Lecture 1: Introduction Introduction

Transcription

1 Module 1: Signals in Natural Domain Lecture 1: Introduction Introduction The intent of this introduction is to give the reader an idea about Signals and Systems as a field of study and its applications. But we must first, at least vaguely define what signals and systems are. Signals are functions of one or more variables. Systems respond to an input signal by producing an output signal. Examples of signals include: 1. A voltage signal: voltage across two points varying as a function of time. 2. A force pattern: force varying as a function of 2-dimensional space. 3. A photograph: color and intensity as a function of 2-dimensional space. 4. A video signal: color and intensity as a function of 2-dimensional space and time. Examples of systems include: 1. An oscilloscope: takes in a voltage signal, outputs a 2-dimensional image characteristic of the voltage signal. 2. A computer monitor: inputs voltage pulses from the CPU and outputs a time varying display. 3 An accelerating mass: force as a function of time may be looked at as the input signal, and velocity as a function of time as the output signal. 4 A capacitance: terminal voltage signal may be looked at as the input, current signal as the output. 1

2 Examples of mechanical and electrical systems You are surely familiar with many of these signals and systems and have probably analyzed them as well, but in isolation. For instance, you must have studied accelerating masses in a mechanics course (see Fig (a)), and capacitances in an electrostatic course (see Fig (b)), separately. Fig (a) Fig (b) As you can see, there is a similarity in the way the input signal is related to the output signal. These similarities will interest us in this course as we may be able to make inferences common to both these systems from these similarities. We will develop very general tools and techniques of analyzing systems, independent of the actual context of their use. Our approach in this course would be to define certain properties of signals and systems (inspired of course by properties real-life examples we have), and then link these properties to consequences. These "links" can then be used directly in connection with a large variety of systems: electrical, mechanical, chemical, biological knowing only how the input and output signal are related! Thus, our focus when dealing with signals and systems will be on the relationship between the input and output signal and not really on the internals of the system. 2

3 Lecture 2: Description of Signals What is a signal? A signal, as stated before is a function of one or more independent variables. Recall that a function defines a correspondence between 2 sets, i.e.: corresponding to each element of one set (called the domain), there exists a unique element of another set (called the co-domain ). Notice that more than one element in the domain may correspond to the same element in the codomain. A function is also sometimes referred to as a mapping. Thus a signal may also be defined as a mapping from one set to another. For example a speech signal would be mathematically represented by acoustic pressure as a function of time. Some more examples of signals are voltage, current or power as functions of time. A monochromatic picture can be described as a signal which is mathematically represented by brightness as a function of two spatial variables. As mentioned earlier, there may be more than one independent variable. For example, the independent variable for a photograph is 2-dimensional space (2 space variables). The variables may also be hybrid, say 2 space variables and 1 time variable (E.g.: a video signal). Note: In this course, we shall focus our attention on signals of only one variable. Also, for convenience, we shall generally refer to the independent variable as time. So don't let the recurring reference to time confuse you. It is symbolic for any independent variable you care to choose. Continuous-time and discrete-time signals: A signal was defined as merely a mapping from one set to another. In certain cases, the independent variable is continuous, i.e. the elements of the domain set have a continuity associated with them. This means the mapping is defined over a continuum of values of the 3

4 independent variable. Such signals are called continuous-time signals. A force pattern (force as a function of 2-dimensional space) or a speech signal would be an example of a continuous-time signal. On the other hand, certain signals are defined for only discrete values of the independent variable; i.e. the elements of the domain are not continuous. Such signals are called discretetime signals. India's population count, done every 10 years is an example of a discrete-time signal. In fact, the image files on your computer are also discrete-time signals; the information is stored pixel-wise, and not over a continuous stretch of 2 spatial co-ordinates. We typically index a discrete time variable by integers. Note that it is not necessary that the co-domain of a discrete-time signal is discrete and that of a continuous-time signal is continuous. Henceforth, we shall represent the independent variable for continuous-time signals by t (enclosed in (.) brackets), and for discrete-time signals by n (enclosed in [.] brackets). As we are familiar with continuous-time signals, we shall now describe discrete-time signals in more detail. Discrete-time signals Discrete variables are those in which there exists a neighborhood around each value in which no other value is present. Intuitively, it means a variable like the natural numbers on the real line - we can isolate each instance of the discrete variable from the other instances. Why should we bother about discrete variables? Discrete variables come up intrinsically in several applications. Take for example, the cost of gold in the market every day. The dependent variable (cost) is a function of discrete time (incremented once every day). Another example is the marks scored by the students in class. Here the dependent variable (marks) is a function of the discrete variable roll number. While it is perfectly fine to talk about marks of , it makes no sense to talk of marks of roll no this system is inherently discrete. Another point that should be noted here is that some results about signals and systems are common to both: continuous as well as discrete signals, but can be grasped more intuitively in one case as compared to the other. So, we shall pursue the study of both these cases simultaneously in this course. Need the discrete variable be uniform? 4

