# How fft works

By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. For five years I tried to understand how Fourier transform works. Read a lot of articles, but nobody could explain it in simple terms.

## Fast Fourier transform

It turns out to be very simple thing! I feel that all the theoretical elements are present in the others' responses, but not the pragmatic reason why FFT reduces the complexity by a significant factor and hence the popularity of this algorithm among other things.

Applying the DFT amounts to multiplying a matrix to a vector your original, sampled, signal, for example, each entry of your vector being one sample.

That matrix is the Fourier matrix. Now the magic happens because of a so-called butterfly property of these matrices, you can decompose them in a product of matrices: each of which is cheap to apply to a vector cf.

Let's look at the first two matrices: their structure is very typical, you could show by recursion how this generalizes but more about this later.

What matters for us now is that there are only 2 non-zero entries per line. Finally, the last matrix in the decomposition is a bit-reversal matrix. Not going in the details it can be computed extremely efficiently. If it is not the case, it is cheaper to pad your vector with zeros this is actually called zero padding. The last thing to clarify is the butterfly idea see also .

For this, nothing better than a small drawing sorry I didn't want to do this in Tikz, if there's a hero out there If you represent the vectors after each stage by dots and you connect the dots from the original vector to the vector after the matrix has been applied, you get this nice diagram ignoring the bit reversal.

Just to make it clearer, starting from the right it means that the first matrix multiplication after the bit-reversal will.

#65: Basics of using FFT on an oscilloscope

Further, observe that, at each level, we can consider blocks of entries independently. Again this leads to further optimization that can make the FFT very very efficient. Ps: I've ignored all multiplication constants since that's an O 1 computation, it's easy to check what they are and it will depend on conventions you're using. As said by user, the FFT is just an efficient computational procedure and does not explain the theory behind the discrete Fourier transform.

This principle is applied recursively, i. You can multiply the sparse matrices and obtain the DFT in fewer operations than you could by simply multiplying the DFT matrix. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered.

How does FFT work? Ask Question. Asked 3 years, 7 months ago. Active 3 years, 7 months ago. Viewed 5k times. Now I want to understand how FFT works.

And again nobody can explain it in simple terms. Can anyone explain it to a non mathematician? As it is it is unclear what you are asking?

Remember that asking for a guided walkthru of a chapter in a book will be closed as too broad. You can easily compute these eigenvectors yourself right now.The Fourier Transform is one of deepest insights ever made. Unfortunately, the meaning is buried within dense equations:. Rather than jumping into the symbols, let's experience the key idea firsthand. Here's a plain-English metaphor:. Time for the equations? Let's get our hands dirty and experience how any pattern can be built with cycles, with live simulations.

If all goes well, we'll have an aha! We'll save the detailed math analysis for the follow-up.

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A math transformation is a change of perspective. We change our notion of quantity from "single items" lines in the sand, tally system to "groups of 10" decimal depending on what we're counting. Scoring a game? Tally it up. Decimals, please. The Fourier Transform changes our perspective from consumer to producer, turning What do I have?

Well, recipes are great descriptions of drinks. A recipe is more easily categorized, compared, and modified than the object itself. Filters must be independent.

The banana filter needs to capture bananas, and nothing else. Adding more oranges should never affect the banana reading. Filters must be complete.

### An Interactive Guide To The Fourier Transform

We won't get the real recipe if we leave out a filter "There were mangoes too! Our collection of filters must catch every possible ingredient. Ingredients must be combine-able. Smoothies can be separated and re-combined without issue A cookie? Not so much. Who wants crumbs? The ingredients, when separated and combined in any order, must make the same result.

The Fourier Transform takes a specific viewpoint: What if any signal could be filtered into a bunch of circular paths? This concept is mind-blowing, and poor Joseph Fourier had his idea rejected at first.Documentation Help Center. If X is a vector, then fft X returns the Fourier transform of the vector. If X is a matrix, then fft X treats the columns of X as vectors and returns the Fourier transform of each column.

If X is a multidimensional array, then fft X treats the values along the first array dimension whose size does not equal 1 as vectors and returns the Fourier transform of each vector.

## An Introduction to the Fast Fourier Transform

If no value is specified, Y is the same size as X. If X is a vector and the length of X is less than nthen X is padded with trailing zeros to length n.

If X is a vector and the length of X is greater than nthen X is truncated to length n. If X is a matrix, then each column is treated as in the vector case. If X is a multidimensional array, then the first array dimension whose size does not equal 1 is treated as in the vector case.

For example, if X is a matrix, then fft X,n,2 returns the n-point Fourier transform of each row. Use Fourier transforms to find the frequency components of a signal buried in noise. Specify the parameters of a signal with a sampling frequency of 1 kHz and a signal duration of 1.

