Cascading sampling rate converters

Given a discrete-time signal x(n) we can want to convert its sampling rate by a non integer factor, in particular, by a rational number. For instance, we can be interested in x(3n/2). That adds a 2-fold up-sampling and a 3-fold down-sampling for a net down sampling by a factor of 1.5 (= 3/2). The sequence x(3n/5) has a 5-fold up-sampling and a 3-fold down-sampling for a net up-sampling by a factor of 1.67 (= 5/3).

Usually, in a cascade of an M-fold down-sampler and an L-fold up-sampler the positions of the two samplers are inter-changeable with no difference in the input-output strcture if and only if M and L are co-prime (relatively prime, that is, M and L do not have a

common factor). The sequence x(3n/2) can be prepared by cascading the up-sampler and the down-sampler in either order, that is, down followed by up or vice versa. However, a cascade of a 6-fold down-sampler (M = 6) followed by a 4-fold up-sampler (L = 4) is not the same as a cascade of a 4-fold up-sampler followed by a 6-fold down-sampler even though in both types M/L = 6/4. This is due to M and L have a common factor, that is, the rational number M/L is not in its reduced form. The ratio M/L could be replaced to 3/2; then the 3-fold down-sampler and the 2-fold up-sampler are interchangeable in position.

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