Found 189 Articles for Signals and Systems

Fourier SeriesConsider a periodic signal 𝑔(𝑡) be periodic with period T, then the Fourier series of the function 𝑔(𝑡) is defined as, $$\mathrm{g(t)=\sum_{n=-\infty}^{\infty}C_{n}e^{jn\omega_{0}t}\:\:\:\:....(1)}$$Where, 𝐶𝑛 is the Fourier series coefficient and is given by, $$\mathrm{C_{n}=\frac{1}{T}\int_{\frac{-T}{2}}^{\frac{T}{2}}g(t)e^{-jn\omega_{0}t}dt\:\:\:\:....(2)}$$Derivation of Fourier Transform from Fourier SeriesLet 𝑥(𝑡) be a non-periodic signal and let the relation between 𝑥(𝑡) and 𝑔(𝑡) is given by, $$\mathrm{X(t)=\lim_{T\rightarrow \infty}g(t)\:\:\:\:.....(3)}$$Where, T is the time period of the periodic signal 𝑔(𝑡).By rearranging eq. (2), we get, $$\mathrm{TC_n=\int_{\frac{-T}{2}}^{\frac{T}{2}}g(t)e^{-jn\omega_{0}t}dt}$$The term 𝐶𝑛 represents the magnitude of the component of frequency nω0.Let nω0 = ω at 𝑇 → ∞. Then, we have, $$\mathrm{\omega_0=\frac{2\pi}{t}|_{T\rightarrow \infty}\rightarrow 0}$$Thus, the discrete ... Read More

Fourier TransformThe Fourier transform of a continuous-time function 𝑥(𝑡) can be defined as, $$\mathrm{X(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt}$$Convolution Property of Fourier TransformStatement – The convolution of two signals in time domain is equivalent to the multiplication of their spectra in frequency domain. Therefore, if$$\mathrm{x_1(t)\overset{FT}{\leftrightarrow}X_1(\omega)\:and\:x_2(t)\overset{FT}{\leftrightarrow}X_2(\omega)}$$Then, according to time convolution property of Fourier transform, $$\mathrm{x_1(t)*x_2(t)\overset{FT}{\leftrightarrow}X_1(\omega)*X_2(\omega)}$$ProofThe convolution of two continuous time signals 𝑥1(𝑡) and 𝑥2(𝑡) is defined as, $$\mathrm{x_1(t)*x_2(t)=\int_{-\infty}^{\infty}x_1(\tau)x_2(t-\tau)d\tau}$$Now, from the definition of Fourier transform, we have, $$\mathrm{X(\omega)=F[x_1(t)*x_2(t)]=\int_{-\infty}^{\infty}[x_1(t)*x_2(t)]e^{-j \omega t}dt}$$$$\mathrm{\Rightarrow F[x_1(t)*x_2(t)]=\int_{-\infty}^{\infty}[\int_{-\infty}^{\infty}x_1(\tau)x_2(t-\tau)d\tau]e^{-j \omega t}dt }$$By interchanging the order of integration, we get, $$\mathrm{\Rightarrow F[x_1(t)*x_2(t)]=\int_{-\infty}^{\infty}x_1(\tau)[\int_{-\infty}^{\infty}x_{2}(t-\tau)e^{-j \omega t}dt]d\tau }$$By replacing (𝑡 − 𝜏) = 𝑢 in the second integration, ... Read More

Fourier SeriesIf 𝑥(𝑡) is a periodic function with period T, then the continuous-time Fourier series of the function is defined as, $$\mathrm{x(t)=\sum_{n=-\infty}^{\infty}C_ne^{jn\omega_{0}t}\:\:\:\:\:.....(1)}$$Where, 𝐶𝑛 is the exponential Fourier series coefficient, that is given by$$\mathrm{C_n=\frac{1}{T}\int_{t_0}^{t_0+T}x(t)e^{-jn\omega_0t}dt\:\:\:\:\:.....(2)}$$Convolution Property of Fourier SeriesAccording to the convolution property, the Fourier series of the convolution of two functions 𝑥1(𝑡) and 𝑥2(𝑡) in time domain is equal to the multiplication of their Fourier series coefficients in frequency domain.If 𝑥1(𝑡) and 𝑥2(𝑡) are two periodic functions with time period T and with Fourier series coefficients 𝐶𝑛 and 𝐷𝑛. Then, if$$\mathrm{x_1(t)\overset{FS}{\leftrightarrow}C_n}$$$$\mathrm{x_2(t)\overset{FS}{\leftrightarrow}D_n}$$Then, the convolution property of continuous time Fourier series states that$$\mathrm{x_1(t)*x_2(t)\overset{FS}{\leftrightarrow}TC_nD_n}$$ProofBy ... Read More

