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## Introduction to the Fourier Transform

The Fourier Transform is a mathematical technique that transforms a function of tim*e, x(t)*, to a function of frequency, *X(&omega)*. It is closely related to the Fourier Series. If you are familiar with the Fourier Series, the following derivation may be helpful. If you are only interested in the mathematical statement of transform, please skip ahead to *Definition of Fourier Transform*.

The Fourier Transform of a function can be derived as a special case of the Fourier Series when the period, *T&rarr&infin* (Note: this derivation is performed in more detail elsewhere) . Start with the Fourier Series synthesis equation

where *c _{n}* is given by the Fourier Series analysis equation,

As *T&rarr&infin* the fundamental frequency, *&omega _{0}=2&pi/T*, becomes extremely small and the quantity

*n&omega*becomes a continuous quantity that can take on any value (since

_{0}*n*has a range of ±&infin) so we define a new variable

*&omega=n&omega*we also let

_{0}*X(&omega)=Tc*. Making these substitutions in the previous equation yields the analysis equation for the Fourier Transform (also called the Forward Fourier Transform).

_{n}Likewise, we can derive the Inverse Fourier Transform (i.e., the synthesis equation) by starting with the synthesis equation for the Fourier Series (and multiply and divide by *T*).

As *T&rarr&infin*, *1/T=&omega0/2&pi*. Since *&omega _{0}* is very small (as

*T*gets large, replace it by the quantity

*d&omega*). As before, we write

*&omega=n&omega0*and

*X(&omega)=Tc*. A little work (and replacing the sum by an integral) yields the synthesis equation of the Fourier Transform.

_{n}## Signal Generation and Phase Shift

If we want to describe a signal, we need three things :

**The frequency of the signal**which shows, how many occurrences in the period we have.**Amplitude**which shows the height of the signal or in other terms the strength of the signal.**Phase shift**as to where does the signal starts.

The very first example that we took was very simple, every signal had the same *frequency* and *phase difference* and only different *amplitudes*.

We will now look at a slightly complex example and we will look at the individual signal from the above example because in order to understand Fourier transform better we need to look at the individual signals closely.

Below is the original signals that we were looking above.

**Frequency:** If we look closely at the three signals, we will notice that the *frequency of all the three signals is different.*

*If for the same period of time, in signal 1 there is n number of waves then there are 2n number of waves in signal 2 and vice versa.*

**Phase:** Also, when we look closely as to where actually the signal starts. We will find that While *signal 1 starts at (0,0)*, *signal 2 starts at (-0.5,0)* if we trace the wave to meet the y-axis at 0. So, at 0 we already have the maximum amplitude of the signal. This is what we call *Phase shift*.

**Amplitude**: All the three signals have different amplitudes, Signal 1 has an amplitude of 1 while signal 2 and signal 3 has an amplitude of 2 and 3 respectively.

This is all captured in an elegant and super simple mathematical formula. So, In the above examples if the x-axis is termed as x and y-axis is termed as y. We can generate y as a function of t such that :

Using this formula, we can generate any type of signal that we want and then we can merge them together and play with them. For example, If we merge signals 1, 2 and 3. we will get a signal like this :

## College Math Teaching

I took a break and watched a 45 minute video on Fourier Transforms:

A few take away points for college mathematics instructors:

1. When one talks about the Laplace Transform, one should distinguish between the one sided and two sided transforms (e. g., the latter integrates over the full real line, instead of 0 to .

2. Engineers care about being able to take limits (e. g., using L’Hopitals rule and about problems such as )

3. Engineers care about DOMAINS they matter a great deal.

4. Sometimes the dabble in taking limits of sequences of functions (in an informal sense) here the Dirac Delta (a generalized function or distribution) is developed (informally) as a limit of Fourier transforms of a pulse function of height 1 and increasing width.

5. Even students at MIT have to be goaded into issuing answers.

6. They care about doing algebra, especially in the case of a change of variable.

So, I am teaching two sections of first semester calculus. I will emphasize things that students (and sometimes, faculty members of other departments) complain about.

