Inverse Z-TransformThe inverse Z-transform is defined as the process of finding the time domain signal $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ from its Z-transform $\mathit{X}\mathrm{\left(\mathit{z}\right)}$. The inverse Z-transform is denoted as −$$\mathrm{\mathit{x}\mathrm{\left(\mathit{n}\right)}\:\mathrm{=}\:\mathit{Z}^{-\mathrm{1}}\mathrm{\left[\mathit{X}\mathrm{\left(\mathit{z}\right)}\right]}}$$Partial Fraction Expansion Method to Find Inverse Z-TransformIn order to determine the inverse Z-transform of $\mathit{X}\mathrm{\left(\mathit{z}\right)}$ using partial fraction expansion method, the denominator of $\mathit{X}\mathrm{\left(\mathit{z}\right)}$ must be in factored form. In this method, we obtained the partial fraction expansion of $\frac{\mathit{X}\mathrm{\left(\mathit{z}\right)}}{\mathit{z}}$ instead of $\mathit{X}\mathrm{\left(\mathit{z}\right)}$. This is because the Z-transform of time-domain sequences have Z in their numerators.The partial fraction expansion method is applied only if $\frac{\mathit{X}\mathrm{\left(\mathit{z}\right)}}{\mathit{z}}$ is a proper rational function, i.e., the order ... Read More

Average PowerThe average power of a signal is defined as the mean power dissipated by the signal such as voltage or current in a unit resistance over a period. Mathematically, the average power is given by, $$\mathit{P}\:\mathrm{=}\:\lim_{T \rightarrow \infty}\frac{1}{\mathit{T}}\int_{\mathrm{-(\mathit{T}/\mathrm{2})}}^{\mathrm{(\mathit{T}/\mathrm{2})}}|\mathit{x}\mathrm{(\mathit{t})}|^\mathrm{2}\:\mathit{dt}$$Parseval's Power TheoremStatement − Parseval's power theorem states that the power of a signal is equal to the sum of square of the magnitudes of various harmonic components present in the discrete spectrum.Mathematically, the Parseval's power theorem is defined as −$$\mathit{P}\:\mathrm{=}\:\displaystyle\sum\limits_{n=-\infty}^\infty |\mathit{C}_\mathit{n}|^2$$ProofConsider a function $\mathit{x}\mathrm{(\mathit{t})}$. Then, the average power of the signal $\mathit{x}\mathrm{(\mathit{t})}$ over one complete cycle is given by, $$\mathit{P}\:\mathrm{=}\:\frac{1}{\mathit{T}}\int_{\mathrm{-(\mathit{T}/\mathrm{2})}}^{\mathrm{(\mathit{T}/\mathrm{2})}}|\mathit{x}\mathrm{(\mathit{t})}|^\mathrm{2}\:\mathit{dt}$$ $$\because|\mathit{x}\mathrm{(\mathit{t})}|^\mathrm{2}\:\mathrm{=}\: ... Read More