5 No. though we imagine natural number or integers when we think of discrete signals, the points need not be equally spaced. For example, if the markets remained closed on Sundays, we would not record a price for gold on that day - so the spacing between the variables on this axis changes. In most common cases, however, the independent variable is uniform - and throughout this course, we shall assume a uniform spacing of the variable unless otherwise stated explicitly. This assumption makes the analysis more intuitive and also yields several good theorems for our use, which we shall see as we proceed. Do discrete signals necessarily come from continuous signals? Although intuition may suggest so, this is not necessarily the case. In one of the example above - we considered the daily rate of gold. Here, time is intrinsically a continuous variable, and we made a discrete variable by taking measurements after certain intervals. However, the marks as a function of roll numbers intrinsically form a discrete system - there is no continuous axis of roll numbers. Then how do we define the neighborhood? Okay, by now it may seem that we are hiding some details here - we defined a discrete variable as one in which no other value exists in a certain neighborhood of one. Now for roll numbers, a neighborhood does not make sense. How do we formally define a discrete signal? A discrete variable is one which can ultimately be indexed by integers. Examples of discrete variables Now that we seem to have an intuitive understanding of what a discrete variable is, let us take some examples of discrete variables: First, the simplest and most intuitive discrete set is the integer axis itself: Then we can consider a set of tuples (a,b) such that a and b are both in the range 0 to 5 - how can we index them by integers? 5

6 Now let s come to something that is discrete alright, but not very intuitive about how we can index it - rational numbers: We represent the rational numbers along the fourth quadrant, as y/x. The repeated areas (like 2/2, 3/3, 4/2 etc) are to be neglected, hence are in gray. Then we go on indexing them diagonally as shown by the animation. Now, we go ahead another step - how do we index a full plane? 6

7 Note the method: we start in expanding circles from the origin. As soon as a circle cuts integer points, we pause and number the points clockwise from the positive y axis. This method is by no means unique - but just one set of indexing is enough for us to call the system discrete. Here we pause to note that although variables like the integer plane above can be indexed by integers, it is far more convenient to use tuples of integers to index them. It can mathematically be proved that any finite set of integers {a 1, a 2, a 3... a n } can be indexed by a single variable. We leave out the proof here, but the interested reader can find it in books on number theory. Representation of discrete variables Let us decide some conventions for use with discrete variables: We shall mostly deal with time as the discrete variable, and shall denote it by n and keep t for continuous time. We will enclose discrete variables in brackets [.] as opposed to parenthesis (.) for continuous variables. A discrete signal is also called a sequence - the word coming from the familiar usage in mathematics. We shall next discuss about systems. 7

8 What is a system? Lecture 3: Description of Systems A signal was defined as a mapping from a set of the independent variable (domain) to the set of the dependent variable (co-domain). A system is also a mapping, but across signals, or across mappings. That is, the domain set and the co-domain set for a system are both sets of signals, and corresponding to each signal in the domain set, there exists a unique signal in the co-domain set. In signals and systems terminology, we say; corresponding to every possible input signal, a system produces an output signal. In that sense, realize that a system, as a mapping is one step hierarchically higher than a signal. While the correspondence for a signal is from one element of one set to a unique element of another, the correspondence for a system is from one whole mapping from a set of mappings to a unique mapping in another set of mappings! Examples of systems Examples of systems are all around us. The speakers that go with your computer can be looked at as systems whose input is voltage pulses from the CPU and output is music (audio signal). A spring may be looked as a system with the input, say, the longitudinal force on it as a function of time, and output signal being its elongation as a function of time. The independent variable for the input and output signal of a system need not even be the same. In fact, it is even possible for the input signal to be continuous-time and the output signal to be discrete-time or vice-versa. For example, our speech is a continuous-time signal, while a digital recording of it is a discrete-time signal! The system that converts any one to the other is an example of this class of systems. As these examples may have made evident, we look at many physical objects/devices as systems, by identifying some variation associated with them as the input signal and some other variation associated with them as the output signal (the relationship between these, that essentially defines the system depends on the laws or rules that govern the system). Thus a capacitance with voltage (as a function of time) considered as the input signal and current 8

