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7 plots the original signal and the up-sampled signal with L=2. Linearity and Time-Dependence Properties Down-sampling and up-sampling are linear time-dependent operations. 6). The time-dependence property has to be examined in more details. The time varying property of a down-sampler (up-sampler) is due to the fact that the delay of n0 samples in the original signal does not result in the same delay of the down-sampled (up-sampled) signal. Let us consider the responses of a down-sampler to the sequences {x[n]}, and {x1[n]} = {x[n–n0]}.

1. The up-sampler is also a time-varying system. To show this, we observe from Eq. 6) that its output y1[m] for an input x1[n]= x[n– n0] is given by  x [m L ], m = 0, ± L, ± 2 L,  y1 [m ] =  1 otherwise  0,  x (m - Ln0 ) / L  , m = 0, ± L, ± 2 L,  . 8. Illustration of the periodical time-invariance of a down-sampler with M=2 Hence, the up-sampling is a time-varying operation. 9 illustrates the time-dependence of upsampling operation for the factor M = 2, and delays n0 = 1 and n0 = 2 samples, respectively.

60) shows that when the sampling is performed in a sufficiently high rate, the spectrum of the discrete signal appears as a periodic repetition of the original spectrum. The original signal can be recovered by selecting with an ideal lowpass filter the baseband spectrum from the periodic spectral function X(ejω). On the contrary, the undersampling causes aliasing in the spectrum X(ejω) thus making the signal recovery impossible. Ideally, the reconstructed signal xr(t), can be expressed in terms of the sample values {x[n]} and the impulse response of the ideal reconstruction filter hr(t), xr (t ) = ∞ ∑ x [n]h (t − nT ).

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