Representation of Signals

There are various methods to represent signals. Few of them are listed below :

1. Graphical representation
2. Tabular Form
3. Sequential Form
4. Functional Form

Graphical Representation

In case of graphical representation x[n] value will be plotted for each value of n(-∞ < n < ∞). It is easy to represent x[n] graphically if x[n] is of finite length.

Graphical Representation of a Discrete time signal x[n]

Tabular Representation

In this type of representation, a table is prepared for all values of x[n]. If the signal length increases it is tedious to prepare table for all values of n and x[n].

n	-2	-5	0	1	-3	4	2	3
X[n]	5	3	4	5	4	1	3	2

Sequential Representation

In this type of representation, a finite duration sequence with time origin (n=0). It is indicated by the symbol ↑ is represented as 

The arrow head represent that at 0 instant the amplitude of the signal is 3 while at -1 instant the signal has an amplitude of 2.

An infinite duration sequence can be represented as

Functional Representation

In case of functional representation, sequence x[n] is expressed in mathematical form such as

Basic Circuit Elements- Resistance

Different types of Resistance
Photo Credit - physics.tutorvista.com

It is defined as the property of a substance due to which it opposes the flow of current (i.e. electrons) through it resistance is denoted by capital letter  R  and its symbol is

Resistance Symbol

Resistance is one of the basic element of an electric circuit . The  practical resistance is called a resistor and its unit is "ohm" () .A conductor is said to have a resistance of one ohm if it permits one ampere current to flow through it when one volt is applied across its terminals.

Metals , acids and salts are good conductor of electricity due to presence of a large number of free or loosely  attached -electrons in their atoms. Hence they have a very large resistance. Resistance is a passive element and hence dissipates power in the form of heat. The power dissipated by a resistance can be given as
P = vi

                          =ⁱ/_Ri       OR   ^v/_Rv

Laws of Resistance :-

The resistance R offered by a conductor depends on the following factors :

1. It varies directly as  length l of conductor.
2. It varies inversely as the cross sectional area A of the conductor. 
3. On the nature of material.
4. On the temperature of the conductor.

R ∝^l/_A

R = Ρ^l/_A

where P is proportionality constant or specific resistance or resistivity.

Classification of Signals

The signals are classified according to their nature.

1. Continuous time signal and Discrete time signals
2. Deterministic and Random Signals 
3. Even and Odd Signals
4. Periodic and Aperiodic (Non Periodic) Signals
5. Energy and Power Signals

Continuous Time and Discrete Time Signals

In case of continuous time signals, the independent variable is continuous and thus these signals are defined for a continuum of values of the independent variable. The symbol "t" is generally used to denote the continuous time independent variable. Continuous time signals encloses the independent variable in parentheses like (t).

Continuous Time Signal Graphical Representation

Discrete time signals are defined only at discrete times and consequently for these signals, the independent variable takes only a discrete set of values. The discrete time signals are denoted by "n" and are enclosed in brackets like this [n].
Continuous time signals and discrete time signals are further classified, which will be discussed in next post.

Discrete Time Signals Graphical Representation

Deterministic and Random Signal

Deterministic Signal

Any signal that can be uniquely described by an explicit mathematical expression, a table of data or well defined rule is called deterministic signal. It can also be defined as a signal about which there is no uncertainty at any time with respect to its value. The term deterministic emphasizes that all past, present and future values of the signal are known precisely without any uncertainty and are completely specified functions of times. 

Examples : x(t) = Ae^-2t
                  x[n] = 2ⁿ
Random Signal
It is defined as a signal about which there is uncertainty before its actual occurrence. Random signals take on random values at any given time instant and must be modelled probabilistically. They cannot be described to any reasonable degree of accuracy by explicit mathematical formulas.
Examples : Seismic signal, speech signal.

Even and Odd Signal

Even signal 
If the time reversal of x(t) or x[n] is identical to it that is with its reflection about the origin, then the signal is referred to as an even signal. in continuous time a signal is even if

x(-t)=x(t)     ....... for all t 

A discrete time signal is even if

(-n)=x[n]     ....... for all n

Odd signal-

A signal x(t) or x[n] is to odd signal if x(-t)= -x(t) an odd signal must necessarily be at 0 at t=0 or n=0 since x(0)= -x(0) and x[0]= -x[0]

Examples-
Functional form :- Even signal - cosω₀t
                               Odd signal - sinω₀t

Fig(b): Odd continuous time signal

Graphical representation :

Fig(a): Even time  continuous signal

Periodic and Non Periodic(Aperiodic) Signal

Periodic Signal

Any signal whose amplitude values repeat after certain time interval is called a periodic continuous signal x(t) has the property that there is a positive value of T for which 

x(t) = x(t + T)          .......for all values of t

The above equation holds for the smallest positive values of T which is the fundamental time period T₀ of x(t). The fundamental period T₀ is the amount of time taken by the signal x(t) to complete its one cycle. The fundamental frequency of signal is the reciprocal of fundamental time period T₀.

