Miscellaneous remarks and derivations

Abstract

As we have seen in Sec. 2.5, the DFT imposes a cyclic behaviour on both of its input and output data sequences, in the sense that each of them should be understood as a single N-element period (or “truncation window”) from a periodic sequence which repeatedly extends to both directions ad infinitum (even if the underlying continuous data before sampling was not periodic). In the informal graphical development of the DFT, as presented in Sec. 2.5, this periodicity is explained by the fact that sampling (i.e. multiplication with an infinite impulse train) in one domain is equivalent to a convolution with the reciprocal infinite impulse train in the other domain. Thus, sampling in the signal domain results in a periodic behaviour in the frequency domain, and sampling in the frequency domain results in a periodic behaviour in the signal domain (see Fig. 2.1).