Inverse Z-TransformThe inverse Z-transform is defined as the process of finding the time domain signal $\mathit{x}\mathrm{(\mathit{n})}$ from its Z-transform $\mathit{X}\mathrm{(\mathit{z})}$. The inverse Z-transform is denoted as:$$\mathit{x}\mathrm{(\mathit{n})}\:\mathrm{=}\:\mathit{Z}^{\mathrm{-1}} [\mathit{X}\mathrm{(\mathit{z})}]$$Long Division Method to Calculate Inverse Z-TransformIf $\mathit{x}\mathrm{(\mathit{n})}$ is a two sided sequence, then its Z-transform is defined as, $$\mathit{X}\mathrm{(z)}\:\mathrm{=}\:\displaystyle\sum\limits_{n=-\infty}^\infty \mathit{x}\mathrm{(n)}\mathit{z}^{-\mathit{n}}$$Where, the Z-transform $\mathit{X}\mathrm{(\mathit{z})}$ has both positive powers of z as well as negative powers of z. Using the long division method, a two sided sequence cannot be obtained. Therefore, if the sequence $\mathit{x}\mathrm{(\mathit{n})}$ is a causal sequence, then$$\mathit{X}\mathrm{(z)}\:\mathrm{=}\:\displaystyle\sum\limits_{n=0}^\infty \mathit{x}\mathrm{(n)}\mathit{z}^{-\mathit{n}}\:\mathrm{=}\:\mathit{x}\mathrm{(0)}+\mathit{x}\mathrm{(1)}\mathit{z}^{\mathrm{-1}}+\mathit{x}\mathrm{(2)}\mathit{z}^{\mathrm{-2}}+\mathit{x}\mathrm{(3)}\mathit{z}^{\mathrm{-3}}+\dotso$$i.e., $\mathit{X}\mathrm{(\mathit{z})}$ has only negative powers of z and its ROC is $|\mathit{z}|>\:\mathit{a}$.And, if the ... 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 $\mathit{x}\mathrm{(\mathit{t})}$ is a time domain function, then its Laplace transform is defined as −$$\mathit{L}\mathrm{[\mathit{x}\mathrm{(\mathit{t})}]}\:\mathrm{=}\:\mathit{X}\mathrm{(\mathit{s})}\:\mathrm{=}\:\int_{-\infty}^{\infty}\mathit{x}\mathrm{(\mathit{t})\mathit{e^{-st}}}\mathit{dt}\:\:\:..(1)$$Equation (1) gives the bilateral Laplace transform of the function $\mathit{x}\mathrm{(\mathit{t})}$. But for the causal signals, the unilateral Laplace transform is applied, which is defined as −$$\mathit{L}\mathrm{[\mathit{x}\mathrm{(\mathit{t})}]}\:\mathrm{=}\:\mathit{X}\mathrm{(\mathit{s})}\:\mathrm{=}\:\int_{\mathrm{0}}^{\infty}\mathit{x}\mathrm{(\mathit{t})\mathit{e^{-st}}}\mathit{dt}\:\:\:..(2)$$Linearity Property of Laplace TransformStatement − The Linearity property of Laplace transform states that the Laplace transform of a weighted sum of two signals is equal to the weighted sum of ... 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 $\mathit{x}\mathrm{(\mathit{t})}$ is a time domain function, then its Laplace transform is defined as−$$\mathit{L}\mathrm{[\mathit{x}\mathrm{(\mathit{t})}]}\:\mathrm{=}\:\mathit{X}\mathrm{(\mathit{s})}\:\mathrm{=}\:\int_{-\infty}^{\infty}\mathit{x}\mathrm{(\mathit{t})\mathit{e^{-st}}}\mathit{dt}\:\:\:..(1)$$Equation (1) gives the bilateral Laplace transform of the function $\mathit{x}\mathrm{(\mathit{t})}$. But for the causal signals, the unilateral Laplace transform is applied, which is defined as −$$\mathit{L}\mathrm{[\mathit{x}\mathrm{(\mathit{t})}]}\:\mathrm{=}\:\mathit{X}\mathrm{(\mathit{s})}\:\mathrm{=}\:\int_{\mathrm{0}}^{\infty}\mathit{x}\mathrm{(\mathit{t})\mathit{e^{-st}}}\mathit{dt}\:\:\:..(2)$$Final Value TheoremThe final value theorem of Laplace transform enables us to find the final value of a function$\mathit{x}\mathrm{(\mathit{t})}$[i.e., $\mathit{x}\mathrm{(\infty)}$] directly from its Laplace transform X(s) without the need for finding the ... 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 $\mathit{x}\mathrm{(\mathit{t})}$ is a time domain function, then its Laplace transform is defined as −$$\mathit{L}\mathrm{[\mathit{x}\mathrm{(\mathit{t})}]}\:\mathrm{=}\:\mathit{X}\mathrm{(\mathit{s})}\:\mathrm{=}\:\int_{-\infty}^{\infty}\mathit{x}\mathrm{(\mathit{t})\mathit{e^{-st}}}\mathit{dt} \:\:\:...(1)$$Equation (1) gives the bilateral Laplace transform of the function $\mathit{x}\mathrm{(\mathit{t})}$. But for the causal signals, the unilateral Laplace transform is applied, which is defined as −$$\mathit{L}\mathrm{[\mathit{x}\mathrm{(\mathit{t})}]}\:\mathrm{=}\:\mathit{X}\mathrm{(\mathit{s})}\:\mathrm{=}\:\int_{\mathrm{0}}^{\infty}\mathit{x}\mathrm{\mathrm{(\mathit{t})}\mathit{e^{-st}}}\mathit{dt} \:\:\:...(2)$$Initial Value TheoremThe initial value theorem of Laplace transform enables us to calculate the initial value of a function $\mathit{x}\mathrm{(\mathit{t})}$[i.e., $\:\:\mathit{x}\mathrm{(0)}$] directly from its Laplace transform X(s) without the ... 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 $\mathit{x}\mathrm{(\mathit{t})}$is a time-domain function, then its Laplace transform is defined as −$$\mathit{L}\mathrm{[\mathit{x}\mathrm{(\mathit{t})}]}\:\mathrm{=}\:\mathit{X}\mathrm{(\mathit{s})}\:\mathrm{=}\:\int_{-\infty}^{\infty}\mathit{x}\mathrm{(t)}\mathit{e^{-st}}\mathit{dt} \:\:...(1)$$Equation (1) gives the bilateral Laplace transform of the function $\mathit{x}\mathrm{(\mathit{t})}$. But for the causal signals, the unilateral Laplace transform is applied, which is defined as −$$\mathit{L}\mathrm{[\mathit{x}\mathrm{(\mathit{t})}]}\:\mathrm{=}\:\mathit{X}\mathrm{(\mathit{s})}\:\mathrm{=}\:\int_{\mathrm{0}}^{\infty}\mathit{x}\mathrm{(t)}\mathit{e^{-st}}\mathit{dt} \:\: ...(2)$$Frequency Derivative Property of Laplace TransformStatement − The differentiation in frequency domain or s-domain property of Laplace transform states that the multiplication of the function by $\mathit{'t'}$ in time domain ... Read More

