### Fourier transform amplitude

By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Signal Processing Stack Exchange is a question and answer site for practitioners of the art and science of signal, image and video processing. It only takes a minute to sign up. I totally understand the concept of fourier transform, but one thing thats bothering me is the amplitude that we plot in the frequency domain. What does that amplitude of each frequency signifies?

Is it the amplitude of that particular signal component in the input signal, as it is for the frequency? Is there any other meaning or any other way we can infer the amplitude? Lucky you if you really do. Some of us me, in first place don't in totality.

Krunker discord linkThe Fourier transform and its avatars is a prototype for duality. Duality here means that you can represent a signal on some primal domain time onto a dual domain here frequency.

This transformation is meant to possess useful properties, to preserve signal information, and to add insight, like a somewhat easier interpretation. Signals are traditionally represented as "amplitudes of something for each time".

These amplitudes are not always absolute. But we can hope they are linearly related to the actual physical values. And that the linearity factor does not change over time. Under the above premises, sines and cosines, or complex exponentials cisoidsare a very good way perhaps the best to model the system. And in turn they can be used to represent the data, as a linear combinaison of shifted sines.

Since the sines are orthogonal, the squared magnitude of the coefficients weighting the sines is proportional to the energy of each corresponding sine frequency. The proportional factor depends on how the coefficients are computed, often up to a scaling factor. The amplitude of a given frequency component can be directly computed with correlation.

There are various flavors physical interpretations of the continuous Fourier Transform. Here is the one that works best for me:. The amplitude of the Fourier Transform is a metric of spectral density. Loosely speaking it's a measure of how much energy per unit of bandwidth you have.

That works both in the time and the frequency domain and the total energy can be calculated the same way squaring and integrating and they are the same. That's Parseval's Theorem. A somewhat different interpretation is to view the Fourier Transform as a representation through orthogonal functions. These orthogonal functions are the complex exponential and they form an ortho-normal base more or less.

Basically that means you can represent any time domain signal as being made up of an infinite number of complex exponential. Similar to how any Lego contraption is made up of the same set of elementary Lego blocks. The Fourier Transform amplitude simply tells you how much of each Logo black are in any contraption.One of the most important uses of the Fourier transform is to find the amplitude and phase of a sinusoidal signal buried in noise. One of the spikes is at index 32 and the other is at N The most important thing is to determine what the frequencies are from the spectrum.

The first spike is at a positive frequency and the second is at a negative frequency. Why negative frequencies?

### Fourier transform

Basically, the reason is that two spikes are required to provide information about both amplitude and phase. We have already seen amplitudes in the previous examples: abs c [32] and abs c [ N ] are both amplitudes. But if you print c [32] and c [ N ], you will see that they have large imaginary parts:. The phase of c [32] is atan These are almost exactlywhich is the phase of. You may have noticed a difference in indexing between equation 1 and the FFT.

This is just something you and the rest of us have to live with. The difference is a matter of context, and one has to be aware of the context at all times in using FFTs. To make this clear, try doing the FFT of the old Sawtooth function. The two spikes produced by the sinusoidal signal will show you precisely how to convert the FFT index into real frequency.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Mathematica Stack Exchange is a question and answer site for users of Wolfram Mathematica. It only takes a minute to sign up. This is a popular question with several answers. This edit organises the answers and gives links.

A method due to Daniel Lichtblau that looks for periodicities in the time history answer below. A method using Prony series. Code from Daniel Lichtblau answer below. Straightforward fitting of sine wave to data. There are pitfalls in formulating this problem which are discussed here. A method using auto-correlation due to Szabolics. Answer below. The original question was how to use the output from Fourier when the computed values do not lie at a single point.

This is answered in two ways below. The problem is that a typical harmonic time history will not have an integer number of cycles in the data. For example, with the following time history you get a numerical Fourier transform. This produces a plot with several points making up the peak.

The frequency resolution is 2 Hz and there is no point at the harmonic frequency of Clearly the frequency is bracketed between the 9 th and 10 th point 16 Hz and 18 Hz. How do we find the exact frequency and amplitude? I have considered extending by zeros but although this increases the resolution it still results in points on either side of the frequency. Also, I have considered calculating points with non-integer values.

