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

System RealizationThe realization of a continuous-time system means obtaining a network corresponding to the differential equation or transfer function of the system.Block DiagramA diagram of a system in which the main parts or functions are represented by blocks connected by the lines that show the relationship of the blocks is called the block diagram of that system.Elements to Construct the Block-Diagram of Continuous- Time SystemThe transfer function of a continuous-time system can be realised either by using integrators or differentiators. Although, due to certain drawbacks, the differentiators are not used to realise the practical systems. The chief drawback of differentiators ... Read More