Plot the noisy signal in the time domain. It is difficult to identify the frequency components by looking at the signal X t. Compute the two-sided spectrum P2. Then compute the single-sided spectrum P1 based on P2 and the even-valued signal length L. Define the frequency domain f and plot the single-sided amplitude spectrum P1. The amplitudes are not exactly at 0. On average, longer signals produce better frequency approximations.

Now, take the Fourier transform of the original, uncorrupted signal and retrieve the exact amplitudes, 0. To use the fft function to convert the signal to the frequency domain, first identify a new input length that is the next power of 2 from the original signal length.

This will pad the signal X with trailing zeros in order to improve the performance of fft. Specify the parameters of a signal with a sampling frequency of 1kHz and a signal duration of 1 second. Create a matrix where each row represents a cosine wave with scaled frequency. The result, Xis a 3-by matrix.

The first row has a wave frequency of 50, the second row has a wave frequency ofand the third row has a wave frequency of Plot the first entries from each row of X in a single figure in order and compare their frequencies. For algorithm performance purposes, fft allows you to pad the input with trailing zeros. In this case, pad each row of X with zeros so that the length of each row is the next higher power of 2 from the current length. Define the new length using the nextpow2 function.The FFT is a complicated algorithm, and its details are usually left to those that specialize in such things.

This section describes the general operation of the FFT, but skirts a key issue: the use of complex numbers.

If you have a background in complex mathematics, you can read between the lines to understand the true nature of the algorithm. Don't worry if the details elude you; few scientists and engineers that use the FFT could write the program from scratch. In complex notation, the time and frequency domains each contain one signal made up of N complex points.

Each of these complex points is composed of two numbers, the real part and the imaginary part. For example, when we talk about complex sample X , it refers to the combination of ReX  and ImX . In other words, each complex variable holds two numbers. When two complex variables are multiplied, the four individual components must be combined to form the two components of the product such as in Eq.

The following discussion on "How the FFT works" uses this jargon of complex notation. That is, the singular terms: signal, point, sampleand valuerefer to the combination of the real part and the imaginary part. The FFT operates by decomposing an N point time domain signal into N time domain signals each composed of a single point. The second step is to calculate the N frequency spectra corresponding to these N time domain signals. Lastly, the N spectra are synthesized into a single frequency spectrum.

Figure shows an example of the time domain decomposition used in the FFT. In this example, a 16 point signal is decomposed through four. The first stage breaks the 16 point signal into two signals each consisting of 8 points.

The second stage decomposes the data into four signals of 4 points. This pattern continues until there are N signals composed of a single point. An interlaced decomposition is used each time a signal is broken in two, that is, the signal is separated into its even and odd numbered samples.

The best way to understand this is by inspecting Fig. There are Log 2 N stages required in this decomposition, i.Fourier analysis converts a signal from its original domain often time or space to a representation in the frequency domain and vice versa. The DFT is obtained by decomposing a sequence of values into components of different frequencies. An FFT rapidly computes such transformations by factorizing the DFT matrix into a product of sparse mostly zero factors.

The difference in speed can be enormous, especially for long data sets where N may be in the thousands or millions. In the presence of round-off errormany FFT algorithms are much more accurate than evaluating the DFT definition directly or indirectly. There are many different FFT algorithms based on a wide range of published theories, from simple complex-number arithmetic to group theory and number theory.

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Fast Fourier transforms are widely used for applications in engineering, music, science, and mathematics. The basic ideas were popularized inbut some algorithms had been derived as early as The development of fast algorithms for DFT can be traced to Carl Friedrich Gauss 's unpublished work in when he needed it to interpolate the orbit of asteroids Pallas and Juno from sample observations. While Gauss's work predated even Joseph Fourier 's results inhe did not analyze the computation time and eventually used other methods to achieve his goal.

Between andsome versions of FFT were published by other authors. Frank Yates in published his version called interaction algorithmwhich provided efficient computation of Hadamard and Walsh transforms. InG. Danielson and Cornelius Lanczos published their version to compute DFT for x-ray crystallographya field where calculation of Fourier transforms presented a formidable bottleneck.

James Cooley and John Tukey published a more general version of FFT in that is applicable when N is composite and not necessarily a power of 2. To analyze the output of these sensors, an FFT algorithm would be needed. In discussion with Tukey, Richard Garwin recognized the general applicability of the algorithm not just to national security problems, but also to a wide range of problems including one of immediate interest to him, determining the periodicities of the spin orientations in a 3-D crystal of Helium The DFT is defined by the formula.

This method and the general idea of an FFT was popularized by a publication of Cooley and Tukey in but it was later discovered  that those two authors had independently re-invented an algorithm known to Carl Friedrich Gauss around  and subsequently rediscovered several times in limited forms. These are called the radix-2 and mixed-radix cases, respectively and other variants such as the split-radix FFT have their own names as well.