However, the resulting spectrum does not have a maximum at the frequency of the time history. My actual data includes noise and other harmonics hence my use of Fourier which is good at concentrating the data I am interested in around a few points in the spectrum. I have worked out one very poor way of finding the frequency and amplitude and will post this when I have worked out how to use StackExchange more fully. I also will give a different method, based heavily on one from this prior period estimation post.

Before getting to the somewhat thorny code let me list its advantages and disadvantages. We now sort differences of consecutive y values, get corresponding differences of x values, and discard those that are too close to be a period apart. This gives good contenders for separation by an integer multiple of periods lengths. And the frequency should be the reciprocal of the period.

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Also we know, from e. So take those values that are in that ballpark. Here is a perhaps safer way, more in line with what is at the referenced link.

We start just prior to where we computed gaps above.This tutorial describes the calculation of the amplitude and the phase from DFT spectra with finite sampling.

The DFT matrix is intuitively understood as a set of probes, each sensitive for a certain frequency and corresponding phase information present in a sequential sample of data, distributed in time or space. It delivers information about the amplitudes, frequencies and phases of the basic cosines spanning the function of which an interval having a length of integer number of periods is being sampled into the vector.

The index r may count samples taken progressing in time or space. During the matrix multiplication the vector of angular orientation corresponding to r is multiplied with the sample of value and the vector sum is calculated over all r, giving a complex number.

This is repeated for all s from 1 to n, each signaling the presence of a contribution in the samples oscillating with a frequency related to s.

The set of all represents then the spectrum. Note the cancellation of the interval tt inside the argument of the exponential. If the samples contain parts changing with the same period as the rotating unit vector this vector sum will exhibit a large magnitude, because the biggest product vectors all point roughly in the same direction. If the changes of the samples are of a different period with respect to the rotating unit vector the sums magnitude will be smaller.

This will be shown in the next four pictures, their production code is also given. Above we see the rotating unit vector multiplied with a sampled sine function, so the magnitude of the vector is proportional to the value of the sine. The sine and the vectors appear in the graphic to lie in the same coordinate system and show their relation in a clear way.

The vectors were drawn starting from the position of the sampled points. All rotating vectors complex numbers are summed and the result see the red line below, beginning in the origin 0,0 and ending in the point pe upper black point is large. It corresponds to the Fourier amplitude of frequency 1Hz. Next the same function is sampled, but the unit vector is rotating more frequently not once but four times!

Above we see the rotating unit vector multiplied with a sampled sine function being out of phase different frequency. The red line representing the sum of rotating vectors has completely disappeared behind the overlapping black points, see below. The scale is enlarged compared to the previous vector sum plot note the length of the edges. The length of the red line is too close to zero to be displayed :. Now we consider more complicated functions.

For example let's define a periodic function g[t]. The lower limit for n lies in this case at. The Drop command is used to get the number of samples right :. The next graph shows the function g[t], the samples and the 'cosine components' of the function. Recall that the first sample sits on the left edge of the interval, whereas the last sample is taken inside the interval near the right edge. Only iff the point on the right edge of the sampling interval of the period is dropped, the correct reconstruction and periodic continuation is ensured From the samples we get the complex Fourier array.

The most relevant bins as well as their mirrors are readily identified offset in s by 1, because the first term in the spectrum is the one describing the zero frequency part of the functionall others being zero. This operation leaves the matrix element, which is just a complex number, unchanged. There will then be pairs of rows with the same row number distance to the middle.

The entries of these paired rows will be complex conjugates of each other. Going further up in s we get the same sequence, but in reverse order. The next statement collects all bin numbers of the spectrum with bin values over threshold into a list s.

The threshold is used to suppress insignificant bins not necessary in this example which is very clean, but add some noise Now using the Fourier spectrum the offset and amplitudes are found see next section to be.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. What I'm trying to understand is an analogy with the fourier series. In the fourier series the coefficients are the amplitude for each of the particular waves that make our signal. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Fourier Transform and amplitude of waves Ask Question. Asked 6 years, 7 months ago. Active 6 years, 7 months ago.

Viewed 1k times. Ambesh Ambesh 2, 1 1 gold badge 18 18 silver badges 42 42 bronze badges. And just like there is a continuum of time values, there is a continuum of frequency values.

