Unit 3.2: Properties and Eigenfunctions of Continuous-Time LTI Systems

Unit 3.2: Properties and Eigenfunctions of Continuous-Time LTI Systems#

This section is based on Sections 2.3 and 2.4 of [Hsu, 2020]

Follow along at cpjobling.github.io/eg-150-textbook/lti_systems/lti2

QR Code for this lecture

Subjects to be covered#

We continue our introduction to continuous-time LTI systems by considering:

Properties of Continuous-Time LTI Systems
Eigenfunctions of Continuous-Time LTI Systems
Exercises 6: Properties of Continuous-Time LTI Systems
Exercises 7: Eigenfunctions of Continuous-Time LTI systems

Properties of Continuous-Time LTI Systems#

A. Systems with or without memory
B. Causality
C. Stability

A. Systems with or without memory#

Since the output $y (t)$ of a memoryless system depends only on the current input $x (t)$ , then, if the system is also linear and time-invariant, this relationship can only be of the form

y (t) = K x (t)

where $K$ is a (gain) constant.

Thus the corresponding impulse response $h (t)$ is simply

h (t) = K δ (t)

Therefore, if $h (t_{0}) \neq 0$ for $t_{0} \neq 0$ , then the continuous-time LTI system has memory.

B. Causality#

Causal continuous-time LTI systems#

As discussed in Section Causal and Non-Causal Systems, a causal system does not respond to an input event until that event actually occurs.

Therefore, for a causal LTI system, we have

h (t) = 0 t < 0

Applying the causality condition to the convolution integral, the output of a causal continuous-time LTI system is expressed as

y (t) = \int_{0}^{\infty} h (τ) x (t - τ) d τ

Alternatively, applying the causality to the convolution integral (defined in Section C. Convolution Integral)

y (t) = \int_{- \infty}^{\infty} x (τ) h (t - τ) d τ

we have

y (t) = \int_{- \infty}^{t} x (τ) h (t - τ) d τ

This equation shows that the only values of the input $x (t)$ used to evaluate the output $y (t)$ are those for $τ \leq t$ .

Causal signals#

Based on the causality condition, any signal is called causal if

x (t) = 0 t < 0

and is called anticausal if

x (t) = 0 t > 0

Combining the definition of a causal signal with a causal system, when the input $x (t)$ is causal, the output of a causal continuous-time LTI system is given by

y (t) = \int_{0}^{t} h (τ) x (t - τ) d τ = \int_{0}^{t} x (τ) h (t - τ) d τ

C. Stability#

The BIBO (bounded-input/bounded-output) stability of an LTI system (Section Stable Systems) is readily ascertained by the impuse response. It can be shown (Exercise 6.5) that a continuous-time LTI system is BIBO stable if its impulse response is absolutely integrable; that is,

(1)#

\int_{- \infty}^{\infty} | h (τ) | d τ < \infty

Eigenfunctions of Continuous-Time LTI Systems#

In Unit 2.4: Systems and Classification of Systems (Example Exercise 4.7) we saw that the eigenfunctions of continuous-time LTI system represented by the complex exponentials $e^{s t}$ , with $s$ a complex variable.

That is

T {e^{s t}} = λ e^{s t}

where $λ$ is the eigenvalue of $T$ associated with $e^{s t}$ .

Setting $x (t) = e^{s t}$ in the convolution integral, we have

y (t) = T {e^{s t}}

y (t) = \int_{- \infty}^{\infty} h (τ) e^{s (t - τ)} d τ

y (t) = [\int_{- \infty}^{\infty} h (τ) e^{- s τ} d τ] e^{s t}

y (t) = H (s) e^{s t} = λ e^{s t}

where

λ = H (s) = \int_{- \infty}^{\infty} h (τ) e^{- s τ} d τ

Thus, the eigenvalue of a continuous-time LTI system associated with the eigenfunction $e^{s t}$ is given by $H (s)$ which is a complex constant whose value is determined by the value of $s$ via the equation

H (s) = \int_{- \infty}^{\infty} h (τ) e^{- s τ} d τ .

Note from the equation

y (t) = H (s) e^{s t}

that $y (0) = H (s)$ .

