Found 189 Articles for Signals and Systems

Solution of Difference Equations Using Z-Transform

Manish Kumar Saini
Updated on 31-Jan-2022 12:09:05

15K+ Views

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, if $\mathrm{\mathit{x\left ( n \right )}}$ is a discrete time function, then its Z-transform is defined as, $$\mathrm{\mathit{Z\left [ x\left ( n \right ) \right ]\mathrm{\, =\, }X\left ( z \right )\mathrm{\, =\, }\sum_{n\mathrm{\, =\, }-\infty }^{\infty }x\left ( n \right )z^{-n}}}$$Solving Difference Equations by Z-TransformIn order to solve the difference equation, first it is converted into the algebraic equation by taking its Z-transform. Then, the solution of the equation is calculated in z-domain and ... Read More

Residue Method to Calculate Inverse Z-Transform

Manish Kumar Saini
Updated on 31-Jan-2022 11:01:48

4K+ Views

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, if $\mathrm{\mathit{x\left ( n \right )}}$ is a discrete time function, then its Z-transform is defined as, $$\mathrm{\mathit{Z\left [ x\left ( n \right ) \right ]\mathrm{\, =\, }X\left ( z \right )\mathrm{\, =\, }\sum_{n\mathrm{\, =\, }-\infty }^{\infty }x\left ( n \right )z^{-n}}}$$Inverse Z-Transform using Residue MethodThe residue method is also known as complex inversion integral method. As the Z-transform of a discrete-time signal $\mathrm{\mathit{x\left ( n \right )}}$ is defined as$$\mathrm{\mathit{Z\left [ x\left ( n ... Read More

Time Expansion Property of Z-Transform

Manish Kumar Saini
Updated on 31-Jan-2022 10:41:10

701 Views

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, if $\mathrm{\mathit{x\left ( n \right )}}$ is a discrete time function, then its Z-transform is defined as, $$\mathrm{\mathit{Z\left [ x\left ( n \right ) \right ]\mathrm{\, =\, }X\left ( z \right )\mathrm{\, =\, }\sum_{n\mathrm{\, =\, }-\infty }^{\infty }x\left ( n \right )z^{-n}}}$$Time Expansion Property of Z-TransformStatement – The time expansion property of Z-transform states that if$$\mathrm{\mathit{x\left ( n \right )\overset{ZT}{\leftrightarrow}X\left ( z \right );\; \; \; \mathrm{ROC}\to \mathit{R}}} $$Then$$\mathrm{\mathit{x_{m}\left ( n \right )\overset{ZT}{\leftrightarrow}X\left ( z^{m} ... Read More

Multiplication Property of Z-Transform

Manish Kumar Saini
Updated on 31-Jan-2022 10:24:45

2K+ Views

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, if $\mathrm{\mathit{x\left ( n \right )}}$ is a discrete time function, then its Z-transform is defined as, $$\mathrm{\mathit{Z\left [ x\left ( n \right ) \right ]\mathrm{\, =\, }X\left ( z \right )\mathrm{\, =\, }\sum_{n\mathrm{\, =\, }-\infty }^{\infty }x\left ( n \right )z^{-n}}}$$Multiplication Property of Z-TransformStatement – The multiplication property of Z-transform states that the multiplication of two signals in time domain corresponds to the complex convolution in z-domain. For this reason, the multiplication property is ... Read More

What is the Frequency Response of Discrete-Time Systems?

Manish Kumar Saini
Updated on 31-Jan-2022 05:22:41

4K+ Views

Frequency Response of Discrete-Time SystemsA spectrum of input sinusoids is applied to a linear time invariant discrete-time system to obtain the frequency response of the system. The frequency response of the discrete-time system gives the magnitude and phase response of the system to the input sinusoids at all frequencies.Now, let the impulse response of an LTI discrete-time system is $\mathit{h}\mathrm{\left(\mathit{n}\right)}$ and the input to the system is a complex exponential function, i.e., $\mathit{x}\mathrm{\left(\mathit{n}\right)}\:\mathrm{=}\:\mathit{e^{\mathit{j\omega n}}}$. Then, the output $\mathit{y}\mathrm{\left(\mathit{n}\right)}$ of the system is obtained by using the convolution theorem, i.e., $$\mathrm{\mathit{y}\mathrm{\left(\mathit{n}\right)}\:\mathrm{=}\:\mathit{h}\mathrm{\left(\mathit{n}\right)}*\mathit{x}\mathrm{\left(\mathit{n}\right)}\:\mathrm{=}\:\sum_{\mathit{k=-\infty} }^{\infty}\mathit{h}\mathrm{\left(\mathit{k}\right)}\mathit{x}\mathrm{\left(\mathit{n-k}\right)}}$$As the input to the system is $\mathit{x}\mathrm{\left(\mathit{n}\right)}\:\mathrm{=}\:\mathit{e^{\mathit{j\omega n}}}$ ,then, ... Read More

