Time Shifting:
Time shifting is an operation performed on a signal where a new signal is defined by adding or subtracting a finite time value from it. This operation changes the position of the signal along the time axis. It can be used to shift a signal forward or backward in time.

Example:
Let's consider a continuous-time signal x(t) and we want to perform a time shift on it. The time-shifted signal y(t) is defined as:

y(t) = x(t - T)

Here, T represents the amount of time by which the signal is shifted. If T is positive, the signal is shifted to the right (forward in time), and if T is negative, the signal is shifted to the left (backward in time).

Illustration:
To illustrate this concept, let's consider a simple example. Suppose we have a continuous-time signal x(t) = cos(t) and we want to shift it by 2 units of time to the right. The time-shifted signal y(t) would be:

y(t) = cos(t - 2)

This means that at any given time t, the value of y(t) would be the same as the value of x(t - 2). In other words, the entire signal x(t) is shifted 2 units of time to the right.

Importance:
Time shifting is an important operation in signal processing and communication systems. It allows us to manipulate the timing of signals, which can be useful in various applications. For example, in audio processing, time shifting can be used to create echo effects or adjust the timing of different audio tracks. In communication systems, time shifting is used to synchronize signals and compensate for delays introduced by transmission channels.

Conclusion:
In summary, time shifting is an operation where a new signal is defined by adding or subtracting a finite time value from it. This operation changes the position of the signal along the time axis. It is an important operation in signal processing and communication systems, allowing us to manipulate the timing of signals for various applications.