1.2 Basic signal operations

We will be considering the following basic operations on signals:

• Time shifting:
  $$y\left(t\right)=x(t-\tau )$$
  The effect that a time shift has on the appearance of a signal is seen in [link] . If $\tau$ is a positive number, the time shifted signal, $x(t-\tau )$ gets shifted to the right, otherwise it gets shifted left.
• Time reversal:
  $$y\left(t\right)=x(-t)$$
  Time reversal flips the signal about $t=0$ as seen in [link] .
• Addition: any two signals can be added to form a third signal,
  $$z\left(t\right)=x\left(t\right)+y\left(t\right)$$
• Time scaling:
  $$y\left(t\right)=x\left(\Omega t\right)$$
  Time scaling “compresses" the signal if $\Omega >1$ or “stretches" it if $\Omega <1$ (see [link] ).
• Multiplication by a constant, $\alpha$ :
  $$y\left(t\right)=\alpha x\left(t\right)$$
• Multiplication of two signals, their product is also a signal.
  $$z\left(t\right)=x\left(t\right)y\left(t\right)$$
  Multiplication of signals has many useful applications in wireless communications.
• Differentiation:
  $$y\left(t\right)=\frac{dx\left(t\right)}{dt}$$
• Integration:
  $$y\left(t\right)=\int x\left(t\right)dt$$

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.