## Contents

This wide applicability stems from many useful properties of the transforms:

- The transforms are linear operators and, with proper normalization, are unitary as well (a property known as Parseval's theorem or, more generally, as the Plancherel theorem, and most generally via Pontryagin duality). [2]
- The transforms are usually invertible.
- The exponential functions are eigenfunctions of differentiation, which means that this representation transforms linear differential equations with constant coefficients into ordinary algebraic ones. [3] Therefore, the behavior of a linear time-invariant system can be analyzed at each frequency independently.
- By the convolution theorem, Fourier transforms turn the complicated convolution operation into simple multiplication, which means that they provide an efficient way to compute convolution-based operations such as polynomial multiplication and multiplying large numbers. [4]
- The discrete version of the Fourier transform (see below) can be evaluated quickly on computers using fast Fourier transform (FFT) algorithms. [5]

In forensics, laboratory infrared spectrophotometers use Fourier transform analysis for measuring the wavelengths of light at which a material will absorb in the infrared spectrum. The FT method is used to decode the measured signals and record the wavelength data. And by using a computer, these Fourier calculations are rapidly carried out, so that in a matter of seconds, a computer-operated FT-IR instrument can produce an infrared absorption pattern comparable to that of a prism instrument. [6]

Fourier transformation is also useful as a compact representation of a signal. For example, JPEG compression uses a variant of the Fourier transformation (discrete cosine transform) of small square pieces of a digital image. The Fourier components of each square are rounded to lower arithmetic precision, and weak components are eliminated entirely, so that the remaining components can be stored very compactly. In image reconstruction, each image square is reassembled from the preserved approximate Fourier-transformed components, which are then inverse-transformed to produce an approximation of the original image.

### Applications in signal processing Edit

When processing signals, such as audio, radio waves, light waves, seismic waves, and even images, Fourier analysis can isolate narrowband components of a compound waveform, concentrating them for easier detection or removal. A large family of signal processing techniques consist of Fourier-transforming a signal, manipulating the Fourier-transformed data in a simple way, and reversing the transformation. [7]

- of audio recordings with a series of bandpass filters
- Digital radio reception without a superheterodyne circuit, as in a modern cell phone or radio scanner to remove periodic or anisotropic artifacts such as jaggies from interlaced video, strip artifacts from strip aerial photography, or wave patterns from radio frequency interference in a digital camera of similar images for co-alignment to reconstruct a crystal structure from its diffraction pattern mass spectrometry to determine the mass of ions from the frequency of cyclotron motion in a magnetic field
- Many other forms of spectroscopy, including infrared and nuclear magnetic resonance spectroscopies
- Generation of sound spectrograms used to analyze sounds
- Passive sonar used to classify targets based on machinery noise.

### (Continuous) Fourier transform Edit

Most often, the unqualified term **Fourier transform** refers to the transform of functions of a continuous real argument, and it produces a continuous function of frequency, known as a *frequency distribution*. One function is transformed into another, and the operation is reversible. When the domain of the input (initial) function is time ( t ), and the domain of the output (final) function is ordinary frequency, the transform of function *s*(*t*) at frequency f is given by the complex number:

Evaluating this quantity for all values of f produces the *frequency-domain* function. Then *s*(*t*) can be represented as a recombination of complex exponentials of all possible frequencies:

which is the inverse transform formula. The complex number, *S*( *f* ) , conveys both amplitude and phase of frequency f .

See Fourier transform for much more information, including:

- conventions for amplitude normalization and frequency scaling/units
- transform properties
- tabulated transforms of specific functions
- an extension/generalization for functions of multiple dimensions, such as images.

