### ComplexToRealEven Initialization

Source: https://docs.rs/realfft/3.5.0/src/realfft/lib.rs.html

Constructor and setup logic for even-length complex-to-real inverse FFTs.

```rust
impl<T: FftNum> ComplexToRealEven<T> {
    /// Create a new ComplexToRealEven inverse FFT for complex input spectra.
    /// The `length` parameter refers to the length of the resulting real-valued signal.
    /// Uses the given FftPlanner to build the inner FFT.
    /// Panics if the length is not even.
    pub fn new(length: usize, fft_planner: &mut FftPlanner<T>) -> Self {
        if length % 2 > 0 {
            panic!("Length must be even, got {}", length,);
        }
        let twiddle_count = if length % 4 == 0 {
            length / 4
        } else {
            length / 4 + 1
        };
        let twiddles: Vec<Complex<T>> = (1..twiddle_count)
            .map(|i| compute_twiddle(i, length).conj())
            .collect();
        let fft = fft_planner.plan_fft_inverse(length / 2);
```

--------------------------------

### Representing Real-to-Complex FFT Data

Source: https://docs.rs/realfft/3.5.0/index.html

Examples of data layouts for real-valued vectors and their corresponding FFT outputs.

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

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

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

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

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

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

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

--------------------------------

### Representing real-valued signal as complex

Source: https://docs.rs/realfft/3.5.0/realfft

Example of a 6-element real vector converted to complex by adding zero-valued imaginary parts.

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

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

--------------------------------

### Generate and run benchmarks

Source: https://docs.rs/realfft/3.5.0/realfft

Commands to generate documentation and run performance benchmarks for the crate.

```bash
cargo doc --open
```

```bash
cargo bench
```

--------------------------------

### RealFftPlanner::new

Source: https://docs.rs/realfft/3.5.0/realfft/struct.RealFftPlanner.html

Creates a new instance of the RealFftPlanner.

```APIDOC
### pub fn new() -> Self

Creates a new planner instance.
```

--------------------------------

### Generate documentation

Source: https://docs.rs/realfft/3.5.0/realfft/index.html

Command to generate and open the crate documentation.

```bash
cargo doc --open
```

--------------------------------

### Run benchmarks

Source: https://docs.rs/realfft/3.5.0/realfft/index.html

Command to execute performance benchmarks.

```bash
cargo bench
```

--------------------------------

### RealFftPlanner

Source: https://docs.rs/realfft/3.5.0/src/realfft/lib.rs.html

A planner used to create and cache FFT instances.

```APIDOC
## RealFftPlanner<T>

### Description
Used to create FFT instances. It caches results internally, so it is recommended to reuse the same planner instance.

### Methods
- **new() -> Self**: Creates a new RealFftPlanner instance.
```

--------------------------------

### Utility Methods

Source: https://docs.rs/realfft/3.5.0/realfft/trait.RealToComplex.html

Methods for retrieving FFT configuration details and creating appropriately sized vectors.

```APIDOC
## Utility Methods

### fn get_scratch_len(&self) -> usize
Returns the minimum length required for the scratch buffer.

### fn len(&self) -> usize
Returns the length of the real signal input.

### fn complex_len(&self) -> usize
Returns the number of complex data points in the output.

### fn make_input_vec(&self) -> Vec<T>
Creates an input vector of the correct type and length.

### fn make_output_vec(&self) -> Vec<Complex<T>>
Creates an output vector of the correct type and length.

### fn make_scratch_vec(&self) -> Vec<Complex<T>>
Creates a scratch vector of the correct type and length.
```

--------------------------------

### RealFftPlanner

Source: https://docs.rs/realfft/3.5.0/realfft/all.html

The primary entry point for creating FFT plans for real-to-complex and complex-to-real transformations.

```APIDOC
## RealFftPlanner

### Description
Provides methods to plan FFT operations for real-to-complex and complex-to-real transformations.

### Usage
Use `RealFftPlanner` to instantiate the appropriate FFT processors based on the desired transform type and length.
```

--------------------------------

### RealToComplexEven Initialization

Source: https://docs.rs/realfft/3.5.0/src/realfft/lib.rs.html

Initialization logic for even-length real-to-complex FFTs, which panics if the length is not even.

```rust
impl<T: FftNum> RealToComplexEven<T> {
    /// Create a new RealToComplex forward FFT for real-valued input data of a given length,
    /// and uses the given FftPlanner to build the inner FFT.
    /// Panics if the length is not even.
    pub fn new(length: usize, fft_planner: &mut FftPlanner<T>) -> Self {
        if length % 2 > 0 {
            panic!("Length must be even, got {}", length,);
        }
```

