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Two intuitively reasonable approaches to developing the equations for the FDM-TDM transmultiplexer are presented in this section. The first emulates [link] . We first develop the equations for a digital counterpart of the analog tuners used in the filter bank and then observe that significant computational improvements can be obtained when the tuning frequencies are linked together in a simple way. The second subsection starts from a different point, that of using the discrete Fourier transform as a spectral channelizer. We ultimately find out that these two approaches yield essentially the same analytical results.
The input FDM signal is assumed to be the continuous-time waveform ${x}_{c}\left(t\right)$ . The analog-to-digital converter shown in [link] samples this waveform at the uniform rate of f _{s} samples per second, producing the discrete-time sequence $x\left(k\right)$ , where $x\left(k\right)\equiv {x}_{c}(t=kT)$ , the integer k is the time index, and T is the sampling interval given by $T=\frac{1}{{f}_{s}}$ . The spectrum of this sequence is shifted down in frequency by multiplying it by a complex exponential of the form ${e}^{-j2\pi {f}_{0}kT}$ , where f _{0} is the desired amount of the frequency downconversion. The product of $x\left(k\right)$ and this exponential is then filtered in discrete time by using the pulse response $h\left(k\right)$ . The duration of the pulse response $h\left(k\right)$ is assumed to be finite and in particular of length no greater than L , an integer. The filter output $\overline{y}\left(k\right)$ is then decimated by a factor of M , yielding the sequence $y\left(r\right)$ , where the integer r is the decimated time index.
These processing steps are shown in graphical form in [link] . Both sides of the two-sided spectrum of the sampled input signal are seen in [link] (a). For the moment, the input signal is assumed to be real-valued and therefore the spectrum is symmetrical around 0 Hz Even though real-valued inputs are assumed here, all of the ensuing analysis applies to complex-valued signals as well. . A channel of interest in this spectrum has been shaded and its center frequency is noted to be f _{0} . Multiplying the input signal by ${e}^{-j2\pi {f}_{0}kT}$ has the effect of shifting the spectrum to the left (assuming $0\le {f}_{0}\le \frac{{f}_{s}}{2}$ ) and centering the desired channel at 0 Hz. The downconverted signal is now complex-valued, and therefore spectral symmetry around 0 Hz is neither required nor expected. The transfer function of the lowpass filter appears in [link] (c). The filter pulse response $h\left(k\right)$ is chosen to attain the desired spectral characteristics. In particular, the filter needs to pass the channel of interest without degradation and suppress all others sufficiently. How to design such a pulse response is discussed in Appendix A. In general, the quality of the filter grows with the value of the parameter L . The filter shown here is symmetrical around 0 Hz and its pulse response $h\left(k\right)$ can therefore be real-valued. This is not required however.
After the application of the shifted signal $\rho \left(k\right)$ to the filter, the spectrum shown in [link] (d) results. The desired channel is isolated from all others. It is sampled, however, at a rate far faster than required by the Nyquist sampling theorem. The filter output is then decimated by the factor M , resulting in the spectrum shown in [link] (e). The channel's bandwidth is the same as before but now its percentage bandwidth, that is, its bandwidth compared to its final sampling rate, is much higher. In a good digital tuner the percentage bandwidth after decimation usually ranges between 0.5 and 0.9, where unity is the theoretical limit imposed by the sampling theorem.
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