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Found 1024 Articles for Digital Electronics
9K+ Views
Invertible SystemIf a system has a unique relationship between its input and output, the system is called the invertible system. In other words, a system is said to be an invertible system only if an inverse system exists which when cascaded with the original system produces an output equal to the input of the first system. The block diagram representation of an invertible system is shown in Figure-1.Mathematically, an invertible system is defined as, Β Β π₯(π‘) = πβ1[π¦(π‘)] = πβ1{π[π₯(π‘)]} Β β¦ for continuous time systemΒ Β π₯(π) = πβ1[π¦(π)] = πβ1{π[π₯(π)]} Β β¦ for discrete time systemNon-Invertible SystemA ... Read More
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Energy SignalA signal is said to be an energy signal if and only if its total energy E is finite, i.e., 0 < πΈ < β. For an energy signal, the average power P = 0. The nonperiodic signals are the examples of energy signals.Power SignalA signal is said to be a power signal if its average power P is finite, i.e., 0 < π < β. For a power signal, the total energy E = β. The periodic signals are the examples of power signals.Continuous Time CaseIn electric circuits, the signals may represent current or voltage. Consider a voltage ... Read More
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Even SignalA signal is said to be an even signal if it is symmetrical about the vertical axis or time origin, i.e., π₯(π‘) = π₯(βπ‘); for all π‘ Β Β β¦ continuous time signalπ₯(π) = π₯(βπ); for all π Β Β β¦ discrete time signalOdd SignalA signal is said to be an odd signal if it is anti-symmetrical about the vertical axis, i.e., π₯(βπ‘) = βπ₯(π‘); for all π‘ Β Β β¦ continuous time signalπ₯(βπ) = βπ₯(π); for all π Β Β β¦ discrete time signalDetermination of Even and Odd Components of a SignalContinuous-time CaseEvery signal need not be either purely ... Read More
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Bounded SignalA signal whose magnitude is a finite value is called the bounded signal. A sine wave is an example of bounded signal.BIBO Stable SystemA system is called BIBO stable (or bounded-input, bounded-output stable) system, if and only if every bounded input to the system produces a bounded output.BIBO Stability CriterionFor a system to be BIBO stable, the necessary condition is given by the expression, $$\mathrm{\int_{-\infty }^{\infty}\left | h(t) \right |dt < \infty \; \;}\;\;...(1)$$Where, h(t) is the impulse response of the system. The condition given in the expression (1) is called the BIBO stability criterion.ProofConsider an LTI (linear time-invariant) ... Read More
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Linear SystemA system is said to be linear if it obeys the principle of homogeneity and principle of superposition.Principle of HomogeneityThe principle of homogeneity says that a system which generates an output y(t) for an input x(t) must produce an output ay(t) for an input ax(t).Superposition PrincipleAccording to the principle of superposition, a system which gives an output π¦1(π‘) for an input π₯1(π‘) and an output π¦2(π‘) for an input π₯2(π‘) must produce an output [π¦1(π‘) + π¦2(π‘)] for an input [π₯1(π‘) + π₯2(π‘)].Therefore, for a continuous-time linear system, [ππ¦1(π‘) + ππ¦2(π‘)] = π[ππ₯1(π‘) + ππ₯2(π‘)] = ππ[π₯1(π‘)] + ππ[π₯2(π‘)]Also, ... Read More
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Linear Time-Invariant (LTI) SystemA system that possesses two basic properties namely linearity and timeinvariant is known as linear time-invariant system or LTI system.There are two major reasons behind the use of the LTI systems βThe mathematical analysis becomes easier.Many physical processes through not absolutely LTI systems can be approximated with the properties of linearity and time-invariance.Continuous-Time LTI SystemThe LTI systems are always considered with respect to the impulse response. That means the input is the impulse signal and the output is the impulse response.Consider a continuous-time LTI system as shown in the block diagram of Figure-1.Here, the input to the ... Read More
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Multiplication of Continuous-Time SignalsThe product of two continuous-time signals can be obtained by multiplying their values at every instant of time. Consider two continuous time signals π₯1(π‘) and π₯2(π‘) as shown in the figure.ExplanationThe multiplication of the two signals can be performed by considering different time intervals as follows βFor π β€ π β€ π: π₯1(π‘) = 3 and π₯2(π‘) = 2, thusπ₯1(π‘)π₯2(π‘) = 3 Γ 2 = 6For 1β€ π β€ π: π₯1(π‘) = 2 and π₯2(π‘) = 2 + (π‘ β 1), hence, π₯1(π‘)π₯2(π‘) = 2[2 + (π‘ β 1)] = 4 + 4(π‘ β 1)For 2β€ π ... Read More
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A signal is said to be periodic signal if it has a definite pattern and repeats itself at a regular interval of time. Whereas, the signal which does not at the regular interval of time is known as an aperiodic signal or non-periodic signal.Continuous Time Periodic SignalA continuous time signal x(t) is said to be periodic if and only ifπ₯(π‘ + π) = π₯(π‘) for β β < π‘ < βWhere, T is a positive constant that represents the time period of the periodic signal. The smallest value of the time period (T) which justifies the definition of the periodic ... Read More
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What is a Discrete Time Impulse Sequence?The discrete time unit impulse sequence πΏ[π], also called the unit sample sequence, is defined as, $$\mathrm{\delta \left [ n \right ]=\left\{\begin{matrix} 1\; for\: n=0\ 0\; for \: neq 0\ \end{matrix}\right.}$$Properties of Discrete Time Unit Impulse SequenceScaling PropertyAccording to the scaling property of discrete time unit impulse sequence, πΏ[ππ] = πΏ[π]Where, k is an integer.Proof β By the definition of the discrete time unit impulse sequence, $$\mathrm{\delta \left [ n \right ]=\left\{\begin{matrix} 1\; for\: n=0\ 0\; for \: neq 0\ \end{matrix}\right.}$$Similarly, for the scaled unit impulse sequence, $$\mathrm{\delta \left [ kn \right ]=\left\{\begin{matrix} 1\; ... Read More
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What is a Power Signal?A signal is said to be a power signal if its average power (P) is finite, i.e., 0 < π < β. The total energy of a power signal is infinity over infinite time, i.e., πΈ = β. The periodic signals are the examples of power signals.Energy of a Power SignalConsider a continuous-time power signal x(t). The power of the signal x(t) is finite and is given by, $$\mathrm{P=\lim_{T\rightarrow \infty }\frac{1}{2T}\int_{-T }^{T }x^{2}(t)dt\; \; ...(1)}$$Therefore, the energy of the signal is given by, $$\mathrm{E=\lim_{T\rightarrow \infty }\int_{-T }^{T }x^{2}(t)dt}$$ $$\mathrm{\Rightarrow E=\lim_{T\rightarrow \infty }\left [2T\cdot \frac{1}{2T}\int_{-T }^{T }x^{2}(t)dt ... Read More