Fast Fourier transform

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## Syntax

`Y = fft(X)`

`Y = fft(X,n)`

`Y = fft(X,n,dim)`

## Description

example

`Y = fft(X)`

computes the discrete Fourier transform (DFT) of `X`

using a fast Fourier transform (FFT) algorithm. `Y`

is the same size as `X`

.

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.

example

`Y = fft(X,n)`

returns the `n`

-point DFT.

If

`X`

is a vector and the length of`X`

is less than`n`

, then`X`

is padded with trailing zeros to length`n`

.If

`X`

is a vector and the length of`X`

is greater than`n`

, then`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.

example

`Y = fft(X,n,dim)`

returns the Fourier transform along the dimension `dim`

. For example, if `X`

is a matrix, then `fft(X,n,2)`

returns the `n`

-point Fourier transform of each row.

## Examples

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### Noisy Signal

Open Live Script

Find the frequency components of a signal buried in noise and find the amplitudes of the peak frequencies by using Fourier transform.

Specify the parameters of a signal with a sampling frequency of 1 kHz and a signal duration of 1.5 seconds.

Fs = 1000; % Sampling frequency T = 1/Fs; % Sampling period L = 1500; % Length of signalt = (0:L-1)*T; % Time vector

Form a signal containing a DC offset of amplitude 0.8, a 50 Hz sinusoid of amplitude 0.7, and a 120 Hz sinusoid of amplitude 1.

S = 0.8 + 0.7*sin(2*pi*50*t) + sin(2*pi*120*t);

Corrupt the signal with zero-mean random noise with a variance of 4.

X = S + 2*randn(size(t));

Plot the noisy signal in the time domain. The frequency components are not visually apparent in the plot.

plot(1000*t,X)title("Signal Corrupted with Zero-Mean Random Noise")xlabel("t (milliseconds)")ylabel("X(t)")

Compute the Fourier transform of the signal.

Y = fft(X);

Because Fourier transforms involve complex numbers, plot the complex magnitude of the `fft`

spectrum.

plot(Fs/L*(0:L-1),abs(Y),"LineWidth",3)title("Complex Magnitude of fft Spectrum")xlabel("f (Hz)")ylabel("|fft(X)|")

The plot shows five frequency peaks including the peak at 0 Hz for the DC offset. In this example, the signal is expected to have three frequency peaks at 0 Hz, 50 Hz, and 120 Hz. Here, the second half of the plot is the mirror reflection of the first half without including the peak at 0 Hz. The reason is that the discrete Fourier transform of a time-domain signal has a periodic nature, where the first half of its spectrum is in positive frequencies and the second half is in negative frequencies, with the first element reserved for the zero frequency.

For real signals, the `fft`

spectrum is a two-sided spectrum, where the spectrum in the positive frequencies is the complex conjugate of the spectrum in the negative frequencies. To show the `fft`

spectrum in the positive and negative frequencies, you can use `fftshift`

. For an even length of `L`

, the frequency domain starts from the negative of the Nyquist frequency `-Fs/2`

up to `Fs/2-Fs/L`

with a spacing or frequency resolution of `Fs/L`

.

plot(Fs/L*(-L/2:L/2-1),abs(fftshift(Y)),"LineWidth",3)title("fft Spectrum in the Positive and Negative Frequencies")xlabel("f (Hz)")ylabel("|fft(X)|")

To find the amplitudes of the three frequency peaks, convert the `fft`

spectrum in `Y`

to the single-sided amplitude spectrum. Because the `fft`

function includes a scaling factor `L`

between the original and the transformed signals, rescale `Y`

by dividing by `L`

. Take the complex magnitude of the `fft`

spectrum. The two-sided amplitude spectrum `P2`

, where the spectrum in the positive frequencies is the complex conjugate of the spectrum in the negative frequencies, has half the peak amplitudes of the time-domain signal. To convert to the single-sided spectrum, take the first half of the two-sided spectrum `P2`

. Multiply the spectrum in the positive frequencies by 2. You do not need to multiply `P1(1)`

and `P1(end)`

by 2 because these amplitudes correspond to the zero and Nyquist frequencies, respectively, and they do not have the complex conjugate pairs in the negative frequencies.

