Z-TransformThe Z-transform is a mathematical tool which is used to convert the difference equations in discrete time domain into the algebraic equations in z-domain.Mathematically, $\mathrm{\mathit{x\left ( n \right )}}$ if is a discrete time function, then its Z-transform is defined as, $$\mathrm{\mathit{Z\left [ x\left ( n \right ) \right ]\mathrm{\, ... Read More

Realization of Continuous-Time SystemRealisation of a continuous-time LTI system means obtaining a network corresponding to the differential equation or transfer function of the system.The transfer function of the system can be realised either by using integrators or differentiators. Due to certain drawbacks, the differentiators are not used to realise the ... Read More

Stability and CausalityThe necessary and sufficient condition for a causal linear time invariant (LTI) discrete-time system to be BIBO stable is given by, $$\mathrm{\mathit{\sum_{n=\mathrm{0}}^{\infty }\left|h\left ( n \right ) \right|< \infty }}$$Therefore, if the impulse response of an LTI discrete-time system is absolutely summable, then the system is BIBO stable.Also, ... Read More

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 $\mathrm{\mathit{x\left ( n \right )}}$ is a discrete-time signal or sequence, then its bilateral or two-sided Z-transform is defined as −$$\mathrm{\mathit{Z\left [ x\left ( n ... Read More

The 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 $\mathrm{\mathit{x\left ( n \right )}}$ is a discrete-time signal or sequence, then its bilateral or two-sided Z-transform is defined as −$$\mathrm{\mathit{Z\left [ x\left ( n ... Read More

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 $\mathrm{\mathit{x\left ( n \right )}}$ is a discrete-time signal or sequence, then its bilateral or two-sided Z-transform is defined as −$$\mathrm{\mathit{Z\left [ x\left ( n ... Read More

The sequences having a finite number of samples are called the finite duration sequences. The finite duration sequences may be of following three types viz. −Right-Hand SequencesLeft-Hand SequencesTwo-Sided SequencesRight-Hand SequenceA sequence for which $\mathrm{\mathit{x\left ( n \right )}}$ = 0 for $\mathit{n}$ < $\mathit{n_{\mathrm{0}}}$ where $\mathit{n_{\mathrm{0}}}$ may be positive or ... Read More

Fourier TransformThe Fourier transform is a transformation technique which is used to transform the signals from continuous-time domain to the corresponding frequency domain.Mathematically, if $\mathit{x}\mathrm{\left(\mathit{t}\right)}$ is a continuous-time domain function, then its Fourier transform is given by, $$\mathrm{\mathit{F}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{t}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{\omega }\right)}\:\mathrm{=}\:\int_{-\infty }^{\infty }\mathit{x}\mathrm{\left(\mathit{t}\right)}\mathit{e^{-\mathit{j\omega t}}\:\mathit{dt}} \:\:\:\:\:\:...(1)}$$Laplace TransformThe Laplace transform is a mathematical tool ... Read More

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 −$$\mathrm{\mathit{Z}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\sum_{\mathit{n=-\infty }}^{\infty}\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{z^{-\mathit{n}}}\:\:\:\:\:\:...(1)}$$Where, z is a complex variable.Also, the unilateral ... 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 −$$\mathrm{\mathit{L}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{t}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{s}\right)}\:\mathrm{=}\:\int_{-\infty }^{\infty }\mathit{x}\mathrm{\left(\mathit{t}\right)}\mathit{e^{-\mathit{st}}\:\mathit{dt}}\:\:\:\:\:\:...(1)}$$Equation (1) gives the bilateral ... Read More