The document discusses the convolution sum/integral for discrete-time and continuous-time linear time-invariant (LTI) systems. For discrete-time LTI systems, the response to a time-shifted unit impulse is a time-shifted version of the response to the original unit impulse. This leads to the convolution sum representation of the output. For continuous-time LTI systems, the convolution integral represents the output as the integral of the input multiplied by a time-shifted impulse response. The convolution operation describes how LTI systems transform inputs to outputs through a shifting and accumulating process.

1. DISCRETE-TIME LTI SYSTEMS: THE CONVOLUTION SUM
• In general, of course, the responses ℎ𝑘[𝑛] need not be related to each other for different values of k. However,
if the linear system is also time invariant, then these responses to time-shifted unit impulses are all time-
shifted versions of each other. Specifically, since 𝛿[𝑛 − 𝑘]is a time-shifted version of 𝛿[𝑛], the response
ℎ𝑘[𝑛] is a time-shifted version of ℎ0[𝑛]; i.e.,
• For notational convenience, we will drop the subscript on ℎ0[𝑛] and define the unit impulse (sample)
response
• That is, h[n] is the output of the LTI system when 𝛿[𝑛] is the input. Then for an LTI system.
This result is referred to as the convolution sum or superposition
sum. We will represent the operation of convolution
symbolically as

3. THE CONVOLUTION SUM: Example

4. THE CONVOLUTION SUM: Example

5. THE CONVOLUTION SUM: Example

6. THE CONVOLUTION SUM: Example
• Suggested examples: S. Palani, Signals and Systems, Chapter 3
• Example 3.45
• Example 3.50
• Example 3.52

8. CONTINUOUS-TIME LTI SYSTEMS: THE CONVOLUTION
INTEGRAL
2. CONTINUOUS-TIME LTI SYSTEMS: THE
CONVOLUTION INTEGRAL
For a CT system
• The unit impulse response of CT system
𝑥 𝑡 = න
−∞
∞
𝑥(𝜏)𝛿 𝑡 − 𝜏 𝑑𝜏
If the system is linear
𝒚 𝒕 = න
−∞
∞
𝒙 𝝉 𝒉𝝉 𝒕 𝒅𝝉
In case of time invariant system
𝒉𝝉 𝒕 = 𝒉 𝒕 − 𝝉
𝑦 𝑡 = ‫׬‬−∞
∞
𝑥 𝜏 ℎ 𝑡 − 𝜏 𝑑𝜏 (the output of LTI system)
This result is referred to as the convolution integral or
superposition integral. We will represent the operation
of convolution symbolically as
𝑥(𝑡) 𝑦(𝑡)
CT system
𝑥 𝑡 = 𝛿(𝑡 − 𝜏) 𝑦 𝑡 = ℎ𝜏(𝑡)
CT system
𝑥 𝑡 = 𝛿(𝑡) 𝑦 𝑡 = ℎ0(𝑡)

9. CONTINUOUS-TIME LTI SYSTEMS: THE CONVOLUTION
INTEGRAL
The convolution integral:
𝑦 𝑡 = න
−∞
∞
𝑥 𝜏 ℎ 𝑡 − 𝜏 𝑑𝜏
LTI System
Impulse Response: ℎ(𝑡)
𝑥(𝑡) 𝑦 𝑡) = 𝑥 𝑡 ∗ ℎ(𝑡 = න
−∞
∞
𝑥 𝜏 ℎ 𝑡 − 𝜏 𝑑𝜏

11. THE CONVOLUTION INTEGRAL
The following are the basic steps involved in convolution integral equation:
1. For the input signal 𝑥 𝑡 , express it as 𝑥 𝜏 .
2. Express impulse response function ℎ(𝑡) as ℎ(𝑡 − 𝜏) by substituting 𝑡 = 𝑡 − 𝜏
3. Integrate ‫׬‬−∞
∞
𝑥 𝜏 ℎ 𝑡 − 𝜏 𝑑𝜏 to get 𝑦(𝑡).
4. The limit of integration depends on the time limit of 𝑥 𝑡 . For causal signals 𝑥 𝑡 and ℎ 𝑡 (causal signal or
system : 𝑥 𝑡 =0 for 𝑥 < 0), the lower limit of convolution integral is zero and the upper limit is t.
5. Find 𝑦 𝑡 = ‫׬‬−∞
∞
𝑥 𝜏 ℎ 𝑡 − 𝜏 𝑑𝜏

12. THE CONVOLUTION INTEGRAL
B.P. Lathi, Linear Systems and Signals, 2005, Oxford University Press (optional)

13. THE CONVOLUTION INTEGRAL: Example

14. THE CONVOLUTION INTEGRAL: Example
• The graphical representation of convolution operation is shown in Fig. 3.8.

15. THE CONVOLUTION INTEGRAL: Example

16. THE CONVOLUTION INTEGRAL: Example

17. THE CONVOLUTION INTEGRAL: Example

18. THE CONVOLUTION INTEGRAL: Example

19. THE CONVOLUTION INTEGRAL: Example

20. THE CONVOLUTION INTEGRAL: Example

21. THE CONVOLUTION INTEGRAL: Example

22. THE CONVOLUTION INTEGRAL: Example
• Suggested examples: S. Palani, Signals and Systems, Chapter 3
• Example 3.2
• Example 3.11
• Example 3.13
• Example 3.16
• Example 3.19
• Example 3.21