Found 189 Articles for Signals and Systems

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, $$\mathrm{\mathit{Z}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty}}^{\infty}\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{z^{-\mathit{n}}}}$$Convolution Method to Find Inverse Z-TransformThe inverse Z-transform can be calculated using the convolution theorem. In the convolution integration method, the given Z-transform X(z) is first split into $\mathit{X}_{\mathrm{1}}\mathrm{\left(\mathit{z}\right)}$ and $\mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{z}\right)}$ such that $\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\:\mathit{X}_{\mathrm{1}}\mathrm{\left(\mathit{z}\right)}\mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{z}\right)}$.The signals $\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{n}\right)}$ and $\mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{n}\right)}$ are then obtained by taking the inverse Z-transform of $\mathit{X}_{\mathrm{1}}\mathrm{\left(\mathit{z}\right)}$ and $\mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{z}\right)}$ respectively. Finally, the function $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is obtained by performing the convolution of ... Read More

The inverse discrete-time Fourier transform (IDTFT) is defined as the process of finding the discrete-time sequence $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ from its frequency response X(ω).Mathematically, the inverse discrete-time Fourier transform is defined as −$$\mathrm{\mathit{x}\mathrm{\left(\mathit{n}\right)}\:\mathrm{=}\: \frac{1}{2\pi}\int_{-\pi}^{\pi}\mathit{X}\mathrm{\left(\mathit{\omega}\right)}\mathit{e}^{\mathit{j\omega n}}\:\mathit{d\omega}\:\:\:\:\:\:...(1)}$$The solution of the equation (1) for $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is useful for the analytical purpose, but it is very difficult to evaluate for typical functional forms of function X(ω). Therefore, an alternate method of determining the values of the discrete-time sequence $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ follows directly from the definition of the Fourier transform, i.e., $$\mathrm{\mathit{X}\mathrm{\left(\mathit{\omega}\right)}\:\mathrm{=}\:\sum_{n=-\infty }^{\infty}\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{e}^{-\mathit{j\omega n}}\:\mathrm{=}\:...\:\mathrm{+}\:\mathit{x}\mathrm{\left(\mathrm{-3}\right)}\mathit{e}^{\mathit{j}\mathrm{3}\omega}\:\mathrm{+}\:\mathit{x}\mathrm{\left(\mathrm{-2}\right)}\mathit{e}^{\mathit{j}\mathrm{2}\omega}\:\mathrm{+}\:\mathit{x}\mathrm{\left(\mathrm{-1}\right)}\mathit{e}^{\mathit{j}\omega}\:\mathrm{+}\:\mathit{x}\mathrm{\left(\mathrm{0}\right)}\:\mathrm{+}\:\mathit{x}\mathrm{\left(\mathrm{1}\right)}\mathit{e}^{\mathit{-j}\omega}\:\mathrm{+}\:\mathit{x}\mathrm{\left(\mathrm{2}\right)}\mathit{e}^{\mathit{-j}2\omega}\:\mathrm{+}\:\mathit{x}\mathrm{\left(\mathrm{3}\right)}\mathit{e}^{\mathit{-j}3\omega}\:\:\:\:\:\:...(2)}$$Hence, from the equation of X(ω) we can say that, if X(ω) can be expressed ... Read More

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, $$\mathrm{\mathit{Z}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty}}^{\infty}\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{z^{-\mathit{n}}}}$$Final Value Theorem of Z-TransformThe final value theorem of Z-transform enables us to calculate the steady state value of a sequence $\mathit{x}\mathrm{\left(\mathit{n}\right)}$, i.e., $\mathit{x}\mathrm{\left(\mathit{\infty}\right)}$ directly from its Z-transform, without the need for finding its inverse Z-transform.Statement - If $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is a causal sequence, then the final value theorem of Z-transform states that if, $$\mathrm{\mathit{x}\mathrm{\left(\mathit{n}\right)}\overset{\mathit{ZT}}{\leftrightarrow}\mathit{X}\mathrm{\left(\mathit{z}\right)}}$$And if the Z-transform X(z) has no poles outside ... Read More

