The process of reflection and shifting is illustrated below.
An impulse response function can be reflected and shifted as shown above. It is possible to obtain the same result by shifting and then reflecting, but that process is not of much help for evaluating the convolution integral using graphical approach.
For the function f(t), shown in Fig. 10, obtain and plot
i. f(-t)
ii. f(t - 2),
iii. f(t + 3),
iv. f(-t + 3), and
v. f(- t - 3).
SOLUTION:
From the waveform in Fig. 10, an equation can be developed to describe the waveform.
The first two lines of equation (2.1) are easy to obtain. For 1 < t < 2 seconds, the equation for the line can be obtained as shown below.
From the second line of equation (2.1) that f(1) = 1. It can be verified from equation (2.2) that f(1) = 1.
i. Reflection
To obtain f (-t), we can replace t by -t in equation (2.1). The first step in obtaining f (-t) is illustrated below.
The second step is easy. The second step is for the period - 1 < t < 0.
The third step, shown below, is for the period - 2 < t < - 1.
The previous steps can be combined and expressed as shown below.
In short, the equation that describes the reflected waveform is presented below.
The plot of the reflected waveform is presented below.
The reflected waveform is obtained by rotating the original waveform by 180o about the y-axis. It is somewhat similar to turning over a page containing an image and looking at the image from the backside.
ii. Shifting to the Right
We can obtain an expression for the shifted pulse as shown below. From equation (2.1), we get that
That is, we can state that
The plot of equation (2.4) is shown in Fig. 12. It can be obtained by shifting the waveform for f (t) to the right by 2 units. Given f (t), f(t - a) is obtained by shifting f (t) to the right by ‘a’ units.
iii. Shifting to the Left
Based on equation (2.1), we can state that
Hence
The plot of equation (2.5) is shown in Fig. 13. It can be obtained by shifting the waveform for f(t) to the left by 3 units. Given f(t) , f(t+ a) is obtained by shifting f(t) to the left by ‘a’ units under the condition that a is positive.
iv. Reflect and Shifting to the Right
Based on equation (2.1), we can state that
Hence
The plot of equation (2.6) is shown in Fig. 14. It is seen that f(-t -3) is equivalent to reflecting f(t) first and then shifting it to the right by 3 units. Given f(t), f(-t + a) is obtained by reflecting f(t) and then shifting it to the right by ‘a’ units with ‘a’ being positive. We can get the same result, by shifting to the right first, and then reflecting it.
v. Reflect and Shifting to the Left
Based on equation (2.1), we can state that
Hence
The plot of equation (2.7) is shown in Fig. 15. It is seen that f(-t +3) is equivalent to reflecting f(t) first and then shifting it to the left by 3 units. Given f(t) , f(-t + a) is obtained by reflecting f(t) and then shifting it to the left by ‘a’ units with ‘a’ being positive.
The example presented above has illustrated how a waveform can be reflected and shifted. The following pages present examples that illustrate how the convolution integral can be evaluated.