Autocorrelation FunctionThe autocorrelation function defines the measure of similarity or coherence between a signal and its time delayed version. The autocorrelation function of a real energy signal $\mathit{x}\mathrm{(\mathit{t})}$ is given by, $$\mathit{R}\mathrm{(\mathit{\tau})} \:\mathrm{=}\: \int_{-\infty}^{\infty}\mathit{x\mathrm(\mathit{t})}\:\mathit{x}\mathrm{(\mathit{t-\tau})}\:\mathit{dt}$$Energy Spectral Density (ESD) FunctionThe distribution of the energy of a signal in the frequency domain is called the energy spectral density.The ESD function of a signal is given by, $$\mathit{\psi}\mathrm{(\mathit{\omega})}\: \mathrm{=}\: \mathrm{|\mathit{X}\mathrm{(\mathit{\omega})}|}^\mathrm{2} \:\mathrm{=}\: \mathit{X}\mathrm{(\mathit{\omega})} \mathit{X}\mathrm{(\mathit{-\omega})}$$Autocorrelation TheoremStatement − The autocorrelation theorem states that the autocorrelation function $\mathit{R}\mathrm{(\mathrm{\tau})}$ and the ESD (Energy Spectral Density) function $\mathit{\psi}\mathrm{(\mathit{\omega})}$ of an energy signal $\mathit{x}\mathrm{(\mathit{t})}$ form a Fourier transform pair, i.e., $$\mathit{R}\mathrm{(\mathit{\tau})} ... Read More

Energy Spectral DensityThe distribution of the energy of a signal in the frequency domain is known as energy spectral density (ESD) or energy density (ED) or energy density spectrum. The ESD function is denoted by $\mathrm{\mathit{\psi \left ( \omega \right )}}$ and is given by, $$\mathrm{\mathit{\psi \left ( \omega \right )\mathrm{=}\left|X\left ( \omega \right ) \right|^{\mathrm{2}}}}$$For an energy signal, the total area under the energy spectral density curve plotted as the function of frequency is equal to the total energy of the signal.ExplanationConsider a linear system having $\mathrm{\mathit{x\left ( \mathit{t} \right )}}$ and $\mathrm{\mathit{y\left ( \mathit{t} \right )}}$ as input ... Read More

What is Autocorrelation?The autocorrelation function of a signal is defined as the measure of similarity or coherence between a signal and its time delayed version. Thus, the autocorrelation is the correlation of a signal with itself.The autocorrelation function is defined separately for energy or aperiodic signals and power or periodic signals.Autocorrelation Function for Energy SignalsThe autocorrelation function of an energy signal $\mathrm{\mathit{x\left ( \mathit{t} \right )}}$ is defined as −$$\mathrm{\mathit{R_{\mathrm{11}}\left ( \tau \right )\mathrm{=}R\left ( \tau \right )\mathrm{=}\int_{-\infty }^{\infty }x\left ( t \right )x^{\ast }\left ( t-\tau \right )dt\mathrm{=}\int_{-\infty }^{\infty }x\left ( t\mathrm{+ }\tau \right )x^{\ast }\left ( t \right ... 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 $\mathrm{\mathit{x\left ( \mathit{t} \right )}}$ is a time domain function, then its Laplace transform is defined as, $$\mathrm{\mathit{L\left [ x\left ( \mathrm{t} \right ) \right ]}\mathrm{=} \mathit{X\left ( s \right )}\mathrm{=}\int_{-\infty }^{\infty}\mathit{x\left ( \mathrm{t} \right )e^{-st}\; dt}\; \; ...\left ( 1 \right )}$$Equation (1) gives the bilateral Laplace transform of the function $\mathrm{\mathit{x\left ( \mathit{t} \right )}}$. But for the causal signals, the unilateral Laplace transform is applied, which is ... Read More

Laplace TransformThe Laplace transform is a mathematical tool which is used to convert the differential equations in time domain into the algebraic equations in the frequency domain or s-domain.Mathematically, if $\mathrm{\mathit{x\left ( \mathit{t} \right )}}$ is a time domain function, then its Laplace transform is defined as −$$\mathrm{\mathit{L\left [ x\left ( \mathrm{t} \right ) \right ]}\mathrm{=} \mathit{X\left ( s \right )}\mathrm{=}\int_{-\infty }^{\infty}\mathit{x\left ( \mathrm{t} \right )e^{-st}\; dt}\; \; ...\left ( 1 \right )}$$Where, 𝑠 is a complex variable and it is given by, $$\mathrm{s = \sigma + j\omega }$$And the operator L is called the Laplace transform operator which transforms ... Read More

