Found 189 Articles for Signals and Systems

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, $$\mathrm{\mathit{L\left [ x\left ( t \right ) \right ]\mathrm{\, =\, }X\left ( s \right )\mathrm{\, =\, }\int_{-\infty }^{\infty }x\left ( t \right )e^{-st}\:dt \; \; \cdot \cdot \cdot \left ( \mathrm{1} \right )}}$$Equation (1) gives the bilateral Laplace transform of the function $\mathrm{\mathit{x\left ( t \right )}}$. But for the causal signals, the ... 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 −$$\mathrm{\mathit{L\left [ x\left ( t \right ) \right ]\mathrm{\, =\, }X\left ( s \right )\mathrm{\, =\, }\int_{-\infty }^{\infty }x\left ( t \right )e^{-st}\:dt }}$$Time Reversal Property of Laplace TransformStatement – The time reversal property of Laplace transform states that if a signal is reversed about the vertical axis at origin in the time ... 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 −$$\mathrm{\mathit{L\left [ x\left ( t \right ) \right ]\mathrm{\, =\, }X\left ( s \right )\mathrm{\, =\, }\int_{-\infty }^{\infty }x\left ( t \right )e^{-st}\:dt\; \; \cdot \cdot \cdot\left ( \mathrm{1} \right ) }}$$Equation (1) gives the bilateral Laplace transform of the function $\mathrm{\mathit{x\left ( t \right )}}$. But for the causal signals, the unilateral ... Read More

The Inverse Z-TransformThe inverse Z-transform is defined as the process of finding the time domain signal $\mathrm{\mathit{x\left ( n \right )}}$ from its Z-transform $\mathrm{\mathit{X\left ( z \right )}}$. The inverse Z-transform is denoted as −$$\mathrm{\mathit{x\left ( n \right )\mathrm{\, =\, }Z^{-\mathrm{1}}\left [ X\left ( z \right ) \right ]}}$$Since the Z-transform is defined as, $$\mathrm{\mathit{X\left ( z \right )\mathrm{\, =\, }\sum_{n\mathrm{\, =\, }-\infty }^{\infty }x\left ( n \right )z^{-n}\; \; \; \cdot \cdot \cdot \left ( \mathrm{1} \right )}}$$Where, z is a complex variable and is given by, $$\mathrm{\mathit{z\mathrm{\, =\, }r\, e^{j\, \omega }}}$$Where, r is the radius of ... 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 sdomain.Mathematically, if $\mathit{x}\mathrm{\left(\mathit{t}\right)}$ is a time domain function, then its Laplace transform is defined as −$$\mathrm{\mathit{L}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{t}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{s}\right)}\:\mathrm{=}\:\int_{-\infty }^{\infty }\mathit{x}\mathrm{\left(\mathit{t}\right)}\mathit{e^{-\mathit{st}}\:\mathit{dt}}}$$Solution of Differential Equations Using Laplace TransformA linear time invariant (LTI) system is described by constant coefficient differential equations which are relating the input and output of the system. The response of the LTI system is obtained by solving these differential equations.The Laplace transformation technique can be used for solving the differential equation describing the ... Read More

Z-TransformThe Z-Transform is a mathematical tool which is used to convert the difference equations in time domain into the algebraic equations in the z-domain. Mathematically, the Z-transform of a discrete-time signal or a sequence $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is defined as −$$\mathrm{\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty }}^{\infty }\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{z}^{-\mathit{n}}}$$Properties of Z-TransformThe following table highlights some of the important properties of Z-Transform −PropertyTime-Domainz-DomainRegion of Convergence (ROC)Notation$\mathrm{\mathit{x}\mathrm{\left(\mathit{n}\right)}}$$\mathrm{\mathit{X}\mathrm{\left(\mathit{z}\right)}}$$\mathrm{\mathit{R}}$$\mathrm{\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{n}\right)}}$$\mathrm{\mathit{X}_{\mathrm{1}}\mathrm{\left(\mathit{z}\right)}}$$\mathrm{\mathit{R}_{\mathrm{1}}}$$\mathrm{\mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{n}\right)}}$$\mathrm{\mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{z}\right)}}$$\mathrm{\mathit{R}_{\mathrm{2}}}$Linearity and Superposition$\mathrm{\mathit{a}\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{n} \right)}\:\mathrm{+}\:\mathit{b}\mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{n} \right)}}$$\mathrm{\mathit{a}\mathit{X}_{\mathrm{1}}\mathrm{\left(\mathit{z} \right)}\:\mathrm{+}\:\mathit{b}\mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{z}\right)}}$$\mathrm{\mathit{R}_{\mathrm{1}}\:\cap \mathit{R}_{\mathrm{2}}}$Time-Shifting$\mathrm{\mathit{x}\mathrm{\left(\mathit{n-k}\right)}}$$\mathrm{\mathit{z}^{-\mathit{k}}\mathit{X}\mathrm{\left(\mathit{z}\right)}}$$\mathrm{\mathrm{same\:as\:}\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{except}\:\mathit{z}\:\mathrm{=}\:\mathrm{0}}$$\mathrm{\mathit{x}\mathrm{\left(\mathit{n\mathrm{+}\mathit{k}}\right)}}$$\mathrm{\mathit{z}^{\mathit{k}}\mathit{X}\mathrm{\left(\mathit{z}\right)}}$$\mathrm{\mathrm{same\:as\:}\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{except}\:\mathit{z}\:\mathrm{=}\:\mathrm{\infty}}$Scaling in zdomain$\mathrm{\mathit{a}^{\mathit{n}}\mathit{x}\mathrm{\left(\mathit{n}\right)}}$$\mathrm{\mathit{X}\mathrm{\left( \frac{\mathit{z}}{\mathit{a}}\right )}}$$\mathrm{\left|\mathit{a}\right|\mathit{R}_{\mathrm{1}}Read More