ƒ = ¹/_T0...........Hertz or cycles/second

The fundamental angular frequency is given by 

ω = 2ϖƒ = ^2ϖ/_T0

Periodic Continuous Time Signal

Note:- Sinusoidal signals are periodic signals. 

           Asinω₀t , Acosω₀t ..............................................with period ^2ϖ/_T0

Periodic Signal are defined analogously in discrete time. A discrete time signal x[n] is said to be periodic if it satisfies the condition,

x[n] = x[n +N]        ............for all N

where N is a positive integer, if it is unchanged by time shift of N. The smallest value of N which satisfies the condition is called as fundamental period N₀ of x[n]. The fundamental angular frequency of x[n] is given by,

Ω = ^2ϖ/_N0.......... radians

A discrete time periodic signals

Non Periodic Signal (Aperiodic Signal)

Any continuous time signal x(t) which does not satisfy the condition,

x(t) ≠ x(t + T)

is called non periodic or aperiodic signal. There is no value of T which satisfies the above condition.

Any discrete time signal x[n] which does not satisfy the condition, 

 x[n] ≠ x[n +N]

is called non periodic or aperiodic discrete time signal.

Examples :- Exponential signals like e^-αt, parabolic signals are non periodic signals.

Control System - Some Important Definitions

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Mathematical Models

Mathematical models may assume different forms. Depending on the particular system and the particular circumstances, one mathematical model may be better suited than other other models, Once a mathematical model of a system is obtained, various analytical and computer tools can be used for analysis and synthesis purpose.

Simplicity vs Accuracy

In obtaining a mathematical model, one must make compromise between the simplicity of the system and the accuracy  of the result of the analysis. In deriving a reasonably simplified model, it is necessary to ignore certain inherent physical properties of the system.

While solving a new problem, it is desirable to build a simplified model so that a general feeling for the solution should be felt as we feel when we have found solution to a very complex mathematical problem. A more complete model then can be built and used for a more accurate analysis. A linear lumped parameter model which may be valid in low frequency operations may not be valid at sufficiently high frequency. Since the neglected property of distributed parameters may become an important factor in the dynamic behaviour of the system.  

Linear System

A system is called linear if the principle of superposition applies. The principle of superposition states that the response produced by the simultaneous application of two different forcing functions is the sum of the two individual response. For the linear system, the response of several inputs can be calculated by treating one input at a time and adding the results. It is this principle that allows one to build up complicated solutions to the linear differential equation from simple solutions. In an experimental investigation of a dynamic system, if cause and effect are proportional thus implying that the principle of superposition holds then the system can be considered linear.

Linear Time Invariant System

A differential equation is linear if the coefficients are constants or functions only of the independent variables. Dynamic systems that are composed of linear time invariant lumped parameter components may be described linear time invariant (constant coefficient) differential equations. Such systems are called linear time invariant system. 

Systems that are represented by differential equation and coefficients are functions of time  then it is called as linear time varying system.

Controlled variable and manipulated variable

The controlled variable is the quantity or condition that is measured and controlled

The manipulated variable is the quantity or condition that is varied by the controlled variable so as to affect the value of the controlled variable.

Normally controlled output is the output of system.

Control means measuring the value of the controlled variable of the system and applying the manipulated variable to the system to correct or limit the deviation of the measured value from a desired value.

Plant

A plant may be a piece of equipment perhaps just a set of machine parts functioning together to perform a particular operation/task.

Processes

The process is defined in various ways depending on the context for which it is used. The Merriam Webster dictionary defines a process to be

a natural, progressively continuing operation or development marked by a series of gradual changes that succeed one another in relatively fixed way and lead toward a particular end or result. It may also be defined as an artificial or voluntary, progressively continuing operation that consists of a series of controlled actions or movements systematically directed toward a particular result or end.

Example :- chemical process, biological process.

System

A system is a combination of components that act together and perform a certain objective. A system is not limited to physical ones. The concept of the system can be applied to abstract, dynamic phenomena such as those encountered in economics.

Disturbances

A disturbance is a signal that tends to adversely affect the value of the output of a system. If disturbances is generate within the system, it is called internal while an external disturbance is generated outside the system and is an input.

Feedback Control

Feedback control refers to an operation that, in the presence of disturbance tends to reduce the difference between the output of a system and some reference input and does so on the basis of this difference.