Time Shifting, Time Scaling & Time Reversal of Signals

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

Operations corresponding to transformation of independent variable (Time)

  1. Time Shifting
  2. Time Scaling
  3. 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 t−t0 or t+t0 . Here tis 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−t0 ,the signal is shifted to the right and the time shift results in a delay of the signal by tunits of time. This type of time shifting is known as Right side shifting. This can be achieved by adding tvalue 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=W0.

right side time shifting of signal-delay
Fig. 1 Right side time shifting

If the independent variable t is replaced by t+t0 ,the signal is shifted to the left and the time shift results in an advancement of the signal by tunits of time. This type of time shifting is known as Left side shifting. This can be achieved by subtracting tvalue 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=W0.

left side time shifting of signal-advancement
Fig. 2 Left side time shifting

Time Scaling

A signal x(t) may be scaled in time by replacing the independent variable t by at . Here ‘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 at and a>1, the signal is compressed. This can be achieved by dividing every time instant in signal x(t) by ‘a‘.

If the independent variable t is replaced by at and 0<a<1, the signal is expanded. This can be achieved by dividing every time instant in signal x(t) by ‘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>W0. 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<W0.

Fig. 3 Time scaling of signal

Time Reversal

If the independent variable t is replaced by ‘−t’ , this operation is known as 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=W0.

Time reversal or folding of signal
Fig. 4 Time reversal of signal

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