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The Fourier series representation of a periodic signal has various important properties which are useful for various purposes during the transformation of signals from one form to another.Consider two periodic signals 𝑥1(𝑡) and 𝑥2(𝑡) which are periodic with time period T and with Fourier series coefficients 𝐶𝑛 and 𝐷𝑛 respectively. With this assumption, let us proceed and check the various properties of a continuoustime Fourier series.Linearity PropertyThe linearity property of continuous-time Fourier series states that, if$$\mathrm{x_{1}(t)\overset{FS}{\leftrightarrow}C_{n}\: and\:x_{2}(t)\overset{FS}{\leftrightarrow}D_{n}}$$Then$$\mathrm{Ax_{1}(t)+Bx_{2}(t)\overset{FS}{\leftrightarrow}AC_{n}+BD_{n}}$$Time Shifting PropertyThe time scaling property of Fourier series states that, if$$\mathrm{x(t)\overset{FS}{\leftrightarrow}C_{n}}$$Then$$\mathrm{x(t-t_{0})\overset{FS}{\leftrightarrow}e^{-jn\omega_{0}t_{0}}C_{n}}$$Time Scaling PropertyThe time scaling property of Fourier series states that, if$$\mathrm{x(t)\overset{FS}{\leftrightarrow}C_{n}}$$Then$$\mathrm{x(at)\overset{FS}{\leftrightarrow}C_{n}\:with\:\omega_{0}\rightarrow a\omega_{0}}$$Time ... Read More
Fourier TransformFourier transform is a transformation technique that transforms signals from the continuous-time domain to the corresponding frequency domain and vice-versa.The Fourier transform of a continuous-time function $x(t)$ is defined as, $$\mathrm{X(\omega)=\int_{-\infty}^{\infty} x(t)e^{-j\omega t}dt… (1)}$$Inverse Fourier TransformThe inverse Fourier transform of a continuous-time function is defined as, $$\mathrm{x(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}X(\omega)\:e^{j\omega t}d\omega… (2)}$$Equations (1) and (2) for $X(\omega)$ and $x(t)$ are known as Fourier transform pair and can be represented as −$$\mathrm{X(\omega)=F[x(t)]}$$And$$\mathrm{x(t)=F^{-1}[X(\omega)]}$$Table of Fourier Transform PairsFunction, x(t)Fourier Transform, X(ω)$\delta(t)$1$\delta(t-t_{0})$$e^{-j \omega t_{0}}$1$2\pi \delta(\omega)$u(t)$\pi\delta(\omega)+\frac{1}{j\omega}$$\sum_{n=−\infty}^{\infty}\delta(t-nT)$$\omega_{0}\sum_{n=−\infty}^{\infty}\delta(\omega-n\omega_{0});\:\:\left(\omega_{0}=\frac{2\pi}{T} \right)$sgn(t)$\frac{2}{j\omega}$$ e^{j\omega_{0}t}$$ 2\pi\delta(\omega-\omega_{0})$$ cos\:\omega_{0}t$$\pi[\delta(\omega-\omega_{0})+\delta(\omega+\omega_{0})]$$sin\:\omega_{0}t$$-j\pi[\delta(\omega-\omega_{0})-\delta(\omega+\omega_{0})]$$e^{-at}u(t);\:\:\:a >0$$\frac{1}{a+j\omega}$$t\:e^{at}u(t);\:\:\:a >0$$\frac{1}{(a+j\omega)^{2}}$$e^{-|at|};\:\:a >0$$\frac{2a}{a^{2}+\omega^{2}}$$e^{-|t|}$$\frac{2}{1+\omega^{2}}$$\frac{1}{\pi t}$$-j\:sgn(\omega)$$\frac{1}{a^{2}+t^{2}}$$\frac{\pi}{a}e^{-a|\omega|}$$\Pi (\frac{t}{τ})$$τ\:sin c(\frac{\omega τ}{2})$$\Delta(\frac{t}{τ})$$\frac{τ}{2}sin C^{2}(\frac{\omega τ}{4})$$\frac{sin\:at}{\pi t}$$P_{a}(\omega)=\begin{cases}1 & for\:|\omega|\:< a\0 & for\:|\omega|\: > a ... Read More
Trigonometric Fourier SeriesA periodic function can be represented over a certain interval of time in terms of the linear combination of orthogonal functions. If these orthogonal functions are the trigonometric functions, then it is known as trigonometric Fourier series.Mathematically, the standard trigonometric Fourier series expansion of a periodic signal is, $$\mathrm{x(t)=a_{0}+ \sum_{n=1}^{\infty}a_{n}\:cos\:\omega_{0}nt+b_{n}\:sin\:\omega_{0}nt\:\:… (1)}$$Exponential Fourier SeriesA periodic function can be represented over a certain interval of time in terms of the linear combination of orthogonal functions, if these orthogonal functions are the exponential functions, then it is known as exponential Fourier series.Mathematically, the standard exponential Fourier series expansion for a periodic ... Read More
Fourier TransformThe Fourier transform of a continuous-time function $x(t)$ is defined as, $$\mathrm{X(\omega)=\int_{−\infty}^{\infty}x(t)e^{-j\omega t}dt}$$Inverse Fourier TransformThe inverse Fourier transform of a continuous-time function is defined as, $$\mathrm{x(t)=\frac{1}{2\pi}\int_{−\infty}^{\infty}X(\omega)e^{j\omega t}d\omega}$$Properties of Fourier TransformThe continuous-time Fourier transform (CTFT) has a number of important properties. These properties are useful for driving Fourier transform pairs and also for deducing general frequency domain relationships. These properties also help to find the effect of various time domain operations on the frequency domain. Some of the important properties of continuous time Fourier transform are given in the table as −Property of CTFTTime Domain x(t)Frequency Domain X(ω)Linearity Property$ax_{1}(t)+bx_{2}(t)$$aX_{1}(\omega)+bX_{2}(\omega)$Time Shifting ... Read More
