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Signals and Systems: Addition and Subtraction of Signals
Addition and Subtraction of Continuous-Time Signals
The sum of two continuous time signals ?1(?) and ?2(?) can be obtained by adding their values at every instant of time. Likewise, the difference of two continuous time signals ?1(?) and ?2(?) can be obtained by subtracting the values of one signal (say ?2(?)) from another signal (say ?1(?)) at every instant of time.
Addition of Signals
Let two continuous time signals ?1(?) and ?2(?) as shown in Figure-1. The sum of these two signals ?1(?) + ?2(?) is also shown in Figure-1.
Explanation
The sum of these two signals can be obtained by considering different time intervals as follows −
For ? ≤ ? ≤ ?− ?1(?) = 3 and ?2(?) is increasing linearly from 0 to 2. Therefore, the sum [i.e., ?1(?) + ?2(?)] will also increase linearly from (3 + 0 = 3) to (3 + 2 = 5).
-
For ? ≤ ? ≤ ?− ?1(?) = 2 and ?2(?) = 2, thus
?1(?) +?2(?) = 2 + 2 = 4
For 3≤ ? ≤ ?− ?1(?) = 3 and ?2(?) decreases linearly from 2 to 0. Hence, the sum of the signals will also fall linearly from (3 + 2 = 5) to (3 + 0 = 3).
Subtraction of Signals
The subtraction [?1(?) − ?2(?)] of two continuous time signals ?1(?) and ?2(?) is shown in Figure-2.
Explanation
The difference of two continuous-time signals ?1(?) and ?2(?) can be obtained by considering different time intervals as follows −
For ? ≤ ? ≤ ?−?1(?) = 3 and ?2(?) is increasing linearly from 0 to 2. Thus, the difference [?1(?) −?2(?)] falls linearly from (3 – 0 = 3) to (3 – 2 = 1).
-
For ? ≤ ? ≤ ?− ?1(?) = 2 and ?2(?) = 2, thus
?1(?) − ?2(?) = 2 − 2 = 0
For 3≤ ? ≤ ?− ?1(?) = 3 and ?2(?) decreases linearly from 2 to 0. Hence, the difference of the signals will rise linearly from (3 – 2 = 1) to (3 – 0 = 3).
Addition and Subtraction of Discrete-Time Signals
In the discrete time case, the sum of two sequences ?1(?) and ?2(?) can be obtained by adding the corresponding sample values. Similarly, the difference of the two sequences ?1(?) and ?2(?) can be obtained by subtracting each sample of one signal from the corresponding sample of the other signal.
Let two discrete-time sequences are given as follows −
?1(?) = {2, 1, 3, 5, 2}
?2(?) = {1, 4, 2, 1, −3}
Then, the addition of discrete-time signals is,
?1(?) + ?2(?) = {2 + 1, 1 + 4, 3 + 2, 5 + 1, 2 − 3} = {3, 5, 5, 6, −1}
Similarly, the subtraction of discrete-time signals is,
?1(?) - ?2(?) = {2 − 1, 1 − 4, 3 − 2, 5 − 1, 2 + 3} = {1, −3, 1, 4, 5}
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