Continuous Time LTI systems: the convolution …...2017/10/06 · system can be computed by a...

Signals and Systems Lecture: 4 (Convolution Integral)

Dr. Ayman Elshenawy Elsefy Page | 1

Continuous Time LTI systems: the convolution Integral

In much the same way as for discrete-time systems, the response of a continuous time LTI

system can be computed by a convolution of the system’s impulse response with the input signal,

using a convolution integral rather than a sum.

Representation of Continuous-Time Signals in Terms of Impulses

Impulse Response and the Convolution Integral Representation of a Continuous-Time

LTI System



The Convolution Operation



Calculation of the Convolution Integral

The calculation of a convolution integral is very similar to the calculation of a convolution

sum. To evaluate the integral in Equation

for a specific value of t, we first obtain the signal viewed as a function of , then we

multiply it by x( )to obtain the function g( ), and finally we integrate g( ) to get y(t).

Example 1: given ℎ(𝜏) as indicated in the below figure, sketch ℎ(𝑡 − 𝜏) for t=-2 and t=2.

At t= -2, ℎ(𝑡 − 𝜏) = ℎ(−2 − 𝜏) = ℎ(−(𝜏 + 2)) which is equal to ℎ(−𝜏) [the time

reverse of ℎ(𝜏)] advanced by 2.

At t= 2, ℎ(𝑡 − 𝜏) = ℎ(2 − 𝜏) = ℎ(−(𝜏 − 2)) which is equal to ℎ(−𝜏) [the time reverse

of ℎ(𝜏)] delay by 2.



Example 2: compute the response of continuous time LTI system described by its

impulse response ℎ(𝑡) = 𝑒−𝑎𝑡𝑢(𝑡), 𝑎 > 0 to the step input signal 𝑥(𝑡) = 𝑢(𝑡)

- Sketch the impulse response ℎ(𝜏) and 𝑥(𝜏) as shown in the figure.

- Sketch ℎ(−𝜏) and ℎ(𝜏 − 𝑡) for both positive (𝑡 > 0) and negative times (𝑡 < 0).

For < 0 : 𝑥(𝜏) and ℎ(𝜏 − 𝑡) does not overlap , and 𝑦(𝑡) = 0.

Example 3: calculate the response of continuous time LTI system described by its impulse

response ℎ(𝑡) = 𝑢(𝑡 + 1) to the input signal 𝑥(𝑡) = −𝑒2(𝑡−1)𝑢(−(𝑡 − 1))

- Sketch the impulse response ℎ(𝜏) and 𝑥(𝜏) as shown in the figure.



- The time-reversed impulse response ℎ(−𝜏) is plotted, ℎ(𝜏 − 𝑡) for both positive

(𝑡 > 0) and negative times (𝑡 < 0).

From the above figure we can see that that there are two distinct cases:

for t ≤ 0, the two functions overlap over the interval−∞ < 𝜏 < 𝑡 + 1.



Properties of linear Time-Invariant systems

LTI systems are completely characterized by their impulse response (a nonlinear system is

not). It should be no surprise that their properties are also characterized by their impulse

response.

The Commutative Property

The output of an LTI system with input 𝑥 and impulse response ℎ is identical to the output

of an LTI system with input ℎ and impulse response 𝑥, as suggested by the block diagrams in

the following figure. This results from the fact that a convolution is commutative, as we have

already seen.

The distributive property of LTI systems



Exercise 3

Solution



Exercise 3



Exercise 4



Solution:



Exercise 5

Solution

Continuous Time LTI systems: the convolution …...2017/10/06 · system can be computed by a...

Documents

Transcript of Continuous Time LTI systems: the convolution …...2017/10/06 · system can be computed by a...