Laplace TransformThe Laplace transform is a mathematical tool which is used to convert the differential equations in time domain into the algebraic equations in the frequency domain or s-domain.Mathematically, the Laplace transform of a time-domain function $\mathit{x}\mathrm{\left(\mathit{t}\right)}$ is defined as −$$\mathrm{\mathit{L}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{t}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{s}\right)}\:\mathrm{=}\:\int_{\mathrm{0} }^{\mathrm{\infty} }\mathit{x}\mathrm{\left(\mathit{t}\right)}\mathit{e^{-\mathit{st}}\:\mathit{dt}}}$$Where, s is a complex variable and it is given by, $$\mathrm{\mathit{s}\:\mathrm{=}\:\sigma \:\mathrm{+}\:\mathit{j\omega}}$$And the operator L is called the Laplace transform operator which transforms the domain function $\mathit{x}\mathrm{\left(\mathit{t}\right)}$ into the frequency domain function X(s).Properties of Laplace TransformThe following table highlights some of the important properties of Laplace transform −PropertyFunction $\mathit{x}\mathrm{\left(\mathit{t}\right)}$Laplace Transform $\mathit{X}\mathrm{\left(\mathit{s}\right)}$Notation$\mathrm{\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{t}\right)}}$$\mathrm{\mathit{X}_{\mathrm{1}}\mathrm{\left(\mathit{s}\right)}}$$\mathrm{\mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{t}\right)}}$$\mathrm{\mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{s}\right)}}$Scalar Multiplication$\mathrm{\mathit{k}\mathit{x}\mathrm{\left(\mathit{t}\right)}}$$\mathrm{\mathit{k}\mathit{X}\mathrm{\left(\mathit{s}\right)}}$Linearity$\mathrm{\mathit{a}\mathit{x}_{\mathrm{1}}\mathrm{\left( \mathit{t}\right)}\:\mathrm{+}\:\mathit{b}\mathit{x}_{\mathrm{2}}\mathrm{\left( \mathit{t}\right)}}$$\mathrm{\mathit{a}\mathit{X}_{\mathrm{1}}\mathrm{\left( \mathit{s }\right)}\:\mathrm{+}\:\mathit{b}\mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{s}\right)}}$Time Shifting$\mathrm{\mathit{x}\mathrm{\left(\mathit{t-t_{\mathrm{0}}}\right)}}$$\mathrm{\mathit{e}^{- ... Read More

