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Notes for Signals and Systems, Table of Contents, 0 Introduction 4. 0 1 Introductory Comments, 0 2 Background in Complex Arithmetic. 0 3 Analysis Background, 1 Signals 10, 1 1 Mathematical Definitions of Signals. 1 2 Elementary Operations on Signals, 1 3 Elementary Operations on the Independent Variable. 1 4 Energy and Power Classifications, 1 5 Symmetry Based Classifications of Signals.

1 6 Additional Classifications of Signals, 1 7 Discrete Time Signals Definitions Classifications and Operations. 2 Continuous Time Signal Classes 23, 2 1 Continuous Time Exponential Signals. 2 2 Continuous Time Singularity Signals, 2 3 Generalized Calculus. 3 Discrete Time Signal Classes 37, 3 1 Discrete Time Exponential Signals. 3 2 Discrete Time Singularity Signals, 4 Systems 43.

4 1 Introduction to Systems, 4 2 System Properties. 4 3 Interconnections of Systems, 5 Discrete Time LTI Systems 50. 5 1 DT LTI Systems and Convolution, 5 2 Properties of Convolution Interconnections of DT LTI Systems. 5 3 DT LTI System Properties, 5 4 Response to Singularity Signals. 5 5 Response to Exponentials Eigenfunction Properties. 5 6 DT LTI Systems Described by Linear Difference Equations. 6 Continuous Time LTI Systems 68, 6 1 CT LTI Systems and Convolution.

6 2 Properties of Convolution Interconnections of DT LTI Systems. 6 3 CT LTI System Properties, 6 4 Response to Singularity Signals. 6 5 Response to Exponentials Eigenfunction Properties. 6 6 CT LTI Systems Described by Linear Difference Equations. 7 Introduction to Signal Representation 82, 7 1 Introduction to CT Signal Representation. 7 2 Orthogonality and Minimum ISE Representation, 7 3 Complex Basis Signals. 7 4 DT Signal Representation, 8 Periodic CT Signal Representation Fourier Series 92. 8 1 CT Fourier Series, 8 2 Real Forms Spectra and Convergence.

8 3 Operations on Signals, 8 4 CT LTI Frequency Response and Filtering. 9 Periodic DT Signal Representation Fourier Series 106. 9 1 DT Fourier Series, 9 2 Real Forms Spectra and Convergence. 9 3 Operations on Signals, 9 4 DT LTI Frequency Response and Filtering. 10 Fourier Transform Representation for CT Signals 118. 10 1 Introduction to CT Fourier Transform, 10 2 Fourier Transform for Periodic Signals. 10 3 Properties of Fourier Transform, 10 4 Convolution Property and LTI Frequency Response.

10 5 Additional Fourier Transform Properties, 10 6 Inverse Fourier Transform. 10 7 Fourier Transform and LTI Systems Described by Differential Equations. 10 8 Fourier Transform and Interconnections of LTI Systems. 11 Unilateral Laplace Transform 143, 11 1 Introduction. 11 2 Properties of the Laplace Transform, 11 3 Inverse Transform. 11 4 Systems Described by Differential Equations, 11 5 Introduction to Laplace Transform Analysis of Systems. 12 Application to Circuits 156, 12 1 Circuits with Zero Initial Conditions.