9 considered as the output signal is not the same system as a capacitance with, say charge considered as the input signal and voltage considered as the output signal. Why? The mappings that define the system are different in these two cases. We shall next discuss what system description means. Example of CRO An input voltage signal f(t) is provided to the CRO by using a function generator. The CRO (the system) transforms this input function into an image that is displayed on the CRO screen. The luminosity of every point on this display (i.e. value of the signal) is dependent on the x and y coordinates. So, the output S(x, y) has its independent variable as space, whereas the input independent variable is time. 9

10 System description The system description specifies the transformation of the input signal to the output signal. In certain cases, a system has a closed form description. E.g. the continuous-time system with description y(t) = x(t) + x(t-1); where x(t) is the input signal and y(t) is the output signal. Not all systems have such a closed form description. Just as certain "pathological" functions can only be specified by tabulating the value of the dependent variable against all values of the independent variable; some systems can only be described by tabulating the output signal against all possible input signals. Explicit and Implicit Description When a closed form system description is provided, it may either be classified as an explicit description or an implicit one. For an explicit description, it is possible to express the output at a point, purely in terms of the input signal. Hence, when the input is known, it is easily possible to find the output of the system, when the system description is Explicit. In case of an Explicit description, it is clear to see the relationship between the input and the output. e.g. y(t) = { x(t) } 2 + x(t-5). In case the system has an implicit description, it is harder to see the input-output relationship. An example of an Implicit description is y(t) - y(t-1) x(t) = 1. So when the input is provided, we are not directly able to calculate the output at that instant (since, the output at 't-1' also needs to be known). Although in this case also, there are methods to obtain the output based solely on the input, or, to convert this implicit description into an explicit one. The description by itself however is in the implicit form. The mapping involved in systems We shall next discuss the idea of mapping in a system in a little more depth. 10

11 A signal maps an element in one set to an element in another. A system, on the other hand maps a whole signal in one set to a signal in another. That is why a system is called a mapping over mappings. Therefore, the value of the output signal at any instant of time (remember "time" is merely symbolic) in general depends on the whole input signal. Thus, even if the independent variable for the input and output signal are the same (say time t), do not assume the value the output signal at, say t = 5 depends on only the value of the input signal at t = 5. For example, consider the system with description: The output at, say t = 5 depends on the values of the input signal for all t <= 5. Henceforth; we shall call systems with both input and output signal being continuous-time as continuous-time systems, and those with both input and output signal being discrete-time as discrete-time systems. Those that do not fall into either of these classes (i.e. input discrete-time and output continuous-time and vice-versa) we shall call hybrid systems. Now that the necessary introductions are done, we can get on to system properties. 11

12 Lecture 4: Properties of Systems Memory: Memory is a property relevant only to systems whose input and output signals has the same independent variable. A system is said to be memory less if its output for each value of the independent variable is dependent only on the input signal at that value of independent variable. For example the system with description: y(t) = 5x(t) ( y(t) is the output signal corresponding to input signal x(t) ) is memory less. In the physical world a resistor can be considered to be a memory less system (with voltage considered to be the input signal, current the output signal). By definition, a system that does not have this property is said to have memory. How can we identify if a system has memory? For a memory less system, changing the input at any instant can change the output only at that instant. If, in some case, a change in input signals at some instant changes the output at some other instant, we can be sure that the system has memory. Note: Consider a system whose output Y(t) depends on input X(t) as: Y(t) = X(t-5) + { X(t) - X(t-5) } While at first glance, the system might appear to have memory, it does not. This brings us to the idea that given the description of a system, it need not at all be the most economical one. The same system may have more than one description. Examples: Assume y[n] and y(t) are respectively outputs corresponding to input signals x[n] and x(t) 1. The identity system y(t) = x(t) is of-course Memoryless 2. System with description y[n] = x[n-5] has memory. The input at any "instant" depends on the input 5 "instants" earlier. 3. System with description inputs. also has memory. The output at any instant depends on all past and present 12