### Fourier series Edit

The Fourier transform of a periodic function, *s*_{P}(*t*) , with period P , becomes a Dirac comb function, modulated by a sequence of complex coefficients**:**

The inverse transform, known as **Fourier series**, is a representation of *s*_{P}(*t*) in terms of a summation of a potentially infinite number of harmonically related sinusoids or complex exponential functions, each with an amplitude and phase specified by one of the coefficients**:**

Any *s*_{P}(*t*) can be expressed as a periodic summation of another function, *s*(*t*) **:**

and the coefficients are proportional to samples of *S*( *f* ) at discrete intervals of 1 / *P* **:**

Note that any *s*(*t*) whose transform has the same discrete sample values can be used in the periodic summation. A sufficient condition for recovering *s*(*t*) (and therefore *S*( *f* ) ) from just these samples (i.e. from the Fourier series) is that the non-zero portion of *s*(*t*) be confined to a known interval of duration P , which is the frequency domain dual of the Nyquist–Shannon sampling theorem.

See Fourier series for more information, including the historical development.

### Discrete-time Fourier transform (DTFT) Edit

The DTFT is the mathematical dual of the time-domain Fourier series. Thus, a convergent periodic summation in the frequency domain can be represented by a Fourier series, whose coefficients are samples of a related continuous time function**:**

which is known as the DTFT. Thus the **DTFT** of the *s*[*n*] sequence is also the **Fourier transform** of the modulated Dirac comb function. [B]

The Fourier series coefficients (and inverse transform), are defined by**:**

Another reason to be interested in *S*_{1/T}( *f* ) is that it often provides insight into the amount of aliasing caused by the sampling process.

Applications of the DTFT are not limited to sampled functions. See Discrete-time Fourier transform for more information on this and other topics, including**:**

- normalized frequency units
- windowing (finite-length sequences)
- transform properties
- tabulated transforms of specific functions

### Discrete Fourier transform (DFT) Edit

Similar to a Fourier series, the DTFT of a periodic sequence, *s*_{N}[*n*] , with period N , becomes a Dirac comb function, modulated by a sequence of complex coefficients (see DTFT § Periodic data):

The *S*[*k*] sequence is what is customarily known as the **DFT** of one cycle of *s*_{N} . It is also N -periodic, so it is never necessary to compute more than N coefficients. The inverse transform, also known as a discrete Fourier series, is given by:

When *s*_{N}[*n*] is expressed as a periodic summation of another function:

Conversely, when one wants to compute an arbitrary number ( N ) of discrete samples of one cycle of a continuous DTFT, *S*_{1/T}( *f* ) , it can be done by computing the relatively simple DFT of *s*_{N}[*n*] , as defined above. In most cases, N is chosen equal to the length of non-zero portion of *s*[*n*] . Increasing N , known as *zero-padding* or *interpolation*, results in more closely spaced samples of one cycle of *S*_{1/T}( *f* ) . Decreasing N , causes overlap (adding) in the time-domain (analogous to aliasing), which corresponds to decimation in the frequency domain. (see DTFT § Sampling the DTFT) In most cases of practical interest, the *s*[*n*] sequence represents a longer sequence that was truncated by the application of a finite-length window function or FIR filter array.

The DFT can be computed using a fast Fourier transform (FFT) algorithm, which makes it a practical and important transformation on computers.

See Discrete Fourier transform for much more information, including:

- transform properties
- applications
- tabulated transforms of specific functions

### Summary Edit

For periodic functions, both the Fourier transform and the DTFT comprise only a discrete set of frequency components (Fourier series), and the transforms diverge at those frequencies. One common practice (not discussed above) is to handle that divergence via Dirac delta and Dirac comb functions. But the same spectral information can be discerned from just one cycle of the periodic function, since all the other cycles are identical. Similarly, finite-duration functions can be represented as a Fourier series, with no actual loss of information except that the periodicity of the inverse transform is a mere artifact.

It is common in practice for the duration of *s*(•) to be limited to the period, P or N . But these formulas do not require that condition.