## Fourier Transformation and Its Mathematics

Active Oldest Votes. For a continuous random variable, we use a probability density function. Fourier transform is a "frequency density function". Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. The Overflow Blog. Q2 Community Roadmap.

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Ivermectin overdose humansAutofilters for Hot Network Questions. Related 7. Hot Network Questions.The Fourier transform FT decomposes a signal into the frequencies that make it up. What does this mean?

Biology unit 1 packetLet we have a signal S1. If we want to measure the strength of this signal at some specific time.

**Fourier Transform Example 05 - Impulses and Constants**

We measure it by its amplitude. So, the amplitude of the signal S1 is 1. If we do the same for another signal and select the same moment in time and we measure its amplitude.

Let we have another signal S2 like this. Also, the amplitude of the signal S2 is 2. Now, what happens when we emit these two signals at the same time S1 and S2? So, when we emit these two signals at the same moment of time, we get a new signal which is the sum of the amplitude of these two signals.

This is so because these two signals are being added together. Now, the interesting question is :. If we are given signal S3 only which is the sum of signals S1 and S2.

Can we recover the original signals S1 and S2? That's what a Fourier transform does. It takes up a signal and decomposes it to the frequencies that made it up.

In our example, a Fourier transform would decompose the signal S3 into its constituent frequencies like signals S1 and S2. But, How can we recover the original signals? What will the Fourier transform do for us? What Fourier transform does is It kind of moves us from the time domain to frequency domain. In case, If anyone is wondering, What if we want to go back from the frequency domain to the time domain? But we won't be discussing this in this article.The Fourier transform FT decomposes a function often a function of time, or a signal into its constituent frequencies.

A special case is the expression of a musical chord in terms of the volumes and frequencies of its constituent notes.

The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time. The Fourier transform of a function of time is itself a complex -valued function of frequency, whose magnitude modulus represents the amount of that frequency present in the original function, and whose argument is the phase offset of the basic sinusoid in that frequency. The Fourier transform is not limited to functions of time, but the domain of the original function is commonly referred to as the time domain.

There is also an inverse Fourier transform that mathematically synthesizes the original function from its frequency domain representation. A sinusoidal curve, with peak amplitude 1peak-to-peak 2RMS 3and wave period 4. Linear operations performed in one domain time or frequency have corresponding operations in the other domain, which are sometimes easier to perform.

The operation of differentiation in the time domain corresponds to multiplication by the frequency, [remark 1] so some differential equations are easier to analyze in the frequency domain. Also, convolution in the time domain corresponds to ordinary multiplication in the frequency domain see Convolution theorem.

After performing the desired operations, transformation of the result can be made back to the time domain. Harmonic analysis is the systematic study of the relationship between the frequency and time domains, including the kinds of functions or operations that are "simpler" in one or the other, and has deep connections to many areas of modern mathematics.

Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle.

The critical case for this principle is the Gaussian functionof substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution e. The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the transform in his study of heat transferwhere Gaussian functions appear as solutions of the heat equation. The Fourier transform can be formally defined as an improper Riemann integralmaking it an integral transformalthough this definition is not suitable for many applications requiring a more sophisticated integration theory.

This idea makes the spatial Fourier transform very natural in the study of waves, as well as in quantum mechanicswhere it is important to be able to represent wave solutions as functions of either position or momentum and sometimes both.

In general, functions to which Fourier methods are applicable are complex-valued, and possibly vector-valued.

The latter is routinely employed to handle periodic functions. Many other characterizations of the Fourier transform exist. InJoseph Fourier showed that some functions could be written as an infinite sum of harmonics. One motivation for the Fourier transform comes from the study of Fourier series.

Autec kba 43494In the study of Fourier series, complicated but periodic functions are written as the sum of simple waves mathematically represented by sines and cosines. The Fourier transform is an extension of the Fourier series that results when the period of the represented function is lengthened and allowed to approach infinity. Due to the properties of sine and cosine, it is possible to recover the amplitude of each wave in a Fourier series using an integral.

This has the advantage of simplifying many of the formulas involved, and provides a formulation for Fourier series that more closely resembles the definition followed in this article.

Re-writing sines and cosines as complex exponentials makes it necessary for the Fourier coefficients to be complex valued. The usual interpretation of this complex number is that it gives both the amplitude or size of the wave present in the function and the phase or the initial angle of the wave.

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