P2 = abs(Y/L);P1 = P2(1:L/2+1);P1(2:end-1) = 2*P1(2:end-1);

Define the frequency domain `f`

for the single-sided spectrum. Plot the single-sided amplitude spectrum `P1`

. As expected, the amplitudes are close to 0.8, 0.7, and 1, but they are not exact because of the added noise. In most cases, longer signals produce better frequency approximations.

f = Fs/L*(0:(L/2));plot(f,P1,"LineWidth",3) title("Single-Sided Amplitude Spectrum of X(t)")xlabel("f (Hz)")ylabel("|P1(f)|")

Now, take the Fourier transform of the original, uncorrupted signal and retrieve the exact amplitudes at 0.8, 0.7, and 1.0.

Y = fft(S);P2 = abs(Y/L);P1 = P2(1:L/2+1);P1(2:end-1) = 2*P1(2:end-1);plot(f,P1,"LineWidth",3) title("Single-Sided Amplitude Spectrum of S(t)")xlabel("f (Hz)")ylabel("|P1(f)|")

### Gaussian Pulse

Open Live Script

Convert a Gaussian pulse from the time domain to the frequency domain.

Specify the parameters of a signal with a sampling frequency of 44.1 kHz and a signal duration of 1 ms. Create a Gaussian pulse with a standard deviation of 0.1 ms.

Fs = 44100; % Sampling frequencyT = 1/Fs; % Sampling periodt = -0.5:T:0.5; % Time vectorL = length(t); % Signal lengthX = 1/(0.4*sqrt(2*pi))*(exp(-t.^2/(2*(0.1*1e-3)^2)));

Plot the pulse in the time domain.

plot(t,X)title("Gaussian Pulse in Time Domain")xlabel("Time (t)")ylabel("X(t)")axis([-1e-3 1e-3 0 1.1])

The execution time of `fft`

depends on the length of the transform. Transform lengths that have only small prime factors result in significantly faster execution time than those that have large prime factors.

In this example, the signal length `L`

is 44,101, which is a very large prime number. To improve the performance of `fft`

, identify an input length that is the next power of 2 from the original signal length. Calling `fft`

with this input length pads the pulse `X`

with trailing zeros to the specified transform length.

n = 2^nextpow2(L);

Convert the Gaussian pulse to the frequency domain.

Y = fft(X,n);

Define the frequency domain and plot the unique frequencies.

f = Fs*(0:(n/2))/n;P = abs(Y/sqrt(n)).^2;plot(f,P(1:n/2+1)) title("Gaussian Pulse in Frequency Domain")xlabel("f (Hz)")ylabel("|P(f)|")

### Cosine Waves

Open Live Script

Compare cosine waves in the time domain and the frequency domain.

Specify the parameters of a signal with a sampling frequency of 1 kHz and a signal duration of 1 second.

Fs = 1000; % Sampling frequencyT = 1/Fs; % Sampling periodL = 1000; % Length of signalt = (0:L-1)*T; % Time vector

Create a matrix where each row represents a cosine wave with scaled frequency. The result, `X`

, is a 3-by-1000 matrix. The first row has a wave frequency of 50, the second row has a wave frequency of 150, and the third row has a wave frequency of 300.

x1 = cos(2*pi*50*t); % First row wavex2 = cos(2*pi*150*t); % Second row wavex3 = cos(2*pi*300*t); % Third row waveX = [x1; x2; x3];

Plot the first 100 entries from each row of `X`

in a single figure in order and compare their frequencies.

for i = 1:3 subplot(3,1,i) plot(t(1:100),X(i,1:100)) title("Row " + num2str(i) + " in the Time Domain")end

Specify the `dim`

argument to use `fft`

along the rows of `X`

, that is, for each signal.

dim = 2;

Compute the Fourier transform of the signals.