13 Linearity: Now we come to one of the most important and revealing properties systems may have - Linearity. Basically, the principle of linearity is equivalent to the principle of superposition, i.e. a system can be said to be linear if, for any two input signals, their linear combination yields as output the same linear combination of the corresponding output signals. Definition: (It is not necessary for the input and output signals to have the same independent variable for linearity to make sense. The definition for systems with input and/or output signal being discretetime is similar.) Example of linearity A capacitor, an inductor, a resistor or any combination of these are all linear systems, if we consider the voltage applied across them as an input signal, and the current through them as an output signal. This is because these simple passive circuit components follow the principle of superposition within their ranges of operation. Additivity and Homogeneity: Linearity can be thought of as consisting of two properties: Additivity 13

14 A system is said to be additive if for any two input signals x 1 (t) and x 2 (t), i.e. the output corresponding to the sum of any two inputs is the sum of the two outputs. Homogeneity (Scaling) A system is said to be homogenous if, for any input signal X(t), i.e. scaling any input signal scales the output signal by the same factor. To say a system is linear is equivalent to saying the system obeys both additivity and homogeneity. a) We shall first prove homogeneity and additivity imply linearity. b) To prove linearity implies homogeneity and additivity. This is easy; put both constants equal to 1 in the definition to get additivity; one of them to 0 to get homogeneity. Additivity and homogeneity are independent properties. We can prove this by finding examples of systems which are additive but not homogeneous, and vice versa. Again, y(t) is the response of the system to the input x(t). Example of a system which is additive but not homogeneous: [It is homogeneous for real constants but not complex ones - consider ] 14

15 Example of a system which is homogeneous but not additive: [From this example can you generalize to a class of such systems?] Examples of Linearity: Assume y[n] and y(t) are respectively outputs corresponding to input signals x[n] and x(t) 1) System with description y(t) = t. x(t) is linear. Consider any two input signals, x 1 (t) and x 2 (t), with corresponding outputs y 1 (t) and y 2 (t). a and b are arbitrary constants. The output corresponding to a.x 1 (t) + b.x 2 (t) is = t (a.x 1 (t) + b.x 2 (t)) = t.a.x 1 (t) + t.b.x 2 (t), which is the same linear combination of y 1 (t) and y 2 (t). Hence proved. 2) The system with description is not linear. See for yourself that the system is neither additive, nor homogenous. Show for yourself that systems with the following descriptions are linear: Shift Invariance This is another important property applicable to systems with the same independent variable for the input and output signal. We shall first define the property for continuous time systems and the definition for discrete time systems will follow naturally. Definition: 15

16 Say, for a system, the input signal x(t) gives rise to an output signal y(t). If the input signal x(t - t 0 ) gives rise to output y(t - t 0 ), for every t 0, and every possible input signal, we say the system is shift invariant. i.e. for every permissible x(t) and every t 0 In other words, for a shift invariant system, shifting the input signal shifts the output signal by the same offset. Note this is not to be expected from every system. x(t) and x(t - t 0 ) are different (related by a shift, but different) input signals and a system, which simply maps one set of signals to another, need not at all map x(t) and x(t - t 0 ) to output signal also shift by t 0 A system that does not satisfy this property is said to be shift variant. 16

17 Examples of Shift Invariance: Assume y[n] and y(t) are respectively outputs corresponding to input signals x[n] and x(t) Stability Let us learn about one more important system property known as stability. Most of us are familiar with the word stability, which intuitively means resistance to change or displacement. Broadly speaking a stable system is a one in which small inputs lead to predictable responses that do not diverge, i.e. are bounded. To get the qualitative idea let us consider the following physical example. Example Consider an ideal mechanical spring (elongation proportional to tension). If we consider tension in the spring as a function of time as the input signal and elongation as a function of time to be the output signal, it would appear intuitively that the system is stable. A small tension leads only to a finite elongation. There are various ideas/notions about stability not all of which are equivalent. We shall now introduce the notion of BIBO Stability, i.e. BOUNDED INPUT-BOUNDED OUTPUT STABILITY. 17

