Communication(chapter 1 Signal & System)

Signals & Systems

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Signal The signals are detectable physical quantities that vary with time, space or any other independent variable or variables.

Classification of Signals Signals can be classified as: Continuous time Signals Discrete time Signals Continuous Time Signals The Signal, which varies with respect to time, is termed as continuous time signal or analog signal.

Discrete Time Signals When signal is specified at certain time instants it is termed as discrete-time Signal.

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Further classification of Signals  Even and Odd Signals  Energy and Power Signals.  Deterministic and Random Signal.  Periodic and Aperiodic Signals

Even Signals If by changing independent axis, without changing dependent axis the magnitude of the signal remains same, the signal is termed as even signal. Odd Signals When independent variable is changed and there is significant change in the amplitude of function, it is called odd signal.

f (t ) = f (− t )

f (t ) = − f (− t )

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Periodic and Aperiodic Signals  Any continuous-time signal that

satisfies the condition: x(t+T) is a periodic Signal.  For Aperiodic Signals

Energy and Power Signals  Any signal, integrated over time where it exists is energy signal. Theses signals contain finite amount of energy and zero power.  Time average of energy signal

is Power signal, they contain infinite energy.

x(t ) = x(t+ T ) x(t ) ≠ x(t+ T ) L im E = T→∞

T

∫

2

x(t ) d t

−T

2 L im 1 T P = x(t ) d t ∫ T → ∞ 2T − T 4

SYSTEM  Any process or any device by which you have certain operations

under certain rules is called a system. For now consider e.g. of a robot arm, which performs certain operations on some control inputs ( signals )under certain limitations can be consider as a system.

Basic System representation

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Classification of Systems Systems can be classified as: • Linear and Non linear Systems • Time Invariant Systems • Causal and Non- Causal Systems • Memory and Memoryless Systems • Stable and Unstable Systems

Linear and Non- Linear Systems Linear System is one which follows the important property of superposition. Additivity The response to x1(t) + x2(t) is y1(t) + y2(t) Homogeneity/Scaling The response to a x1(t) is a y1(t)  So a system that follows both the properties is a linear system.

Non Linear system is one which do not follow the law of superposition. 6

Example: Consider a system S whose input x(t) and output y(t) are related by y(t) = tx(t) To determine whether S is linear or not, we consider two arbitrary inputs x1(t) and x2(t). x1(t) → y1(t) = tx1(t) x2(t) → y2(t) = tx2(t) Let x3(t) be a linear combination of x1(t) and x2(t).That is, x3(t) = a x1(t) + b x2(t) where a and b are arbitrary scalars. If x3(t) is the input to S, then the corresponding output may be expressed as y3(t) = tx3(t) = t(a x1(t) + b x2(t)) = at x1(t) + bt x2(t) = a y1(t) + b y2(t) we conclude that the system S is linear one.

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Numerical If the input x(t) and output y(t) are related to system S by y(t) = x2(t) then determine S is linear or not.

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Time Invariant Systems  Time Invariant System condition is based on two

steps:  Give delay in input.  Give delay in output  If delay in input is equal to delay in output then

the system is Time Invariant. If not equal then the system is Time Variant.

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Numerical Time Variant System:

consider the system defined by y(t) = 4tx(t)

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Example Time Invariant System: Consider the continuous-time system defined by y(t) = sin [x(t)] To check that this system is time invariant, we must determine whether the time-invariance property holds for any input and any time shift t0. Thus, let x1(t) be an arbitrary input to this system, and the output is y1(t) = sin [x1(t)] Let us consider a second input obtained by shifting x1(t) in time: x2(t) = x1(t - t0) The output corresponding to this input is y2(t) = sin [x 2(t)] = sin [x1(t – t0)] Similarly y1(t – t0) = sin [x1(t – t0)] Then y2(t) = y1(t – t0) , and therefore, this system is time invariant.

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Causal and Non Causal Systems  System which outputs are dependent on present and past input, system is called causal system. y [n] = x [-n]  Non causal systems are those systems, the output of which depends on future inputs.

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System with and Without Memory  A system is said to be memory less, or instantaneous, if

their present value of the output depends only on the present value of the input  y(t) = x(t)

 A system which depends upon past and future values is

called a Memory System.  y [n] = x [n - 1]

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Transformation of Independent Variable “t”  Transformation of Independent variable ‘t’ is defined in

three ways:  Time Shifting  Time Scaling  Time Reversal

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Time Shifting, Time Reversal,Time Scaling  Suppose we have a signal x(t) and we say we want to shift a

signal such as x(t-2) or x(t+2) so ‘-’ values indicate the past values while the ‘+’ values indicate the future value  Time reversal is the mirror image of the given signal as x(t)

= x(-t)  Time Scaling is the scaled time according to input for e.g

x(2t) will be a compact signal as compared to x(t).

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Unit Impulse  The signal which exist at t = 0 (time =0) is called a Unit

Impulse or delta function.  It is represented by

δ (t ) = 1 t = 0;

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Unit Step Function  The Signal which exist in single step is called a Unit step

function.  It is represented by

 (t )

 If we integrate the area in which all unit impulses exist

then we get Unit step function.

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Continuous time Exponential Functions  The continuous time exponential function is:

f (t ) =

at

Ae

 For this we have conditions:  When “a” is positive.  When “a” is negative.

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When ‘a’ is positive 1 0

 When ‘a’ is positive we

get exponentially growing function as illustrated with the graph.

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When ‘a’ is negative 4

 When ‘a’ is negative we

get exponentially decaying function as illustrated in graph.

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