Signals and Systems Notes (PDF)

Notes for Signals and Systems Version 1.0 Wilson J. Rugh These notes were developed for use in 520.214, Signals and Systems, Department of Electrical and Computer Engineering, Johns Hopkins...

Notes for Signals and Systems Version 1.0 Wilson J. Rugh These notes were developed for use in 520.214, Signals and Systems, Department of Electrical and Computer Engineering, Johns Hopkins University, over the period 2000 – 2005. As indicated by the Table of Contents, the notes cover traditional, introductory concepts in the time domain and frequency domain analysis of signals and systems. Not as complete or polished as a book, though perhaps subject to further development, these notes are offered on an as is or use at your own risk basis. 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 equations is not required, though some very simple differential equations appear in the material. The material includes links to demonstrations of various concepts. These and other demonstrations can be found at http://www.jhu.edu/~signals/. Email comments to [email protected] are welcome. Copyright 2000 -2005, Johns Hopkins University and Wilson J. Rugh, all rights reserved. Use of this material is permitted for personal or non-profit educational purposes only. Use of this material for business or commercial purposes is prohibited.  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 Exercises 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 Exercises 2. Continuous-Time Signal Classes ………………………………………………………..23 2.1. Continuous-Time Exponential Signals 2.2. Continuous-Time Singularity Signals 2.3. Generalized Calculus Exercises 3. Discrete-Time Signal Classes ………………………………………………..…………..37 3.1. Discrete-Time Exponential Signals 3.2. Discrete-Time Singularity Signals Exercises 4. Systems ………………………………………………………..………………………….43 4.1. Introduction to Systems 4.2. System Properties 4.3. Interconnections of Systems Exercises 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 Exercises 6. Continuous-Time LTI Systems ……………………………………………….………….68 6.1. CT LTI Systems and Convolution 2 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 Exercises 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 Exercises 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 Exercises 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 Exercises 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 Exercises 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 Exercises 12. Application to Circuits ………………………………………………………..………….156 12.1. Circuits with Zero Initial Conditions 12.2. Circuits with Nonzero Initial Conditions Exercises 3  Notes for Signals and Systems 0.1 Introductory Comments What is “Signals and Systems?” Easy, but perhaps unhelpful answers, include the α and the ω , 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 world! 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 c =| c | e j∠c 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 c = a + jb 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, 4  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) = − π / 4 while ∠(−1 + j ) = tan −1 (1/(−1)) = 3π / 4 It is important to be able to mentally compute the sine, cosine, and tangent of angles that are integer multiples of π / 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 jθ = cos(θ ) + j sin(θ ) and the complex exponential representation for trigonometric functions: e jθ + e− jθ e jθ − e− jθ cos(θ ) = , sin(θ ) = 2 2j 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 course. 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, 5  3 ∑ x[k ] = x + x + x k =1 is extensively used. Of course, addition is commutative, and so we conclude that 3 1 ∑ x[k ] = ∑ x[k ] k =1 k =3 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: 3 1 ∫ x(t ) dt = − ∫ x(t ) dt 1 3 It is important to manage summation indices to avoid collisions. For example, 3 z[k ] ∑ x[k ] k =1 is not the same thing as 3 ∑ z[k ] x[k ] k =1 But it is the same thing as 3 ∑ z[k ] x[ j ] j =1 All these observations are involved in changes of variables of summation. A typical case is 3 ∑ x[n − k ] k =1 Let j = n − k (relying on context to distinguish the new index from the imaginary unit j ) to rewrite the sum as n −3 n −1 ∑ x[ j ] = ∑ x[ j ] 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 ⎛ 4 ⎞⎛ 5 ⎞ ⎛ 4 ⎞⎛ 5 ⎞ ⎜ ∑ 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 6  t ∫ x(t − τ ) dτ 0 to σ = t − τ gives 0 t ∫ x(σ ) (−dσ ) = ∫ x(σ ) dσ t 0 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, ⎛3 ⎞⎛ 3 ⎞ ⎛3 ⎞⎛ 3 ⎞ ⎜⎜ ∫ x(t ) dt ⎟⎟ ⎜⎜ ∫ z (t ) dt ⎟⎟ = ⎜⎜ ∫ x(t ) dt ⎟⎟ ⎜⎜ ∫ z (τ ) dτ ⎟⎟ ⎝0 ⎠ ⎝ −1 ⎠ ⎝0 ⎠ ⎝ −1 ⎠ 33 = ∫ ∫ x(t ) z (τ ) dt dτ 0 −1 The Fundamental Theorem of Calculus arises frequently: d t ∫ x(τ ) dτ = x(t ) dt −∞ 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 ∞ ∑ x[k ] k =−∞ 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 ∞ ∫ x(t ) dt −∞ 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 7  t t t ∫ x(τ ) dτ = ∫ xR (τ ) dτ + j ∫ xI (τ ) dτ −∞ −∞ −∞ 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 ⎧1, t = 0 x(t ) = ⎨ ⎩ 0, else 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 ⎧1, t >0 u (t ) = ⎨ ⎩0, t