Y = fft(X,L,dim);

Calculate the double-sided spectrum and single-sided spectrum of each signal.

P2 = abs(Y/L);P1 = P2(:,1:L/2+1);P1(:,2:end-1) = 2*P1(:,2:end-1);

In the frequency domain, plot the single-sided amplitude spectrum for each row in a single figure.

for i=1:3 subplot(3,1,i) plot(0:(Fs/L):(Fs/2-Fs/L),P1(i,1:L/2)) title("Row " + num2str(i) + " in the Frequency Domain")end

### Phase of Sinusoids

Open Live Script

Create a signal that consists of two sinusoids of frequencies 15 Hz and 40 Hz. The first sinusoid is a cosine wave with phase $-\pi /4$, and the second is a cosine wave with phase $\pi /2$. Sample the signal at 100 Hz for 1 s.

Fs = 100;t = 0:1/Fs:1-1/Fs;x = cos(2*pi*15*t - pi/4) + cos(2*pi*40*t + pi/2);

Compute the Fourier transform of the signal. Plot the magnitude of the transform as a function of frequency.

y = fft(x);z = fftshift(y);ly = length(y);f = (-ly/2:ly/2-1)/ly*Fs;stem(f,abs(z))title("Double-Sided Amplitude Spectrum of x(t)")xlabel("Frequency (Hz)")ylabel("|y|")grid

Compute the phase of the transform, removing small-magnitude transform values. Plot the phase as a function of frequency.

tol = 1e-6;z(abs(z) < tol) = 0;theta = angle(z);stem(f,theta/pi)title("Phase Spectrum of x(t)")xlabel("Frequency (Hz)")ylabel("Phase/\pi")grid

### Interpolation of FFT

Open Live Script

Interpolate the Fourier transform of a signal by padding with zeros.

Specify the parameters of a signal with a sampling frequency of 80 Hz and a signal duration of 0.8 s.

Fs = 80;T = 1/Fs;L = 65;t = (0:L-1)*T;

Create a superposition of a 2 Hz sinusoidal signal and its higher harmonics. The signal contains a 2 Hz cosine wave, a 4 Hz cosine wave, and a 6 Hz sine wave.

X = 3*cos(2*pi*2*t) + 2*cos(2*pi*4*t) + sin(2*pi*6*t);

Plot the signal in the time domain.

plot(t,X)title("Signal superposition in time domain")xlabel("t (ms)")ylabel("X(t)")

Compute the Fourier transform of the signal.

Y = fft(X);

Compute the single-sided amplitude spectrum of the signal.

f = Fs*(0:(L-1)/2)/L;P2 = abs(Y/L);P1 = P2(1:(L+1)/2);P1(2:end) = 2*P1(2:end);

In the frequency domain, plot the single-sided spectrum. Because the time sampling of the signal is quite short, the frequency resolution of the Fourier transform is not precise enough to show the peak frequency near 4 Hz.

plot(f,P1,"-o") title("Single-Sided Spectrum of Original Signal")xlabel("f (Hz)")ylabel("|P1(f)|")

To better assess the peak frequencies, you can increase the length of the analysis window by padding the original signal with zeros. This method automatically interpolates the Fourier transform of the signal with a more precise frequency resolution.