*s*(

*t*) transforms (continuous-time)

Continuous frequency | Discrete frequencies | |
---|---|---|

Transform | S ( f ) ≜ ∫ − ∞ ∞ s ( t ) ⋅ e − i 2 π f t d t | 1 P ⋅ S ( k P ) ⏞ S [ k ] ≜ 1 P ∫ − ∞ ∞ s ( t ) ⋅ e − i 2 π k P t d t ≡ 1 P ∫ P s P ( t ) ⋅ e − i 2 π k P t d t >cdot Sleft( >
ight)> ^ >int _<-infty >^ >t>,dtequiv >int _ s_ (t)cdot e^<-i2pi >t>,dt> |

Inverse | s ( t ) = ∫ − ∞ ∞ S ( f ) ⋅ e i 2 π f t d f | s P ( t ) = ∑ k = − ∞ ∞ S [ k ] ⋅ e i 2 π k P t ⏟ Poisson summation formula (Fourier series) (t)=sum _ >t>> _< ext |

*s*(

*nT*) transforms (discrete-time)

### Symmetry properties Edit

When the real and imaginary parts of a complex function are decomposed into their even and odd parts, there are four components, denoted below by the subscripts RE, RO, IE, and IO. And there is a one-to-one mapping between the four components of a complex time function and the four components of its complex frequency transform**:** [8]

From this, various relationships are apparent, for example**:**

- The transform of a real-valued function (
*s*_{RE}+*s*_{RO}) is the even symmetric function S_{RE}+ i S_{IO}. Conversely, an even-symmetric transform implies a real-valued time-domain. - The transform of an imaginary-valued function ( i
*s*_{IE}+ i*s*_{IO}) is the odd symmetric function S_{RO}+ i S_{IE}, and the converse is true. - The transform of an even-symmetric function (
*s*_{RE}+ i*s*_{IO}) is the real-valued function S_{RE}+ S_{RO}, and the converse is true. - The transform of an odd-symmetric function (
*s*_{RO}+ i*s*_{IE}) is the imaginary-valued function i S_{IE}+ i S_{IO}, and the converse is true.

### Fourier transforms on arbitrary locally compact abelian topological groups Edit

The Fourier variants can also be generalized to Fourier transforms on arbitrary locally compact Abelian topological groups, which are studied in harmonic analysis there, the Fourier transform takes functions on a group to functions on the dual group. This treatment also allows a general formulation of the convolution theorem, which relates Fourier transforms and convolutions. See also the Pontryagin duality for the generalized underpinnings of the Fourier transform.

More specific, Fourier analysis can be done on cosets, [9] even discrete cosets.

### Time–frequency transforms Edit

In signal processing terms, a function (of time) is a representation of a signal with perfect *time resolution*, but no frequency information, while the Fourier transform has perfect *frequency resolution*, but no time information.

As alternatives to the Fourier transform, in time–frequency analysis, one uses time–frequency transforms to represent signals in a form that has some time information and some frequency information – by the uncertainty principle, there is a trade-off between these. These can be generalizations of the Fourier transform, such as the short-time Fourier transform, the Gabor transform or fractional Fourier transform (FRFT), or can use different functions to represent signals, as in wavelet transforms and chirplet transforms, with the wavelet analog of the (continuous) Fourier transform being the continuous wavelet transform.

An early form of harmonic series dates back to ancient Babylonian mathematics, where they were used to compute ephemerides (tables of astronomical positions). [10] [11] [12] [13]

The classical Greek concepts of deferent and epicycle in the Ptolemaic system of astronomy were related to Fourier series (see Deferent and epicycle § Mathematical formalism).

In modern times, variants of the discrete Fourier transform were used by Alexis Clairaut in 1754 to compute an orbit, [14] which has been described as the first formula for the DFT, [15] and in 1759 by Joseph Louis Lagrange, in computing the coefficients of a trigonometric series for a vibrating string. [15] Technically, Clairaut's work was a cosine-only series (a form of discrete cosine transform), while Lagrange's work was a sine-only series (a form of discrete sine transform) a true cosine+sine DFT was used by Gauss in 1805 for trigonometric interpolation of asteroid orbits. [16] Euler and Lagrange both discretized the vibrating string problem, using what would today be called samples. [15]

An early modern development toward Fourier analysis was the 1770 paper *Réflexions sur la résolution algébrique des équations* by Lagrange, which in the method of Lagrange resolvents used a complex Fourier decomposition to study the solution of a cubic: [17] Lagrange transformed the roots *x*_{1}, *x*_{2}, *x*_{3} into the resolvents:

where ζ is a cubic root of unity, which is the DFT of order 3.

