Found 189 Articles for Signals and Systems

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 −$$\mathrm{\mathit{Z\left [ x\left ( n \right ) \right ]\mathrm{\, =\, }X\left ( z \right )\mathrm{\, =\, }\sum_{n\mathrm{\, =\, }-\infty }^{\infty }x\left ( n \right )z^{-n}}}$$Where, z is a complex variable.Also, the unilateral or one-sided z-transform is defined as −$$\mathrm{\mathit{Z\left [ x\left ( n \right ) \right ]\mathrm{\, =\, }X\left ( z \right )\mathrm{\, =\, }\sum_{n\mathrm{\, ... 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 −$$\mathrm{\mathit{Z\left [ x\left ( n \right ) \right ]\mathrm{\, =\, }X\left ( z \right )\mathrm{\, =\, }\sum_{n\mathrm{\, =\, }-\infty }^{\infty }x\left ( n \right )z^{-n}}}$$Where, z is a complex variable.Also, the unilateral or one-sided z-transform is defined as −$$\mathrm{\mathit{Z\left [ x\left ( n \right ) \right ]\mathrm{\, =\, }X\left ( z \right )\mathrm{\, =\, }\sum_{n\mathrm{\, ... Read More

What is Z-Transform?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.The Z-transform is a very useful tool in the analysis of a linear shift invariant (LSI) system. An LSI discrete time system is represented by difference equations. To solve these difference equations which are in time domain, they are converted first into algebraic equations in z-domain using the Z-transform, then the algebraic equations are manipulated in z-domain and the result obtained is converted back into time domain using the inverse Z-transform.The Z-transform may be of ... 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 negative but finite, is called the right hand sequence. If $\mathit{n_{\mathrm{0}}}$ ≥ 0, the resulting sequence is a causal sequence. The ROC of a causal sequence is the entire z-plane except at 𝑧 = 0.Numerical Example (1)Find the ROC and Z-Transform of the causal sequence.$$\mathrm{\mathit{x\left ( n \right )}\mathrm{\, =\, ... 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 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 −$$\mathrm{\mathit{L}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{t}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{s}\right)}\:\mathrm{=}\:\int_{-\infty }^{\infty }\mathit{x}\mathrm{\left(\mathit{t}\right)}\mathit{e^{-\mathit{st}}\:\mathit{dt}}\:\:\:\:\:\:...(2)}$$Where, s is a complex variable and it is given by, $$\mathrm{\mathit{s}\:\mathrm{=}\:\sigma \:\mathrm{+}\:\mathit{j\omega}}$$Relation ... 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 or one-sided z-transform is defined as −$$\mathrm{\mathit{Z}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\sum_{\mathit{n=\mathrm{0} }}^{\infty}\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{z^{-\mathit{n}}}\:\:\:\:\:\:...(2)}$$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 ... 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 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, $$\mathrm{\mathit{L}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{t}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{s}\right)}\:\mathrm{=}\:\int_{\mathrm{0}}^{\infty }\mathit{x}\mathrm{\left(\mathit{t}\right)}\mathit{e^{-\mathit{st}}\:\mathit{dt}}\:\:\:\:\:\:...(2)}$$Time Shifting Property of Laplace TransformStatement - The time shifting property of Laplace transform states that a shift of t0 in time domain corresponds to the multiplication by ... 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 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 −$$\mathrm{\mathit{L}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{t}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{s}\right)}\:\mathrm{=}\:\int_{\mathrm{0}}^{\infty }\mathit{x}\mathrm{\left(\mathit{t}\right)}\mathit{e^{-\mathit{st}}\:\mathit{dt}}\:\:\:\:\:\:...(2)}$$Time Scaling Property of Laplace TransformStatement - The time scaling property of Laplace transform states that if, $$\mathrm{\mathit{x}\mathrm{\left(\mathit{t}\right)}\overset{\mathit{LT}}{\leftrightarrow}\mathit{X}\mathrm{\left(\mathit{s}\right)}}$$Then$$\mathrm{\mathit{x}\mathrm{\left(\mathit{at}\right)}\overset{\mathit{LT}}{\leftrightarrow}\frac{1}{\left|\mathit{a}\right|}\mathit{X}\mathrm{\left( \frac{\mathit{s}}{\mathit{a}}\right )}}$$ProofFrom the definition of Laplace transform, we have, $$\mathrm{\mathit{L}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{t}\right)}\right]}\:\mathrm{=}\:\int_{\mathrm{0}}^{\infty ... 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 −$$\mathrm{\mathit{L}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{t}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{s}\right)}\:\mathrm{=}\:\int_{-\infty }^{\infty }\mathit{x}\mathrm{\left(\mathit{t}\right)}\mathit{e^{-\mathit{st}}\:\mathit{dt}}\:\:\:\:\:\:...(1)}$$Integration in Time Domain Property of Laplace TransformStatement - The time integration property of Laplace transform states that if$$\mathrm{\mathit{x}\mathrm{\left(\mathit{t}\right)}\overset{\mathit{LT}}{\leftrightarrow}\mathit{X}\mathrm{\left(\mathit{s}\right)}}$$Then$$\mathrm{\int_{-\infty}^{\mathit{t}}\mathit{x}\mathrm{\left(\mathit{\tau }\right)}\mathit{d\tau}\overset{\mathit{LT}}{\leftrightarrow}\frac{\mathit{x}\mathrm{\left(\mathit{s}\right)}}{\mathit{s}}\:\mathrm{+}\:\int_{-\infty}^{\mathrm{0}}\frac{\mathit{x}\mathrm{\left(\mathit{\tau }\right)}}{\mathit{s}}\:\mathit{d\tau}}$$ProofConsider a function $\mathit{y}\mathrm{\left(\mathit{t}\right)}$ as, $$\mathrm{\mathit{y}\mathrm{\left(\mathit{t}\right)}\:\mathrm{=}\:\int_{-\infty }^{\mathit{t}}\mathit{x}\mathrm{\left(\mathit{\tau }\right)}\:\mathit{d\tau}}$$Taking differentiation on both sides with respect to time, we have, $$\mathrm{\frac{\mathit{d\mathit{y}\mathrm{\left(\mathit{t}\right)}}}{\mathit{dt}}\:\mathrm{=}\:\mathit{x}\mathrm{\left(\mathit{t}\right)}\:\:\:\:\:\:...(2)}$$Also, $$\mathrm{\mathit{y}\mathrm{\left(\mathrm{0}^{-}\right)}\:\mathrm{=}\:\int_{-\infty }^{\mathrm{0}}\mathit{x}\mathrm{\left(\mathit{\tau }\right)}\:\mathit{d\tau}\:\:\:\:\:\:...(3)}$$Taking the Laplace transform of equation (2), we get, $$\mathrm{\mathit{L}\mathrm{\left[ \frac{\mathit{d\mathit{y}\mathrm{\left(\mathit{t}\right)}}}{\mathit{dt}}\right ]}\:\mathrm{=}\:\mathit{L}\mathrm{\left [ \mathit{x}\mathrm{\left(\mathit{t}\right)} ... Read More

Z-TransformZ-transform is a mathematical tool which is used to convert the difference equations in time domain into the algebraic equations in the frequency domain.Mathematically, if $\mathrm{\mathit{x\left ( n \right )}}$ is a discrete-time sequence, then its 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. The z-transform defined in eq. (1) is called bilateral or two-sided z-transform.The unilateral or one-sided z-transform is defined as −$$\mathrm{\mathit{X\left ( z \right )\mathrm{\, =\, }\sum_{n\mathrm{\, =\, }\mathrm{0}}^{\infty }x\left ( ... Read More