# dsp.FFT

Discrete Fourier transform

## Description

The `dsp.FFT` System object™ computes the discrete Fourier transform (DFT) of an input using fast Fourier transform (FFT). The object uses one or more of the following fast Fourier transform (FFT) algorithms depending on the complexity of the input and whether the output is in linear or bit-reversed order:

• Double-signal algorithm

• Half-length algorithm

• An algorithm chosen from FFTW [1] , [2]

To compute the DFT of an input:

1. Create the `dsp.FFT` object and set its properties.

2. Call the object with arguments, as if it were a function.

## Creation

### Syntax

``ft = dsp.FFT``
``ft = dsp.FFT(Name,Value)``

### Description

````ft = dsp.FFT` returns a `FFT` object, `ft`, that computes the DFT of an N-D array. For column vectors or multidimensional arrays, the `FFT` object computes the DFT along the first dimension. If the input is a row vector, the `FFT` object computes a row of single-sample DFTs and issues a warning.```

example

````ft = dsp.FFT(Name,Value)` returns a `FFT` object with each specified property set to the specified value. Enclose each property name in single quotes. Unspecified properties have default values.```

## Properties

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Unless otherwise indicated, properties are nontunable, which means you cannot change their values after calling the object. Objects lock when you call them, and the `release` function unlocks them.

If a property is tunable, you can change its value at any time.

Specify the implementation used for the FFT as one of `Auto`, `Radix-2`, or `FFTW`. When you set this property to `Radix-2`, the FFT length must be a power of two.

Designate order of output channel elements relative to order of input elements. Set this property to `true` to output the frequency indices in bit-reversed order. The default is `false`, which corresponds to a linear ordering of frequency indices.

Set this property to `true` if the output of the FFT should be divided by the FFT length. This option is useful when you want the output of the FFT to stay in the same amplitude range as its input. This is particularly useful when working with fixed-point data types.

The default value of this property is `false` with no scaling.

Specify how to determine the FFT length as `Auto` or `Property`. When you set this property to `Auto`, the FFT length equals the number of rows of the input signal.

FFT length, specified as an integer greater than or equal to 2.

This property must be a power of two if any of these conditions apply:

#### Dependencies

This property applies when you set the `FFTLengthSource` property to `Property`.

Data Types: `single` | `double` | `int8` | `int16` | `int32` | `int64` | `uint8` | `uint16` | `uint32` | `uint64`

Wrap input data when FFT length is shorter than input length. If this property is set to true, modulo-length data wrapping occurs before the FFT operation, given FFT length is shorter than the input length. If this property is set to false, truncation of the input data to the FFT length occurs before the FFT operation.

### Fixed-Point Properties

Specify the rounding method.

Specify the overflow action as `Wrap` or `Saturate`.

Specify the sine table data type as `Same word length as input` or `Custom`.

Specify the sine table fixed-point type as an unscaled `numerictype` (Fixed-Point Designer) object with a `Signedness` of `Auto`.

#### Dependencies

This property applies when you set the `SineTableDataType` property to `Custom`.

Specify the product data type as `Full precision`, ```Same as input```, or `Custom`.

Specify the product fixed-point type as a scaled `numerictype` (Fixed-Point Designer) object with a `Signedness` of `Auto`.

#### Dependencies

This property applies when you set the `ProductDataType` property to `Custom`.

Specify the accumulator data type as `Full precision`, `Same as input`, `Same as product`, or `Custom`.

Specify the accumulator fixed-point type as a scaled `numerictype` (Fixed-Point Designer) object with a `Signedness` of `Auto`.

#### Dependencies

This property applies when you set the `AccumulatorDataType` property to `Custom`.

Specify the output data type as one of `Full precision`, `Same as input`, `Custom`.

Specify the output fixed-point type as a scaled `numerictype` (Fixed-Point Designer) object with a `Signedness` of `Auto`.

#### Dependencies

This property applies when you set the OutputDataType property to `Custom`.

## Usage

### Syntax

``y = ft(x)``

### Description

example

````y = ft(x)` computes the DFT, `y`, of the input `x` along the first dimension of `x`.```

### Input Arguments

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Time-domain input signal, specified as a vector, matrix, or N-D array.

When the `FFTLengthSource` property is set to `'Auto'`, the length of `x` along the first dimension must be a positive integer power of two. This length is also the FFT length. When the `FFTLengthSource` property is `'Property'`, the value you specify in `FFTLength` property must be a positive integer power of two.

Variable-size input signals are only supported when the `FFTLengthSource` property is set to `'Auto'`.

Data Types: `single` | `double` | `int8` | `int16` | `int32` | `int64` | `uint8` | `uint16` | `uint32` | `uint64` | `fi`
Complex Number Support: Yes

### Output Arguments

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Discrete Fourier transform of input signal, returned as a vector, matrix, or an N-D array. When `FFTLengthSource` property is set to `'Auto'`, the FFT length is same as the number of rows in the input signal. When `FFTLengthSource` property is set to `'Property'`, the FFT length is specified through the `FFTLength` property.

To support non-power-of-two transform lengths with variable-size data, set the `FFTImplementation` property to `'FFTW'`.

Data Types: `single` | `double` | `int8` | `int16` | `int32` | `int64` | `uint8` | `uint16` | `uint32` | `uint64` | `fi`
Complex Number Support: Yes

## Object Functions

To use an object function, specify the System object as the first input argument. For example, to release system resources of a System object named `obj`, use this syntax:

`release(obj)`

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 `step` Run System object algorithm `release` Release resources and allow changes to System object property values and input characteristics `reset` Reset internal states of System object

## Examples

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Find frequency components of a signal in additive noise.

Note: This example runs only in R2016b or later. If you are using an earlier release, replace each call to the function with the equivalent `step` syntax. For example, myObject(x) becomes step(myObject,x).