A number of authors, notably Jean le Rond d'Alembert, and Carl Friedrich Gauss used trigonometric series to study the heat equation, [18] but the breakthrough development was the 1807 paper *Mémoire sur la propagation de la chaleur dans les corps solides* by Joseph Fourier, whose crucial insight was to model *all* functions by trigonometric series, introducing the Fourier series.

Historians are divided as to how much to credit Lagrange and others for the development of Fourier theory: Daniel Bernoulli and Leonhard Euler had introduced trigonometric representations of functions, and Lagrange had given the Fourier series solution to the wave equation, so Fourier's contribution was mainly the bold claim that an arbitrary function could be represented by a Fourier series. [15]

The subsequent development of the field is known as harmonic analysis, and is also an early instance of representation theory.

The first fast Fourier transform (FFT) algorithm for the DFT was discovered around 1805 by Carl Friedrich Gauss when interpolating measurements of the orbit of the asteroids Juno and Pallas, although that particular FFT algorithm is more often attributed to its modern rediscoverers Cooley and Tukey. [16] [14]

In signal processing, the Fourier transform often takes a time series or a function of continuous time, and maps it into a frequency spectrum. That is, it takes a function from the time domain into the frequency domain it is a decomposition of a function into sinusoids of different frequencies in the case of a Fourier series or discrete Fourier transform, the sinusoids are harmonics of the fundamental frequency of the function being analyzed.

When the function f is a function of time and represents a physical signal, the transform has a standard interpretation as the frequency spectrum of the signal. The magnitude of the resulting complex-valued function F at frequency ω represents the amplitude of a frequency component whose initial phase is given by the phase of F .

Fourier transforms are not limited to functions of time, and temporal frequencies. They can equally be applied to analyze *spatial* frequencies, and indeed for nearly any function domain. This justifies their use in such diverse branches as image processing, heat conduction, and automatic control.

### Example #1

#### Find the Fourier transform of exp $ left(-a x^ ight) $.

By fourier transform formula we have,

Here is the graph of fourier transform

### Example #2

#### Find the Fourier transform of a below non periodic function

The above function is not a periodic function.

A non periodic function cannot be represented as fourier series.But can be represented as **Fourier integral**.

Then,using Fourier integral formula we get,

This is the Fourier transform of above function.

We can find Fourier integral representation of above function using **fourier inverse transform**.

This is the **fourier integral representation** of our non periodic function.

## A New Twist to Fourier Transforms

Making use of the inherent helix in the Fourier transform expression, this book illustrates both Fourier transforms and their properties in the round. The author draws on elementary complex algebra to manipulate the transforms, presenting the ideas in such a way as to avoid pages of complicated mathematics. Similarly, abbreviations are not used throughout and the language is kept deliberately clear so that the result is a text that is accessible to a much wider readership.

The treatment is extended with the use of sampled data to finite and discrete transforms, the fast Fourier transform, or FFT, being a special case of a discrete transform. The application of Fourier transforms in statistics is illustrated for the first time using the examples operational research and later radar detection. In addition, a whole chapter on tapering or weighting functions is added for reference. The whole is rounded off by a glossary and examples of diagrams in three dimensions made possible by today's mathematics programs.

## Moving The Time Spike

Not everything happens at t=0. Can we change our spike to (0 4 0 0) ?

It seems the cycle ingredients should be similar to (4 0 0 0) , but the cycles must align at t=1 (one second in the future). Here's where phase comes in.

Imagine a race with 4 runners. Normal races have everyone lined up at the starting line, the (4 0 0 0) time pattern. Boring.

