Z-TransformThe Z-transform (ZT) is a mathematical tool which is used to convert the difference equations in time domain into the algebraic equations in z-domain.Mathematically, if $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is a discrete-time signal or sequence, then its bilateral or two-sided Z-transform is defined as −$$\mathrm{\mathit{Z}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty }}^{\infty }\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{z}^{-\mathit{n}}}$$Where, z is a complex variable.Region of Convergence (ROC) of Z-TransformThe set of points in z-plane for which the Z-transform of a discrete-time sequence $\mathit{x}\mathrm{\left(\mathit{n}\right)}$, i.e., $\mathit{X}\mathrm{\left(\mathit{z}\right)}$ converges is called the region of convergence (ROC) of $\mathit{X}\mathrm{\left(\mathit{z}\right)}$.Properties of ROC of Z-TransformThe region of convergence (ROC) of Z-transform has the following properties −The ROC of the Z-transform is ... 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 $\mathit{x}\mathrm{\left(\mathit{t}\right)}$ is a time domain function, then its Laplace transform is defined as −$$\mathrm{\mathit{L}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{t}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{s}\right)}\:\mathrm{=}\:\int_{-\infty}^{\infty}\mathit{x}\mathrm{\left(\mathit{t}\right)}\mathit{e^{-st}}\:\mathit{dt}}$$Inverse Laplace TransformThe inverse Laplace transform is the method for obtaining the time domain function from its Laplace transform and mathematically, it is defined as −$$\mathrm{\mathit{L}^{-\mathrm{1}}\mathrm{\left[\mathit{X}\mathrm{\left(\mathit{s}\right)}\right]}\:\mathrm{=}\:\mathit{x}\mathrm{\left(\mathit{t}\right)}\:\mathrm{=}\:\frac{1}{2\pi \mathit{j}}\int_{\mathrm{\left ( \sigma -\mathit{j\infty} \right )}}^{\mathrm{\left (\mathit{\sigma \mathrm{+}\mathit{j}\infty}\right )}}\mathit{X}\mathrm{\left(\mathit{s}\right)}\mathit{e^{st}}\:\mathit{ds}}$$Parseval’s Theorem for Laplace TransformStatement - The Parseval’s theorem or Parseval’s relation for Laplace transform states that if, $$\mathrm{\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{t}\right)}\overset{\mathit{LT}}{\leftrightarrow}\mathit{X}_{\mathrm{1}}\mathrm{\left(\mathit{s}\right)}\:\mathrm{and}\:\mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{t}\right)}\overset{\mathit{LT}}{\leftrightarrow}\mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{s}\right)}}$$Where, $\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{t}\right)}$ and $\mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{t}\right)}$ are ... Read More

Data ReconstructionThe data reconstruction is defined as the process of obtaining the analog signal $\mathrm{\mathit{x\left ( t \right )}}$ from the sampled signal $\mathrm{\mathit{x_{s}\left ( t \right )}}$. The data reconstruction is also known as interpolation.The sampled signal is given by, $$\mathrm{\mathit{x_{s}\left ( t \right )\mathrm{=}x\left ( t \right )\sum_{n\mathrm{=}-\infty }^{\infty }\delta \left ( t-nT \right )}}$$$$\mathrm{\Rightarrow \mathit{x_{s}\left ( t \right )\mathrm{=}\sum_{n\mathrm{=}-\infty }^{\infty }x\left ( nT \right )\delta \left ( t-nT \right )}}$$Where, $\mathrm{\mathit{\delta \left ( t-nT \right )}}$ is zero except at the instants $\mathrm{\mathit{t\mathrm{=}nT}}$. A reconstruction filter which is assumed to be linear and time invariant has unit ... 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 −$$\mathrm{\mathit{L\left [ x\left ( t \right ) \right ]\mathrm{\mathrm{=}}X\left ( s \right )\mathrm{\mathrm{=}}\int_{-\infty }^{\infty }x\left ( t \right )e^{-st}\:dt}}$$Laplace Transform of Full-Wave Rectified Sine Wave FunctionThe full-wave rectified sine wave function is shown in Figure-1 and is given by, $$\mathrm{\mathit{x\left ( t \right )=\mathrm{sin}\: \omega t;\; \; \mathrm{for\: 0}< \mathit{t}< \frac{\pi }{\omega }}}$$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 −$$\mathrm{\mathit{L\left [ x\left ( t \right ) \right ]\mathrm{\mathrm{=}}X\left ( s \right )\mathrm{\mathrm{=}}\int_{-\infty }^{\infty }x\left ( t \right )e^{-st}\; dt\; \; \; \cdot \cdot \cdot \left ( \mathrm{1} \right )}}$$Also, the inverse Laplace transform of the function is defined as, $$\mathrm{\mathit{L^{-\mathrm{1}}\left [X\left ( s \right ) \right ]\mathrm{\mathrm{=}}x\left ( t \right ) \mathrm{\mathrm{=}}\int_{\sigma ... Read More

An electric circuit consisting of a resistance (R) and a capacitor (C), connected in series, is shown in Figure-1. Consider the switch (S) is closed at $\mathrm{\mathit{t=\mathrm{0}}}$.Step Response of Series RC Circuit Using Laplace TransformTo obtain the step response of the series RC circuit, the applied input is given by, $$\mathrm{\mathit{x\left ( t \right )\mathrm{=}Vu\left ( t \right )}}$$By applying KVL to the circuit, the following equation describing the series RC circuit is obtained −$$\mathrm{\mathit{Vu\left ( t \right )\mathrm{=}Ri\left ( t \right )\mathrm{\: +\: }\frac{\mathrm{1}}{C}\int_{-\infty }^{t}i\left ( t \right )dt}}$$This equation can be written as, $$\mathrm{\mathit{Vu\left ( t \right )\mathrm{=}Ri\left ... Read More

