SIGNAL OPERATIONS - II

EDITOR: B. SOMANATHAN NAIR

Example 5 (Combined Time Scaling and Shifting): – Generalized Method of Solution 1): Figure 9 shows a rectangular function x(t) having a base width of 2τ units of time and an amplitude of one unit. Plot the function y(t) = x(at+b).     

Solution: This is a combined operation of time shifting and scaling.                 

                                                                 

Step 1: We rewrite the given function by taking the factor a outside the brackets and then putting the remaining terms inside a single bracket to get:

                                                        y(t) = x[a(t+b/a)]  (1)

                                                                      

Step 2 (Time shifting): The next step is to find the function x(t+b/a). This is the function x(t) shifted to the left by b/a units of time (Fig. 10). It may be noted that in the figure, the pulse limits are given as c = –b/a +τ and d = –b/a–τ, respectively.

Step 3 (Compression): Finally, we find that the function x[a(t+b/a)] is equivalent to the function x(t+b/a) compressed by a units. This is shown in Fig. 11.

Example 6: Assume that in Fig. 9, the base width is 2 units of time and the amplitude one unit. Plot the function y(t) = x(2t+3).

Solution:  Here, we have a = 2, b = 3, and τ = 1. Substituting these values in the relevant expressions we get the center point of the compressed and time-shifted pulse as:

 ‒b/a = ‒3/2= ‒1.5

The limits of the pulse are given by

c = ‒b/a + τ = ‒3/2+1 = ‒0.5

d = ‒b/a ‒ τ = ‒3/2‒1 = ‒2.5

Using the above results, we get the desired function y(t) as shown in Fig. 12.         

Example 7 (Combined Time Scaling and Shifting – Generalized Method of Solution 2): Using the function shown in Fig.9, plot the function y(t) = x(at+b), where a = 2, b = 3, and τ = 1.

Solution: This second method can be seen to be simpler than the first method. We use the following steps to perform the operation of x(at+b) = x(2t+3).

                                                                 

Solution:  

Step 1: In this method, we first perform the compression operation x(at).  

Step 2: Next we shift x(at) by b/a units of time to the left or right depending on the requirement.                    

Here, we have a = 2, b = 3, and τ = 1. In step 1, we compress x(t) by a factor 2. This gives the waveform shown in Fig. 13. After compression, the next step is to shift the signal x(2t) by the factor b/a = 3/2 = 1.5 unit  to the left to get the function  y(t) = x(2t+3). as shown in Fig. 14.