--------------------------------

### Plan Real-to-Complex and Complex-to-Real FFTs

Source: https://docs.rs/realfft/3.5.0/src/realfft/lib.rs.html

Methods for planning forward and inverse FFTs with caching to reuse existing references for the same length.

```rust
pub fn plan_fft_forward(&mut self, len: usize) -> Arc<dyn RealToComplex<T>> {
        if let Some(fft) = self.r2c_cache.get(&len) {
            Arc::clone(fft)
        } else {
            let fft = if len % 2 > 0 {
                Arc::new(RealToComplexOdd::new(len, &mut self.planner)) as Arc<dyn RealToComplex<T>>
            } else {
                Arc::new(RealToComplexEven::new(len, &mut self.planner))
                    as Arc<dyn RealToComplex<T>>
            };
            self.r2c_cache.insert(len, Arc::clone(&fft));
            fft
        }
    }

    /// Plan a complex-to-real inverse FFT. Returns the FFT in a shared reference.
    /// If requesting a second inverse FFT of the same length,
    /// the planner will return a new reference to the already existing one.
    pub fn plan_fft_inverse(&mut self, len: usize) -> Arc<dyn ComplexToReal<T>> {
        if let Some(fft) = self.c2r_cache.get(&len) {
            Arc::clone(fft)
        } else {
            let fft = if len % 2 > 0 {
                Arc::new(ComplexToRealOdd::new(len, &mut self.planner)) as Arc<dyn ComplexToReal<T>>
            } else {
                Arc::new(ComplexToRealEven::new(len, &mut self.planner))
                    as Arc<dyn ComplexToReal<T>>
            };
            self.c2r_cache.insert(len, Arc::clone(&fft));
            fft
        }
    }
```

--------------------------------

### Perform Forward and Inverse FFT with RealFftPlanner

Source: https://docs.rs/realfft/3.5.0/realfft

Use RealFftPlanner to plan and execute forward and inverse FFTs on real-valued data.

```rust
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();
```

--------------------------------

### RealToComplexOdd Initialization and Processing

Source: https://docs.rs/realfft/3.5.0/src/realfft/lib.rs.html

Handles creation and processing logic for odd-length real-to-complex FFTs.

```rust
impl<T: FftNum> RealToComplexOdd<T> {
    /// Create a new RealToComplex forward FFT for real-valued input data of a given length,
    /// and uses the given FftPlanner to build the inner FFT.
    /// Panics if the length is not odd.
    pub fn new(length: usize, fft_planner: &mut FftPlanner<T>) -> Self {
        if length % 2 == 0 {
            panic!("Length must be odd, got {}", length,);
        }
        let fft = fft_planner.plan_fft_forward(length);
        let scratch_len = fft.get_inplace_scratch_len() + length;
        RealToComplexOdd {
            length,
            fft,
            scratch_len,
        }
    }
}

impl<T: FftNum> RealToComplex<T> for RealToComplexOdd<T> {
    fn process(&self, input: &mut [T], output: &mut [Complex<T>]) -> Res<()> {
        let mut scratch = self.make_scratch_vec();
        self.process_with_scratch(input, output, &mut scratch)
    }

    fn process_with_scratch(
        &self,
        input: &mut [T],
        output: &mut [Complex<T>],
        scratch: &mut [Complex<T>],
    ) -> Res<()> {
        if input.len() != self.length {
            return Err(FftError::InputBuffer(self.length, input.len()));
        }
        let expected_output_buffer_size = self.complex_len();
        if output.len() != expected_output_buffer_size {
            return Err(FftError::OutputBuffer(
                expected_output_buffer_size,
                output.len(),
            ));
        }
        if scratch.len() < (self.scratch_len) {
            return Err(FftError::ScratchBuffer(self.scratch_len, scratch.len()));
        }
        let (buffer, fft_scratch) = scratch.split_at_mut(self.length);

        for (val, buf) in input.iter().zip(buffer.iter_mut()) {
            *buf = Complex::new(*val, T::zero());
        }
        // FFT and store result in buffer_out
        self.fft.process_with_scratch(buffer, fft_scratch);
        output.copy_from_slice(&buffer[0..self.complex_len()]);
        if let Some(elem) = output.first_mut() {
            elem.im = T::zero();
        }
        Ok(())
    }

    fn get_scratch_len(&self) -> usize {
        self.scratch_len
    }

    fn len(&self) -> usize {
        self.length
    }

    fn make_input_vec(&self) -> Vec<T> {
        vec![T::zero(); self.len()]
    }

    fn make_output_vec(&self) -> Vec<Complex<T>> {
        vec![Complex::zero(); self.complex_len()]
    }

    fn make_scratch_vec(&self) -> Vec<Complex<T>> {
        vec![Complex::zero(); self.get_scratch_len()]
    }
}
```