An electric circuit consisting of a resistance (R) and an inductor (L), connected in series, is shown in Figure-1. Consider the switch (S) is closed at time $\mathrm{\mathit{ t=\mathrm{0}}}$.Step Response of Series RL CircuitTo obtain the step response of the series RL circuit, the input $\mathrm{\mathit{x\left ( t \right )}}$ applied to the circuit is given by, $$\mathrm{\mathit{x\left ( t \right )\mathrm{=}Vu\left ( t \right )}}$$Now by applying KVL in the loop, we obtain the following differential equation, $$\mathrm{\mathit{Vu\left ( t \right )\mathrm{=}Ri\left ( t \right )\mathrm{+}L\frac{di\left ( t \right )}{dt}}}$$Taking the Laplace transform on both sides, we get, $$\mathrm{\mathit{\frac{V}{s}\mathrm{=}RI\left ... 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 $\mathit{x}\mathrm{\left(\mathit{t}\right)}$ is a time-domain function, then its Laplace transform is defined as −$$\mathrm{\mathit{L}\mathrm{\left[ \mathit{x}\mathrm{\left(\mathit{t}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{s}\right)}\:\mathrm{=}\:\int_{-\infty}^{\infty}\mathit{x}\mathrm{\left(\mathit{t}\right)}\mathit{e^{-st}}\:\mathit{dt}\:\:\:\:\:\:...(1)}$$Equation (1) gives the bilateral Laplace transform of the function $\mathit{x}\mathrm{\left(\mathit{t}\right)}$. But for the causal signals, the unilateral Laplace transform is applied, which is defined as, $$\mathrm{\mathit{L}\mathrm{\left[ \mathit{x}\mathrm{\left(\mathit{t}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{s}\right)}\:\mathrm{=}\:\int_{\mathrm{0}}^{\infty }\mathit{x}\mathrm{\left(\mathit{t}\right)}\mathit{e^{-st}}\:\mathit{dt}\:\:\:\:\:\:...(2)}$$Laplace Transform of Impulse FunctionThe impulse function is defined as, $$\mathrm{\mathit{\delta}\mathrm{\left(\mathit{t}\right)}\mathrm{=}\begin{cases} 1& \text{ for } t= 0 \ 0 & \text{ for } teq 0 \end{cases}}$$Thus, from the definition ... 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 $\mathit{x}\mathrm{\left(\mathit{t}\right)}$ is a time domain function, then its Laplace transform is defined as −$$\mathrm{\mathit{L}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{t}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{s}\right)}\:\mathrm{=}\:\int_{-\infty}^{\infty}\mathit{x}\mathrm{\left(\mathit{t}\right)}\mathit{e^{-st}}\:\mathit{dt}\:\:\:\:\:\:...(1)}$$Equation (1) gives the bilateral Laplace transform of the function $\mathit{x}\mathrm{\left(\mathit{t}\right)}$. But for the causal signals, the unilateral Laplace transform is applied, which is defined as, $$\mathrm{\mathit{L}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{t}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{s}\right)}\:\mathrm{=}\:\int_{\mathrm{0}}^{\infty}\mathit{x}\mathrm{\left(\mathit{t}\right)}\mathit{e^{-st}}\:\mathit{dt}\:\:\:\:\:\:...(2)}$$Laplace Transform of Damped Sine FunctionThe Damped Sine Function is given by, $$\mathrm{\mathit{x}\mathrm{\left(\mathit{t}\right)}\:\mathrm{=}\:\mathit{e^{-at}}\:\mathrm{sin}\:\mathit{\omega t\:\mathit{u}\mathrm{\left( \mathit{t}\right)}}\:\mathrm{=}\:\mathit{e^{-at}}\mathrm{\left( \frac{\mathit{e^{j\omega t}-e^{-j\omega t}}}{2\mathit{j}} \right )}\mathit{u}\mathrm{\left(\mathit{t}\right )}}$$Now, from the definition of the Laplace transform, we get, ... Read More

What is Region of Convergence?Region of Convergence (ROC) is defined as the set of points in s-plane for which the Laplace transform of a function $\mathit{x}\mathrm{\left(\mathit{t}\right)}$ converges. In other words, the range of $\mathit{Re}\mathrm{\left(\mathit{s} \right)}$ (i.e., σ) for which the function $\mathit{X}\mathrm{\left(\mathit{s}\right)}$ converges is called the region of convergence.ROC of Right-Sided SignalsA signal $\mathit{x}\mathrm{\left(\mathit{t}\right)}$ is said to be a right-sided signal if the signal $\mathit{x}\mathrm{\left(\mathit{t}\right)}$ = 0 for t < $\mathit{T}_{\mathrm{1}}$ for some finite time $\mathit{T}_{\mathrm{1}}$ as shown in Figure-1.For a right-sided signal $\mathit{x}\mathrm{\left(\mathit{t}\right)}$, the ROC of the Laplace transform $\mathit{X}\mathrm{\left(\mathit{s}\right)}$ is $\mathit{Re}\mathrm{\left(\mathit{s} \right )}>\mathrm{\sigma _{\mathrm{1}}}$, where $\mathrm{\sigma _{\mathrm{1}}}$ is ... Read More