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, $$\mathrm{\mathit{Z}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty}}^{\infty}\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{z^{-\mathit{n}}}}$$Initial Value Theorem of Z-TransformThe initial value theorem enables us to calculate the initial value of a signal $\mathit{x}\mathrm{\left(\mathit{n}\right)}$, i.e., $\mathit{x}\mathrm{\left(\mathrm{0}\right)}$ directly from its Z-transform X(z) without the need for finding the inverse Z-transform of X(z).Statement - The initial value theorem of Z-transform states that if$$\mathrm{\mathit{x}\mathrm{\left(\mathit{n}\right)}\overset{\mathit{ZT}}{\leftrightarrow}\mathit{X}\mathrm{\left(\mathit{z}\right)}}$$Where, $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is a causal sequence. Then, $$\mathrm{\mathit{x}\mathrm{\left(\mathrm{0}\right)}\:\mathrm{=}\:\displaystyle \lim_{\mathit{n} \to 0}\mathit{x}\mathrm{\left(\mathit{n}\right)}\:\mathrm{=}\:\displaystyle \lim_{\mathit{z} \to \infty }\mathit{X}\mathrm{\left(\mathit{z}\right)}}$$ProofFrom the definition ... Read More

Discrete-Time Fourier TransformA discrete-time signal can be represented in the frequency domain using discrete-time Fourier transform. Therefore, the Fourier transform of a discretetime sequence is called the discrete-time Fourier transform (DTFT).Mathematically, if $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is a discrete-time sequence, then its discrete-time Fourier transform is defined as −$$\mathrm{\mathit{F}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n }\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{\omega}\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty}}^{\infty}\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{e^{-\mathit{j\omega n}}}}$$The discrete-time Fourier transform X(ω) of a discrete-time sequence $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ represents the frequency content of the sequence $\mathit{x}\mathrm{\left(\mathit{n}\right)}$. Therefore, by taking the Fourier transform of the discrete-time sequence, the sequence is decomposed into its frequency components. For this reason, the DTFT X(ω) is also called the signal spectrum.Condition for Existence of Discrete-Time Fourier ... Read More

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 (DTFT) of a discrete-time sequence $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is defined as −$$\mathrm{\mathit{F}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n }\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{\omega}\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty}}^{\infty}\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{e^{-\mathit{j\omega n}}}}$$Time Shifting Property of Discrete-Time Fourier TransformStatement - The time-shifting property of discrete-time Fourier transform states that if a signal $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is shifted by k in time domain, then its DTFT is multiplied by $\mathit{e^{-j\omega k }}$. Therefore, if$$\mathrm{\mathit{x}\mathrm{\left(\mathit{n}\right)}\overset{\mathit{FT}}{\leftrightarrow}\mathit{X}\mathrm{\left(\mathit{\omega }\right)}}$$Then$$\mathrm{\mathit{x}\mathrm{\left(\mathit{n-k}\right)}\overset{\mathit{FT}}{\leftrightarrow}\mathit{e^{-j\omega k }}\mathit{X}\mathrm{\left(\mathit{\omega }\right)}}$$Where, k is an integer.ProofFrom the definition of discrete-time Fourier transform, we have, $$\mathrm{\mathit{F}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{\omega }\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty}}^{\infty}\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{e^{-\mathit{j\omega n}}}}$$$$\mathrm{\therefore\mathit{F}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n-k}\right)}\right]}\:\mathrm{=}\:\sum_{\mathit{n=-\infty}}^{\infty}\mathit{x}\mathrm{\left(\mathit{n-k}\right)}\mathit{e^{-\mathit{j\omega n}}}}$$Substituting $\mathrm{\left(\mathit{n-k}\right)}\:\mathrm{=}\:\mathit{m}$ then $\mathit{n}\:\mathrm{=}\:\mathrm{\left(\mathit{m\mathrm{+}k}\right)}$ in the above summation, we get, ... Read More

