## Expand description

## §RealFFT: Real-to-complex forward FFT and complex-to-real inverse FFT based on RustFFT

This library is a wrapper for RustFFT that enables fast and convenient 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 real-valued signal of length N into an N/2 long complex buffer, which is transformed using a standard complex-to-complex FFT. The FFT result is then post-processed to give only the first half of the complex spectrum, as an N/2+1 long complex vector.

The inverse FFT goes through the same steps backwards, to transform a complex spectrum of length N/2+1 to a real-valued signal of length N.

The speed increase compared to just converting the input to a length N complex vector and using a length N complex-to-complex FFT depends on the length of the input data. The largest improvements are for longer 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 transform a real valued signal is to convert it to complex, and then use a complex-to-complex FFT.

Let’s assume `x`

is a 6 element long real vector:

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

We now convert `x`

to complex by adding an imaginary part with value zero.
Using the notation `(xNr, xNi)`

for the complex value `xN`

, this becomes:

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

Performing a normal complex FFT, the result of `FFT(x_c)`

is:

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

But because our `x_c`

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

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

The last two values are the complex conjugates of `X1`

and `X2`

. Additionally, `X0i`

and `X3i`

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.

If the length of `x`

instead had been 7, result would have been:

`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 `X3i`

.
Also in this case we got the same number of independent values as we started with.

#### §Real-to-complex

Using a real-to-complex FFT removes the need for converting the input data to complex. It also avoids calculating 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 inverse FFT.

### §Scaling

RealFFT matches the behaviour of RustFFT and does not normalize the output of either forward or inverse FFT.
To get normalized results, each element must be scaled by `1/sqrt(length)`

,
where `length`

is the length of the real-valued signal.
If the processing involves both a forward and an inverse FFT step,
it is advisable to merge the two normalization steps to a single, by scaling by `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 `target/criterion/report/index.html`

in a browser.

### §Example

Transform a signal, 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 a dummy real-valued signal (filled with zeros)
let mut indata = r2c.make_input_vec();
// make a vector for storing the spectrum
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 signal
r2c.process(&mut indata, &mut spectrum).unwrap();
// create an inverse FFT
let c2r = real_planner.plan_fft_inverse(length);
// create a vector for storing the output
let mut outdata = c2r.make_output_vec();
assert_eq!(outdata.len(), length);
// inverse transform the spectrum back to a real-valued signal
c2r.process(&mut spectrum, &mut outdata).unwrap();
```

#### §Versions

- 3.4.0: Fix undefined behavior reported by Miri. Update to latest RustFFT.
- 3.3.0: Add method for getting the length of the complex input/output. Bugfix: clean up numerical noise in the zero imaginary components.
- 3.2.0: Allow scratch buffer to be larger than needed.
- 3.1.0: Update to RustFFT 6.1 with Neon support.
- 3.0.2: Fix confusing typos in errors about scratch length.
- 3.0.1: More helpful error messages, fix confusing typos.
- 3.0.0: Improved error reporting.
- 2.0.1: Minor bugfix.
- 2.0.0: Update RustFFT to 6.0.0 and num-complex to 0.4.0.
- 1.1.0: Add missing Sync+Send.
- 1.0.0: First version with new api.

#### §Compatibility

The `realfft`

crate has the same rustc version requirements as RustFFT.
The minimum rustc version is 1.61.

## Re-exports§

`pub use rustfft::num_complex;`

`pub use rustfft::num_traits;`

## Structs§

- A planner is used to create FFTs. It caches results internally, so when making more than one FFT it is advisable to reuse the same planner.

## Enums§

- Custom error returned by FFTs

## Traits§

- An inverse FFT that takes a complex spectrum of length N/2+1 and transforms it to a real-valued signal of length N.
- Generic floating point number, implemented for f32 and f64
- A forward FFT that takes a real-valued input signal of length N and transforms it to a complex spectrum of length N/2+1.