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Show 92 not so-constrained are, nevertheless, also reverse mappable onto the line by members of a set of functions, called retracts, of which B. 2 and B.3 are examples. The way in which this reverse mapping occurs is determined by the form of the retract, and is of importance when considering the effects of spectral domain modifications which violate time or frequency resolution constraints placed on spectral domain signals by h(t). The OLA and FBS synthesis do not exhaust the possible forms of a more general class of retracts useable for short-time Fourier synthesis. Rather, they are special cases of the general retract, 1 f(t) = ff F(W,T)g(t-r)ejWt dw dT (B.4) 2n<g,h> where <g,h> is the inner product, <g,h> = f g(t)h(t) dt (B.5) The FBS and OLA synthesis are easily seen to be special forms of B.4. In particular, the FBS synthesis is obtained given the condition, g(t) = 6(t) (B.6) where 6(t) is the Dirac delta function. Similarly, the OLA synthesis is obtained if g(t) = 1 (B.7) and if the area of the analysis window is unity. We now show that B.1 and B.4 form an ... . --.. . . . . . -" i iI I I . .. |
