Found 189 Articles for Signals and Systems

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 with time is called the linear time invariant (LTI) system.Properties of LTI SystemA continuous-time LTI system can be represented in terms of its unit impulse response. It takes the form of convolution integral. Hence, the properties followed by the continuous time convolution are also followed by the LTI system. The impulse response of an LTI system is very important because it can completely determine the characteristics of an LTI system.In this article, we will highlight some ... Read More

Fourier series is a branch of Fourier analysis of periodic signals. Fourier series splits a periodic signal into a sum of sines and cosines with different amplitudes and frequencies. Fourier series was introduced by a French mathematician Joseph Fourier. On the other hand, the Fourier Transform is a mathematical operation that decompose a signal into its constituent frequencies. The Fourier transform is also called frequency domain representation of a signal because it depends on the frequency of the signal. Read through this article to find out more about Fourier Series and Fourier Transform and how they are different from each ... Read More

What is a Linear System?System − An entity which acts on an input signal and transforms it into an output signal is called the system.Linear System − A linear system is defined as a system for which the principle of superposition and the principle of homogeneity are valid.Superposition PrincipleThe principle of superposition states that the response of the system to a weighted sum of input signals is equal to the corresponding weighted sum of the outputs of the system to each of the input signals.Therefore, if an input signal x1(t) produces an output signal y1(t) and another input signal x2(t) ... Read More

What is Convolution?Convolution is a mathematical tool to combining two signals to form a third signal. Therefore, in signals and systems, the convolution is very important because it relates the input signal and the impulse response of the system to produce the output signal from the system. In other words, the convolution is used to express the input and output relationship of an LTI system.ExplanationConsider a continuous-time LTI system which is relaxed at t = 0, i.e., initially, no input is applied to it. Now, if the impulse signal [δ(t)] is input to the system, then output of the system ... Read More

Hilbert TransformWhen the phase angles of all the positive frequency spectral components of a signal are shifted by (-90°) and the phase angles of all the negative frequency spectral components are shifted by (+90°), then the resulting function of time is known as Hilbert transform of the given signal.In case of Hilbert transformation of a signal, the magnitude spectrum of the signal does not change, only phase spectrum of the signal is changed. Also, Hilbert transform of a signal does not change the domain of the signal.Let a signal x(t) with Fourier transform X(ω). The Hilbert transform of x(t) is ... Read More

For a continuous-time function 𝑥(𝑡), the Fourier transform of 𝑥(𝑡) can be defined as, $$\mathrm{X\left ( \omega \right )=\int_{-\infty }^{\infty }x\left ( t \right )e^{-j\omega t}\: dt}$$Time-Shifting Property of Fourier TransformStatement – The time shifting property of Fourier transform states that if a signal 𝑥(𝑡) is shifted by 𝑡0 in time domain, then the frequency spectrum is modified by a linear phase shift of slope (−𝜔𝑡0). Therefore, if, $$\mathrm{x\left ( t \right )\overset{FT}{\leftrightarrow}X\left ( \omega \right )}$$Then, according to the time-shifting property of Fourier transform, $$\mathrm{x\left ( t -t_{0}\right )\overset{FT}{\leftrightarrow}e^{-j\omega t_{0}}X\left ( \omega \right )}$$ProofFrom the definition of Fourier transform, ... Read More

Signal BandwidthThe spectral components of a signal extends from (−∞) to ∞ and any practical signal has finite energy. Consequently, the spectral components approach zero when the frequency 𝜔 tends to ∞. Therefore, those spectral components can be neglected which have negligible energy and hence only a band of frequency components is selected which have most of the signal energy. This band of frequency components which contains most of the signal energy is called the signal bandwidth.Normally, this band has the frequency components that contains around 95% of the total energy depending upon the precision.System BandwidthA system which has infinite ... Read More

ConvolutionThe convolution is a mathematical operation for combining two signals to form a third signal. In other words, the convolution is a mathematical way which is used to express the relation between the input and output characteristics of an LTI system.Mathematically, the convolution of two signals is given by, $$\mathrm{x_{1}\left ( t \right )\ast x_{2}\left ( t \right )=\int_{-\infty }^{\infty }x_{1}\left ( \tau \right )x_{2}\left ( t-\tau \right )d\tau =\int_{-\infty }^{\infty }x_{2}\left ( \tau \right )x_{1}\left ( t-\tau \right )d\tau}$$CorrelationThe correlation is defined as the measure of similarity between two signals or functions or waveforms. The correlation is of two ... Read More

For a continuous-time function 𝑥(𝑡), the Fourier transform of 𝑥(𝑡) can be defined as, $$\mathrm{X\left ( \omega \right )=\int_{-\infty }^{\infty }x\left ( t \right )e^{-j\omega t}\: dt}$$Time Reversal Property of Fourier TransformStatement – The time reversal property of Fourier transform states that if a function 𝑥(𝑡) is reversed in time domain, then its spectrum in frequency domain is also reversed, i.e., if$$\mathrm{x\left ( t \right )\overset{FT}{\leftrightarrow}X\left ( \omega \right )}$$Then, according to the time-reversal property of Fourier transform, $$\mathrm{x\left ( -t \right )\overset{FT}{\leftrightarrow}X\left ( -\omega \right )}$$ProofForm the definition of Fourier transform, we have, $$\mathrm{F\left [ x\left ( t \right ... Read More

Fourier TransformThe Fourier transform of a continuous-time function $x(t)$ can be defined as, $$\mathrm{X(\omega)=\int_{−\infty}^{\infty}x(t)e^{-j\omega t}dt}$$Fourier Transform of Unit Impulse FunctionThe unit impulse function is defined as, $$\mathrm{\delta(t)=\begin{cases}1 & for\:t=0 \0 & for\:t ≠ 0 \end{cases}}$$If it is given that$$\mathrm{x(t)=\delta(t)}$$Then, from the definition of Fourier transform, we have, $$\mathrm{X(\omega)=\int_{−\infty}^{\infty}x(t)e^{-j\omega t}dt=\int_{−\infty}^{\infty}\delta(t)e^{-j\omega t}dt}$$As the impulse function exists only at t= 0. Thus, $$\mathrm{X(\omega)=\int_{−\infty}^{\infty}\delta(t) e^{-j\omega t}dt=\int_{−\infty}^{\infty}1\cdot e^{-j\omega t}dt=e^{-j\omega t}|_{t=0}=1}$$$$\mathrm{\therefore\:F[\delta(t)]=1\:\:or\:\:\delta(t) \overset{FT}{\leftrightarrow}1}$$That is, the Fourier transform of a unit impulse function is unity.The magnitude and phase representation of the Fourier transform of unit impulse function are as follows −$$\mathrm{Magnitude, |X(\omega)|=1;\:\:for\:all\:\omega}$$$$\mathrm{Phase, \angle X(\omega)=0;\:\:for\:all\:\omega}$$The graphical representation of the ... Read More