Manish Kumar Saini has Published 751 Answers

Frequency Convolution Theorem

Manish Kumar Saini
Published on 17-Dec-2021 10:34:46
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 ... Read More

Signals and Systems – Fourier Transform of Periodic Signals

Manish Kumar Saini
Published on 17-Dec-2021 10:32:40
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 ... Read More

Signals and Systems – Response of Linear Time Invariant (LTI) System

Manish Kumar Saini
Published on 17-Dec-2021 10:28:02
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 ... Read More

Signals and Systems – Transfer Function of Linear Time Invariant (LTI) System

Manish Kumar Saini
Published on 17-Dec-2021 07:49:01
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 ... Read More

Signals and Systems – Causality and Paley-Wiener Criterion for Physical Realization

Manish Kumar Saini
Published on 17-Dec-2021 07:43:05
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};\; \; ... Read More

Parseval’s Theorem & Parseval’s Identity of Fourier Transform

Manish Kumar Saini
Published on 17-Dec-2021 07:38:13
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 ... Read More

Signals and Systems – Symmetric Impulse Response of Linear-Phase System

Manish Kumar Saini
Published on 17-Dec-2021 07:26:03
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 ... Read More

Signals and Systems – Multiplication Property of Fourier Transform

Manish Kumar Saini
Published on 17-Dec-2021 07:20:33
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 ... Read More

Fourier Transform of a Gaussian Signal

Manish Kumar Saini
Published on 17-Dec-2021 07:09:18
For a continuous-time function $\mathit{x(t)}$, the Fourier transform of $\mathit{x(t)}$ can be defined as, $$\mathrm{\mathit{X\left(\omega\right )\mathrm{=}\int_{-\infty }^{\infty} x\left(t\right)\:e^{-j\omega t}\:dt} }$$Fourier Transform of Gaussian SignalGaussian Function - The Gaussian function is defined as, $$\mathrm{\mathit{g_{a}\left(t\right)\mathrm{=} e^{-at^{\mathrm{2}}} ;\:\:\mathrm{for\:all} \:t} }$$Therefore, from the definition of Fourier transform, we have, $$\mathrm{\mathit{X\left(\omega\right)\mathrm{=} F\left [e^{-at^\mathrm{2}} \right ]=\int_{-\infty ... Read More

Power Spectral Density (PSD) and Autocorrelation Function

Manish Kumar Saini
Published on 17-Dec-2021 07:03:42
Power Spectral DensityThe distribution of average power of a signal in the frequency domain is called the power spectral density (PSD) or power density (PD) or power density spectrum. The power spectral density is denoted by $\mathit{S\left (\omega \right )}$ and is given by, $$\mathrm{\mathit{S\left (\omega \right )\mathrm{=}\lim_{\tau \rightarrow \infty ... Read More
Advertisements