Signals And Systems Linearity

[Carlito Austin]

Especiallythree properties are critical characteristics of any system. Linear systems are most easy to analyze analytically, time-invariant systems allow to treat the systems input-output-relation independent of the absolute time and causal systems ensure that the system can be realized in real-time, since the system does not use information from the future.

Clearly, this system is linear, since the red and black curve overlap. (Actually, we cannot say yet that it's linear, because we have just found one example where it is linear. To really prove linearity, one would need to do this mathematically based on the input-output-relation. Despite not being too complicated, it is out of scope here).

Obviously, this system is not linear, since red and black do not overlap. (Here we can really say it is not linear, because we have found one example where the linearity condition does not hold).

For linear systems powerful mathematical tools have been developed. In particular, the superposition technique in conjunction with signal decompositions such as Fourier Series or Fourier Transform are valuable methods to analyze the input-output relation of a system.

How can we see, if a system is linear, just from looking at its transformation expression? As a rule of thumb, a system is linear, if the operations on the input signal are all linear and no signal-independent terms are contained. What are linear operations?

The property of causality is a requirement for a system to be realizable in reality. Causality means that the output of the system does not depend on future inputs, but only on past input. In particular, this means that if the input signal is zero for all $t

A time-invariant system can be recognized from the fact that the transformation expression does not depend on the absolute time $t$, but $t$ is only used as an argument to the input functions. Let us look at some examples. First, let's define an exponential impulse as the input signal.

Clearly, the system is not time-invariant: When the inputs of the system are time-shifted exponential impulses, the outputs of the system are not just time-shifted versions of each other. Hence, the system is not time-invariant, but it is time-variant.

We could go on for ever and find examples for each combination of properties. However, one particular combination is especially important in signal processing: The class of Linear Time-Invariant (LTI) systems. All these systems can be described by their response to a Dirac input, which is called the impulse response. The class of LTI systems is so important that it deserves a dedicated article, which I'll write soon. Subscribe to the newsletter to be first to know about new content!

I am currently using Scipy's signal processing module scipy.signals to examine linear time invariant (LTI) systems. I would like to know how best to connect the systems together. For example, say I want to connection two systems

to get the resulting system. This notation is not very elegant though, especially if we are dealing with more than two systems. Also, connecting two systems like this in parallel or feeding back a signal through another system is not as simple.

The course is designed to provide the fundamental concepts in signals and systems. By the end of the course, students should be able to use signal transforms, system convolution and describe linear operations on these.

We draw a distinction between the fundamentals of signal modelling in time and frequency domains, and indicate the signficance of alternative descriptions. The basic concepts of Fourier series, Fourier transforms, Laplace transforms and related areas are developed. The idea of convolution for linear time-variant systems are introduced and expanded on from a range of perspectives. The transfer function for continuous and discrete tiem systems is used in this context. Stability is duscussed with respect to the pole locations. Some elements of statistical signal description are introduced as signal comparision methods. The Discrete Fourier Transform is discussed as a z-transform evaluation and its consequences examined. Some basic filtering operatings for both continuous and discrete signals are developed.

Textbook & Key References "Linear Systems and Signals", B.P. Lathi, 2nd Edition, Oxford University Press (Main Textbook) "Signals and Systems" , A. Oppenheim, A. Wilsky, Prentice HallMatlab LicenceThis course includes the use of Matlab for tutorial problems. Two Matlab tutorial sessions will be given at the beginning of the course. It is important that you have a copy of Matlab installed and properly licensed under Imperial College's Licensing Scheme. You can find the instruction about how to obtain Matlab here (Imperial login required). Since you are a full-time member of the College, if you wish to install Matlab on a personally owned system, please complete a licence form available here. For installation instructions, please click here.

E E 200 Undergraduate Research Exploration Seminar (1)

Weekly seminar featuring research primarily from within the Department of Electrical and Computer Engineering. Speakers include senior PhD students and postdocs as well as faculty from within the department. Provides students with an opportunity to connect with the broader research community in electrical and computer engineering. Credit/no-credit only.

