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Debt rebalancing is a process of rebalancing the debt while calculating the Weighted Average Cost of Capital (WACC). The concept of WACC is based on the assumption that WACC remains constant throughout the lifetime of a project. It also depends on the fact that debt proportionality remains the same over the course of years of a project.As WACC remains constant throughout the lifetime of a project, the debt will go down each year according to WACC.As WACC remains constant over the years, to keep the debt proportionality constant, debt has to be rebalanced to keep the WACC constant.This change in ... 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)$$Final Value TheoremThe final value theorem of Laplace transform enables us to find the final value of a function$\mathit{x}\mathrm{(\mathit{t})}$[i.e., $\mathit{x}\mathrm{(\infty)}$] directly from its Laplace transform X(s) without the need for finding the ... Read More
Laplace TransformThe Laplace transform is a mathematical tool which is used to convert the differential equations in time domain into the algebraic equations in the frequency domain or s-domain.Mathematically, if $\mathrm{\mathit{x\left ( \mathit{t} \right )}}$ is a time domain function, then its Laplace transform is defined as −$$\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 }$$And the operator L is called the Laplace transform operator which transforms ... Read More
Laplace TransformThe Laplace transform is a mathematical tool which is used to convert the differential equation in time domain into the algebraic equations in the frequency domain or s-domain.Mathematically, if $\mathit{x}\mathrm{(\mathit{t})}$ is a time domain function, then its Laplace transform is defined as −$$\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