18 Statement: Note: This should be true for all bound inputs x(t) It is not necessary for the input and output signal to have the same independent variable for this property to make sense. It is valid for continuous time, discrete time and hybrid systems. Examples Consider systems with the following descriptions. y(t) is the output signal corresponding to the input signal x(t). 18

19 CONCLUSION BIBO Stable system: In a BIBO stable system, every bounded input is assured to give a bounded output. An unbounded input can give us either a bounded or an unbounded output, i.e. nothing can be said for sure. BIBO Unstable system: In a BIBO unstable system, there exists at least one bounded input for which output is unbounded. Again, nothing can be said about the system's response to an unbounded input. Causality Causality refers to cause and effect relationship (the effect follows the cause). In a causal system, the value of the output signal at any instant depends only on "past" and "present" values of the input signal (i.e. only on values of the input signal at "instants" less than or equal to that "instant"). Such a system is often referred to as being non-anticipative; as the system output does not anticipate future values of the input (remember again the reference to time is merely symbolic). As you might have realized, causality as a property is relevant only for systems whose input and output signals have the same independent variable. Further, this independent variable must be ordered (it makes no sense to talk of "past" and "future" when the independent variable is not ordered). What this means mathematically is that If two inputs to a causal (continuous-time) system are identical up to some time to, the corresponding outputs must also be equal up to this same time (we'll define the property for continuous-time systems; the definition for discrete-time systems will then be obvious). Definition Let x 1 (t) and x 2 (t) be two input signals to a system and y 1 (t) and y 2 (t) be their respective outputs. The system is said to be causal if and only if: This of course is only another way of stating what we said before: for any t 0 : y( t 0 ) depends only on values of x(t) for t <= t 0 As an example of the behavior of causal systems, consider the figure below: 19

20 The two input signals in the figure above are identical to the point t = t 0, and the system being a causal system, their corresponding outputs are also identical till the point t = t 0. Examples of Causal systems Assume y[n] and y(t) are respectively the outputs corresponding to input signals x[n] and x(t) 1. System with description y[n] = x[n-1] + x[n] is clearly causal, as output "at" n depends on only values of the input "at instants" less than or equal to n ( in this case n and n-1 ). 2. Similarly, the continuous-time system with description is causal, as value of output at any time t 0 depends on only value of the input at t 0 and before. 3. But system with description y[n] = x[n+1] is not causal as output at n depends on input one instant later. Note: If you think the idea of non-causal systems is counter intuitive, i.e: if you think no system can "anticipate the future", remember the independent variable need not be time. Visualizing non- 20

21 causal systems with, say one-dimensional space as the independent variable is not difficult at all! Even if the independent variable is time, we need not always be dealing with real-time, i.e. with the time axes of the input and output signals synchronized. The input signal may be a recorded audio signal and the output may be the same signal played backwards. This is clearly not causal! Deductions from System Properties Now that we have defined a few system properties, let us see how powerful inferences can be drawn about systems having one or more of these properties. Theorem Statement: If a system is additive or homogeneous, then x(t)=0 implies y(t)=0. Proof: This completes the proof. Theorem: Statement: If a causal system is either additive or homogeneous, then y(t) cannot be non zero before x(t) is non-zero. Proof: 21

22 Say x(t) = 0 for all t less than or equal to t 0. We have to show that the system response y(t) = 0 for all t less than or equal to t 0. Since the system is either additive or homogeneous the response to the zero input signal is the zero output signal. The zero input signal and x(t) are identical for all t less than or equal to t 0. Hence, from causality, their output signals are identical for all t less than or equal to t 0. We conclude the discussion on system properties by noting that this is not an end, but merely a beginning! Through much of our further discussions, we will be looking at an important class of systems - Linear Shift-Invariant (LSI) Systems. 22