--------------------------------

### Default Implementation for RealFftPlanner

Source: https://docs.rs/realfft/3.5.0/src/realfft/lib.rs.html

Provides a default implementation for the RealFftPlanner using its new method.

```rust
impl<T: FftNum> Default for RealFftPlanner<T> {
    fn default() -> Self {
        Self::new()
    }
}
```

--------------------------------

### ComplexToRealEven::process_with_scratch

Source: https://docs.rs/realfft/3.5.0/realfft/struct.ComplexToRealEven.html

Performs an inverse transform using a provided scratch buffer.

```APIDOC
## fn process_with_scratch(&self, input: &mut [Complex<T>], output: &mut [T], scratch: &mut [Complex<T>]) -> Result<(), FftError>

### Description
Inverse transform a complex spectrum, similar to `process()`. This method uses the provided scratch buffer instead of allocating new scratch space, which is more efficient for multiple calls.
```

--------------------------------

### Initialize RealToComplexEven

Source: https://docs.rs/realfft/3.5.0/src/realfft/lib.rs.html

Calculates twiddle factors and plans the forward FFT for the RealToComplexEven struct.

```rust
let twiddle_count = if length % 4 == 0 {
            length / 4
        } else {
            length / 4 + 1
        };
        let twiddles: Vec<Complex<T>> = (1..twiddle_count)
            .map(|i| compute_twiddle(i, length) * T::from_f64(0.5).unwrap())
            .collect();
        let fft = fft_planner.plan_fft_forward(length / 2);
        let scratch_len = fft.get_outofplace_scratch_len();
        RealToComplexEven {
            twiddles,
            length,
            fft,
            scratch_len,
        }
    }
}
```

--------------------------------

### ComplexToRealOdd Implementation

Source: https://docs.rs/realfft/3.5.0/src/realfft/lib.rs.html

Methods for handling odd-length inverse FFTs, including buffer management and scratch space allocation.

```rust
fn get_scratch_len(&self) -> usize {
        self.scratch_len
    }

    fn len(&self) -> usize {
        self.length
    }

    fn make_input_vec(&self) -> Vec<T> {
        vec![T::zero(); self.len()]
    }

    fn make_output_vec(&self) -> Vec<Complex<T>> {
        vec![Complex::zero(); self.complex_len()]
    }

    fn make_scratch_vec(&self) -> Vec<Complex<T>> {
        vec![Complex::zero(); self.get_scratch_len()]
    }
}

impl<T: FftNum> ComplexToRealOdd<T> {
    /// Create a new ComplexToRealOdd inverse FFT for complex input spectra.
    /// The `length` parameter refers to the length of the resulting real-valued signal.
    /// Uses the given FftPlanner to build the inner FFT.
    /// Panics if the length is not odd.
    pub fn new(length: usize, fft_planner: &mut FftPlanner<T>) -> Self {
        if length % 2 == 0 {
            panic!("Length must be odd, got {}", length,);
        }
        let fft = fft_planner.plan_fft_inverse(length);
        let scratch_len = length + fft.get_inplace_scratch_len();
        ComplexToRealOdd {
            length,
            fft,
            scratch_len,
        }
    }
}
```

--------------------------------

### ComplexToRealOdd::process_with_scratch

Source: https://docs.rs/realfft/3.5.0/realfft/struct.ComplexToRealOdd.html

Performs the inverse transform using a provided scratch buffer for improved performance.

```APIDOC
## fn process_with_scratch(&self, input: &mut [Complex<T>], output: &mut [T], scratch: &mut [Complex<T>]) -> Result<(), FftError>

### Description
Inverse transform a complex spectrum, similar to `process()`. The difference is that this method uses the provided scratch buffer instead of allocating new scratch space. This is faster if the same scratch buffer is used for multiple calls.
```

