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Fourier TransformFor a continuous-time function x(t), the Fourier transform of x(t) can be defined as, $$\mathrm{X(\omega)=\int_{-\infty }^{\infty}x(t)\:e^{-jwt}\:dt}$$And the inverse Fourier transform is defined as, $$\mathrm{x(t)=\frac{1}{2\pi}\int_{-\infty }^{\infty}X(\omega)\:e^{jwt}\:d\omega}$$Time Integration Property of Fourier TransformStatementThe time integration property of continuous-time Fourier transform states that the integration of a function x(t) in time domain is equivalent to the division of its Fourier transform by a factor jω in frequency domain. Therefore, if, $$\mathrm{x(t)\overset{FT}{\leftrightarrow}X(\omega )}$$Then, according to the time integration property$$\mathrm{\int_{-\infty }^{t}x(\tau )\:\overset{FT}{\leftrightarrow}\frac{X(\omega )}{j\omega };\:\:(if\:X(0)=0)}$$ProofWhen X(0)=0; then the time integration property of CTFT can be proved by using integration by parts.Therefore, from the definition of inverse ... Read More
Linear Time Invariant SystemA system for which the principle of superposition and the principle of homogeneity are valid and the input/output characteristics do not with time is called the linear time invariant (LTI) system.Properties of LTI SystemA continuous-time LTI system can be represented in terms of its unit impulse response. It takes the form of convolution integral. Hence, the properties followed by the continuous time convolution are also followed by the LTI system. The impulse response of an LTI system is very important because it can completely determine the characteristics of an LTI system.In this article, we will highlight some ... Read More
For a continuous-time function 𝑥(𝑡), the Fourier transform of 𝑥(𝑡) can be defined as, $$\mathrm{X\left ( \omega \right )=\int_{-\infty }^{\infty }x\left ( t \right )e^{-j\omega t}\: dt}$$Time-Shifting Property of Fourier TransformStatement – The time shifting property of Fourier transform states that if a signal 𝑥(𝑡) is shifted by 𝑡0 in time domain, then the frequency spectrum is modified by a linear phase shift of slope (−𝜔𝑡0). Therefore, if, $$\mathrm{x\left ( t \right )\overset{FT}{\leftrightarrow}X\left ( \omega \right )}$$Then, according to the time-shifting property of Fourier transform, $$\mathrm{x\left ( t -t_{0}\right )\overset{FT}{\leftrightarrow}e^{-j\omega t_{0}}X\left ( \omega \right )}$$ProofFrom the definition of Fourier transform, ... Read More
For a continuous-time function 𝑥(𝑡), the Fourier transform of 𝑥(𝑡) can be defined as, $$\mathrm{X\left ( \omega \right )=\int_{-\infty }^{\infty }x\left ( t \right )e^{-j\omega t}\: dt}$$Time Reversal Property of Fourier TransformStatement – The time reversal property of Fourier transform states that if a function 𝑥(𝑡) is reversed in time domain, then its spectrum in frequency domain is also reversed, i.e., if$$\mathrm{x\left ( t \right )\overset{FT}{\leftrightarrow}X\left ( \omega \right )}$$Then, according to the time-reversal property of Fourier transform, $$\mathrm{x\left ( -t \right )\overset{FT}{\leftrightarrow}X\left ( -\omega \right )}$$ProofForm the definition of Fourier transform, we have, $$\mathrm{F\left [ x\left ( t \right ... Read More
Signal BandwidthThe spectral components of a signal extends from (−∞) to ∞ and any practical signal has finite energy. Consequently, the spectral components approach zero when the frequency 𝜔 tends to ∞. Therefore, those spectral components can be neglected which have negligible energy and hence only a band of frequency components is selected which have most of the signal energy. This band of frequency components which contains most of the signal energy is called the signal bandwidth.Normally, this band has the frequency components that contains around 95% of the total energy depending upon the precision.System BandwidthA system which has infinite ... Read More
ConvolutionThe convolution is a mathematical operation for combining two signals to form a third signal. In other words, the convolution is a mathematical way which is used to express the relation between the input and output characteristics of an LTI system.Mathematically, the convolution of two signals is given by, $$\mathrm{x_{1}\left ( t \right )\ast x_{2}\left ( t \right )=\int_{-\infty }^{\infty }x_{1}\left ( \tau \right )x_{2}\left ( t-\tau \right )d\tau =\int_{-\infty }^{\infty }x_{2}\left ( \tau \right )x_{1}\left ( t-\tau \right )d\tau}$$CorrelationThe correlation is defined as the measure of similarity between two signals or functions or waveforms. The correlation is of two ... Read More
The scipy.interpolate.interp1d(x, y, kind, axis, copy, bounds_error, fill_value, assumesorted) class of SciPy library, as name implies, is used to interpolate a 1-Dimensional function. Here, x and y are the arrays of values which are used to approximate some function, say f; y=f(x). The output of this class is a function whose call method uses interpolation to find the value of new points.Below is given the detailed explanation of its parameters −Parametersx − (N, ) array_likeIt is a 1-dimensional array of real values.y − (…, N, …) array_likeIt is a N-dimensional array of real values. The condition is that the length ... Read More
NumPy, stands for Numerical Python, is used for the manipulation of elements of numerical array data. SciPy, stands for Scientific Python, is used for numerical computations in Python. Both these packages provide extended functionality to work with Python. Let’s understand some basic differences between NumPy and SciPy −Functional differences − NumPy has a faster processing speed than SciPy. The functions defined in NumPy library are not in depth whereas SciPy library consists of detailed versions of the functions. SciPy is built on NumPy and it is recommended to use both libraries altogether for fast and efficient scientific and mathematical computations.Array concept − ... Read More
SciPy is built upon the following core packages −Python − Python, a general-purpose programming language, is dynamically typed and interpreted. It is well suited for interactive work and quick prototyping. It is also powerful to write AI and ML applications.NumPy − NumPy is a base N-dimensional array package for SciPy that allows us to efficiently work with data in numerical arrays. It is the fundamental package for numerical computation.Matplotlib − Matplotlib is used to create comprehensive 2-dimensional charts and plots from data. It also provides us basic 3-dimensional plotting.The SciPy library − It is one of the core packages providing us many user-friendly and ... Read More
We can install Python SciPy with the help of following methods −Scientific Python Distributions − There are various scientific Python distributions that provide the language itself along with the most used packages. The advantage of using these distributions is that they require little configuration and work on almost all the setups. Here we will be discussing three most useful distributions −Anaconda − Anaconda, a free Python distribution, works well on MS Windows, Mac OS, and Linux. It provides us over 1500 Python and R packages along with a large collection of libraries. This Python distribution is best suited for beginners.WinPython − It ... Read More