23 Lecture 5: Discrete-Time Convolution Discrete time convolution As the name suggests the two basic properties of a LTI system are: 1) Linearity A linear system (continuous or discrete time) is a system that possesses the property of SUPERPOSITION. The principle of superposition states that the response of sum of two or more weighted inputs is the sum of the weighted responses of each of the signals. Mathematically y[n] = Σ a k y k [n] = a 1 y 1 [n] + a 2 y 2 [n] +... Superposition combines in itself the properties of ADDITIVITY and HOMOGENEITY. This is a powerful property and allows us to evaluate the response for an arbitrary input, if it can be expressed as a sum of functions whose responses are known. 2) Time Invariance It allows us to find the response to a function which is delayed or advanced in time; but similar in shape to a function whose response is known. Given the response of a system to a particular input, these two properties enable us to find the response to all its delays or advances and their linear combination. Discrete Time LTI Systems Consider any discrete time signal x[n]. It is intuitive to see how the signal x[n] can be represented as sum of many delayed/advanced and scaled Unit Impulse Signals. 23

24 Mathematically, the above function can be represented as More generally any discrete time signal x[n] can be represented as The above expression corresponds to the representation of any arbitrary sequence as a linear combination of shifted Unit Impulses which are scaled by x[n]. Consider for example the Unit Step function. As shown earlier it can be represented as 24

25 Now if we knew the response of a system for a Unit Impulse Function, we can obtain the response of any arbitrary input. To see why this is so, we invoke the properties of Linearity, Homogeneity (Superposition ) and Time Invariance. The left hand side can be identified as any arbitrary input, while the right hand side can be identified as the total output to the signal. The total response of the system is referred to as the CONVOLUTION SUM or superposition sum of the sequences x[n] and h[n]. The result is more concisely stated as y[n] = x[n] * h[n], where Therefore, as we said earlier a LTI system is completely characterized by its response to a single signal i.e. response to the Unit Impulse signal. Example Related to Discrete Time LTI Systems 25

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29 Recall that the convolution sum is given by Now we plot x[k] and h[n-k] as functions of k and not n because of the summation over k. Functions x[k] and h[k] are the same as x[n] and h[n] but plotted as functions of k. Then, the convolution sum is realized as follows 1. Invert h[k] about k=0 to obtain h[-k]. 2. The function h[n-k] is given by h[-k] shifted to the right by n (if n is positive) and to the left (if n is negative). It may appear contradictory but think a while to verify this (note the sign of the independent variable). In the figure below n=1 3. Multiply x[k] and h[n-k] for same coordinates on the k axis. The value obtained is the response at n i.e. Value of y[n] at a particular n the value chosen in step 2. Now we demonstrate the entire procedure taking n=0,1 thereby obtaining the response at n=0,1. The input signal x[n] and for this example is taken as : Case 1: For n=0 29

30 Remember the independence axis has k as the independent variable. Then taking the product x[k] h[-k] for same k and summing it we get the value of the response at n=0. Let h[-k] = g[k] y[0] =...x[-1]g[-1] + x[0] g[0] +... = (-2) (1) +(1) (2) = 0 Case 2: For n=1 h[1-k] =g[k] y[1] =...+ x[0]g[0] + x[1]g[1] +... = (1)(1) +(2)(2) = 5 The values are the same as that obtained previously. The total response referred to as the Convolution sum need not always be found graphically. The formula can directly be applied if the input and the impulse response are some mathematical functions. We show this by an example next. Example Find the total response when the input function is. And the impulse response is given by. Applying the convolution formula we get 30

31 We now give an alternative method for calculating the convolution of the given signal x[n] and the response to the unit impulse function. Discrete Time Convolution Let the given signal x[n] be 31

32 Let the Impulse Response be Now we break the signal in its components i.e. expressed as a sum of unit impulses scaled and delayed or advanced appropriately. Simultaneously we show the output as sum of responses of unit impulses function scaled by the same multiplying factor and appropriately delayed or advanced. Summing the left and the right hand sides of the above figures we get the input x[n] and the total response on the left and the right sides respectively. Thus we see the graphical analog the above formula. 32

33 The total response referred to as the Convolution sum need not always be found graphically. The formula can directly be applied if the input and the impulse response are some mathematical functions. We show this by a example. 33