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

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 time unit impulse sequence, 𝛿[𝑘𝑛] = 𝛿[𝑛]Where, k is an integer.Proof − 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

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 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

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 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

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 −𝑥(𝑡) = 𝐴𝑒𝛼𝑡Where, A and 𝛼 both are real. Here the parameter A is the amplitude of the exponential signal measured at t = 0 and the parameter 𝛼 can be either positive or negative.Depending upon the value of 𝛼, we obtain different exponential signals as −When 𝛼 = ... Read More

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 −𝑐(𝑡) = ... Read More

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 ‘t’ by ‘𝛼t’ in the function. Mathematically, it is ... Read More

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 −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

An ideal impulse signal is a signal that is zero everywhere but at the origin (t = 0), it is infinitely high. Although, the area of the impulse is finite. The unit impulse signal is the most widely used standard signal used in the analysis of signals and systems.Continuous-Time Unit Impulse SignalThe continuous-time unit impulse signal is denoted by δ(t) and is defined as −$$\mathrm{\delta (t)=\left\{\begin{matrix} 1\; \; for\: t=0\ 0\; \; for\:teq 0 \ \end{matrix}\right.}$$Hence, by the definition, the unit impulse signal has zero amplitude everywhere except at t = 0. At the origin (t = 0) the amplitude ... Read More