Time Shifting: Delay and Advance

Time Shifting: Delay and Advance

Delay (Right Shift)

\(y[n] = x[n - k], \quad k > 0\)

The signal moves right by $k$ samples. The output at time $n$ depends on the input at earlier time $n-k$.

x[n]:     [•] [•] [•] [•] [•]       (indices: -1, 0, 1, 2, 3)
              ↑              ↑
             n=0            n=2

x[n-2]:            [•] [•] [•] [•] [•]   (shifted right by 2)
                       ↑              ↑
                      n=2            n=4

Advance (Left Shift)

\(y[n] = x[n + k], \quad k > 0\)

The signal moves left by $k$ samples. The output at time $n$ depends on the input at future time $n+k$.

Causality note: $x[n+k]$ requires future input values. Non-causal systems (not realizable in real-time). Causal systems only use $x[n-k]$ for $k \ge 0$.

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