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The next problem encountered, however, is that the choice of f i n as 8000 Hz and f s as 3840 Hz means that the tuner's decimation factor M t is not an integer. In particular, with these sampling rate choices, M t is given by 25 12 . As a result, a simple one-step decimating tuner of the type shown in Figure 1 from "Derivation of the equations for a Basic FDM-TDM Transmux" cannot be used directly.

The solution to this problem comes with the use of digital interpolation and decimation techniques. These are described in [link] and we refer to it here as digital resampling, the process of creating new digital samples at the desired rate from a sequence sampled at a different rate. The block diagram of this process is shown in the top portion of [link] . The incoming real-valued signal is first quadrature downconverted to move the band of interest into the passband of the digital lowpass filter and to register the filter bank's filters with the mark and space frequencies of the VFT signal. Conceptually, the downconverted quadrature signal is then zero-filled The zero-filling factor is 6 for input signals sampled at 16 kHz. by a factor of 12, lowpass filtered, and then decimated by a factor of 25. The zero-filling artificially increases the sampling rate to 96 kHz, creating 11 extra images of the input signal in the process. The lowpass filter removes these images and bandlimits the zero-filled signal to just the 2880 Hz band of interest. The decimation leads to an output rate of 96000 25 = 3840 Hz, exactly the desired value. In fact, the signal is never physically zero-filled. Pointers in the hardware keep track of where the non-zero data points lie and use that information to avoid doing unnecessary multiplication.

The bottom portion of [link] shows a circuit card assembly built to perform the downconversion and resampling processes for 24 input voice channels. An multiplier chip was used for the downconversion of all 24 channels and a pair of MACs performs the filtering needed to resample all 24 inputs. Programmable ROMs were used to generate the sequencing signals needed for the resampler. The extra MAC and ALU visible on the card are used to spectrum-analyze all 24 input channels with 60 Hz resolution at about 40 times a second. This spectral data is D/A-converted and provided to an oscilloscope for use by the equipment's operator.

Note that even though resampling is being performed, the equations used to choose the optimum value of f s are still valid. The fundamental reason for this is that the filter segment of the tuner is still of the FIR variety and that one-step decimation is still employed. As a result, the average computation for the tuner remains as predicted by [link] .

We might note in passing that the resampler used here is termed a synchronous resampler since the ratio of the number of input samples to output samples is rigidly fixed. It is also possible to employ a so-called asynchronous resampler to produce the desired samples. This is usually done when the input sampling rate varies slightly over time and it is desired to have the output rate locked to some frequency standard. The control of such resamplers is more complicated than the synchronous variety but the amount of computation needed for the downconversion and filtering is essentially the same.

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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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