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Basic Signal Operations

Module by: Carlos E. Davila. E-mail the author

Summary: Overview of basic continuous-time signal operations.

We will be considering the following basic operations on signals:

  • Time shifting:
    y(t)=x(t-τ)y(t)=x(t-τ)(1)
    The effect that a time shift has on the appearance of a signal is seen in Figure 1. If ττ is a positive number, the time shifted signal, x(t-τ)x(t-τ) gets shifted to the right, otherwise it gets shifted left.
  • Time reversal:
    y(t)=x(-t)y(t)=x(-t)(2)
    Time reversal flips the signal about t=0t=0 as seen in Figure 1.
  • Addition: any two signals can be added to form a third signal,
    z(t)=x(t)+y(t)z(t)=x(t)+y(t)(3)
  • Time scaling:
    y(t)=x(Ωt)y(t)=x(Ωt)(4)
    Time scaling “compresses" the signal if Ω>1Ω>1 or “stretches" it if Ω<1Ω<1 (see Figure 2).
  • Multiplication by a constant, αα:
    y(t)=αx(t)y(t)=αx(t)(5)
  • Multiplication of two signals, their product is also a signal.
    z(t)=x(t)y(t)z(t)=x(t)y(t)(6)
    Multiplication of signals has many useful applications in wireless communications.
  • Differentiation:
    y(t)=dx(t)dty(t)=dx(t)dt(7)
  • Integration:
    y(t)=∫x(t)dty(t)=∫x(t)dt(8)

There is another very important signal operation called convolution which we will look at in detail in Chapter 3. As we shall see, convolution is a combination of several of the above operations.

Figure 1: (a) original signal, (b) time-shift, (c) time-reversal.
Figure 1 (time_shift.png)
Figure 2: (a) original signal, (b) Ω>1Ω>1, (c) Ω<1Ω<1.
Figure 2 (time_scale.png)

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