Identity #1 If we use the notation ↓M{.} to mean the down-sampling of the signal in braces, then we have

↓M{a x1(n) + b x2(n)} = ↓M{a x1(n)} + ↓M{b x2(n)} → (1) = a ↓M{x1(n)} + b ↓M{x2(n)} → (2)

In words, the result of down-sampling the weighted sum of signals equals the weighted sum of the down-sampled signals. In other words, the two block diagrams below are equivalent.

Equations (A) and (B) are identical.

Eventually, by virtue of Example 6.1, the down-samplers are moved to the upstream side of the multipliers which would correspond to Eq. (2).

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