--------------------------------

### fn process_with_scratch

Source: https://docs.rs/realfft/3.5.0/realfft/trait.RealToComplex.html

Transforms a signal of N real-valued samples using a provided scratch buffer to avoid internal allocations.

```APIDOC
## fn process_with_scratch(&self, input: &mut [T], output: &mut [Complex<T>], scratch: &mut [Complex<T>]) -> Result<(), FftError>

### Description
Similar to process(), but uses the provided scratch buffer instead of allocating new space, which is more efficient for repeated calls.

### Parameters
- **input** (&mut [T]) - Required - The real-valued input signal buffer.
- **output** (&mut [Complex<T>]) - Required - The buffer to store the resulting complex spectrum.
- **scratch** (&mut [Complex<T>]) - Required - A pre-allocated scratch buffer.

### Returns
- **Result<(), FftError>** - Returns Ok(()) on success, or an FftError if slice lengths are incorrect.
```

--------------------------------

### RealFFT Buffer and Vector Management

Source: https://docs.rs/realfft/3.5.0/src/realfft/lib.rs.html

Methods for managing scratch space and allocating input/output vectors for FFT operations.

```rust
    fn get_scratch_len(&self) -> usize {
        self.scratch_len
    }

    fn len(&self) -> usize {
        self.length
    }

    fn make_input_vec(&self) -> Vec<Complex<T>> {
        vec![Complex::zero(); self.complex_len()]
    }

    fn make_output_vec(&self) -> Vec<T> {
        vec![T::zero(); self.len()]
    }

    fn make_scratch_vec(&self) -> Vec<Complex<T>> {
        vec![Complex::zero(); self.get_scratch_len()]
    }
```

--------------------------------

### ComplexToRealEven::new

Source: https://docs.rs/realfft/3.5.0/realfft/struct.ComplexToRealEven.html

Creates a new ComplexToRealEven inverse FFT instance for a specified signal length.

```APIDOC
## pub fn new(length: usize, fft_planner: &mut FftPlanner<T>) -> Self

### Description
Create a new ComplexToRealEven inverse FFT for complex input spectra. The `length` parameter refers to the length of the resulting real-valued signal. Uses the given FftPlanner to build the inner FFT. Panics if the length is not even.
```