12 2 Circuits with Nonzero Initial Conditions, Notes for Signals and Systems. 0 1 Introductory Comments, What is Signals and Systems Easy but perhaps unhelpful answers include. the and the, the question and the answer, the fever and the cure. calculus and complex arithmetic for fun and profit. More seriously signals are functions of time continuous time signals or sequences in time. discrete time signals that presumably represent quantities of interest Systems are operators that. accept a given signal the input signal and produce a new signal the output signal Of course. this is an abstraction of the processing of a signal. From a more general viewpoint systems are simply functions that have domain and range that are. sets of functions of time or sequences in time It is traditional to use a fancier term such as. operator or mapping in place of function to describe such a situation However we will not be so. formal with our viewpoints or terminologies Simply remember that signals are abstractions of. time varying quantities of interest and systems are abstractions of processes that modify these. quantities to produce new time varying quantities of interest. These notes are about the mathematical representation of signals and systems The most. important representations we introduce involve the frequency domain a different way of looking. at signals and systems and a complement to the time domain viewpoint Indeed engineers and. scientists often think of signals in terms of frequency content and systems in terms of their effect. on the frequency content of the input signal Some of the associated mathematical concepts and. manipulations involved are challenging but the mathematics leads to a new way of looking at the. 0 2 Background in Complex Arithmetic, We assume easy familiarity with the arithmetic of complex numbers In particular the polar form. of a complex number c written as, is most convenient for multiplication and division e g.

c1 c2 c1 e j c1 c2 e j c2 c1 c2 e j c1 c2, The rectangular form for c written. where a and b are real numbers is most convenient for addition and subtraction e g. c1 c2 a1 jb1 a2 jb2 a1 a2 j b1 b2, Of course connections between the two forms of a complex number c include. c a jb a 2 b 2 c a jb tan 1 b a, and the other way round. a Re c c cos c b Im c c sin c, Note especially that the quadrant ambiguity of the inverse tangent must be resolved in making. these computations For example, 1 j tan 1 1 1 4, 1 j tan 1 1 1 3 4.

It is important to be able to mentally compute the sine cosine and tangent of angles that are. integer multiples of 4 since many problems will be set up this way to avoid the distraction of. calculators, You should also be familiar with Euler s formula. e j cos j sin, and the complex exponential representation for trigonometric functions. e j e j e j e j, Notions of complex numbers extend to notions of complex valued functions of a real variable in. the obvious way For example we can think of a complex valued function of time x t in the. rectangular form, x t Re x t j Im x t, In a simpler notation this can be written as. x t xR t j xI t, where xR t and xI t are real valued functions of t.

Or we can consider polar form, x t x t e j x t, where x t and x t are real valued functions of t with of course x t nonnegative for. all t In terms of these forms multiplication and addition of complex functions can be carried. out in the obvious way with polar form most convenient for multiplication and rectangular form. most convenient for addition, In all cases signals we encounter are functions of the real variable t That is while signals that. are complex valued functions of t or some other real variable will arise as mathematical. conveniences we will not deal with functions of a complex variable until near the end of the. 0 3 Analysis Background, We will use the notation x n for a real or complex valued sequence discrete time signal. defined for integer values of n This notation is intended to emphasize the similarity of our. treatment of functions of a continuous variable time and our treatment of sequences in time. But use of the square brackets is intended to remind us that the similarity should not be overdone. Summation notation for example, x k x 1 x 2 x 3, is extensively used Of course addition is commutative and so we conclude that. Care must be exercised in consulting other references since some use the convention that a. summation is zero if the upper limit is less than the lower limit And of course this summation. limit reversal is not to be confused with the integral limit reversal formula. x t dt x t dt, It is important to manage summation indices to avoid collisions For example.

is not the same thing as, But it is the same thing as. All these observations are involved in changes of variables of summation A typical case is. Let j n k relying on context to distinguish the new index from the imaginary unit j to. rewrite the sum as, j n 1 j n 3, Sometimes we will encounter multiple summations often as a result of a product of summations. for example, 4 5 4 5 5 4, x k z j x k z j x k z j, k 1 j 0 k 1 j 0 j 0 k 1. The order of summations here is immaterial But again look ahead to be sure to avoid index. collisions by changing index names when needed For example write. x k z k x k z j, k 1 k 0 k 1 j 0, before proceeding as above. These considerations also arise in slightly different form when integral expressions are. manipulated For example changing the variable of integration in the expression. to t gives, We encounter multiple integrals on rare occasions usually as a result of a product of integrals.

