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

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, $$\mathrm{\mathit{Z}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty}}^{\infty}\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{z^{-\mathit{n}}}}$$Transform Analysis of Discrete-Time SystemThe Z-transform plays a vital role in the design and analysis of discrete-time LTI (Linear Time Invariant) systems.Transfer Function of a Discrete-Time LTI SystemThe figure shows a discrete-time LTI system having an impulse response $\mathit{h}\mathrm{\left(\mathit{n}\right)}$.Consider the system gives an output $\mathit{y}\mathrm{\left(\mathit{n}\right)}$ for an input $\mathit{x}\mathrm{\left(\mathit{n}\right)}$. Then, $$\mathrm{\mathit{y}\mathrm{\left(\mathit{n}\right)}\:\mathrm{=}\:\mathit{h}\mathrm{\left(\mathit{n}\right)}*\mathit{x}\mathrm{\left(\mathit{n}\right)}}$$Taking Z-transform on both the sides, we get, $$\mathrm{\mathit{Z}\mathrm{\left[ \mathit{y}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{\mathrm{=}}\mathit{Z}\mathrm{\left[\mathit{h}\mathrm{\left(\mathit{n}\right)}*\mathit{x}\mathrm{\left(\mathit{n}\right)} \right ]}}$$$$\mathrm{\therefore ... 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 practical systems. Therefore, only integrators are used for the realization of continuous-time systems. The adder and multipliers are other two elements which are used realise the continuous-time systems.Parallel Form Realisation of Continuous-Time SystemsIn the parallel form realisation of continuous-time systems, the transfer function of the system is expressed into its ... 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 practical systems. Therefore, only integrators are used for the realization of continuous-time systems. The adder and multipliers are other two elements which are used realise the continuous-time systems.Direct Form-II Realization of CT SystemsThe advantage of the direct form-II realization of continuous-time systems is that it uses minimum number of integrators. ... 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 practical systems. Therefore, only integrators are used for the realization of continuous-time systems. The adder and multipliers are other two elements which are used realise the continuous-time systems.Direct Form-I Realization of CT SystemsThe direct form-I realization is the simplest and most straight forward structure for the realization of a continuous-time ... Read More

SamplingThe process of converting a continuous-time signal into a discrete-time signal is called the sampling.After sampling, the signal is defined at discrete-instants of time and the time interval between two successive sampling instants is called the sampling period.Sampling TechniquesThe sampling of a signal is done in several ways. Generally, there are three types of sampling techniques viz. −Instantaneous Sampling or Impulse SamplingNatural SamplingFlat Top SamplingHere, the instantaneous sampling or impulse sampling is also called the ideal sampling, whereas the natural sampling and flat top sampling are called the practical sampling techniques. These three sampling methods are explained as follows −Ideal ... 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, $\mathrm{\mathit{x\left ( n \right )}}$ if is a discrete time function, then its Z-transform is defined as, $$\mathrm{\mathit{Z\left [ x\left ( n \right ) \right ]\mathrm{\, =\, }X\left ( z \right )\mathrm{\, =\, }\sum_{n\mathrm{\, =\, }-\infty }^{\infty }x\left ( n \right )z^{-n} }}$$Conjugation Property of Z-TransformStatement – The conjugation property of Z-transform states that if$$\mathrm{\mathit{x\left ( n \right )\overset{ZT}{\leftrightarrow}X\left ( z \right );\; \; \mathrm{ROC}\mathrm{\, =\, }\mathit{R} }}$$Then, $$\mathrm{\mathit{x^{*}\left ( n \right )\overset{ZT}{\leftrightarrow}X^{*}\left ( z^{*} \right ... Read More