Nyquist Rate of SamplingThe theoretical minimum sampling rate at which a signal can be sampled and still can be reconstructed from its samples without any distortion is called the Nyquist rate of sampling.Mathematically, $$\mathrm{Nyquist\: Rate, \mathit{f_{N}}\mathrm{=}2\mathit{f_{m}}}$$Where, $\mathit{f_{m}}$is the maximum frequency component present in the signal.If the signal is sampled at the rate greater than the Nyquist rate, then the signal is called over sampled.If the signal is sampled at the rate less than its Nyquist rate, then it is said to be under sampled.Nyquist IntervalWhen the rate of sampling is equal to the Nyquist rate, then the time interval between ... Read More

Laplace TransformThe Laplace transform is a mathematical tool which is used to convert the differential equation in time domain into the algebraic equations in the frequency domain or s-domain.Mathematically, if $\mathrm{\mathit{x\left(t\right)}}$ is a time domain function, then its Laplace transform is defined as −$$\mathrm{\mathit{L\left[\mathit{x}\mathrm{\left(\mathit{t} \right )}\right ]\mathrm{=}X\mathrm{\left( \mathit{s}\right)}\mathrm{=}\int_{-\infty }^{\infty}x\mathrm{\left (\mathit{t} \right )}e^{-st} \:dt}}$$Circuit Analysis Using Laplace TransformThe Laplace transform can be used to solve the different circuit problems. In order to solve the circuit problems, first the differential equations of the circuits are to be written and then these differential equations are solved by using the Laplace transform. Also, the ... Read More

ConvolutionThe convolution of two signals $\mathit{x\left ( t \right )}$ and $\mathit{h\left ( t \right )}$ is defined as, $$\mathrm{\mathit{y\left(t\right)\mathrm{=}x\left(t\right)*h\left(t\right)\mathrm{=}\int_{-\infty }^{\infty}x\left(\tau\right)\:h\left(t-\tau\right)\:d\tau}}$$This integral is also called the convolution integral.Frequency Convolution TheoremStatement - The frequency convolution theorem states that the multiplication of two signals in time domain is equivalent to the convolution of their spectra in the frequency domain.Therefore, if the Fourier transform of two signals $\mathit{x_{\mathrm{1}}\left ( t \right )}$ and $\mathit{x_{\mathrm{2}}\left ( t \right )}$ is defined as$$\mathrm{\mathit{x_{\mathrm{1}}\left(t\right)\overset{FT}{\leftrightarrow} X_{\mathrm{1}}\left(\omega\right)} }$$And$$\mathrm{\mathit{x_{\mathrm{2}}\left(t\right)\overset{FT}{\leftrightarrow} X_{\mathrm{2}}\left(\omega\right)}}$$Then, according to the frequency convolution theorem, $$\mathrm{\mathit{x_{\mathrm{1}}\left(t\right).x_{\mathrm{2}}\left(t\right)\overset{FT}{\leftrightarrow}\frac{\mathrm{1}}{\mathrm{2}\pi }\left [ X_{\mathrm{1}}\left(\omega\right)* X_{\mathrm{2}}\left(\omega\right)\right ]}}$$ProofFrom the definition of Fourier transform, we have, ... Read More

The Fourier series can be used to analyse only the periodic signals, while the Fourier transform can be used to analyse both periodic as well as non-periodic functions. Therefore, the Fourier transform can be used as a universal mathematical tool in the analysis of both periodic and aperiodic signals over the entire interval. The Fourier transform of periodic signals can be found using the concept of impulse function.Now, consider a periodic signal $\mathit{x\left(t\right )}$ with period $\mathit{T}$. Then, the expression of $\mathit{x\left(t\right )}$ in terms of exponential Fourier series is given by, $$\mathrm{\mathit{x\left(t\right)=\sum_{n=-\infty }^{\infty } C_{n}\:e^{jn\omega _{\mathrm{0}}t}}}$$Where $\mathit{C_{n}}$ be the ... Read More

