Transformation of the independent variable

Time scaling Given the sequence x(n), the sequence y(n) = x(2n) is calculated by skipping odd-numbered samples in x(n) and obtaining the even-numbered ones. The expansion to y(n) = x(Mn) means we have sample numbers 0, M, 2M, 3M, ..., and skip the intervening M-1 samples between those we save. The original sequence x(n) is calculated by sampling a continuous signal x(t) at a different rate

(perhaps over-sampling). The signal y(n) = x(Mn) is then calculated by reducing the sampling rate by a factor of M on the continuous-time signal x(t). That is called as down-sampling or decimation or sampling rate compression.

 Similarly the function of taking the sequence y(n) = x(n/L) from the sequence x(n) seems we derive y(n) by inserting (L-1) sequence points with zero value between points of x(n). That is known as up-sampling or interpolation or sampling rate extension. (Inserting (L-1) zeros is just one side of interpolating. It is also required for the up-sampler to be taken by a digital system that reproduced the inserted zeros with more related values lied on a linear combination of the x(n) samples.) 

 Usually, the result of time scaling a discrete-time signal is not just a compressed or stretched  version of the real but possibly a totally different waveform.

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