Fourier SeriesIf $x(t)$ is a periodic function with period $T$, then the continuous-time exponential Fourier series of the function is defined as, $$\mathrm{x(t)=\sum_{n=−\infty}^{\infty}C_{n}\:e^{jn\omega_{0} t}\:\:… (1)}$$Where, $C_{n}$ is the exponential Fourier series coefficient, which is given by, $$\mathrm{C_{n}=\frac{1}{T}\int_{t_{0}}^{t_{0}+T}x(t)e^{-jn\omega_{0} t}dt\:\:… (2)}$$Modulation or Multiplication PropertyLet $x_{1}(t)$ and $x_{2}(t)$ two periodic signals with time period $T$ and with Fourier series coefficient $C_{n}$ and $D_{n}$. If$$\mathrm{x_{1}(t)\overset{FS}{\leftrightarrow}C_{n}}$$$$\mathrm{x_{2}(t)\overset{FS}{\leftrightarrow}D_{n}}$$Then, the modulation or multiplication property of continuous time Fourier series states, that$$\mathrm{x_{1}(t)\cdot x_{2}(t)\overset{FS}{\leftrightarrow}\sum_{k=−\infty}^{\infty}C_{k}\:D_{n-k}}$$Proof From the definition of continuous time Fourier series, we get, $$\mathrm{FS[x_{1}(t)\cdot x_{2}(t)]=\frac{1}{T}\int_{t_{0}}^{t_{0}+T}[x_{1}(t)\cdot x_{2}(t)]e^{-jn\omega_{0} t}dt}$$$$\mathrm{\Rightarrow\:FS[x_{1}(t)\cdot x_{2}(t)]=\frac{1}{T}\int_{t_{0}}^{t_{0}+T}x_{1}(t)\left (\sum_{k=−\infty}^{\infty} C_{k} e^{jk\omega_{0} t}\right )e^{-jn\omega_{0} t}dt}$$$$\mathrm{\Rightarrow\:FS[x_{1}(t)\cdot x_{2}(t)]=\frac{1}{T}\int_{t_{0}}^{t_{0}+T}x_{1}(t)\left (\sum_{k=−\infty}^{\infty} C_{k} e^{-j(n-k)\omega_{0} t}\right )e^{-jn\omega_{0} ... Read More
Inflation is an important parameter in investments, as it erodes the value of money. It must be included in capital budgeting so that cash flows are appropriately included in the evaluation of an investment. In doing so, the inflation targets must be forecasted accurately over a considerably longer duration.Here are some considerations that must be made while creating inflation in an investment evaluation process.The forecasts must be spread over a periodInflation is an ever-existing truth nowadays. Investors face this reality day-in and day-out. They want to gain a return that is more than the inflation rates so that the value ... Read More
Free cash flow is the capital retained by a company after it has paid all its expenses, including building, rent, tax, payroll, inventory, etc. Companies may use the free cash flow for anything it sees fit.Free cash flow is a true measure of a company’s profitability.Businesses usually calculate free cash flow to take critical business decisions, such as whether to invest the money for expansion or to invest the money in ways to reduce the costs of operations.Investors use the free cash flow metric to check the frauds in accounting, as these measures are stringent and less manipulable than net ... Read More
Product cannibalization is the process in which a new project or product eats away the cash flows earned by an already existing product. That means, when a new project or product is launched, it may take the cash flow away from an already running product or project. Business decisions must include the cannibalization effect into consideration because ignoring this may lead to huge losses sometimes.Capital budgeting decisions are taken keeping profitability and market share of products and they should include a holistic approach.A new product or project must not negatively impact an already available product or project which is running ... Read More
Financial distress is a situation in which a company cannot meet its obligations. In fact, companies borrow debts from investors for operations and growth. When the company performs badly, it may be unable to repay back the loans and funds it has borrowed from the lenders. This is called financial distress of the company.There are many costs attached to financial distress. These may include costs related to employees, managers, customers, suppliers, and shareholders.Costs Related to EmployeesThe employees of a distressed company are not confident. They are demoralized and worried about their future. This affects the quality of the products. The ... Read More
To better understand the two concepts of risk-adjusted and certainty equivalent methods, let’s understand what they mean and how they are useful for investors.What is Risk-Adjusted Discount RateThe risk-adjusted method combines an expected risk premium with a risk-free rate to calculate the present value of an investment.Risky investments include investments in real estate and other high-level risk projects. Although the market rate is considered as the discount rate, in some cases, a risk-adjusted rate for the investments becomes crucial for the investors.The risk-adjusted discount rate method correlates risks and returns while signifying the requisite returns of an investment. This means ... Read More