Unit Impulse and Unit Step Funtions Solution Chapter 1.5

You can find here the solutions of the chapter 1 questions from 41 to 56.























In this section, we introduce several other basic signals-specifically, the unit impulse and step function, in continuous and discrete time-that are also of considerable impertance in signal and system analysis. In Chapter 2, we will see how we can use unit impulse signals as basic building blocks for the construction and representation of other signatti. We begin with the discrete-time case.


Although our discussion of the unit impulse in this section has been somewhat informal, it does provide us with some important intuition about this signal that will be useful throughout the book. As we have stated, the unit impulse should be viewed as an idealization. As we illustrate and discuss in more detail it Section 2.5, any real physical system has some inertia associated with it and thus does not respond instantaneously to inputs. Consequently, if a pulse of sufficiently short duration is applied to such a system, the system response will not be noticeably influenced by the pulse's duration or by the details of the shape of the pulse, for that matter. Instead, the primary characteristic of the pulse that will matter it the net, integrated effect of the pulse-i.e. its area. For systems that respond much more quickly than others, the pulse will have to be of much shorter duration before the details of the pulse shape or its duration no longer matter Nevertheless, for any physical system, we can always find a pulse that is "short enough." The unit impulse then is an idealization of this concept-the pulse that is short enough
for any system. As we will see in Chapter 2. the response of a system to this idealized pulse plays a crucial role in signal and system analysis, and in the process of developing and understanding this role, we will develop additional insight into the idealized signal.

CONTINUOUS-TIME AND DISCRETE TIME SYSTEMS

Physical systems in the broadest sense are an interconnection of components, devices,
or subsystems- In contexts ranging from signal processing and communications to electromechanical motors, automouve vehicles, and chemical-processing plants, a system can be viewed as a process in which input signals are transformed by the system or cause the system to respond in sonic way, resulting in other signals as outputs. For example, a highfidelity system takes a recorded audio signal and generates a reproduction of that signal. If the hi-fi system has tone controls, we can change the tonal quality of the reproduced signal.

In most of this book, we will treat discrete-time systems and continuous-time systems separately but in parallel. In Chapter 7, we will bring continuous-time and discrete-time systems together through the concept of sampling, and we will develop some insights into the use of discrete-time systems to process continuous-time signals that have been sampled.


The mathematical descriptions of systems from a wide variety of applications frequently hatie a great deal in common, and it is this fact that provides considerable motivation for the development of broadly applicable tool, for signal and system analysis. The key to doing this successfully is identifying classes of systems that have two important characteristics: (1) The systems in this class have properties and structures that we can exploit to gain insight into their behavior and to develop effective tools for their analysis; and (2) many systems of practical importance can be accurately modeled using systems in this class. It is on the first of these charactenstics that most of this book focuses, as we develop tools for a particular class of systems referred to as linear, time-invariant systems. In the next section, we will introduce the propertie, that characterize this class, as well as a number of other very important basic system properties.

The second characteristic mentioned in the preceding paragraph is of obvious importance for any system analysis technique to be of value in practice. It is a well-established fact that a wide range of physical systems can be well modeled within the class of systems on which we focus in this book. However, a critical point is that any model used in describing or analyzing a physical system represents an idealization of that system, and thus, any resulting analysis is only as good as the model itselfels are valid.

Interconnections of Systems

An important idea that we will use throughout this book is the concept of the interconnection of systems. Many real systems are built as interconnections of several subsystems. One example is an audio system, which involves the interconnection of a radio receiver, compact disc player, or tapo deck with an amplifier and one or more speakers. Another is a digitally controlled aircraft, which is an interconnection of the aircraft, described by its equaions of motion and the aerodynamic forces affecting it, the sensors, which measure various aircraft variables such as accelerations, rotation rates, and heading; a digital autopilot, which responds to the measured variables and to command inputs from the pilot (e.g., the desired course, altitude, and speed); and the aircraft's actuators, which respond to inputs provided by the autopilot in order to use the aircraft control surfaces (rudder, tail, ailerons) to change the aerodynamic forces on the aircraft.

Invertibility and Inverse Systems

A system is said to he invertible if distinct inputs lead to distinct outputs.

The concept of invertibility is important in mangy contexts. One example arises in
systems for encoding used in a wide variety of communications applications. In such a system, a signal that we wish to transmit is first applied as the input to a system !sown as an encoder. There are many reasons for doing this, ranging from the desire to encrypt the original message for secure or private comet mnication to the objective of providing some redundancy in the signal (for example, by adding what are known as panty bits) so that any errors that occur in transmission can be detected and, possibly, corrected For lossless coding, the input to the encoder must be exactly recoverable from the output: i.e., the encoder must be invertible.

Causality

A system is causal if the output at any time depends only on values of the input at the present time and in the past. Such a system is often referred to as being nonanticiparive, as the system output does not anticipate future values of the input. Consequently, if two inputs to a causal system are identical up to some point in time to or no, the corresponding outputs must also be equal up to this same time.


Although causal systems are of great importance, they do not by any means constitute
the only systems that are of practical significance. For example, causality is not often an essential constraint in applications in which the independent variable is not time, such as in image processing. Furthermore, in processing data that have been recorded previously, as often happens with speech, geophysical, or meteorological signals, to name a few, we are by no means constrained to causal processing. As another example, in many applications, including historical stock market analysis and demographic studies, we may be interested in determining a slowly varying trend in data that also contain high-frequency fluctuations
about that trend. In this case, a commonly used approach is to average data over an interval in order to smooth out the fluctuations and keep only the trend.

Stability

Stability is another important system property. Informally a stable system is one in which small inputs lead to responses that do not diverge. For example, consider the pendulum in which the input is the applied force x(1) and the output is the angular deviation y(t) from the vertical. In this case, gravity applies a restoring force that tends to return the pendulum to the vertical position, and frictional losses due to drag tend to slow it down. Consequently, if a small force x(t) is applied, the resulting deflection from vertical will also be small. In contrast, for the inverted pendulum, the effect of gravity is to apply a force that tends to increase the deviation from vertical. Thus, a small applied force leads to a large vertical deflection causing the pendulum to topple over, despite any retarding forces due to friction.

Time Invariance

Conceptually, a system is time invariant if the behavior and characteristics of the system are fixed over time. For example, the RC circuit is tine invariant if the resistance and capacitance values R and C are constant over time: We would expect to get the same results from an experiment with this circuit today as we would if we ran the identical experiment tomorrow. On the other hand, if the values of R and C are changed or fluctuate over time, then we would expect the results of our experiment to depend on the time at which we run it. Similarly, if the frictional coefficient b and mass m of the automobile are constant, we would expect the vehicle to respond identically independently of when we drive it. On the outer hand, if we load the auto's trunk with heavy suitcases one day, thus increasing in. we would expect the car to behave differently than at other times when it is not so heavily loaded.

Linearity

A linear system, in continuous time or discrete time, is a system that possesses the important property of superposition: If an input consists of the weighted sum of several signals, then the output is the superposition-that is, the weighted sum-of the responses of the system to each of those signals.