In signal processing, signal undergoes several manipulations of independent variable i.e. time. Based upon that we can classify operation as

**Contents**show

Operations corresponding to transformation of independent variable (Time)

- Time Shifting
- Time Scaling
- Time Reversal

Let us see each operation one by one.

## Time Shifting

A signal x(t) may be shifted in time by replacing the independent variable ** t** by either

**or**

*t−t*_{0}*. Here t*

**t+t**_{0 }_{0 }is called as the

*shifting factor*. Shifting in time may results in time delay or time advancement.

If the independent variable t is replaced by t−t_{0} ,the signal is shifted to the right and the time shift results in a delay of the signal by t_{0 }units of time. This type of time shifting is known as **Right side shifting**. This can be achieved by adding t_{0 }value to every time instant in signal x(t).

Consider a signal x(t) as shown in figure 1(i). Figure 1(ii) shows the signal x(t-4). During this process, the width of the signal x(t) does not change. Therefore W=W_{0}.

If the independent variable t is replaced by t+t_{0} ,the signal is shifted to the left and the time shift results in an advancement of the signal by t_{0 }units of time. This type of time shifting is known as **Left side shifting**. This can be achieved by subtracting t_{0 }value to every time instant in signal x(t).

Consider a signal x(t) as shown in figure 2(i). Figure 2(ii) shows the signal x(t+2). During this process, the width of the signal x(t) does not change. Therefore W=W_{0}.

## Time Scaling

A signal x(t) may be scaled in time by replacing the independent variable ** t** by

*. Here ‘*

**at**_{ }

*a’*_{ }is called as the

*scaling factor*. Time scaling may results in signal compression or signal expansion.

If the independent variable ** t** is replaced by

*and*

**at****a>1**, the signal is

**. This can be achieved by dividing every time instant in signal x(t) by ‘**

*compressed**‘.*

**a**If the independent variable ** t** is replaced by

*and*

**at****0<a<1**, the signal is

**. This can be achieved by dividing every time instant in signal x(t) by ‘**

*expanded**‘.*

**a**Consider a signal x(t) as shown in figure 3(i). Figure 3(ii) shows the signal x(3t). It is compressed since **a>1.** During this process, the width of the signal x(t) changes. Therefore W>W_{0}. Figure 3(iii) shows the signal x((1/2.5)t). It is expanded since **0<a<1****.** During this process, the width of the signal x(t) changes. Therefore W<W_{0}.

## Time Reversal

If the independent variable ** t** is replaced by ‘−

*, this operation is known as*

**t’****time reversal**of the signal about the y-axis or amplitude axis. This can be achieved by taking mirror image of the signal x(t) about y-axis or by rotating x(t) by 180° about y-axis. Hence, time reversal is known as

*folding*or

*reflection*.

Consider a signal x(t) as shown in figure 4(i). The time reversal of this signal can be achieved by taking mirror image in the y-axis as shown in figure 4(ii). During this process, the width of the signal x(t) does not change. Therefore W=W_{0}.