--------------------------------

### Process FFT with Scratch Space

Source: https://docs.rs/realfft/3.5.0/src/realfft/lib.rs.html

Validates input/output buffers and executes the FFT processing logic with scratch space.

```rust
impl<T: FftNum> RealToComplex<T> for RealToComplexEven<T> {
    fn process(&self, input: &mut [T], output: &mut [Complex<T>]) -> Res<()> {
        let mut scratch = self.make_scratch_vec();
        self.process_with_scratch(input, output, &mut scratch)
    }

    fn process_with_scratch(
        &self,
        input: &mut [T],
        output: &mut [Complex<T>],
        scratch: &mut [Complex<T>],
    ) -> Res<()> {
        if input.len() != self.length {
            return Err(FftError::InputBuffer(self.length, input.len()));
        }
        let expected_output_buffer_size = self.complex_len();
        if output.len() != expected_output_buffer_size {
            return Err(FftError::OutputBuffer(
                expected_output_buffer_size,
                output.len(),
            ));
        }
        if scratch.len() < (self.scratch_len) {
            return Err(FftError::ScratchBuffer(self.scratch_len, scratch.len()));
        }

        let fftlen = self.length / 2;
        let buf_in = unsafe {
            let ptr = input.as_mut_ptr() as *mut Complex<T>;
            let len = input.len();
            std::slice::from_raw_parts_mut(ptr, len / 2)
        };

        // FFT and store result in buffer_out
        self.fft
            .process_outofplace_with_scratch(buf_in, &mut output[0..fftlen], scratch);
        let (mut output_left, mut output_right) = output.split_at_mut(output.len() / 2);

        // The first and last element don't require any twiddle factors, so skip that work
        match (output_left.first_mut(), output_right.last_mut()) {
            (Some(first_element), Some(last_element)) => {
                // The first and last elements are just a sum and difference of the first value's real and imaginary values
                let first_value = *first_element;
                *first_element = Complex {
                    re: first_value.re + first_value.im,
                    im: T::zero(),
                };
                *last_element = Complex {
                    re: first_value.re - first_value.im,
                    im: T::zero(),
                };

                // Chop the first and last element off of our slices so that the loop below doesn't have to deal with them
                output_left = &mut output_left[1..];
                let right_len = output_right.len();
                output_right = &mut output_right[..right_len - 1];
            }
            _ => {
                return Ok(());
            }
        }
        // Loop over the remaining elements and apply twiddle factors on them
        for (twiddle, out, out_rev) in zip3(
            self.twiddles.iter(),
            output_left.iter_mut(),
            output_right.iter_mut().rev(),
        ) {
            let sum = *out + *out_rev;
            let diff = *out - *out_rev;
            let half = T::from_f64(0.5).unwrap();
            // Apply twiddle factors. Theoretically we'd have to load 2 separate twiddle factors here, one for the beginning
            // and one for the end. But the twiddle factor for the end is just the twiddle for the beginning, with the
            // real part negated. Since it's the same twiddle, we can factor out a ton of math ops and cut the number of
            // multiplications in half.
            let twiddled_re_sum = sum * twiddle.re;
            let twiddled_im_sum = sum * twiddle.im;
            let twiddled_re_diff = diff * twiddle.re;
            let twiddled_im_diff = diff * twiddle.im;
            let half_sum_re = half * sum.re;
            let half_diff_im = half * diff.im;

            let output_twiddled_real = twiddled_re_sum.im + twiddled_im_diff.re;
            let output_twiddled_im = twiddled_im_sum.im - twiddled_re_diff.re;

            // We finally have all the data we need to write the transformed data back out where we found it.
            *out = Complex {
                re: half_sum_re + output_twiddled_real,
                im: half_diff_im + output_twiddled_im,
            };

            *out_rev = Complex {
                re: half_sum_re - output_twiddled_real,
                im: half_diff_im - output_twiddled_im,
            };
        }
        Ok(())
    }
}
```

--------------------------------

### Handle errors with the ? operator

Source: https://docs.rs/realfft/3.5.0/src/realfft/lib.rs.html

Demonstrates that the RealFFT process methods are compatible with the ? operator for error propagation.

```rust
#[allow(dead_code)]
    fn test_error() -> Result<(), Box<dyn Error>> {
        let mut real_planner = RealFftPlanner::<f64>::new();
        let r2c = real_planner.plan_fft_forward(100);
        let mut out_a = r2c.make_output_vec();
        let mut indata = r2c.make_input_vec();
        r2c.process(&mut indata, &mut out_a)?;
        Ok(())
    }
```

--------------------------------

### fn process

Source: https://docs.rs/realfft/3.5.0/realfft/trait.RealToComplex.html

Transforms a signal of N real-valued samples into a complex spectrum. The input buffer is used as scratch space and its contents should be considered garbage after the call.

```APIDOC
## fn process(&self, input: &mut [T], output: &mut [Complex<T>]) -> Result<(), FftError>

### Description
Transforms a signal of N real-valued samples, storing the resulting complex spectrum in the N/2+1 element long output slice. The input buffer is used as scratch space.

### Parameters
- **input** (&mut [T]) - Required - The real-valued input signal buffer.
- **output** (&mut [Complex<T>]) - Required - The buffer to store the resulting complex spectrum.

### Returns
- **Result<(), FftError>** - Returns Ok(()) on success, or an FftError if slice lengths are incorrect.
```

--------------------------------

### FFT Comparison Utilities

Source: https://docs.rs/realfft/3.5.0/src/realfft/lib.rs.html

Helper functions to calculate the maximum difference between complex or scalar arrays.

```rust
    fn compare_complex<T: Float + Sub>(a: &[Complex<T>], b: &[Complex<T>]) -> T {
        a.iter()
            .zip(b.iter())
            .fold(T::zero(), |maxdiff, (val_a, val_b)| {
                let diff = (val_a - val_b).norm();
                if maxdiff > diff {
                    maxdiff
                } else {
                    diff
                }
            })
    }

    fn compare_scalars<T: Float + Sub>(a: &[T], b: &[T]) -> T {
        a.iter()
            .zip(b.iter())
            .fold(T::zero(), |maxdiff, (val_a, val_b)| {
                let diff = (*val_a - *val_b).abs();
                if maxdiff > diff {
                    maxdiff
                } else {
                    diff
                }
            })
    }
```