What if we want everyone to *finish* at the same time? Easy. Just move people forward or backwards by the appropriate distance. Maybe granny can start 2 feet in front of the finish line, Usain Bolt can start 100m back, and they can cross the tape holding hands.

Phase shifts, the starting angle, are delays in the cycle universe. Here's how we adjust the starting position to delay every cycle 1 second:

- A 0Hz cycle doesn't move, so it's already aligned
- A 1Hz cycle goes 1 revolution in the entire 4 seconds, so a 1-second delay is a quarter-turn. Phase shift it 90 degrees backwards (-90) and it gets to phase=0, the max value, at t=1.
- A 2Hz cycle is twice as fast, so give it twice the angle to cover (-180 or 180 phase shift -- it's across the circle, either way).
- A 3Hz cycle is 3x as fast, so give it 3x the distance to move (-270 or +90 phase shift)

If time points (4 0 0 0) are made from cycles [1 1 1 1] , then time points (0 4 0 0) are made from [1 1:-90 1:180 1:90] . (Note: I'm using "1Hz", but I mean "1 cycle over the entire time period").

Whoa -- we're working out the cycles in our head!

The interference visualization is similar, except the alignment is at t=1.

Test your intuition: Can you make (0 0 4 0) , i.e. a 2-second delay? 0Hz has no phase. 1Hz has 180 degrees, 2Hz has 360 (aka 0), and 3Hz has 540 (aka 180), so it's [1 1:180 1 1:180] .

## Elementary Fourier Transforms - Mathematics

It turns out the complex form of the equations makes things a lot simpler and more elegant. As such, everyone uses complex numbers, from physicists, to engineers, and mathematicians. So get used to it, it is actually a very beautiful thing.

On this page we'll start by introducing complex numbers and some simple properties, useful in the study of the Fourier Transform.

A complex number *z* can be written in standard form as:

The complex number *z* has a real part given by *x* and an *imaginary part* given by y. The real part of *z* is written as:

The imaginary part of *z* is written as:

In equations [1,2,3], *i* is given as:

**Example**. Z = 4 + i5 ==> Then Re[Z]=4, Im[Z]=5.

### Addition and Multiplication

Addition and subtraction are straightforward. The addition of two complex numbers (z1 and z2) are the sums of their real and imaginary parts:

Subtraction can be performed in a similar manner to equation [5]. Multiplication of complex numbers follows algebra-style rules:

Division will be discussed after the polar representation for complex numbers.

### Complex Conjugate and Magnitude

The magnitude of a complex number z is given by:

### Polar Form

Equation [9] can be derived by expanding the left side in a Taylor series (with variable theta). Then expand the right side using the Taylor series expansions for cosine and sine and the results are identical.

The polar form of a complex number is written with a magnitude and angle:

Using equation [9], the polar form can be converted back into it's real and imaginary parts:

The angle theta of the complex number can be determined from the real and imaginary part:

The angle theta is zero when the real part of a complex number is positive and the imaginary part is zero. The angle theta is 90 degrees when the imaginary part is positive and the real part is zero.

Hence, we can convert between the rectangular form (real and imaginary part) and the polar form (magnitude and angle).

As an example, consider the complex number z=3+i4. This number can be plotted along the x- and y- axis, as shown in Figure 1. Note that x represents the real part of z, and y represents the imaginary part of z.

Figure 1. Illustration of a Complex Number in the Complex Plane.

In Figure 1, |z|=5 (from equation [8]), and theta=53.13 degrees (from equation [12]).

### Division

The polar form makes division very simple.

Example Suppose z1=1+i3, and z2 = -1 - i1. What is (z1 * z2) and z1/z2?

## Discrete Fourier Transform

The **discrete Fourier transform (DFT)** is a method for converting a sequence of N N N complex numbers x 0 , x 1 , … , x N − 1 x_0,x_1,ldots,x_

The DFT is useful in many applications, including the simple signal spectral analysis outlined above. Knowing how a signal can be expressed as a combination of waves allows for manipulation of that signal and comparisons of different signals:

Digital files (jpg, mp3, etc.) can be shrunk by eliminating contributions from the least important waves in the combination.

