Thermal Power PlantA generating station which converts the heat energy of combustion of coal into electrical energy is called a thermal power plant or steam power station.Schematic Arrangement of Thermal Power PlantThe thermal power plant has many arrangements for proper and efficient working. The schematic of a modern thermal power plant is shown in the figure. This whole schematic arrangement can be divided into the following segments −Coal and Ash Handling PlantThe coal is transported to the site of power plant from the coal mines by rail or road and it is stored in the coal storage plant. From the ... Read More

The mean square error (MSE) is defined as mean or average of the square of the difference between actual and estimated values.Mathematically, the mean square error is, $$\mathrm{\varepsilon =\frac{1}{t_{2}-t_{1}}\int_{t_{1}}^{t_{2}}\left [ x(t) -\sum_{r=1}^{n}C_{r}g_{r}(t)\right ]^{2}dt}$$$$\mathrm{\varepsilon =\frac{1}{t_{2}-t_{1}}\left [ \int_{t_{1}}^{t_{2}}x^{2}(t)dt+\sum_{r=1}^{n}C_{r}^{2}\int_{t_{1}}^{t_{2}}g_{r}^{2}(t)dt-2\sum_{r=1}^{n}C_{r}\int_{t_{1}}^{t_{2}}x(t)g_{r}(t)dt\right ]\; ...(1)}$$$$\mathrm{\therefore C_{r}=\frac{\int_{t_{1}}^{t_{2}}x(t)g_{r}(t)dt}{\int_{t_{1}}^{t_{2}}g_{r}^{2}(t)dt}=\frac{1}{K_{r}}\int_{t_{1}}^{t_{2}}x(t)g_{r}(t)dt\; \; ...(2)}$$$$\mathrm{\therefore \int_{t_{1}}^{t_{2}}x(t)g_{r}(t)dt=C_{r}\int_{t_{1}}^{t_{2}}g_{r}^{2}(t)dt=C_{r}K_{r}\; \; ...(3)}$$Using equations (1) and (3), we have, $$\mathrm{\varepsilon =\frac{1}{t_{2}-t_{1}}\left [\int_{t_{1}}^{t_{2}} x^{2}(t)dt +\sum_{r=1}^{n}C^{2}_{r}K_{r}-2\sum_{r=1}^{n}C^{2}_{r}K_{r}\right ]}$$$$\mathrm{\Rightarrow \varepsilon =\frac{1}{t_{2}-t_{1}}\left [\int_{t_{1}}^{t_{2}} x^{2}(t)dt -\sum_{r=1}^{n}C^{2}_{r}K_{r}\right ]\; \; ...(4)}$$$$\mathrm{\Rightarrow \varepsilon =\frac{1}{t_{2}-t_{1}}\left [ \int_{t_{1}}^{t_{2}}x^{2}(t)dt-(C_{1}^{2}K_{1}+C_{2}^{2}K_{2}+\cdot \cdot \cdot +C_{n}^{2}K_{n}) \right ]\; \; \cdot \cdot \cdot (5)}$$Therefore, the mean square error can be evaluated using eqn. (5).Numerical ExampleA rectangular function is defined as, $$\mathrm{x(t)=\left\{\begin{matrix} 1\; \; for\, 0< t< ... Read More

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,     𝑥(𝑡) = 𝑇−1[𝑦(𝑡)] = 𝑇−1{𝑇[𝑥(𝑡)]}   … for continuous time system    𝑥(𝑛) = 𝑇−1[𝑦(𝑛)] = 𝑇−1{𝑇[𝑥(𝑛)]}   … for discrete time systemNon-Invertible SystemA ... Read More

Energy SignalA signal is said to be an energy signal if and only if its total energy E is finite, i.e., 0 < 𝐸 < ∞. 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 < 𝑃 < ∞. For a power signal, the total energy E = ∞. 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

Even SignalA signal is said to be an even signal if it is symmetrical about the vertical axis or time origin, i.e., 𝑥(𝑡) = 𝑥(−𝑡); for all 𝑡     … continuous time signal𝑥(𝑛) = 𝑥(−𝑛); for all 𝑛     … discrete time signalOdd SignalA signal is said to be an odd signal if it is anti-symmetrical about the vertical axis, i.e., 𝑥(−𝑡) = −𝑥(𝑡); for all 𝑡    … continuous time signal𝑥(−𝑛) = −𝑥(𝑛); for all 𝑛     … discrete time signalDetermination of Even and Odd Components of a SignalContinuous-time CaseEvery signal need not be either purely ... Read More

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

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 𝑦1(𝑡) for an input 𝑥1(𝑡) and an output 𝑦2(𝑡) for an input 𝑥2(𝑡) must produce an output [𝑦1(𝑡) + 𝑦2(𝑡)] for an input [𝑥1(𝑡) + 𝑥2(𝑡)].Therefore, for a continuous-time linear system, [𝑎𝑦1(𝑡) + 𝑏𝑦2(𝑡)] = 𝑇[𝑎𝑥1(𝑡) + 𝑏𝑥2(𝑡)] = 𝑎𝑇[𝑥1(𝑡)] + 𝑏𝑇[𝑥2(𝑡)]Also, ... Read More

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 −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

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 𝑥1(𝑡) and 𝑥2(𝑡) as shown in the figure.ExplanationThe multiplication of the two signals can be performed by considering different time intervals as follows −For 𝟎 ≤ 𝒕 ≤ 𝟏: 𝑥1(𝑡) = 3 and 𝑥2(𝑡) = 2, thus𝑥1(𝑡)𝑥2(𝑡) = 3 × 2 = 6For 1≤ 𝒕 ≤ 𝟐: 𝑥1(𝑡) = 2 and 𝑥2(𝑡) = 2 + (𝑡 − 1), hence, 𝑥1(𝑡)𝑥2(𝑡) = 2[2 + (𝑡 − 1)] = 4 + 4(𝑡 − 1)For 2≤ 𝒕 ... Read More

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 if𝑥(𝑡 + 𝑇) = 𝑥(𝑡) for − ∞ < 𝑡 < ∞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