--------------------------------

### ComplexToRealEven::process

Source: https://docs.rs/realfft/3.5.0/realfft/struct.ComplexToRealEven.html

Performs an inverse transform on a complex spectrum to produce a real-valued signal.

```APIDOC
## fn process(&self, input: &mut [Complex<T>], output: &mut [T]) -> Result<(), FftError>

### Description
Inverse transform a complex spectrum corresponding to a real-valued signal of length N. The input is a slice of complex values with length N/2+1. The resulting real-valued signal is stored in the output slice of length N. The input buffer is used as scratch space.
```

--------------------------------

### RealFftPlanner::plan_fft_forward

Source: https://docs.rs/realfft/3.5.0/realfft/struct.RealFftPlanner.html

Plans a real-to-complex forward FFT of a specified length.

```APIDOC
### pub fn plan_fft_forward(&mut self, len: usize) -> Arc<dyn RealToComplex<T>>

Plans a real-to-complex forward FFT. Returns the FFT in a shared reference. If requesting a second forward FFT of the same length, the planner will return a new reference to the already existing one.
```

--------------------------------

### RealToComplexEven::process_with_scratch

Source: https://docs.rs/realfft/3.5.0/realfft/struct.RealToComplexEven.html

Transforms a signal of N real-valued samples using a provided scratch buffer.

```APIDOC
## fn process_with_scratch(&self, input: &mut [T], output: &mut [Complex<T>], scratch: &mut [Complex<T>]) -> Result<(), FftError>

### Description
Transform a signal of N real-valued samples, similar to process(). This method uses the provided scratch buffer instead of allocating new scratch space, which is faster for multiple calls.
```

--------------------------------

### Compare RealToComplex with Standard FFT

Source: https://docs.rs/realfft/3.5.0/src/realfft/lib.rs.html

Validates the forward RealToComplex transformation by comparing results against a standard FFT implementation for various lengths.

```rust
#[test]
    fn real_to_complex_64() {
        for length in 1..1000 {
            let mut real_planner = RealFftPlanner::<f64>::new();
            let r2c = real_planner.plan_fft_forward(length);
            let mut out_a = r2c.make_output_vec();
            let mut indata = r2c.make_input_vec();
            let mut rng = rand::rng();
            for val in indata.iter_mut() {
                *val = rng.random::<f64>();
            }
            let mut rustfft_check = indata
                .iter()
                .map(Complex::from)
                .collect::<Vec<Complex<f64>>>();
            let mut fft_planner = FftPlanner::<f64>::new();
            let fft = fft_planner.plan_fft_forward(length);
            fft.process(&mut rustfft_check);
            r2c.process(&mut indata, &mut out_a).unwrap();
            assert_eq!(out_a[0].im, 0.0, "First imaginary component must be zero");
            if length % 2 == 0 {
                assert_eq!(
                    out_a.last().unwrap().im,
                    0.0,
                    "Last imaginary component for even lengths must be zero"
                );
            }
            let maxdiff = compare_complex(&out_a, &rustfft_check[0..r2c.complex_len()]);
            assert!(
                maxdiff < 1.0e-9,
            );
        }
    }
```