Looking Ahead#

The above results underline the definition of the Laplace transform and Fourier transform. The Unit 4: Laplace Transforms and their Applications will be discussed later in this course. The Fourier Transform will be introduced in EG-247 Digital Signal Processing next year.

Exercises 6: Properties of Continuous-Time LTI Systems#

Exercise 6.1#

The signals in Fig. 35(a) and (b) are the input $x (t)$ and the output $y (t)$ , respectively, of a certain continuous-time LTI system.

Fig. 35 Input and output signals for a continuous-time LTI system#

Sketch the output to the following inputs:

(a) $x (t - 2)$ ;

(b) $\frac{1}{2} x (t)$ .

For the answer, refer to the lecture recording or see solved problem 2.9 in in [Hsu, 2020].

Exercise 6.2#

Consider a continuous-time LTI system whose step response is given by

y (t) = e^{- t} u_{0} (t)

Determine and sketch the output of the system to the input $x (t)$ shown in Fig. 36.

For the answer, refer to the lecture recording or see solved problem 2.2 in in {cite}schaum.

syms t y(t) 
u_0(t) = heaviside(t);
y(t) = exp(-(t-1))*u_0(t-1) - exp(-(t-3))*u_0(t-3);
fplot(y(t),[0, 6]),ylim([-1,1.25]),grid,xlabel('t'),ylabel('y(t)')

../_images/900aaecae1f947da6ac7b9184f82eef59d8ae54d82c5f0cb83d7557c1ab9962a.png

Exercise 6.3#

Note

Interesting result but not examined further.

Consider a continuous-time LTI system system described by

y (t) = T {x (t)} = \frac{1}{T} \int_{t - \frac{T}{2}}^{t + \frac{T}{2}} x (τ) d τ

(a) Find and sketch the impulse response $h (t)$ of the system.

(b) Is this system causal?

For the answer, see the solved problem 2.11 in in [Hsu, 2020].

Exercise 6.4#

Note

Interesting result but not examined further.

Let $y (t)$ be the output of a continuous-time LTI system with input $x (t)$ . Find the output of the system if ths input is $x^{'} (t)$ , where $x^{'} (t)$ is the first derivative of $x (t)$ .

For the answer, refer to the solved problem 2.12 in in [Hsu, 2020].

Exercise 6.5#

Note

Interesting result but not examined further.

Verify the BIBO stability condition (C. Stability) for continuous-time LTI systems.

For the answer, see the solved problem 2.13 in [Hsu, 2020].

Exercise 6.6#

The system shown in Fig. 37(a) is formed by connecting two systems in cascade. The impulse responses of the two systems are $h_{1} (t)$ and $h_{2} (t)$ , respectively, and

h_{1} (t) = e^{- 2 t} u_{0} (t) h_{2} (t) = 2 e^{- t} u_{0} (t)

(a) Find the impulse response $h (t)$ of the overall system shown in Fig. 36(b).

(b) Determine if the overall system is BIBO stable.

Fig. 37 A cascade connection of two continuous-time LTI systems#

For the answer, refer to the lecture recording or see solved problem 2.14 in [Hsu, 2020].

Exercises 7: Eigenfunctions of Continuous-Time LTI systems#

Exercise 7.1#

Note

This is a foretaste of the Laplace Transform.

Consider a continuous-time LTI system with the input-output relation given by

y (t) = \int_{- \infty}^{t} e^{- (t - τ)} x (τ) d τ

(a) Find the impulse response $h (t)$ of this system.

(b) Show that the complex exponential $e^{s t}$ is an eigenfunction of the system.

(c) Find the eigenvalue of the system corresponding to $e^{s t}$ using the impulse response $h (t)$ obtained in part (a).

For the answer, refer to the lecture recording or see solved problem 2.15 in [Hsu, 2020].

Exercise 7.2#

Note

Interesting result which will not be examined.

Consider the continuous-time LTI system described by

y (t) = \frac{1}{T} \int_{t - \frac{T}{2}}^{t + \frac{T}{2}} x (τ) d τ

(a) Find the eigenvalue of the system corresponding to the eigenfunction $e^{s t}$ .

(b) Repeat part (a) by using the impulse response $h (t)$ of the system.

