Signal time shift and scaling are core concepts in a signals and systems class . The following describes how the signal transformation variables
affect the input signal
The time-scaling factor
is analogous to "play-back speed." When
, the signal is replayed at two times the speed and so takes half as long. When
, the signal is replayed at one-quarter the speed and so takes four times as long. When
, the signal is expanded. When
, the signal is compressed.
The time-shifting factor
is the time delay. The input signal
seconds after the signal .
, the signal is shifted right on the
, the signal is shifted left on the
When the function has both time-scaling and time-shifting factors, the output is
You can display the input signal
, the output signal
, and two intermediate signals
. For an overview of signal transformations, see .
 F. T. Ulaby and A. E. Yagle, Engineering Signals and Systems in Continuous and Discrete Time
, 2nd ed., Allendale, NJ: NTS Press, 2016.