--------------------------------

### ComplexToRealOdd Processing

Source: https://docs.rs/realfft/3.5.0/src/realfft/lib.rs.html

Core processing logic for odd-length complex-to-real inverse FFTs.

```rust
impl<T: FftNum> ComplexToReal<T> for ComplexToRealOdd<T> {
    fn process(&self, input: &mut [Complex<T>], output: &mut [T]) -> Res<()> {
        let mut scratch = self.make_scratch_vec();
        self.process_with_scratch(input, output, &mut scratch)
    }

    fn process_with_scratch(
        &self,
        input: &mut [Complex<T>],
        output: &mut [T],
        scratch: &mut [Complex<T>],
    ) -> Res<()> {
        let expected_input_buffer_size = self.complex_len();
        if input.len() != expected_input_buffer_size {
            return Err(FftError::InputBuffer(
                expected_input_buffer_size,
                input.len(),
            ));
        }
        if output.len() != self.length {
            return Err(FftError::OutputBuffer(self.length, output.len()));
        }
        if scratch.len() < (self.scratch_len) {
            return Err(FftError::ScratchBuffer(self.scratch_len, scratch.len()));
        }

        let first_invalid = if input[0].im != T::from_f64(0.0).unwrap() {
            input[0].im = T::from_f64(0.0).unwrap();
            true
        } else {
            false
        };

        let (buffer, fft_scratch) = scratch.split_at_mut(self.length);

        buffer[0..input.len()].copy_from_slice(input);
        for (buf, val) in buffer
            .iter_mut()
            .rev()
            .take(self.length / 2)
            .zip(input.iter().skip(1))
        {
            *buf = val.conj();
        }
        self.fft.process_with_scratch(buffer, fft_scratch);
        for (val, out) in buffer.iter().zip(output.iter_mut()) {
            *out = val.re;
        }
        if first_invalid {
            return Err(FftError::InputValues(true, false));
        }
        Ok(())
    }

    fn get_scratch_len(&self) -> usize {
        self.scratch_len
    }

    fn len(&self) -> usize {
        self.length
    }

    fn make_input_vec(&self) -> Vec<Complex<T>> {
        vec![Complex::zero(); self.complex_len()]
    }

    fn make_output_vec(&self) -> Vec<T> {
        vec![T::zero(); self.len()]
    }

    fn make_scratch_vec(&self) -> Vec<Complex<T>> {
        vec![Complex::zero(); self.get_scratch_len()]
    }
}
```

--------------------------------

### RealToComplexEven::new

Source: https://docs.rs/realfft/3.5.0/realfft/struct.RealToComplexEven.html

Creates a new RealToComplex forward FFT instance for real-valued input data of a specified length.

```APIDOC
## pub fn new(length: usize, fft_planner: &mut FftPlanner<T>) -> Self

### Description
Creates a new RealToComplex forward FFT for real-valued input data of a given length, and uses the given FftPlanner to build the inner FFT. Panics if the length is not even.
```

--------------------------------

### RealToComplexOdd::process_with_scratch

Source: https://docs.rs/realfft/3.5.0/realfft/struct.RealToComplexOdd.html

Transforms a signal of N real-valued samples using a provided scratch buffer.

```APIDOC
## fn process_with_scratch(&self, input: &mut [T], output: &mut [Complex<T>], scratch: &mut [Complex<T>]) -> Result<(), FftError>

### Description
Transform a signal of N real-valued samples, similar to process(). This method uses the provided scratch buffer instead of allocating new scratch space, which is more efficient for multiple calls.
```

--------------------------------

### RealFftPlanner::plan_fft_inverse

Source: https://docs.rs/realfft/3.5.0/realfft/struct.RealFftPlanner.html

Plans a complex-to-real inverse FFT of a specified length.

```APIDOC
### pub fn plan_fft_inverse(&mut self, len: usize) -> Arc<dyn ComplexToReal<T>>

Plans a complex-to-real inverse FFT. Returns the FFT in a shared reference. If requesting a second inverse FFT of the same length, the planner will return a new reference to the already existing one.
```

--------------------------------

### ComplexToRealOdd::new

Source: https://docs.rs/realfft/3.5.0/realfft/struct.ComplexToRealOdd.html

Creates a new ComplexToRealOdd inverse FFT instance. The length parameter specifies the length of the resulting real-valued signal and must be odd.

```APIDOC
## pub fn new(length: usize, fft_planner: &mut FftPlanner<T>) -> Self

### Description
Create a new ComplexToRealOdd inverse FFT for complex input spectra. The `length` parameter refers to the length of the resulting real-valued signal. Uses the given FftPlanner to build the inner FFT. Panics if the length is not odd.
```

--------------------------------

### ComplexToRealEven::get_scratch_len

Source: https://docs.rs/realfft/3.5.0/realfft/struct.ComplexToRealEven.html

Returns the required length for the scratch buffer.

```APIDOC
## fn get_scratch_len(&self) -> usize

### Description
Get the minimum length of the scratch space needed for `process_with_scratch`.
```

