Some essential operations for manipulating signals include shifting, scaling, and reflection. When we talk about a signal in time domain, those are called time shifting, time scaling, and time reversal (reflection).

Shifting

To shift a signal by , we replace with in the signal and get

  • shift to right (delay)
  • shift to left (advance)

Shifting won’t change the size of the signal.

Scaling

The compression or expansion of a signal in time is known as time scaling. To time scale a signal by a factor of , we replace by . If , the result is a compression. And if , the result is an expansion.

Reflection

Reflection or time reversal can be seen as a special case of scaling with . To time-reverse , we replace with .

Combined Operations

Signal operations often combine the above basic operations as . It can be understand in two different sequences of operations:

  • Time-shift by to obtain and then time-scale the shifted signal by to obtain .
  • Time-scale by to obtain and then time-shift by to obtain

Reference

  • Linear Systems and Signals, 1.2