Signals and Systems: Addition and Subtraction of Signals

Addition and Subtraction of Continuous-Time Signals

The sum of two continuous time signals 𝑥₁(𝑡) and 𝑥₂(𝑡) can be obtained by adding their values at every instant of time. Likewise, the difference of two continuous time signals 𝑥₁(𝑡) and 𝑥₂(𝑡) can be obtained by subtracting the values of one signal (say 𝑥₂(𝑡)) from another signal (say 𝑥₁(𝑡)) at every instant of time.

Addition of Signals

Let two continuous time signals 𝑥₁(𝑡) and 𝑥₂(𝑡) as shown in Figure-1. The sum of these two signals 𝑥₁(𝑡) + 𝑥₂(𝑡) is also shown in Figure-1.

Explanation

The sum of these two signals can be obtained by considering different time intervals as follows −

• For 𝟎 ≤ 𝒕 ≤ 𝟏− 𝑥₁(𝑡) = 3 and 𝑥₂(𝑡) is increasing linearly from 0 to 2. Therefore, the sum [i.e., 𝑥₁(𝑡) + 𝑥₂(𝑡)] will also increase linearly from (3 + 0 = 3) to (3 + 2 = 5).

• For 𝟏 ≤ 𝒕 ≤ 𝟑− 𝑥₁(𝑡) = 2 and 𝑥₂(𝑡) = 2, thus

  𝑥₁(𝑡) +𝑥₂(𝑡) = 2 + 2 = 4

• For 3≤ 𝒕 ≤ 𝟒− 𝑥₁(𝑡) = 3 and 𝑥₂(𝑡) 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 [𝑥₁(𝑡) − 𝑥₂(𝑡)] of two continuous time signals 𝑥₁(𝑡) and 𝑥₂(𝑡) is shown in Figure-2.

Explanation

The difference of two continuous-time signals 𝑥₁(𝑡) and 𝑥₂(𝑡) can be obtained by considering different time intervals as follows −

• For 𝟎 ≤ 𝒕 ≤ 𝟏−𝑥₁(𝑡) = 3 and 𝑥₂(𝑡) is increasing linearly from 0 to 2. Thus, the difference [𝑥₁(𝑡) −𝑥₂(𝑡)] falls linearly from (3 – 0 = 3) to (3 – 2 = 1).

• For 𝟏 ≤ 𝒕 ≤ 𝟑− 𝑥₁(𝑡) = 2 and 𝑥₂(𝑡) = 2, thus

  𝑥₁(𝑡) − 𝑥₂(𝑡) = 2 − 2 = 0

• For 3≤ 𝒕 ≤ 𝟒− 𝑥₁(𝑡) = 3 and 𝑥₂(𝑡) 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 𝑥₁(𝑛) and 𝑥₂(𝑛) can be obtained by adding the corresponding sample values. Similarly, the difference of the two sequences 𝑥₁(𝑛) and 𝑥₂(𝑛) 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 −

𝑥₁(𝑛) = {2, 1, 3, 5, 2}

𝑥₂(𝑛) = {1, 4, 2, 1, −3}

Then, the addition of discrete-time signals is,

 𝑥₁(𝑛) + 𝑥₂(𝑛) = {2 + 1, 1 + 4, 3 + 2, 5 + 1, 2 − 3} = {3, 5, 5, 6, −1}

Similarly, the subtraction of discrete-time signals is,

 𝑥₁(𝑛) - 𝑥₂(𝑛) = {2 − 1, 1 − 4, 3 − 2, 5 − 1, 2 + 3} = {1, −3, 1, 4, 5}