View course details in MyPlan: E E 200

E E 233 Circuit Theory (5)

Electric circuit theory. Analysis of circuits with sinusoidal signals. Phasors, system functions, and complex frequency. Frequency response. Computer analysis of electrical circuits. Power and energy. Two port network theory. Laboratory in basic electrical engineering topics. Prerequisite: E E 215.

View course details in MyPlan: E E 233

E E 280 Exploring Devices (4)

Overview of modern electronic and photonic devices underlying modern electronic products including smartphones, traffic lights, lasers, solar cells, personal computers, and chargers. Introduction to modeling and principles of physics relevant to the analysis of electrical and optical/photonic devices. Prerequisite: either PHYS 122 or PHYS 142; recommended: either Python programming or Matlab; and Linux. Offered: AWSp.

View course details in MyPlan: E E 280

E E 331 Devices and Circuits I (5)

Physics, characteristics, applications, analysis, and design of circuits using semiconductor diodes and field-effect transistors with an emphasis on large-signal behavior and digital logic circuits. Classroom concepts are reinforced through laboratory experiments and design exercises. Prerequisite: 1.0 in E E 233.

View course details in MyPlan: E E 331

E E 332 Devices and Circuits II (5)

Characteristics of bipolar transistors, large- and small- signal models for bipolar and field effect transistors, linear circuit applications, including low and high frequency analysis of differential amplifiers, current sources, gain stages and output stages, internal circuitry of op-amps, op-amp configurations, op-amp stability and compensation. Weekly laboratory. Prerequisite: 1.0 in E E 331.

View course details in MyPlan: E E 332

E E 342 Signals, Systems, and Data II (4)

Review of basic signal processing concepts. Two-sided Laplace and z -transforms and connection to Fourier transforms. Modulation, sampling and the fast Fourier transform. Short-time Fourier transform. Multi-rate signal processing. Applications including inference and machine learning. Computer laboratory. Prerequisite: E E 242.

View course details in MyPlan: E E 342

E E 351 Energy Systems (5)

Develops understanding of modern energy systems through theory and analysis of the system and its components. Discussions of generation, transmission, and utilization are complemented by environmental and energy resources topics as well as electromechanical conversion, power electronics, electric safety, renewable energy, and electricity blackouts. Prerequisite: a minimum grade of 1.0 in E E 215.

View course details in MyPlan: E E 351

E E 391 Probability for Information and Communication Engineering (4)

Introduces probabilistic concepts for Electrical and Computer Engineering majors with applications to information/data science, signal processing, and communication systems. Includes accompanying Python labs that apply probabilistic concepts to these application domains. Prerequisite: E E 235 or E E 241; and MATH 126 or MATH 136.

View course details in MyPlan: E E 391

E E 393 Advanced Technical Communication (4)

Practical skills for day-to-day engineering communication as well as an advanced exploration of how to prepare persuasive documents and presentations for technical and non-technical audiences.

View course details in MyPlan: E E 393

E E 398 Introduction to Professional Issues (1)

Covers topics of interest to students planning their educational and professional path, including salaries, the value of advanced degrees, societal expectations of engineering professionals, the corporate enterprise, ethical dilemmas, patents and trade secrets, outsourcing, and the global market.

View course details in MyPlan: E E 398

E E 400 Advanced Topics in Electrical Engineering (1-5, max. 10)

Contemporary topics at the advanced undergraduate elective level. Faculty presents advanced elective topics not included in the established curriculum.

View course details in MyPlan: E E 400

E E 406 Teaching Engineering (3) DIV

Explores effective and inclusive teaching techniques in engineering and related STEM fields. Includes active and problem-based learning with attention to how racial, ethnicity, gender, and socioeconomic differences affect how students learn and interact with teachers (including faculty and teaching assistants).

View course details in MyPlan: E E 406

3a8082e126