and collisions of integration variables must be avoided by renaming For example. x t dt z t dt x t dt z d, x t z dt d, The Fundamental Theorem of Calculus arises frequently. For finite sums or integrals of well behaved e g continuous functions with finite integration. limits there are no particular technical concerns about existence of the sum or integral or. interchange of order of integration or summation However for infinite sums or improper. integrals over an infinite range we should be concerned about convergence and then about. various manipulations involving change of order of operations However we will be a bit cavalier. about this For summations such as, a rather obvious necessary condition for convergence is that x k 0 as k Typically. we will not worry about general sufficient conditions rather we leave consideration of. convergence to particular cases, For integrals such as. an obvious necessary condition for convergence is that x t 0 as t but again. further details will be ignored We especially will ignore conditions under which the order of a. double infinite summation can be interchanged or the order of a double improper integral can. be interchanged Indeed many of the mathematical magic tricks that appear in our subject are. explainable only by taking a very rigorous view of these issues Such rigor is beyond our scope. For complex valued functions of time operations such as differentiation and integration are. carried out in the usual fashion with j viewed as a constant It sometimes helps to think of the. function in rectangular form to justify this view for example if x t xR t j xI t then. x d xR d j xI d, Similar comments apply to complex summations and sequences. Pathologies that sometimes arise in the calculus such as everywhere continuous but nowhere. differentiable functions signals are of no interest to us On the other hand certain generalized. notions of functions particularly the impulse function will be very useful for representing special. types of signals and systems Because we do not provide a careful mathematical background for. generalized functions we will take a very formulaic approach to working with them Impulse. functions aside fussy matters such as signals that have inconvenient values at isolated points will. be handled informally by simply adjusting values to achieve convenience. Example Consider the function, Certainly the integral of x t between any two limits is zero there being no area under a single.

point The derivative of x t is zero for any t 0 but the derivative is undefined at t 0 there. being no reasonable notion of slope How do we deal with this The answer is to view x t as. equivalent to the identically zero function Indeed we will happily adjust the value of a function. at isolated values of t for purposes of convenience and simplicity. In a similar fashion consider, which probably is familiar as the unit step function What value should we assign to u 0. Again the answer is that we choose u 0 for convenience For some purposes setting. u 0 1 2 is most suitable for other purposes u 0 1 is best But in every instance we freely. choose the value of u 0 to fit the purpose at hand The derivative of u t is zero for all t 0. but is undefined in the usual calculus sense at t 0 However there is an intuitive notion that a. jump upward has infinite slope and a jump downward has slope We will capture this. notion using generalized functions and a notion of generalized calculus in the sequel By. comparison the signal x t in the example above effectively exhibits two simultaneous jumps. and there is little alternative than to simplify x t to the zero signal. Except for generalized functions to be discussed in the sequel we typically work in the context. of piecewise continuous functions and permit only simple finite jumps as discontinuities. 1 Compute the polar form of the complex numbers e j 1 j and 1 j e j 2. 2 Compute the rectangular form of the complex numbers 2 e j 5 4 and e j e j 6. 3 Evaluate the easy way the magnitude 2 j 2 and the angle 1 j 2. 4 Using Euler s relation e j cos j sin derive the expression. cos 1 e j 1 e j, 5 If z1 and z2 are complex numbers and a star denotes complex conjugate express the. following quantities in terms of the real and imaginary parts of z1 and z2. Re z1 z1 Im z1z2 Re z1 z2, 6 What is the relationship among the three expressions below. x d x d 2 x 2 d, 7 Simplify the three expressions below. x d x d x d, Notes for Signals and Systems, 1 1 Mathematical Definitions of Signals.

A continuous time signal is a quantity of interest that depends on an independent variable where. we usually think of the independent variable as time Two examples are the voltage at a particular. node in an electrical circuit and the room temperature at a particular spot both as funct. Prerequisites for the material are the arithmetic of complex numbers differential and integral calculus and a course in electrical circuits Circuits are used as examples in the material and the last section treats circuits by Laplace transform Concurrent study of multivariable calculus is helpful for on occasion a double integral or partial derivative appears A course in differential