Signals And Linear Systems

[Vikki Nagindas]

Theprinciple of superposition states that the response of the system to a weighted sum of input signals is equal to the corresponding weighted sum of the outputs of the system to each of the input signals.

A system which is both linear and time-invariant is called the linear timeinvariant system. In other words, a system for which both the superposition principle and the homogeneity principle are valid and the input-output characteristics of the system do not change with time is called linear timeinvariant (LTI) system.

It means that if the input to the system is delayed by (t0) units, then thecorresponding output will also be delayed by(to) units. Also, for a linear timeinvariant system, all the coefficients of the differential equation describing thesystem are constants.

A system which is linear but time-variant is called the linear time-variantsystem. In other words, a system for which the principle of superposition and homogeneity are valid but the input-output characteristics change with time is called the linear time-variant (LTV) system.

In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator.Linear systems typically exhibit features and properties that are much simpler than the nonlinear case.As a mathematical abstraction or idealization, linear systems find important applications in automatic control theory, signal processing, and telecommunications. For example, the propagation medium for wireless communication systems can often bemodeled by linear systems.

A system is linear if and only if it satisfies the superposition principle, or equivalently both the additivity and homogeneity properties, without restrictions (that is, for all inputs, all scaling constants and all time.)[1][2][3][4]

The superposition principle means that a linear combination of inputs to the system produces a linear combination of the individual zero-state outputs (that is, outputs setting the initial conditions to zero) corresponding to the individual inputs.[5][6]

In a system that satisfies the homogeneity property, scaling the input always results in scaling the zero-state response by the same factor.[6] In a system that satisfies the additivity property, adding two inputs always results in adding the corresponding two zero-state responses due to the individual inputs.[6]

The behavior of the resulting system subjected to a complex input can be described as a sum of responses to simpler inputs. In nonlinear systems, there is no such relation. This mathematical property makes the solution of modelling equations simpler than many nonlinear systems.For time-invariant systems this is the basis of the impulse response or the frequency response methods (see LTI system theory), which describe a general input function x(t) in terms of unit impulses or frequency components.

Typical differential equations of linear time-invariant systems are well adapted to analysis using the Laplace transform in the continuous case, and the Z-transform in the discrete case (especially in computer implementations).

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.

We consider a rather general class of infinite-dimensional linear systems, called regular linear systems, for which convenient representations are known to exist both in time and in the frequency domain. We introduce and study the concept of admissible feedback operator for such a system and of well-posedness radius. We show that the closed-loop system obtained from a regular linear system with an admissible feedback operator is again regular and we describe the relationship between the generating operators of the open-loop and closed-loop systems.

Part of the results reported here were obtained while the author was visiting FUNDP Namur, under the Belgian Program on Inter-University Poles of Attraction initiated by the Belgian state, Prime Minister's Office, Science Policy Programming. The scientific responsibility is assumed by the author.

I'm not entirely sure about how to move on from this point, I'm trying to find the superposition of the responses of the two individual signals so I can later check if that's equal to the response of their (the two signals') superpositions. The intervals in each response involve different signals and thus I'm unable to find the superposition of responses.

As said by @Juancho, the expression let us suspect that the system could be non-linear. So we could look for a counter-example, yet it is not evident how to find a good one. Anyway, lt us try to better understand the system. Since there is something around sign change.

Develops the basic theory of continuous and discrete systems, with emphasis on linear time-invariant

systems. Discusses the representation of signals and systems in both the time and frequency domain.

Topics include linearity, time-invariance, causality, stability, convolution, system interconnection, and

sinusoidal response. The Fourier and Laplace transforms are developed for the discussion of frequencydomain

applications. Sampling and quantization of continuous waveforms (A/D and D/A conversion) are

analyzed, leading to the discussion of discrete-time FIR and IIR systems, recursive analysis, and

realization. The Z-transform and the discrete-time Fourier transform are developed, and applied to the

analysis of discrete-time signals and systems.

Topics Covered:

1. Basic signals and systems

a. Continuous and discrete time signals

b. Signal manipulation

c. Basic system properties

2. Linear time invariant (LTI) systems

a. Discrete time convolution

b. Continuous time convolution

c. Relationship of generic system properties to the impulse response for an LTI system

d. Use of differential and difference equations as models for LTI systems

3. Continuous time Fourier transform (CTFT)

a. Definition and derivation of the CTFT

b. Fourier transform representation of periodic signals using the CTFT

c. Properties of the CTFT

d. Convolution-multiplication duality and the CTFT

4. Discrete time Fourier transform (DTFT)

a. Definition and derivation of the DTFT

b. Fourier transform representation of periodic signals using the DTFT

c. Properties of the DTFT

d. Convolution-multiplication duality and the DTFT

5. Sampling

a. Derivation and application of the Sampling Theorem for bandlimited signals

b. Derivation and application of bandlimited (sinc) interpolation

c. Aliasing

6. The Laplace transform

a. Definition and relationship of Laplace transform to CTFT

b. Region of convergence

c. Inverse Laplace transform via partial fraction expansion method

d. Geometry evaluation of the CTFT via the pole zero plot.

e. Properties of the Laplace transform

f. Relationship of causality and stability to structure in the Laplace s plane

7. Z transform

a. Derivation of Z transform from Laplace assuming ideal, delta function sampling

b. Relationship of Z transform to DTFT

c. Region of convergence

d. Inverse Z transform via partial fraction expansion method

e. Geometry evaluation of the DTFT via the pole zero plot.

f. Properties of the Z transform

g. Relationship of causality and stability to structure in the Z transform z plane

Course Outcomes:

Students should:

1. Demonstrate the ability to recognize, analyze, and manipulate basic continuous time (CT) and

discrete time (DT) signals and to classify continuous and discrete time systems as to their linearity,

time invariance, causality, and stability.

2. Analyze both continuous and discrete linear time invariant (LTI) systems in the time domain

including leveraging the use of the impulse response, setting up and carrying out convolution

integrals and sums, using mathematical properties of the convolution operator to manipulate,

combine, and decompose systems and sub-systems, determine stability and causality from the

impulse response, and use linear constant coefficient differential / difference equations (LCCDEs) as

models for LTI systems.

3. Analyze both CT and DT LTI signals and systems in the frequency domain by using the appropriate

Fourier representation, including calculation of forward and inverse Fourier representation,

determining outputs using the frequency response, characterizing systems based on their frequency

response characteristics, and applying relevant properties of these representations.

4. Apply the Shannon sampling theorem and the sinc interpolation formula and quantify the effects of

aliasing.

5. Analyze CT and DT systems using the bilateral Laplace and Z transforms, including calculating

regions of convergence (ROCs) and interpreting the implications of those regions for the forms of

time domain behavior, determining specific time signals from their transform and ROCs using partial

fraction expansion, relating LCCDEs to the corresponding transform and vice-versa, determining

causality and stability from ROCs, and interpreting the relationship between pole / zero locations and

system frequency response.

Contribution of course to meeting

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