Time Shifting and Time Scaling in Signal Processing
Transforming an input signal by time scaling and time shifting is a fundamental concept in signal processing. Drag the sliders to scale the time and time shift the signal.
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 begins playing seconds after the signal . When , the signal is shifted right on the axis. When , the signal is shifted left on the axis.
When the function has both time-scaling and time-shifting factors, the output is , where .
You can display the input signal , the output signal , and two intermediate signals and . 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.
 E. Riskin, E. Ferré, W. S. Lau, and B. Ngo. EE235: Continuous Time Linear Systems. "Transformations of Continuous-Time Signals." (Jun 20, 2016) www.ee.washington.edu/class/235dl/EE235/Project/lesson2/lesson2.html.