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, $$\mathrm{\mathit{Z}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty}}^{\infty}\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{z^{-\mathit{n}}}}$$Differentiation in z-Domain Property of Z-TransformStatement - The differentiation in z-domain property of Z-transform states that the multiplication by n in time domain corresponds to the differentiation in zdomain. This property is also called the multiplication by n property of Ztransform. 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 differentiation in z-domain property, $$\mathrm{\mathit{n}\mathit{x}\mathrm{\left(\mathit{n}\right)}\overset{\mathit{ZT}}{\leftrightarrow}\mathit{-z}\frac{\mathit{d}}{\mathit{dz}}\mathit{X}\mathrm{\left(\mathit{z}\right)};\:\:\mathrm{ROC}\:\mathrm{=}\:\mathit{R}}$$ProofFrom the definition of Z-transform, we have, $$\mathrm{\mathit{Z}\mathrm{\left [\mathit{x}\mathrm{\left(\mathit{n}\right)} \right ]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty}}^{\infty}\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{z}^{-\mathit{n}}}$$Differentiating the above ... Read More

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 (DTFT) of a discrete-time sequence $\mathit{x}\mathrm{\left(\mathit{n}\right)}$is defined as −$$\mathrm{\mathit{F}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{\omega }\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty}}^{\infty}\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{e^{-\mathit{j\omega n}}}}$$Differentiation in Frequency Domain Property of DTFTStatement - The differentiation in frequency domain property of discrete-time Fourier transform states that the multiplication of a discrete-time sequence $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ by n is equivalent to the differentiation of its discrete-time Fourier transform in frequency domain. Therefore, if, $$\mathrm{\mathit{x}\mathrm{\left(\mathit{n}\right)}\overset{\mathit{FT}}{\leftrightarrow}\mathit{X}\mathrm{\left(\mathit{\omega }\right)}}$$Then$$\mathrm{\mathit{n}\mathit{x}\mathrm{\left(\mathit{n}\right)}\overset{\mathit{FT}}{\leftrightarrow}\mathit{j}\frac{\mathit{d}}{\mathit{d\omega }}\mathit{X}\mathrm{\left(\mathit{\omega }\right)}}$$ProofFrom the definition of DTFT, we have, $$\mathrm{\mathit{F}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{\omega }\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty}}^{\infty}\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{e^{-\mathit{j\omega n}}}}$$Differentiating both sides with respect to ω, we get, $$\mathrm{\frac{\mathit{d}}{\mathit{d\omega }}\mathit{X}\mathrm{\left(\mathit{\omega }\right)}\:\mathrm{=}\:\frac{\mathit{d}}{\mathit{d\omega}}\mathrm{\left[\sum_{\mathit{n=-\infty}}^{\infty}\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{e^{-\mathit{j\omega n}}} ... Read More