Discrete Time Fourier TransformThe discrete time Fourier transform is a mathematical tool which is used to convert a discrete time sequence into the frequency domain. Therefore, the Fourier transform of a discrete time signal or sequence is called the discrete time Fourier transform (DTFT).Mathematically, if $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is a discrete time sequence, then the discrete time Fourier transform of the sequence is defined as −$$\mathrm{\mathit{F}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{\omega }\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty }}^{\infty }\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{e}^{-\mathit{j\omega n}}}$$Properties of Discrete-Time Fourier TransformFollowing table gives the important properties of the discrete-time Fourier transform −PropertyDiscrete-Time SequenceDTFTNotation$\mathrm{\mathit{x}\mathrm{\left(\mathit{n}\right)}}$$\mathrm{\mathit{X}\mathrm{\left(\mathit{\omega}\right)}}$$\mathrm{\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{n}\right)}}$$\mathrm{\mathit{X}_{\mathrm{1}}\mathrm{\left(\mathit{\omega}\right)}}$$\mathrm{\mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{n}\right)}}$$\mathrm{\mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{\omega}\right)}}$Linearity$\mathrm{\mathit{a}\mathit{x}_{\mathrm{1}}\mathrm{\left( \mathit{n}\right)}\:\mathrm{+}\:\mathit{b}\mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{n}\right)}}$$\mathrm{\mathit{a}\mathit{X}_{\mathrm{1}}\mathrm{\left( \mathit{\omega }\right)}\:\mathrm{+}\:\mathit{b}\mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{\omega}\right)}}$Time Shifting$\mathrm{\mathit{x}\mathrm{\left(\mathit{n-k}\right)}}$$\mathrm{\mathit{e}^{\mathit{-j\omega k}}\mathit{X}\mathrm{\left(\mathit{\omega }\right)}}$Frequency Shifting$\mathrm{\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{e}^{\mathit{j\omega} _{\mathrm{0}}\mathit{n}}}$$\mathrm{\mathit{X}\mathrm{\left(\mathit{\omega -\omega _{\mathrm{0}}}\right)}}$Time Reversal$\mathrm{\mathit{x}\mathrm{\left(\mathit{-n}\right)}}$$\mathrm{\mathit{X}\mathrm{\left(\mathit{-\omega}\right)}}$Frequency Differentiation$\mathrm{\mathit{n}\mathit{x}\mathrm{\left(\mathit{n}\right)}}$$\mathrm{\mathit{j}\frac{\mathit{d}}{\mathit{d\omega}}\mathit{X}\mathrm{\left(\mathit{\omega }\right)}}$Time Convolution$\mathrm{\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{n}\right)}\:*\:\mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{n}\right)}}$$\mathrm{\mathit{X}_{\mathrm{1}}\mathrm{\left(\mathit{\omega }\right)}\mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{\omega }\right)}}$Frequency ... Read More

Inverse Z-TransformThe inverse Z-transform is defined as the process of finding the time domain signal $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ from its Z-transform $\mathit{X}\mathrm{\left(\mathit{z}\right)}$. The inverse Z-transform is denoted as −$$\mathrm{\mathit{x}\mathrm{\left(\mathit{n}\right)}\:\mathrm{=}\:\mathit{Z}^{-\mathrm{1}}\mathrm{\left[\mathit{X}\mathrm{\left(\mathit{z}\right)}\right]}}$$Partial Fraction Expansion Method to Find Inverse Z-TransformIn order to determine the inverse Z-transform of $\mathit{X}\mathrm{\left(\mathit{z}\right)}$ using partial fraction expansion method, the denominator of $\mathit{X}\mathrm{\left(\mathit{z}\right)}$ must be in factored form. In this method, we obtained the partial fraction expansion of $\frac{\mathit{X}\mathrm{\left(\mathit{z}\right)}}{\mathit{z}}$ instead of $\mathit{X}\mathrm{\left(\mathit{z}\right)}$. This is because the Z-transform of time-domain sequences have Z in their numerators.The partial fraction expansion method is applied only if $\frac{\mathit{X}\mathrm{\left(\mathit{z}\right)}}{\mathit{z}}$ is a proper rational function, i.e., the order ... Read More

Average PowerThe average power of a signal is defined as the mean power dissipated by the signal such as voltage or current in a unit resistance over a period. Mathematically, the average power is given by, $$\mathit{P}\:\mathrm{=}\:\lim_{T \rightarrow \infty}\frac{1}{\mathit{T}}\int_{\mathrm{-(\mathit{T}/\mathrm{2})}}^{\mathrm{(\mathit{T}/\mathrm{2})}}|\mathit{x}\mathrm{(\mathit{t})}|^\mathrm{2}\:\mathit{dt}$$Parseval's Power TheoremStatement − Parseval's power theorem states that the power of a signal is equal to the sum of square of the magnitudes of various harmonic components present in the discrete spectrum.Mathematically, the Parseval's power theorem is defined as −$$\mathit{P}\:\mathrm{=}\:\displaystyle\sum\limits_{n=-\infty}^\infty |\mathit{C}_\mathit{n}|^2$$ProofConsider a function $\mathit{x}\mathrm{(\mathit{t})}$. Then, the average power of the signal $\mathit{x}\mathrm{(\mathit{t})}$ over one complete cycle is given by, $$\mathit{P}\:\mathrm{=}\:\frac{1}{\mathit{T}}\int_{\mathrm{-(\mathit{T}/\mathrm{2})}}^{\mathrm{(\mathit{T}/\mathrm{2})}}|\mathit{x}\mathrm{(\mathit{t})}|^\mathrm{2}\:\mathit{dt}$$ $$\because|\mathit{x}\mathrm{(\mathit{t})}|^\mathrm{2}\:\mathrm{=}\: ... Read More