Although the basic idea is recursive, most traditional implementations rearrange the algorithm to avoid explicit recursion. Danielson The Rader—Brenner algorithm  is a Cooley—Tukey-like factorization but with purely imaginary twiddle factors, reducing multiplications at the cost of increased additions and reduced numerical stability ; it was later superseded by the split-radix variant of Cooley—Tukey which achieves the same multiplication count but with fewer additions and without sacrificing accuracy.How does the FFT work?

The FFT works by requiring a power of two length for the transform, and splitting the the process into cascading groups of two that's why it's sometimes called a radix-2 FFT. Similarly, there are patterns for when the sine and cosine are at 1. See Full Answer.

Discrete Fourier Transform DFT is the discrete version of the Fourier Transform FT that transforms a signal or discrete sequence from the time domain representation to its representation in the frequency domain. Deal Fell Through. Don't Fault the Teacher. Linear convolution is the basic operation to calculate the output for any linear time invariant system given its input and its impulse response.

Circular convolution is the same thing but considering that the support of the signal is periodic as in a circle, hance the name. Why is the need of FFT? In DSP we convert a signal into its frequency components, so that we can have a better analysis of that signal. Fourier Transform FT is used to convert a signal into its corresponding frequency domain. More formally, it decomposes any periodic function or periodic signal into the sum of a possibly infinite set of simple oscillating functions, namely sines and cosines or, equivalently, complex exponentials.

The discrete-time Fourier transform is a periodic function, often defined in terms of a Fourier series. An MRI term for a very fast and efficient computational method of performing a Fourier transform—the mathematical process by which raw data is processed into a usable image.

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If X is a vector, then fft X returns the Fourier transform of the vector. If X is a matrix, then fft X treats the columns of X as vectors and returns the Fourier transform of each column.

What is an FFT spectrum? The FFT or Fast Fourier Transform spectrum analyser uses digital signal processing techniques to analyser a waveform with Fourier transforms to provide in depth analysis of signal waveform spectra. A spectrum analyzer measures the magnitude of an input signal versus frequency within the full frequency range of the instrument. The primary use is to measure the power of the spectrum of known and unknown signals. A fast Fourier transform FFT is an algorithm that samples a signal over a period of time or space and divides it into its frequency components.

What is the use of Fast Fourier Transform?

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What is the meaning of FFT? What is the use of Fourier Transform? The Fourier transform FT decomposes a function of time a signal into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the frequencies or pitches of its constituent notes.

The Fourier transform is called the frequency domain representation of the original signal. What is meant by fast Fourier transform? It shows the signal's spectral content, divided into discrete bins frequency bands.

The Fast Fourier Transform is a common algorithm for Fourier transforms. What is the FFT size? Larger FFT sizes provide higher spectral resolution but take longer to compute. FFT is based on divide and conquer algorithm where you divide the signal into two smaller signals, compute the DFT of the two smaller signals and join them to get the DFT of the larger signal. What is meant by Radix 2 FFT? The implementation is based on a well- known algorithm, called the Radix 2 FFTand requires that its' input data be an integral power of two in length.

What does FFT data mean?The results of the FFT are the same as with the DFT; the only difference is that the algorithm is optimized to remove redundant calculations. In general, the FFT can make these optimizations when the number of samples to be transformed is an exact power of two, for which it can eliminate many unnecessary operations. From Fourier we know that periodic waveforms can be modeled as the sum of harmonically-related sine waves.

The Fourier Transform aims to decompose a cycle of an arbitrary waveform into its sine components; the Inverse Fourier Transform goes the other way—it converts a series of sine components into the resulting waveform. You can essentially run the transform twice to convert from one form to the other and back!

How do we find its component sine waves? We start with an interesting property of sine waves. Take a look: To multiply two waves, simply multiply their values sample by sample to build the result. Our probe is a sine wave, traveling between The best part is that the target need not be a sine wave. The reason this works is that multiplying a sine wave by another sine wave is balanced modulation, which yields the sum and difference frequency sine waves. Any sine wave averaged over an integral number of cycles is zero.

Since the Fourier transform looks for components that are whole number multiples of the waveform section it is analyzing, and that section is also presumed to be a single cycle, the sum and difference results are always integral to the period. In that case the difference is 0 Hz, or DC though DC stands for Direct Current, the term is often used to describe steady-state offsets in any kind of waveform. Further, when the two waves are identical in phase, the DC value is a direct product of the multiplied sine waves.

If the phases differ, the DC value is proportional to the cosine of the phase difference. At first it might seem that we need to probe at many phases and take the best match; this would result in the ESFT—the Extremely Slow Fourier Transform. However, if we take a second measurement, this time with a cosine wave as a probe, we get a similar result except that the cosine measurement results are exactly in phase where the sine measurement is at its worst.

And when the target phase lies between the sine and cosine phase, both measurements get a partial match. Using the identity. This is it!