Signals and Systems: Linear Time-Invariant Systems

Updated on 13-Nov-2021 09:56:49
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 βThe 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

Signals and Systems: Multiplication of Signals

Updated on 12-Nov-2021 11:55:04
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 βFor 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

Signals and Systems: Periodic and Aperiodic Signals

Updated on 12-Nov-2021 11:48:56
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‘ < βWhere, 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

Properties of Discrete Time Unit Impulse Signal

Updated on 12-Nov-2021 11:44:22

Power of an Energy Signal over Infinite Time

Updated on 12-Nov-2021 11:28:02
What is an Energy Signal?A signal is said to be an energy signal if and only if its total energy (E) is finite. That means 0 < πΈ < β. The average power of an energy signal is zero over infinite time (i.e., P = 0). The non-periodic signals are examples of energy signals.Power of an Energy SignalConsider a continuous-time energy signal x(t). The energy of the signal x(t) is finite, i.e., $$\mathrm{E=\int_{-\infty }^{\infty }x^{2}(t)dt=finite\; \; ...(1)}$$Hence, the power of the signal x(t) is, $$\mathrm{P=\lim_{T\rightarrow \infty }\frac{1}{2T}\int_{-T}^{T}x^{2}(t)dt}$$ $$\mathrm{\Rightarrow P=\lim_{T\rightarrow \infty }\frac{1}{2T}\left [ \lim_{T\rightarrow \infty }\int_{-T}^{T}x^{2}(t)dt \right ]}$$ \mathrm{\mathrm{\Rightarrow P=\lim_{T\rightarrow \infty ... Read More

Signals and Systems: Real and Complex Exponential Signals

Updated on 12-Nov-2021 10:59:41
Real Exponential SignalsAn exponential signal or exponential function is a function that literally represents an exponentially increasing or decreasing series.Continuous-Time Real Exponential SignalA real exponential signal which is defined for every instant of time is called continuous time real exponential signal. A continuous time real exponential signal is defined as follows βπ₯(π‘) = π΄ππΌπ‘Where, A and πΌ both are real. Here the parameter A is the amplitude of the exponential signal measured at t = 0 and the parameter πΌ can be either positive or negative.Depending upon the value of πΌ, we obtain different exponential signals as βWhen πΌ = ... Read More

Signals and Systems: Static and Dynamic System

Updated on 12-Nov-2021 10:55:07
Static SystemA system whose response or output is due to present input alone is known as static system. The static system is also called the memoryless system. For a static or memoryless system, the output of the system at any instant of time (t for continuous-time system or n for discrete-time system) depends only on the input applied at that instant of time (t or n), but not on the past or future values of the input.A purely resistive electrical circuit is an example of static system. Some examples of continuous-time and discrete-time static systems are given below βπ(π‘) = ... Read More

Signals and Systems β Time Scaling of Signals

Updated on 12-Nov-2021 10:52:52
What is Time Scaling?The process of multiplying a constant to the time axis of a signal is known as time scaling of the signal. The time scaling of signal may be time compression or time expansion depending upon the value of the constant or scaling factor. The time scaling operation of signals is very useful when data is to be fed at some rate and is to be taken out at a different rate.Time scaling of continuous-time signalThe time scaling of a continuous time signal x(t) can be accomplished by replacing βtβ by βπΌtβ in the function. Mathematically, it is ... Read More

Signals and Systems: Time Variant and Time-Invariant Systems

Updated on 12-Nov-2021 10:46:53
The property of a system which makes the behaviour of the system independent of time is known as time invariance. Time invariance means that the behaviour of the system does not depend on the time at which the input is applied to the system.Time-Invariant SystemIf the input and output characteristics of a system do not change with time, the system is called the time-invariant system.Continuous-time CaseThe time-invariance property of a continuous time system can be tested as follows βLet x(t) is the input and x(t-t0) is the delayed input by t0 units. Then, the output of the system for the ... Read More
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