Fourier TransformFor a continuous-time function x(t), the Fourier transform of x(t) can be defined as, $$\mathrm{X(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt}$$Conjugation Property of Fourier TransformStatement − The conjugation property of Fourier transform states that the conjugate of function x(t) in time domain results in conjugation of its Fourier transform in the frequency domain and ω is replaced by (−ω), i.e., if$$\mathrm{x(t)\overset{FT}{\leftrightarrow}X(\omega)}$$Then, according to conjugation property of Fourier transform, $$\mathrm{x^*(t)\overset{FT}{\leftrightarrow}X^*(-\omega)}$$ProofFrom the definition of Fourier transform, we have$$\mathrm{X(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt}$$Taking conjugate on both sides, we get$$\mathrm{X^*(\omega)=[\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt]^*}$$$$\mathrm{\Rightarrow X^*(\omega)=\int_{-\infty}^{\infty}x^*(t)e^{j\omega t}dt}$$Now, by replacing (ω) by (−ω), we obtain, $$\mathrm{X^*(-\omega)=\int_{-\infty}^{\infty}x^*(t)e^{-j\omega t}dt=F[x^*(t)]}$$$$\mathrm{\therefore F[x^*(t)]=X^*(-\omega)}$$Or, it can also be represented as, $$\mathrm{x^*(t)\overset{FT}{\leftrightarrow}X^*(-\omega)}$$Autocorrelation Property ... Read More

Exponential Fourier SeriesPeriodic signals are 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 the Fourier series representation of the function is called the exponential Fourier series.The exponential Fourier series is the most widely used form of the Fourier series. In this representation, the periodic function x(t) is expressed as a weighted sum of the complex exponential functions. The complex exponential Fourier series is the convenient and compact form of the Fourier series, hence, its findsextensive application in communication theory.ExplanationLet a set of complex ... Read More

When a voltage of V volts is applied across a resistance of R Ω, then a current I flows through it. The power dissipated in the resistance is given by, $$\mathrm{P=I^2R=\frac{V^2}{R}\:\:\:\:\:\:....(1)}$$But when the voltage and current signals are not constant, then the power varies at every instant, and the equation for the instantaneous power is given by, $$\mathrm{P=i^2(t)R=\frac{V^2(t)}{R}\:\:\:\:\:\:....(2)}$$Where, 𝑖(𝑡) and 𝑣(𝑡) are the corresponding instantaneous values of current and voltage respectivelyNow, if the value of the resistance (R) is 1 Ω, then the instantaneous power can be represented as, $$\mathrm{p=i^2(t)=v^2(t)\:\:\:\:\:\:....(3)}$$Therefore, the instantaneous power of a signal x(t) can be given ... Read More

Fourier TransformFor a continuous-time function x(t), the Fourier transform of x(t) can be defined as$$\mathrm{X(\omega)= \int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt}$$Duality Property of Continuous-Time Fourier TransformStatement – If a function x(t) has a Fourier transform X(ω) and we form a new function in time domain with the functional form of the Fourier transform as X(t), then it will have a Fourier transform X(ω) with the functional form of the original time function, but it is a function of frequency.Mathematically, the duality property of CTFT states that, if$$\mathrm{x(t)\overset{FT}{\leftrightarrow}X(\omega)}$$Then, according to duality property, $$\mathrm{X(t)\overset{FT}{\leftrightarrow}2\pi x(-\omega)}$$ProofFrom the definition of inverse Fourier transform, we have$$\mathrm{x(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}X(\omega)e^{j\omega t}d\omega }$$$$\mathrm{\Rightarrow 2\pi.x(t)=\int_{-\infty}^{\infty}X(\omega)e^{j\omega ... Read More

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

Importance of Wave SymmetryIf a periodic signal 𝑥(𝑡) has some type of symmetry, then some of the trigonometric Fourier series coefficients may become zero and hence the calculation of the coefficients becomes simple.Even or Mirror SymmetryWhen a periodic function is symmetrical about the vertical axis, it is said to have even symmetry or mirror symmetry. The even symmetry is also called the reflection symmetry. Mathematically, a periodic function x(t) is said to have even symmetry, if$$\mathrm{𝑥(𝑡) = 𝑥(−𝑡)\:\:\:\:\:\: ...(1)}$$Some examples of functions having even symmetry are shown in the figure. The even functions are always symmetrical about the vertical axis.ExplanationAs ... 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