Time Reversal Property of Z-Transform

Manish Kumar Saini
Updated on 29-Jan-2022 08:20:32

3K+ Views

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, if $\mathrm{\mathit{x\left ( n \right )}}$ is a discrete time function, then its Z-transform is defined as, $$\mathrm{\mathit{Z\left [ x\left ( n \right ) \right ]\mathrm{\, =\, }X\left ( z \right )\mathrm{\, =\, }\sum_{n\mathrm{\, =\, }-\infty }^{\infty }x\left ( n \right )z^{-n}}}$$Time Reversal Property of Z-TransformStatement – The time reversal property of Z-transform states that the reversal or reflection of the sequence in time domain corresponds to the inversion in z-domain. Therefore, if$$\mathrm{\mathit{x\left ( n ... Read More

Time Shifting Property of Z-Transform

Manish Kumar Saini
Updated on 29-Jan-2022 08:14:46

8K+ Views

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, if $\mathrm{\mathit{x\left ( n \right )}}$ is a discrete time function, then its Z-transform is defined as, $$\mathrm{\mathit{Z\left [ x\left ( n \right ) \right ]=X\left ( z \right )=\sum_{n=-\infty }^{\infty }x\left ( n \right )z^{-n}}}$$Time Shifting Property of Z-TransformStatement – The time shifting property of Z-transform states that if the sequence $\mathrm{\mathit{x\left ( n \right )}}$ is shifted by n0 in time domain, then it results in the multiplication by $\mathrm{\mathit{z^{-n_{\mathrm{0}}}}}$ in the z-domain. ... Read More

Time Convolution and Frequency Convolution Properties of Discrete-Time Fourier Transform

Manish Kumar Saini
Updated on 29-Jan-2022 08:07:49

3K+ Views

Discrete-Time Fourier TransformThe Fourier transform of a discrete-time sequence is known as the discrete-time Fourier transform (DTFT). Mathematically, the discrete-time Fourier transform of a discrete-time sequence $\mathrm{\mathit{x\left ( n \right )}}$ is defined as −$$\mathrm{\mathit{F\left [ x\left ( n \right ) \right ]=X\left ( \omega \right )=\sum_{n=-\infty }^{\infty }x\left ( n \right )e^{-j\, \omega n}}}$$Time Convolution Property of DTFTStatement – The time convolution property of DTFT states that the discretetime Fourier transform of convolution of two sequences in time domain is equivalent to multiplication of their discrete-time Fourier transforms. Therefore, if$$\mathrm{\mathit{x_{\mathrm{1}}\left ( n \right )\overset{FT}{\leftrightarrow}X_{\mathrm{1}}\left ( \omega \right )\: \: ... Read More

Linearity, Periodicity and Symmetry Properties of Discrete-Time Fourier Transform

Manish Kumar Saini
Updated on 29-Jan-2022 08:01:09

4K+ Views

Discrete-Time Fourier TransformThe Fourier transform of a discrete-time sequence is known as the discrete-time Fourier transform (DTFT). Mathematically, the discrete-time Fourier transform of a discrete-time sequence $\mathrm{\mathit{x\left ( n \right )}}$ is defined as −$$\mathrm{\mathit{F\left [ x\left ( n \right ) \right ]\mathrm{\, =\, }X\left ( \omega \right )\mathrm{\, =\, }\sum_{n\mathrm{\, =\, }-\infty }^{\infty }x\left ( n \right )e^{-j\, \omega n}}}$$Linearity Property of Discrete-Time Fourier TransformStatement – The linearity property of discrete-time Fourier transform states that, the DTFT of a weighted sum of two discrete-time sequences is equal to the weighted sum of individual discrete-time Fourier transforms. Therefore, if$$\mathrm{\mathit{F\left [ ... Read More

Multiplication by Exponential Sequence Property of Z-Transform

Manish Kumar Saini
Updated on 29-Jan-2022 06:57:01

757 Views

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, if $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is a discrete time function, then its Z-transform is defined as, $$\mathrm{\mathit{Z}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty}}^{\infty}\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{z^{-\mathit{n}}}}$$Multiplication by Exponential Sequence Property of Z-TransformStatement - The exponential multiplication property of Z-transform states that the exponential sequence multiplied in time domain corresponds to the scaling in z-domain. The exponential multiplication property is also known as scaling in z-domain property of the Z-transform. Therefore, if$$\mathrm{\mathit{x}\mathrm{\left(\mathit{n}\right)}\overset{\mathit{ZT}}{\leftrightarrow}\mathit{X}\mathrm{\left(\mathit{z}\right)};\:\mathrm{ROC}\:\mathrm{=}\:\mathit{R}}$$Then, according to the exponential multiplication property, $$\mathrm{\mathit{a^{\mathit{n}}}\mathit{x}\mathrm{\left(\mathit{n}\right)}\overset{\mathit{ZT}}{\leftrightarrow}\mathit{X}\mathrm{\left(\mathit{\frac{\mathit{z}}{\mathit{a}}}\right)};\:\mathrm{ROC}\:\mathrm{=}\:\left| \mathit{a}\right|\mathit{R}}$$Where, a is a complex number.ProofFrom the definition of ... Read More

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