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, $$\mathrm{\mathit{Z}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty}}^{\infty}\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{z^{-\mathit{n}}}}$$Correlation Property of Z-TransformStatement - The correlation property of Z-transform states that if, $$\mathrm{\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{n}\right)}\overset{\mathit{ZT}}{\leftrightarrow}\mathit{X}_{\mathrm{1}}\mathrm{\left(\mathit{z}\right)}\:\mathrm{and}\:\mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{n}\right)}\overset{\mathit{ZT}}{\leftrightarrow}\mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{z}\right)}}$$Then$$\mathrm{\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{n}\right)}\otimes \mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{n}\right)}\overset{\mathit{ZT}}{\leftrightarrow}\mathit{X}_{\mathrm{1}}\mathrm{\left(\mathit{z}\right)}\mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{z}^{-\mathrm{1}}\right)}}$$Where$$\mathrm{\mathit{R}_{\mathrm{12}}\mathrm{\left ( \mathit{n} \right )}\:\mathrm{=}\:\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{n}\right)}\otimes \mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{n}\right)}}$$ProofFrom the definition of Z-transform, we have, $$\mathrm{\mathit{Z}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty}}^{\infty}\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{z^{-\mathit{n}}}}$$$$\mathrm{\mathit{\therefore \mathit{Z}\mathrm{\left[ \mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{n}\right)}\otimes \mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{n}\right)}\right ]}\:\mathrm{=}\:\sum_{\mathit{n=-\infty}}^{\infty}\mathrm{\left[ \mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{n}\right)}\otimes \mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{n}\right)}\right ]}\mathit{z}^{-n}}\:\:\:\:\:\:...(1)}$$The correlation of two signals is defined as, $$\mathrm{\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{n}\right)}\otimes \mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{n}\right)}\:\mathrm{=}\:\sum_{\mathit{k=-\infty}}^{\infty}\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{k}\right)}\mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{k-n}\right)}\:\mathrm{=}\:\sum_{\mathit{k=-\infty}}^{\infty}\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{k-n}\right)}\mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{k}\right)}\:\:\:\:\:\:...(2)}$$Therefore, from eqns.(1)&(2), we get, $$\mathrm{\mathit{Z}\mathrm{\left[\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{n}\right)}\otimes \mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{n}\right)} \right ]}\:\mathrm{=}\:\sum_{\mathit{n=-\infty}}^{\infty}}\mathrm{\left[\sum_{\mathit{k=-\infty}}^{\infty}\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{k}\right)}\mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{k-n}\right)} \right ]}\mathit{z}^{-n}$$Rearranging the order of summations, we get, $$\mathrm{\mathit{Z}\mathrm{\left[\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{n}\right)}\otimes \mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{n}\right)} \right ]}\:\mathrm{=}\:\sum_{\mathit{k=-\infty}}^{\infty}\:\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{k}\right)}\mathrm{\left[\sum_{\mathit{n=-\infty}}^{\infty}\mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{k-n}\right)}\mathit{z}^{-n} ... Read More

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, $$\mathrm{\mathit{Z}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty}}^{\infty}\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{z^{-\mathit{n}}}}$$Convolution in Time Domain Property of Z-TransformStatement - The convolution in time domain property of Z-transform states that the Z-transform of the convolution of two discrete time sequences is equal to the multiplication of their Z-transforms. Therefore, if, $$\mathrm{\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{n}\right)}\overset{\mathit{ZT}}{\leftrightarrow}\mathit{X}_{\mathrm{1}}\mathrm{\left(\mathit{z}\right)};\:\:\mathrm{ROC}\:\mathrm{=}\:\mathit{R}_{1}}$$$$\mathrm{\mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{n}\right)}\overset{\mathit{ZT}}{\leftrightarrow}\mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{z}\right)};\:\:\mathrm{ROC}\:\mathrm{=}\:\mathit{R}_{2}}$$Then, according to the convolution property, $$\mathrm{\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{n}\right)}*\mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{n}\right)}\overset{\mathit{ZT}}{\leftrightarrow}\mathit{X}_{\mathrm{1}}\mathrm{\left(\mathit{z}\right)}\mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{z}\right)};\:\:\mathrm{ROC}\:\mathrm{=}\:\mathit{R}_{\mathrm{1}}\cap\mathit{R}_{\mathrm{2}} }$$ProofThe convolution of two sequences is defined as, $$\mathrm{\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{n}\right)}*\mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{n}\right)}\:\mathrm{=}\:\sum_{\mathit{k=-\infty}}^{\infty}\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{k}\right)}\mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{n-k}\right)}}$$Now, from the definition of Z-transform, we have, $$\mathrm{\mathit{Z}\mathrm{\left [\mathit{x}\mathrm{\left(\mathit{n}\right)} ... Read More