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, ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... Read More

What is a Discrete Time Impulse Sequence?The discrete time unit impulse sequence 𝛿[𝑛], 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 ... Read More

What is a Power Signal?A signal is said to be a power signal if its average power (P) is finite, i.e., 0 < 𝑃 < ∞. The total energy of a power signal is infinity over infinite time, i.e., 𝐸 = ∞. The periodic signals are the examples of power ... Read More

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 < 𝐸 < ∞. The average power of an energy signal is zero over infinite time (i.e., P = 0). The non-periodic signals are ... Read More