Identify a new input length that is the next power of 2 from the original signal length. Pad the signal `X`

with trailing zeros to extend its length. Compute the Fourier transform of the zero-padded signal.

n = 2^nextpow2(L);Y = fft(X,n);

Compute the single-sided amplitude spectrum of the padded signal. Because the signal length `n`

increased from 65 to 128, the frequency resolution becomes `Fs/n`

, which is 0.625 Hz.

f = Fs*(0:(n/2))/n;P2 = abs(Y/L);P1 = P2(1:n/2+1);P1(2:end-1) = 2*P1(2:end-1);

Plot the single-sided spectrum of the padded signal. This new spectrum shows the peak frequencies near 2 Hz, 4 Hz, and 6 Hz within the frequency resolution of 0.625 Hz.

plot(f,P1,"-o") title("Single-Sided Spectrum of Padded Signal")xlabel("f (Hz)")ylabel("|P1(f)|")

## Input Arguments

collapse all

`X`

— Input array

vector | matrix | multidimensional array

Input array, specified as a vector, matrix, or multidimensional array.

If `X`

is an empty 0-by-0 matrix, then `fft(X)`

returns an empty 0-by-0 matrix.

**Data Types: **`double`

| `single`

| `int8`

| `int16`

| `int32`

| `uint8`

| `uint16`

| `uint32`

| `logical`

**Complex Number Support: **Yes

`n`

— Transform length

`[]`

(default) | nonnegative integer scalar

Transform length, specified as `[]`

or a nonnegative integer scalar. Specifying a positive integer scalar for the transform length can improve the performance of `fft`

. The length is typically specified as a power of 2 or a value that can be factored into a product of small prime numbers (with prime factors not greater than 7). If `n`

is less than the length of the signal, then `fft`

ignores the remaining signal values past the `n`

th entry and returns the truncated result. If `n`

is `0`

, then `fft`

returns an empty matrix.

**Example: **`n = 2^nextpow2(size(X,1))`

**Data Types: **`double`

| `single`

| `int8`

| `int16`

| `int32`

| `uint8`

| `uint16`

| `uint32`

| `logical`

`dim`

— Dimension to operate along

positive integer scalar

Dimension to operate along, specified as a positive integer scalar. If you do not specify the dimension, then the default is the first array dimension of size greater than 1.

`fft(X,[],1)`

operates along the columns of`X`

and returns the Fourier transform of each column.`fft(X,[],2)`

operates along the rows of`X`

and returns the Fourier transform of each row.

If `dim`

is greater than `ndims(X)`

, then `fft(X,[],dim)`

returns `X`

. When `n`

is specified, `fft(X,n,dim)`

pads or truncates `X`

to length `n`

along dimension `dim`

.

**Data Types: **`double`

| `single`

| `int8`

| `int16`

| `int32`

| `uint8`

| `uint16`

| `uint32`

| `logical`

## Output Arguments

collapse all

`Y`

— Frequency domain representation

vector | matrix | multidimensional array

Frequency domain representation returned as a vector, matrix, or multidimensional array.

If `X`

is of type `single`

, then `fft`

natively computes in single precision, and `Y`

is also of type `single`

. Otherwise, `Y`

is returned as type `double`

.

The size of `Y`

is as follows:

For

`Y = fft(X)`

or`Y = fft(X,[],dim)`

, the size of`Y`

is equal to the size of`X`

.For

`Y = fft(X,n,dim)`

, the value of`size(Y,dim)`

is equal to`n`

, while the size of all other dimensions remains as in`X`

.

If `X`

is real, then `Y`

is conjugate symmetric, and the number of unique points in `Y`

is `ceil((n+1)/2)`

.

**Data Types: **`double`

| `single`

## More About

collapse all

### Discrete Fourier Transform of Vector

`Y = fft(X)`

and `X = ifft(Y)`

implement the Fourier transform and inverse Fourier transform, respectively. For `X`

and `Y`

of length `n`

, these transforms are defined as follows:

$$\begin{array}{l}Y(k)={\displaystyle \sum _{j=1}^{n}X}(j)\text{\hspace{0.17em}}{W}_{n}^{(j-1)\text{}(k-1)}\\ X(j)=\frac{1}{n}{\displaystyle \sum _{k=1}^{n}Y}(k)\text{\hspace{0.17em}}{W}_{n}{}^{-(j-1)\text{}(k-1)},\end{array}$$

where

$${W}_{n}={e}^{(-2\pi i)/n}$$

is one of *n* roots of unity.