Laplace TransformThe linear time invariant (LTI) system is described by differential equations. The Laplace transform is a mathematical tool which converts the differential equations in time domain into algebraic equations in the frequency domain (or s-domain).If $\mathrm{\mathit{x\left ( t \right )}}$ is a time function, then the Laplace transform of the function is defined as −$$\mathrm{\mathit{L\left [ x\left ( t \right ) \right ]\mathrm{=}X\left ( s \right )\mathrm{=}\int_{-\infty }^{\infty }x\left ( t \right )e^{-st}\:dt\; \; \cdot \cdot \cdot\left ( \mathrm{1} \right ) }}$$Where, s is a complex variable and it is given by, $$\mathrm{\mathit{s\mathrm{=}\sigma \mathrm{+ }j\omega }}$$Inverse Laplace TransformThe inverse ... 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 continuous-time function, then its Laplace transform is defined as −$$\mathrm{\mathit{L}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{t}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{s}\right)}\:\mathrm{=}\:\int_{-\infty }^{\infty }\mathit{x}\mathrm{\left(\mathit{t}\right)}\mathit{e^{-\mathit{st}}\:\mathit{dt}}\:\:\:\:\:\:...(3)}$$Equation ... Read More

Discrete-Time Fourier TransformThe Fourier transform of the discrete-time signals is known as the discrete-time Fourier transform (DTFT). The DTFT converts a time domain sequence into frequency domain signal. The DTFT of a discrete time sequence $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is given by, $$\mathrm{\mathit{F}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{\omega}\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty}}^{\infty}\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{e^{-j\omega n}}\:\:\:\:\:\:...(1)}$$Z-TransformThe Z-transform is a mathematical which is used to convert the difference equations in time domain into the algebraic equations in z-domain. Mathematically, the Z-transform of a discrete time sequence $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is given by, $$\mathrm{\mathit{Z}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty}}^{\infty}\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{z^{-\mathit{n}}}\:\:\:\:\:\:...(2)}$$Relation between DTFT and Z-TransformSince the DTFT of a discrete time sequence $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is given by, $$\mathrm{\mathit{X}\mathrm{\left(\mathit{\omega }\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty}}^{\infty}\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{e^{-\mathit{j\omega n}}}\:\:\:\:\:\:...(3)}$$For the existence of the DTFT, the sequence ... Read More

Energy of a SignalThe energy of a signal $\mathit{x}\mathrm{\left(\mathit{t}\right)}$ is defined as the area under the curve of square of magnitude of that signal, i.e., $$\mathrm{\mathit{E}\:\mathrm{=}\:\int_{-\infty}^{\infty}\left|\mathit{x}\mathrm{\left(\mathit{t}\right)} \right|^{\mathrm{2}}\:\mathit{dt}}$$The energy signal exists only of the energy (E) of the signal is finite, i.e., only if 0 < E < $\infty$.Rayleigh’s Energy TheoremStatement - The Rayleigh’s energy theorem states that the integral of the square of magnitude of a function (i.e., energy of the function) is equal to the integral of the square of magnitude of its Fourier transform, i.e., $$\mathrm{\mathit{E}\:\mathrm{=}\:\int_{-\infty}^{\infty}\left|\mathit{x}\mathrm{\left(\mathit{t}\right)} \right|^{\mathrm{2}}\:\mathit{dt}\:\mathrm{=}\:\frac{1}{2\pi}\int_{-\infty}^{\infty}\left|\mathit{X}\mathrm{\left(\mathit{\omega }\right)} \right|^{\mathrm{2}}\:\mathit{d\omega }}$$ProofConsider a function $\mathit{x}\mathrm{\left(\mathit{t}\right)}$ such that its Fourier transform ... 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 −$$\mathrm{\mathit{Z}\mathrm{\left[\mathit{x}\mathrm{\left(\mathit{n}\right)}\right]}\:\mathrm{=}\:\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty }}^{\infty }\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{z}^{-\mathit{n}}}$$Where, z is a complex variable.Region of Convergence (ROC) of Z-TransformThe set of points in z-plane for which the Z-transform of a discrete-time sequence $\mathit{x}\mathrm{\left(\mathit{n}\right)}$, i.e., $\mathit{X}\mathrm{\left(\mathit{z}\right)}$ converges is called the region of convergence (ROC) of $\mathit{X}\mathrm{\left(\mathit{z}\right)}$.Properties of ROC of Z-TransformThe region of convergence (ROC) of Z-transform has the following properties −The ROC of the Z-transform is ... Read More