Different sound files can be compared by comparing the coefficients X k X_k X k of the DFT.

Radio waves can be filtered to avoid "noise" and listen to the important components of the signal.

Other applications of the DFT arise because it can be computed very efficiently by the fast Fourier transform (FFT) algorithm. For example, the DFT is used in state-of-the-art algorithms for multiplying polynomials and large integers together instead of working with polynomial multiplication directly, it turns out to be faster to compute the DFT of the polynomial functions and convert the problem of multiplying polynomials to an analogous problem involving their DFTs.

#### Contents

## Fourier Transforms

The Fourier transform is a mathematical formula that relates a signal sampled in time or space to the same signal sampled in frequency. In signal processing, the Fourier transform can reveal important characteristics of a signal, namely, its frequency components.

The Fourier transform is defined for a vector x with n uniformly sampled points by

y k + 1 = ∑ j = 0 n - 1 ω j k x j + 1 .

ω = e - 2 π i / n is one of n complex roots of unity where i is the imaginary unit. For x and y , the indices j and k range from 0 to n - 1 .

The fft function in MATLAB® uses a fast Fourier transform algorithm to compute the Fourier transform of data. Consider a sinusoidal signal x that is a function of time t with frequency components of 15 Hz and 20 Hz. Use a time vector sampled in increments of 1 50 of a second over a period of 10 seconds.

Compute the Fourier transform of the signal, and create the vector f that corresponds to the signal's sampling in frequency space.

When you plot the magnitude of the signal as a function of frequency, the spikes in magnitude correspond to the signal's frequency components of 15 Hz and 20 Hz.

The transform also produces a mirror copy of the spikes, which correspond to the signal's negative frequencies. To better visualize this periodicity, you can use the fftshift function, which performs a zero-centered, circular shift on the transform.

### Noisy Signals

In scientific applications, signals are often corrupted with random noise, disguising their frequency components. The Fourier transform can process out random noise and reveal the frequencies. For example, create a new signal, xnoise , by injecting Gaussian noise into the original signal, x .

Signal power as a function of frequency is a common metric used in signal processing. Power is the squared magnitude of a signal's Fourier transform, normalized by the number of frequency samples. Compute and plot the power spectrum of the noisy signal centered at the zero frequency. Despite noise, you can still make out the signal's frequencies due to the spikes in power.

### Computational Efficiency

Using the Fourier transform formula directly to compute each of the n elements of y requires on the order of n 2 floating-point operations. The fast Fourier transform algorithm requires only on the order of n log n operations to compute. This computational efficiency is a big advantage when processing data that has millions of data points. Many specialized implementations of the fast Fourier transform algorithm are even more efficient when n is a power of 2.

Consider audio data collected from underwater microphones off the coast of California. This data can be found in a library maintained by the Cornell University Bioacoustics Research Program. Load and format a subset of the data in bluewhale.au , which contains a Pacific blue whale vocalization. Because blue whale calls are low-frequency sounds, they are barely audible to humans. The time scale in the data is compressed by a factor of 10 to raise the pitch and make the call more clearly audible. You can use the command sound(x,fs) to listen to the entire audio file.

Specify a new signal length that is the next power of 2 greater than the original length. Then, use fft to compute the Fourier transform using the new signal length. fft automatically pads the data with zeros to increase the sample size. This padding can make the transform computation significantly faster, particularly for sample sizes with large prime factors.

Plot the power spectrum of the signal. The plot indicates that the moan consists of a fundamental frequency around 17 Hz and a sequence of harmonics, where the second harmonic is emphasized.

## Watch the video: Intro to Fourier transforms: how to calculate them (May 2022).

Continuous frequency | Discrete frequencies |
---|---|

Transform | 1 T S 1 T ( f ) ≜ ∑ n = − ∞ ∞ s ( n T ) ⋅ e − i 2 π f n T ⏟ Poisson summation formula (DTFT) |