Linear Time-Invariant SystemA system for which the principle of superposition and the principle of homogeneity are valid and the input/output characteristics do not change with time is called the linear time-invariant (LTI) system.Impulse Response of LTI SystemWhen the impulse signal is applied to a linear system, then the response of the system is called the impulse response. The impulse response of the system is very important for understanding the behaviour of the system.Therefore, if$$\mathrm{\mathit{\mathrm{Input}, x\left(t\right)=\delta\left(t\right)}}$$Then, $$\mathrm{\mathit{\mathrm{Output}, y\left(t\right)=h\left(t\right)}}$$As the Laplace transform and Fourier transform of the impulse function is given by, $$\mathrm{\mathit{L\left [\delta\left(t\right) \right ]\mathrm{=}\mathrm{1}\:\:\mathrm{and} \:\:F\left [\delta\left(t\right) \right ]\mathrm{=}\mathrm{1}}}$$Hence, once the ... Read More

The transfer function of a continuous-time LTI system may be defined using Laplace transform or Fourier transform. Also, the transfer function of the LTI system can only be defined under zero initial conditions. The block diagram of a continuous-time LTI system is shown in the following figure.Transfer Function of LTI System in Frequency DomainThe transfer function 𝐻(𝜔) of an LTI system can be defined in one of the following ways −The transfer function of an LTI system is defined as the ratio of the Fourier transform of the output signal to the Fourier transform of the input signal provided that ... Read More

Condition of CausalityA causal system is the one which does not produce an output before the input is applied. Therefore, for an LTI (Linear Time-Invariant) system to be causal, the impulse response of the system must be zero for t less than zero, i.e., $$\mathrm{\mathit{h\left ( t \right )\mathrm{=}\mathrm{0};\; \; \mathrm{for}\: \: t< 0}}$$The term physical realization denotes that it is physically possible to construct that system in real time. A system which is physically realizable cannot produce an output before the input is applied. This is called the condition of causality for the system.Therefore, the time domain criterion for ... Read More

Fourier TransformFor a continuous-time function $\mathrm{\mathit{x\left ( t \right )}}$ , the Fourier transform of $\mathrm{\mathit{x\left ( t \right )}}$ can be defined as, $$\mathrm{\mathit{X\left ( \omega \right )\mathrm{=}\int_{-\infty }^{\infty }x\left ( t \right )e^{-j\omega t}dt }}$$And the inverse Fourier transform is defined as, $$\mathrm{\mathit{x\left ( t \right )\mathrm{=}\frac{\mathrm{1}}{\mathrm{2}\pi }\int_{-\infty }^{\infty }X\left ( \omega \right )e^{j\omega t}d\omega }} $$Parseval’s Theorem of Fourier TransformStatement – Parseval’s theorem states that the energy of signal $\mathrm{\mathit{x\left ( t \right )}}$ [if $\mathrm{\mathit{x\left ( t \right )}}$ is aperiodic] or power of signal $\mathrm{\mathit{x\left ( t \right )}}$ [if $\mathrm{\mathit{x\left ( t \right )}}$ ... Read More

Distortion-less TransmissionWhen a signal is transmitted through a system and there is a change in the shape of the signal, it called the distortion. If the output of the system is an exact replica of the input signal, then the transmission of the signal through the system is called distortionless transmission.Linear Phase SystemFor distortionless transmission through a system, there should not be any phase distortion, i.e., the phase of the system should be linear. For the linear phase system, the impulse response of the system is symmetrical about the delay time $\mathit{(t_{d})}$.ProofFor a linear phase system, we have, $$\mathrm{ \mathit{H\left ... Read More

For a continuous-time function $\mathit{x(t)}$, the Fourier transform of $\mathit{x(t)}$ can be defined as$$\mathrm{\mathit{X\left ( \omega \right )\mathrm{\mathrm{=}}\int_{-\infty }^{\infty }x\left ( t \right )e^{-j\omega t}dt}}$$And the inverse Fourier transform is defined as, $$\mathrm{\mathit{F^{\mathrm{-1}}\left [ X\left ( \omega \right ) \right ]\mathrm{\mathrm{=}}x\left ( t \right )\mathrm{\mathrm{=}}\frac{\mathrm{1}}{\mathrm{2}\pi }\int_{-\infty }^{\infty }X\left ( \omega \right )e^{j\omega t}d\omega }}$$Multiplication Property of Fourier TransformStatement – The multiplication property of continuous-time Fourier transform (CTFT) states that the multiplication of two functions in time domain is equivalent to the convolution of their spectra in the frequency domain. The multiplication property is also called frequency convolution theorem of Fourier ... Read More