For the answer, see the solved problem 2.16 in [Hsu, 2020].

Exercise 7.3#

Note

Interesting result that is a foretaste of the Fourier transform which will not be examined.

Consider a stable continuous-time LTI system with impulse response $h (t)$ that is real and even. Show that $\cos ω t$ and $\sin ω t$ are eigenfunctions of this system with the same eigenvalue.

For the answer, see the solved problem 2.17 in [Hsu, 2020].

Summary#

We have continued our introdiuction the continuous-time LTI systems by considering

Properties of Continuous-Time LTI Systems
Eigenfunctions of Continuous-Time LTI Systems

Unit 3.2: Take aways#

Properties of continuous-time LTI systems#

The input to an LTI system is $x (t)$ , the impulse response of the LTI system is $h (t)$ and the output is $y (t)$ .

The output of any LTI system is computed from the convolution of $h (t)$ with $x (t)$ :

y (t) = \int_{- \infty}^{\infty} x (τ) h (t - τ) d τ = \int_{- \infty}^{\infty} h (τ) x (t - τ) d τ

The only LTI system that has no memory is the simple gain $y (t) = K x (t)$ for which the impulse response $h (t) = K δ (t)$ .

Causal LTI systems have an impulse response $h (t)$ that satisfies $h (t) = 0$ $t < 0$ . Therefore, for a causal LTI system the output is given by the convolution integral with adjsuted limits:

y (t) = \int_{- \infty}^{t} x (τ) h (t - τ) d τ

An LTI system is BIBO stable if it’s impulse response is integrable. That is:

\int_{- \infty}^{\infty} | h (τ) | d τ < \infty

Linear systems in series can be combined by convolution.

Eignenfunctions of continuous-time LTI systems#

The exponential signal $x (t) = e^{s t}$ (where $s$ is a complex variable, $s = σ + j ω$ ) is an eigenfunction of a continuous-time LTI system with impulse response $h (t)$ . The eigenvalue of the system is

H (s) = \int_{- \infty}^{\infty} h (τ) e^{- s τ} d τ .

From the equation $y (t) = H (s) e^{s t}$ , $H (s) = y (0)$ .

As we will see in Unit 4.1: The Laplace Transformation, $H (s)$ is called the Laplace transform of $h (t)$ . It is also known as the transfer function of the continuous-time LTI system that has impulse response $h (t)$ .

By a similar argument, we can show that if $s = j ω$ , then $e^{j ω}$ is also an eigenfunction of $h (t)$ , and the frequency response of an LTI system is

H (j ω) = \int_{- \infty}^{\infty} h (τ) e^{- j ω τ} d τ .

This is called the Fourier transform and we will return to it in EG-247 Digital Signal Processing.

Next Time#

We will conclude our look at continuous-time LTI systems by considering

Unit 3.3: Systems Described by Differential Equations

References#

[Hsu20] (1,2,3,4,5,6,7,8,9)

Hwei P. Hsu. Schaums outlines signals and systems. McGraw-Hill, New York, NY, 2020. ISBN 9780071634724. Available as an eBook. URL: https://www.accessengineeringlibrary.com/content/book/9781260454246.

Unit 3.2: Properties and Eigenfunctions of Continuous-Time LTI Systems

Contents

Unit 3.2: Properties and Eigenfunctions of Continuous-Time LTI Systems#

Subjects to be covered#

Properties of Continuous-Time LTI Systems#

A. Systems with or without memory#

B. Causality#

Causal continuous-time LTI systems#

Causal signals#

C. Stability#

Eigenfunctions of Continuous-Time LTI Systems#

Looking Ahead#

Exercises 6: Properties of Continuous-Time LTI Systems#

Exercise 6.1#

Exercise 6.2#

Exercise 6.3#

Exercise 6.4#

Exercise 6.5#

Exercise 6.6#

Exercises 7: Eigenfunctions of Continuous-Time LTI systems#

Exercise 7.1#

Exercise 7.2#

Exercise 7.3#

Summary#

Unit 3.2: Take aways#

Properties of continuous-time LTI systems#

Eignenfunctions of continuous-time LTI systems#

Next Time#

References#