## Tips

The execution time of

`fft`

depends on the length of the transform. Transform lengths that have only small prime factors (not greater than 7) result in significantly faster execution time than those that are prime or have large prime factors.For most values of

`n`

, real-input DFTs require roughly half the computation time of complex-input DFTs. However, when`n`

has large prime factors, there is little or no speed difference.You can potentially increase the speed of

`fft`

using the utility function fftw. This function controls the optimization of the algorithm used to compute an FFT of a particular size and dimension.

## Algorithms

The FFT functions (`fft`

, `fft2`

, `fftn`

, `ifft`

, `ifft2`

, `ifftn`

) are based on a library called FFTW [1] [2].

## References

[1] FFTW (https://www.fftw.org)

[2] Frigo, M., and S. G. Johnson. “FFTW: An Adaptive Software Architecture for the FFT.” *Proceedings of the International Conference on Acoustics, Speech, and Signal Processing*. Vol. 3, 1998, pp. 1381-1384.

## Extended Capabilities

### C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

For limitations related to variable-size data, see Variable-Sizing Restrictions for Code Generation of Toolbox Functions (MATLAB Coder).

For MEX output, MATLAB

^{®}Coder™ uses the library that MATLAB uses for FFT algorithms. For standalone C/C++ code, by default, the code generator produces code for FFT algorithms instead of producing FFT library calls. To generate calls to a specific installed FFTW library, provide an FFT library callback class. For more information about an FFT library callback class, see coder.fftw.StandaloneFFTW3Interface (MATLAB Coder).For simulation of a MATLAB Function block, the simulation software uses the library that MATLAB uses for FFT algorithms. For C/C++ code generation, by default, the code generator produces code for FFT algorithms instead of producing FFT library calls. To generate calls to a specific installed FFTW library, provide an FFT library callback class. For more information about an FFT library callback class, see coder.fftw.StandaloneFFTW3Interface (MATLAB Coder).

Using the Code Replacement Library (CRL), you can generate optimized code that runs on ARM

^{®}Cortex^{®}-A processors with Neon extension. To generate this optimized code, you must install the Embedded Coder® Support Package for ARM Cortex-A Processors (Embedded Coder). The generated code for ARM Cortex-A uses the Ne10 library. For more information, see Ne10 Conditions for MATLAB Functions to Support ARM Cortex-A Processors (Embedded Coder).Using the Code Replacement Library (CRL), you can generate optimized code that runs on ARM Cortex-M processors. To generate this optimized code, you must install the Embedded Coder Support Package for ARM Cortex-M Processors (Embedded Coder). The generated code for ARM Cortex-M uses the CMSIS library. For more information, see CMSIS Conditions for MATLAB Functions to Support ARM Cortex-M Processors (Embedded Coder).

### GPU Code Generation

Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.

### Thread-Based Environment

Run code in the background using MATLAB® `backgroundPool`

or accelerate code with Parallel Computing Toolbox™ `ThreadPool`

.

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

### GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

The `fft`

function partially supports GPU arrays. Some syntaxes of the function run on a GPU when you specify the input data as a gpuArray (Parallel Computing Toolbox). Usage notes and limitations:

The output

`Y`

is always complex even if all the imaginary parts are zero.

For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

### Distributed Arrays

Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

Usage notes and limitations:

For distributed arrays, instead of using a parallel FFT algorithm,

`fft`

gathers vectors on a single worker to perform prime length FFTs. For large prime-length vector FFTs, out-of-memory errors can result.

For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).

## Version History

**Introduced before R2006a**

## See Also

### Functions

- fft2 | fftn | fftw | fftshift | ifft | interpft

### Topics

- Fourier Transforms

### External Websites

- Fourier Analysis (MathWorks Teaching Resources)
- Convolution in Digital Signal Processing (MathWorks Teaching Resources)

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