Time Scaling: Compression and Expansion

Time Scaling: Compression and Expansion

Time scaling changes the rate at which samples are taken.

Compression / Down-Sampling: $y[n] = x[Mn]$, $M > 1$ (integer)

Keep every $M$-th sample, discard the rest. The signal speeds up (compressed in time).

x[n]:   [0] [1] [2] [3] [4] [5] [6] [7]
          ↑           ↑           ↑
        n=0         n=3         n=6

y[n] = x[3n]:  [0]      [3]      [6]
              ↑        ↑        ↑
             n=0      n=1      n=2

Expansion / Up-Sampling: $y[n] = x[n/M]$, $M > 1$ (integer)

Insert $M-1$ zeros between samples. The signal slows down (expanded in time).

x[n]:  [0] [1] [2]
        ↑   ↑   ↑
       n=0 n=1 n=2

y[n] = x[n/3]:  [0] [0] [0] [1] [0] [0] [2] [0] [0]
                  ↑       ↑       ↑
                 n=0     n=3     n=6

Note: After up-sampling, interpolation (lowpass filtering) fills in the zeros with meaningful values.

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Worked Example: Multiple Operations on $x[n] = {2, 1, 0, -1, -2}$

Let $x[n] = {2, 1, 0, -1, -2}$ with $n = 0, 1, 2, 3, 4$ (arrow under 2 at $n=0$).

Operation	Result	Description
$y_1[n] = x[n-2]$	${0, 0, 2, 1, 0, -1, -2}$	Delay by 2
$y_2[n] = x[n+1]$	${1, 0, -1, -2, 0}$	Advance by 1
$y_3[n] = x[2n]$	${2, 0, -2}$	Down-sample by 2 (keep $n=0,2,4$)
$y_4[n] = x[n/2]$	${2, 0, 1, 0, 0, 0, -1, 0, -2}$	Up-sample by 2 (insert zeros)
$y_5[n] = x[-n]$	${-2, -1, 0, 1, 2}$	Time reversal (folding)

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