```Fs = 800; L = 1000; t = (0:L-1)'/Fs; x = sin(2*pi*250*t) + 0.75*cos(2*pi*340*t); y = x + .5*randn(size(x)); % noisy signal ft = dsp.FFT('FFTLengthSource','Property', ... 'FFTLength',1024); Y = ft(y);```

Plot the single-sided amplitude spectrum

```plot(Fs/2*linspace(0,1,512), 2*abs(Y(1:512)/1024)) title('Single-sided amplitude spectrum of noisy signal y(t)') xlabel('Frequency (Hz)'); ylabel('|Y(f)|')```

Compute the FFT of a noisy sinusoidal input signal. The energy of the signal is stored as the magnitude square of the FFT coefficients. Determine the FFT coefficients which occupy 99.99% of the signal energy and reconstruct the time-domain signal by taking the IFFT of these coefficients. Compare the reconstructed signal with the original signal.

`Note``: `If you are using R2016a or an earlier release, replace each call to the object with the equivalent `step` syntax. For example, `obj(x)` becomes `step(obj(x))`.

Consider a time-domain signal $\mathit{x}\left[\mathit{n}\right]$, which is defined over the finite time interval $0\le \mathit{n}\le \mathit{N}-1$. The energy of the signal $\mathit{x}\left[\mathit{n}\right]$ is given by the following equation:

`${\mathit{E}}_{\mathit{N}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\sum }_{\mathit{n}=0}^{\mathit{N}-1}{|\mathit{x}\left[\mathit{n}\right]|}^{2}$`

FFT coefficients, $\mathit{X}\left[\mathit{k}\right]$, are considered signal values in the frequency domain. The energy of the signal $\mathit{x}\left[\mathit{n}\right]$ in the frequency-domain is therefore the sum of the squares of the magnitude of the FFT coefficients:

`${\mathit{E}}_{\mathit{N}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{1}{\mathit{N}}{\sum }_{\mathit{k}=0}^{\mathit{N}-1}{|\mathit{X}\left[\mathit{k}\right]|}^{2}$`

According to Parseval's theorem, the total energy of the signal in time or frequency-domain is the same.

`${\mathit{E}}_{\mathit{N}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\sum }_{\mathit{n}=0}^{\mathit{N}-1}{|\mathit{x}\left[\mathit{n}\right]|}^{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{1}{\mathit{N}}{\sum }_{\mathit{k}=0}^{\mathit{N}-1}{|\mathit{X}\left[\mathit{k}\right]|}^{2}\text{\hspace{0.17em}}$`

Initialization

Initialize a `dsp.SineWave` System object to generate a sine wave sampled at 44.1 kHz and has a frequency of 1000 Hz. Construct a `dsp.FFT` and `dsp.IFFT` objects to compute the FFT and the IFFT of the input signal.

The `'FFTLengthSource'` property of each of these transform objects is set to `'Auto'`. The FFT length is hence considered as the input frame size. The input frame size in this example is 1020, which is not a power of 2, so select the `'FFTImplementation'` as `'FFTW'`.

```L = 1020; Sineobject = dsp.SineWave('SamplesPerFrame',L,... 'PhaseOffset',10,... 'SampleRate',44100,... 'Frequency',1000); ft = dsp.FFT('FFTImplementation','FFTW'); ift = dsp.IFFT('FFTImplementation','FFTW',... 'ConjugateSymmetricInput',true); rng(1);```

Streaming

Stream in the noisy input signal. Compute the FFT of each frame and determine the coefficients that constitute 99.99% energy of the signal. Take IFFT of these coefficients to reconstruct the time-domain signal.

```numIter = 1000; for Iter = 1:numIter Sinewave1 = Sineobject(); Input = Sinewave1 + 0.01*randn(size(Sinewave1)); FFTCoeff = ft(Input); FFTCoeffMagSq = abs(FFTCoeff).^2; EnergyFreqDomain = (1/L)*sum(FFTCoeffMagSq); [FFTCoeffSorted, ind] = sort(((1/L)*FFTCoeffMagSq),... 1,'descend'); CumFFTCoeffs = cumsum(FFTCoeffSorted); EnergyPercent = (CumFFTCoeffs/EnergyFreqDomain)*100; Vec = find(EnergyPercent > 99.99); FFTCoeffsModified = zeros(L,1); FFTCoeffsModified(ind(1:Vec(1))) = FFTCoeff(ind(1:Vec(1))); ReconstrSignal = ift(FFTCoeffsModified); end```

99.99% of the signal energy can be represented by the number of FFT coefficients given by `Vec(1)`:

`Vec(1)`
```ans = 296 ```

The signal is reconstructed efficiently using these coefficients. If you compare the last frame of the reconstructed signal with the original time-domain signal, you can see that the difference is very small and the plots match closely.

`max(abs(Input-ReconstrSignal))`
```ans = 0.0431 ```
```plot(Input,'*'); hold on; plot(ReconstrSignal,'o'); hold off;```

## Algorithms

This object implements the algorithm, inputs, and outputs described on the FFT block reference page. The object properties correspond to the block parameters.

## References

[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.

## Version History

Introduced in R2012a