Why fourier analysis

Create a free Team What is Teams? Learn more. Why is the Fourier transform so important? Ask Question. Asked 10 years, 2 months ago. Active 3 years, 3 months ago. Viewed k times. Does it only apply to digital signal processing or does it apply to analog signals as well? Improve this question. Lorem Ipsum 5, 3 3 gold badges 31 31 silver badges 41 41 bronze badges.

SE, and I thought that people on this site might find some of it worthwhile, and might even want to participate. Fourier series date at least as far back as Ptolemy's epicyclic astronomy. Adding more eccentrics and epicycles, akin to adding more terms to a Fourier series, one can account for any continuous motion of an object in the sky.

Add a comment. Active Oldest Votes. History: To understand the importance of the Fourier transform, it is important to step back a little and appreciate the power of the Fourier series put forth by Joseph Fourier. The Fourier transform: The Fourier transform can be viewed as an extension of the above Fourier series to non-periodic functions.

Thus, the transform is energy preserving. While this falls under the elementary property category, this is a widely used property in practice, especially in imaging and tomography applications, Example: When a wave travels through a heterogenous medium, it slows down and speeds up according to changes in the speed of wave propagation in the medium.

Digital signal processing DSP vs. Analog signal processing ASP The theory of Fourier transforms is applicable irrespective of whether the signal is continuous or discrete, as long as it is "nice" and absolutely integrable. Improve this answer. Lorem Ipsum Lorem Ipsum 5, 3 3 gold badges 31 31 silver badges 41 41 bronze badges. Note that a Taylor series is not an expansion in terms of the constituent frequencies. For e. The latter is the correct frequency representation, so I'm not sure if any comparisons with Taylor series is apt here.

Signals can be represented with respect to many different bases. Sines and cosines are special because they are the eigenfunctions of LTI systems. Show 4 more comments. Community Bot 1. Peter K. I think that following the philosophy of choice on the academic correctness over "popularity" of an answer, your answer should be be integrated into the above answer provideded by Lorem Ipsum, which despite being selected as the answer with 96 points by the users, lacks this very important point of view.

Edit: Since people asked me to write why the FFT is fast It's because it cleverly avoids doing extra work. It has everything to do with discrete Fourier transforms. Without the FFT, your Fourier transforms would take time proportional to the square of the input size, which would make them a lot less useful. But with the FFT, they take time proportional the size of the input times its logarithm , which makes them much more useful, and which speeds up a lot of calculations.

Also this might be an interesting read. Where its fast and why do we care that its fast? It's fast because the algorithm is fast It should be paraphrased - "Beside all the nice characteristics explained in other peoples answer, FFT allows it to become a feasible approach in real-time applications". Show 3 more comments. I want to calculate the TWA osha.

We can use the formal mathematical definition too. Scott Scott 4 4 silver badges 7 7 bronze badges. David David 61 1 1 silver badge 1 1 bronze badge. The Overflow Blog. Podcast Explaining the semiconductor shortage, and how it might end. Does ES6 make JavaScript frameworks obsolete? Featured on Meta.

Now live: A fully responsive profile. Linked 1. See more linked questions. Time delay in computer-based signal processing plays an analogous role to differentiation in the physical world. Many signal-processing operations involve weighted combinations of delayed signals. It turns out that delay of a discrete-time complex sinusoid results in a discrete-time sinusoid of the identical frequency. Only the amplitude and phase are changed. Similarly, a delay of a sum of sinusoids of different frequencies results in a sum of sinusoids of the same frequencies.

Linear, time-invariant LTI systems are widely used to model effects in the physical world and are also widely used to manipulate signals in signal processing. A continuous-time LTI system is described as a weighted sum of derivatives of the signals. A discrete-time LTI system is described as a weighted sum of delayed signals. Hence, due to the differentiation and time delay properties, the output of an LTI system in response to a complex sinusoid input is a complex sinusoid of the same frequency.

This is the so-called "eigenfunction" property. It follows that an input expressed as an arbitrary sum of complex sinusoids of different frequencies produces an output given by a sum of complex sinusoids of the same frequencies. The system only modifies the amplitude and phase of the individual sinusoids in the sum. The manner in which the system modifies the amplitudes and phase of the input sinusoids is called the frequency response of the system.

Frequency response is a very intuitive description for the action of an LTI system on a signal. The magnitude of the frequency response tells us how the amplitudes of the input sinusoidal components are changed by the system. This gives rise to the idea of using systems to "filter" signals. A filter separates out certain frequency components of the input signal. The figure at left illustrates the magnitude of the frequency response in dB for a low-pass filter.

Lower frequency sinusoids in the input signal see a gain of 0 dB or unity and are not attenuated. Fourier methods are natural tools for understanding and modeling the effects of the physical world on signals, and for designing and characterizing common signal processing systems.

The discrete Fourier transform may be computed very efficiently using an FFT algorithm. The computational efficiency of FFT algorithms is a direct consequence of the properties of complex sinusoids.

Computational power was quite limited in those early days of digital computing, and the FFT opened up possibilities for computer analysis of signals that were previously unimaginable. Thus, FFT algorithms fueled the rapid growth of the new field of signal processing. Financial Analysis. Tools for Fundamental Analysis. Technical Analysis Basic Education. Fundamental Analysis. Actively scan device characteristics for identification. Use precise geolocation data. Select personalised content.

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What Is Fourier Analysis?