### 7 releases (2 stable)

1.1.0 | Jan 20, 2021 |
---|---|

1.0.0 | Jan 17, 2021 |

0.4.0 | Jan 5, 2021 |

0.3.0 | Oct 10, 2020 |

0.1.0 | Jun 17, 2020 |

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# realfft

## RealFFT: Real-to-complex FFT and complex-to-real iFFT based on RustFFT

This library is a wrapper for RustFFT that enables performing FFT of real-valued data. The API is designed to be as similar as possible to RustFFT.

Using this library instead of RustFFT directly avoids the need of converting real-valued data to complex before performing a FFT. If the length is even, it also enables faster computations by using a complex FFT of half the length. It then packs a 2N long real vector into an N long complex vector, which is transformed using a standard FFT. It then post-processes the result to give only the first half of the complex spectrum, as an N+1 long complex vector.

The iFFT goes through the same steps backwards, to transform an N+1 long complex spectrum to a 2N long real result.

The speed increase compared to just converting the input to a 2N long complex vector and using a 2N long FFT depends on the length f the input data. The largest improvements are for long FFTs and for lengths over around 1000 elements there is an improvement of about a factor 2. The difference shrinks for shorter lengths, and around 30 elements there is no longer any difference.

### Why use real-to-complex FFT?

#### Using a complex-to-complex FFT

A simple way to get the FFT of a rea values vector is to convert it to complex, and using a complex-to-complex FFT.

Let's assume

is a 6 element long real vector:`x`

`x ``=` `[`x0r`,` x1r`,` x2r`,` x3r`,` x4r`,` x5r`]`

We now convert

to complex by adding an imaginary part with value zero. Using the notation `x`

for the complex value `(`xNr`,` xNi`)`

, this becomes:`xN`

`x_c ``=` `[``(`x0r`,` `0``)``,` `(`x1r`,` `0``)``,` `(`x2r`,` `0``)``,` `(`x3r`,` `0``)``,` `(`x4r`,` `0``,` `(`x5r`,` `0``)`]

Performing a normal complex FFT, the result of

is:`FFT``(`x_c`)`

`FFT``(`x_c`)` `=` `[``(`X0r`,` X0i`)``,` `(`X1r`,` X1i`)``,` `(`X2r`,` X2i`)``,` `(`X3r`,` X3i`)``,` `(`X4r`,` X4i`)``,` `(`X5r`,` X5i`)``]`

But because our

is real-valued (all imaginary parts are zero), some of this becomes redundant:`x_c`

`FFT``(`x_c`)` `=` `[``(`X0r`,` `0``)``,` `(`X1r`,` X1i`)``,` `(`X2r`,` X2i`)``,` `(`X3r`,` `0``)``,` `(`X2r`,` `-`X2i`)``,` `(`X1r`,` `-`X1i`)``]`

The last two values are the complex conjugates of

and `X1`

. Additionally, `X2`

and `X0i`

are zero.
As we can see, the output contains 6 independent values, and the rest is redundant.
But it still takes time for the FFT to calculate the redundant values.
Converting the input data to complex also takes a little bit of time.`X3i`

If the length of

instead had been 7, result would have been:`x`

`FFT``(`x_c`)` `=` `[``(`X0r`,` `0``)``,` `(`X1r`,` X1i`)``,` `(`X2r`,` X2i`)``,` `(`X3r`,` X3i`)``,` `(`X3r`,` `-`X3i`)``,` `(`X2r`,` `-`X2i`)``,` `(`X1r`,` `-`X1i`)``]`

The result is similar, but this time there is no zero at

. Also in this case we got the same number of indendent values as we started with.`X3i`

#### Real-to-complex

Using a real-to-complex FFT removes the need for converting the input data to complex. It also avoids caclulating the redundant output values.

The result for 6 elements is:

`RealFFT``(`x`)` `=` `[``(`X0r`,` `0``)``,` `(`X1r`,` X1i`)``,` `(`X2r`,` X2i`)``,` `(`X3r`,` `0``)``]`

The result for 7 elements is:

`RealFFT``(`x`)` `=` `[``(`X0r`,` `0``)``,` `(`X1r`,` X1i`)``,` `(`X2r`,` X2i`)``,` `(`X3r`,` X3i`)``]`

This is the data layout output by the real-to-complex FFT, and the one expected as input to the complex-to-real iFFT.

### Scaling

RealFFT matches the behaviour of RustFFT and does not normalize the output of either FFT of iFFT. To get normalized results, each element must be scaled by

. If the processing involves both an FFT and an iFFT step, it is advisable to merge the two normalization steps to a single, by scaling by `1``/``sqrt``(`length`)`

.`1``/`length

### Documentation

The full documentation can be generated by rustdoc. To generate and view it run:

`cargo`` doc`` --`open

### Benchmarks

To run a set of benchmarks comparing real-to-complex FFT with standard complex-to-complex, type:

`cargo`` bench`

The results are printed while running, and are compiled into an html report containing much more details.
To view, open

in a browser.`target /criterion/report/index.html`

### Example

Transform a vector, and then inverse transform the result.

`use` `realfft``::`RealFftPlanner`;`
`use` `rustfft``::``num_complex``::`Complex`;`
`use` `rustfft``::``num_traits``::`Zero`;`
`let` length `=` `256``;`
`//` make a planner
`let` `mut` real_planner `=` `RealFftPlanner``::``<``f64``>``::`new`(``)``;`
`//` create a FFT
`let` r2c `=` real_planner`.``plan_fft_forward``(`length`)``;`
`//` make input and output vectors
`let` `mut` indata `=` r2c`.``make_input_vec``(``)``;`
`let` `mut` spectrum `=` r2c`.``make_output_vec``(``)``;`
`//` Are they the length we expect?
`assert_eq!``(`indata`.``len``(``)``,` length`)``;`
`assert_eq!``(`spectrum`.``len``(``)``,` length`/``2``+``1``)``;`
`//` Forward transform the input data
r2c`.``process``(``&``mut` indata`,` `&``mut` spectrum`)``.``unwrap``(``)``;`
`//` create an iFFT and an output vector
`let` c2r `=` real_planner`.``plan_fft_inverse``(`length`)``;`
`let` `mut` outdata `=` c2r`.``make_output_vec``(``)``;`
`assert_eq!``(`outdata`.``len``(``)``,` length`)``;`
c2r`.``process``(``&``mut` spectrum`,` `&``mut` outdata`)``.``unwrap``(``)``;`

### Compatibility

The

crate requires rustc version 1.37 or newer.`realfft`

License: MIT

#### Dependencies

~1MB

~23K SLoC