as a dft-based filter bank

We have just developed an FDM-TDM transmultiplexer by first writing the equations for a single, decimated digital tuner. The equations for a bank of tuners come from then assuming that (1) they all use the same filter pulse response and (2) their center frequencies are all integer multiples of some basic frequency step. In this section, we develop an alternate view, which happens to yield the same equations. It produces a different set of insights, however, making its presentation worthwhile.

Using the dft as a filter bank

Instead of building a bank of tuners and then constraining their tuning frequencies to be regularly spaced, suppose we start with a structure known to provide equally-spaced spectral measurements and then manipulate it to obtain the desired performance.

Consider the structure shown in [link] . The sampled input signal x ( k ) enters a tapped delay line of length N . At every sampling instant, all N current and delayed samples are weighted by constant coefficients w ( i ) (where w ( i ) scales x ( k - i ) , for i between 0 and N - 1 ), and then applied to an inverse discrete Fourier transform Whether or not it is implemented with an FFT is irrelevant at this point. Also, we happen to use the inverse DFT to produce a result consistent with that found in the proceeding subsction, but the forward DFT could also be used. . The complete N-point DFT is computed for every value of k and produces N outputs. The output sample stream from the m-th bin of the DFT is denoted as X m ( k ) .

Figure six is a flow chart involving three rows of labeled shapes. The first row is labeled Input data, followed by an arrow pointing to the right labeled x(k) at a box containing the caption z^-1. To the right is another arrow pointing right labeled x(k-1) at a box containing the caption z^-1. To the right of this is a longer unlabeled arrow pointing to the right at a third box containing the caption z^-1. Above this box is a caption that reads x(k-N+1). In the middle of each arrow, and after the last box, there are four arrows pointing down at four circles containing a large x. From left to right, these circles have arrows pointing at them from the left side that are labeled w_0, w_1, w_2, and w_N-1. Each circle also has an arrow below it pointing down at a long rectangle containing the caption N-point DFT. Aligned with the arrows above are four more arrows below this rectangle that point at the expressions below them that read from left to right, x_0(k), x_1(k), x_2(k), x_N-1(k).
Processing Weighted, Delayed Signals with Discrete Fourier Transform

Since DFTs are often associated with spectrum analysis, it may seem counterintuitive to consider the output bins as time samples. It is strictly legal from an analytical point of view, however, since the DFT is merely an N-input, N-output, memoryless, linear transformation. Even so, the relationship of this scheme and digital spectrum analysis will be commented upon later. We continue by first examining the path from the input to a specific output bin, the m-th one, say. For every input sample x ( k ) there is an output sample X m ( k ) . By inspection we can write an equation relating the input and chosen output:

X m ( k ) = p = 0 N - 1 x ( k - p ) w ( p ) e j 2 π m p N

the m-th bin of an N-point DFT of the weighted, delayed data. We can look at this equation another way by defining w ¯ m ( p ) by the expression

w ¯ m ( p ) w ( p ) · e j 2 π n p N

and observing that [link] can be written as

X m ( k ) = p = 0 N - 1 x ( k - p ) · w ¯ m ( p ) .

From this equation it is clear X m ( k ) is the output of the FIR digital filter that has x ( k ) as its input and w ¯ m ( p ) as its pulse response. Since the pulse response does not depend on the time index k , the filtering is linear and shift-invariant. For such a filter we can compute its transfer function, using the expression

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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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