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We now develop some equations that describe the steps just traversed. Starting with [link] we replace N with ${N}^{\text{'}}=QN$ , obtaining an expression for the time sequence seen at the m-th DFT bin.
Suppose, as discussed above, that we eliminate the filter overlapping by evaluating only every Q-th DFT bin. Thus we compute ${X}_{m}\left(k\right)$ only for those values of m that are integer multiples of Q . Specifically, if n is assumed to be an integer, then we only compute ${X}_{m}\left(k\right)$ for values of m given by $m=Qn$ . This is leads to
Since we have achieved the goal of constructing spectrally concentrated bandpass filters (albeit at the cost of expanding the size of all steps preceding the final DFT computation), we can now consider decimating the filter outputs. Since the filter bandwidths are nominally $\frac{{f}_{s}}{N}$ Hz, decimation by up to N is possible without violating the sampling theorem. Suppose we decimate by the factor M , where $0<M\le N$ . This means evaluating the integer time index k only at integer multiples of M . If we allow the integer to be the decimated time index, the decimated version of the n-th DFT bin output is
At this point we can start making comparisons. [link] closely resembles [link] and [link] closely resembles [link] . In fact, if we use the definition of $v(r,p)$ developed earlier, then [link] becomes
which differs from the equation for ${y}_{n}\left(r\right)$ developed in [link] only in the absence of a residual carrier term. If, for example, we want to compute ${y}_{n}\left(r\right)$ , we can do it by selecting the right DFT output bin ( n in this case) and multiplying it by the residual carrier term, if any. Thus for all practical purposes, the bank of tuners viewpoint and the DFT-based filter bank viewpoint yield the same structure and same results.
More insight into the relationship between the DFT-based filter bank and the basic FDM-to-TDM transmultiplexer shown in [link] can be gained by considering the common situation where an FFT is used to compute the DFT. In the preceding section, it was shown that the DFT filter implicitly uses a QN-point DFT but in fact only N output bins are computed. Consider now the FFT flow graph shown in [link] (a). The input is $QN$ (8, in this case) weighted input samples $\overline{x}\left(p\right)$ and the output is $QN$ bins. Suppose now that all we want is the odd numbered output bins. Careful examination of the flow graph shows that more than just the output points can be deleted. Look at $x\left(0\right)$ , for example. It is computed using numbers from the previous stage which are only used to compute undesired outputs . Thus these intermediate terms need not be computed either. This process can continue until the point where the intermediate points are needed. To see how this works, examine [link] (b). Removing all unneeded nodes reveals something very interesting. The FFT processing naturally breaks into two sections. The second section, the QN-point FFT pruned of all unneeded nodes, is recognized to have the flow graph of an N-point FFT. In fact, if the bin decimation is not offset from bin 0, then the twiddle factors are exactly those of an N-point FFT as well. The section preceding the N-point FFT can be written as N Q-point sums of weighted, delayed input data samples. These sums can be recognized as the $\left\{v\right(r,p\left)\right\}$ . Thus by pruning out the unneeded nodes in a QN-point FFT taken over the weighted input data, the computation of the filter bank gracefully separates into the cascade of a preprocessor that computes the $\left\{v\right(r,p\left)\right\}$ and an N-point FFT. The resulting block diagram is exactly the same as that shown in [link] .
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