Detection of Periodic Signals in the Presence of NoiseThe noise signal is an unwanted signal which has random amplitude variation. The noise signals are uncorrelated with any periodic signal.Detection of the periodic signals masked by noise signals is of great importance in signal processing. It is mainly used in the detection of radar and sonar signals, the detection of periodic components in brain signals, in the detection of periodic components in sea wave analysis and in many other areas of geophysics etc. The solution of these problems can be easily provided by thecorrelation techniques. The cross-correlation function, therefore can be ... Read More

Detection of Periodic Signals in the Presence of NoiseThe noise signal is an unwanted signal which has random amplitude variation. The noise signals are uncorrelated with any periodic signal.Detection of the periodic signals masked by noise signals is of great importance in signal processing. It is mainly used in the detection of radar and sonar signals, the detection of periodic components in brain signals, in the detection of periodic components in sea wave analysis and in many other areas of geophysics etc. The solution of these problems can be easily provided by the correlation techniques. The autocorrelation function, therefore can ... Read More

Cross Correlation FunctionThe cross correlation function between two different signals is defined as the measure of similarity or coherence between one signal and the time delayed version of another signal.The cross correlation function is defined separately for energy (or aperiodic) signals and power or periodic signals.Cross Correlation of Energy SignalsConsider two energy signals $\mathit{x_{\mathrm{1}}}\mathrm{(\mathit{t})}$ and $\mathit{x_{\mathrm{2}}}\mathrm{(\mathit{t})}$. The cross correlation of these two energy signals is defined as −$$\mathit{R_{\mathrm{12}}}\mathrm{(\tau)}\:\mathrm{=}\:\int_{-\infty}^{\infty}\mathit{x_{\mathrm{1}}}\mathrm{(\mathit{t})}x_{\mathrm{2}}^{*}\mathrm{(\mathit{t-\tau})}\mathit{dt} \:\mathrm{=}\: \int_{-\infty}^{\infty}\mathit{x_{\mathrm{1}}}\mathrm{(\mathit{t+\tau})}\mathit{x_\mathrm{2}^*}\mathrm{(\mathit{t})}\mathit{dt}$$Where, the variable $\tau$ is called the delay parameter or scanning parameter or searching parameter.The cross correlation of two energy signals is defined in another form as −$$\mathit{R_{\mathrm{12}}}\mathrm{(\mathit{\tau})} \:\mathrm{=}\: \int_{-\infty}^{\infty}\mathit{x_\mathrm{2}}\mathrm{(t)}\mathit{x_\mathrm{1}^*}\mathrm{(t-\tau)}\:\mathit{dt}$$Properties of ... Read More