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dsp.FFT System object

Discrete Fourier transform

Description

The `FFT` object computes the discrete Fourier transform (DFT) of an input. 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 by FFTW [1] , [2]

To compute the DFT of an input:

1. Define and set up your `FFT` object. See Construction.

2. Call `step` to compute the DFT of the input according to the properties of `dsp.FFT`. The behavior of `step` is specific to each object in the toolbox.

Note

Starting in R2016b, instead of using the `step` method to perform the operation defined by the System object™, you can call the object with arguments, as if it were a function. For example, ```y = step(obj,x)``` and `y = obj(x)` perform equivalent operations.

Construction

`fft = dsp.FFT` returns a `FFT` object, `H`, 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.

```H = dsp.FFT('PropertyName',PropertyValue, ...)``` returns a `FFT` object, `H`, with each property set to the specified value.

Properties

 `FFTImplementation` FFT implementation Specify the implementation used for the FFT as one of `Auto` | `Radix-2` | `FFTW`. When you set this property to `Radix-2`, the FFT length must be a power of two. `BitReversedOutput` Order of output elements relative to input elements 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. `Normalize` Divide butterfly outputs by two 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. `FFTLengthSource` Source of FFT length 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. The default is `Auto`. `FFTLength` FFT length Specify the FFT length. This property applies when you set the `FFTLengthSource` property to `Property`. The default is `64`. This property must be a power of two when the input is a fixed-point data type, or when you set the BitReversedOutput property to `true`, or when you set the `FFTImplementation` property to `Radix-2`. `WrapInput` Boolean value of wrapping or truncating input 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. The default is `true`.

Methods

 step Discrete Fourier transform of input
Common to All System Objects
`clone`

Create System object with same property values

`getNumInputs`

Expected number of inputs to a System object

`getNumOutputs`

Expected number of outputs of a System object

`isLocked`

Check locked states of a System object (logical)

`release`

Allow System object property value changes

Examples

expand all

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 , which is defined over the finite time interval . The energy of the signal is given by the following equation:

FFT Coefficients, are considered as signal values in the frequency domain. The energy of the signal in the frequency-domain is hence the sum of the squares of the magnitude of the FFT coefficients:

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

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. Hence, 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 which 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.