--------------------------------

### Compare RealToComplex with standard FFT

Source: https://docs.rs/realfft/3.5.0/src/realfft/lib.rs.html

Validates the RealToComplex transformation by comparing output against a standard complex FFT implementation across various lengths.

```rust
#[test]
    fn real_to_complex_32() {
        for length in 1..1000 {
            let mut real_planner = RealFftPlanner::<f32>::new();
            let r2c = real_planner.plan_fft_forward(length);
            let mut out_a = r2c.make_output_vec();
            let mut indata = r2c.make_input_vec();
            let mut rng = rand::rng();
            for val in indata.iter_mut() {
                *val = rng.random::<f32>();
            }
            let mut rustfft_check = indata
                .iter()
                .map(Complex::from)
                .collect::<Vec<Complex<f32>>>();
            let mut fft_planner = FftPlanner::<f32>::new();
            let fft = fft_planner.plan_fft_forward(length);
            fft.process(&mut rustfft_check);
            r2c.process(&mut indata, &mut out_a).unwrap();
            assert_eq!(out_a[0].im, 0.0, "First imaginary component must be zero");
            if length % 2 == 0 {
                assert_eq!(
                    out_a.last().unwrap().im,
                    0.0,
                    "Last imaginary component for even lengths must be zero"
                );
            }
            let maxdiff = compare_complex(&out_a, &rustfft_check[0..r2c.complex_len()]);
            assert!(
                maxdiff < 5.0e-4,
                "Length: {}, too large error: {}",
                length,
                maxdiff
            );
        }
    }
```

--------------------------------

### RealFftPlanner Structure Definition

Source: https://docs.rs/realfft/3.5.0/realfft/struct.RealFftPlanner.html

The internal structure definition for RealFftPlanner, which requires the FftNum trait.

```rust
pub struct RealFftPlanner<T: FftNum> { /* private fields */ }
```

--------------------------------

### Re-exporting dependencies

Source: https://docs.rs/realfft/3.5.0/realfft

Exposes num_complex and num_traits from the rustfft crate.

```rust
pub use rustfft::num_complex;
```

```rust
pub use rustfft::num_traits;
```

--------------------------------

### ComplexToRealOdd::process

Source: https://docs.rs/realfft/3.5.0/realfft/struct.ComplexToRealOdd.html

Performs the inverse transform on a complex spectrum to produce a real-valued signal.

```APIDOC
## fn process(&self, input: &mut [Complex<T>], output: &mut [T]) -> Result<(), FftError>

### Description
Inverse transform a complex spectrum corresponding to a real-valued signal of length N. The input is a slice of complex values with length N/2+1 (with N/2 rounded down). The resulting real-valued signal is stored in the output slice of length N. The input buffer is used as scratch space, so the contents of input should be considered garbage after calling.
```

--------------------------------

### Traits

Source: https://docs.rs/realfft/3.5.0/realfft/all.html

Core traits defining the interface for FFT operations.

```APIDOC
## Traits

### ComplexToReal
Trait for complex-to-real FFT operations.

### RealToComplex
Trait for real-to-complex FFT operations.

### FftNum
Trait defining the numeric requirements for FFT operations.
```

--------------------------------

### RealToComplexOdd::new

Source: https://docs.rs/realfft/3.5.0/realfft/struct.RealToComplexOdd.html

Creates a new RealToComplex forward FFT instance for real-valued input data of a specified odd length.

```APIDOC
## pub fn new(length: usize, fft_planner: &mut FftPlanner<T>) -> Self

### Description
Creates a new RealToComplex forward FFT for real-valued input data of a given length, and uses the given FftPlanner to build the inner FFT. Panics if the length is not odd.
```

--------------------------------

### ComplexToReal Inverse FFT Validation

Source: https://docs.rs/realfft/3.5.0/src/realfft/lib.rs.html

Unit tests comparing RealFFT inverse operations against standard rustfft inverse FFT for f64 and f32 precision.

```rust
    #[test]
    fn complex_to_real_64() {
        for length in 1..1000 {
            let mut real_planner = RealFftPlanner::<f64>::new();
            let c2r = real_planner.plan_fft_inverse(length);
            let mut out_a = c2r.make_output_vec();
            let mut indata = c2r.make_input_vec();
            let mut rustfft_check: Vec<Complex<f64>> = vec![Complex::zero(); length];
            let mut rng = rand::rng();
            for val in indata.iter_mut() {
                *val = Complex::new(rng.random::<f64>(), rng.random::<f64>());
            }
            indata[0].im = 0.0;
            if length % 2 == 0 {
                indata[length / 2].im = 0.0;
            }
            for (val_long, val) in rustfft_check
                .iter_mut()
                .take(c2r.complex_len())
                .zip(indata.iter())
            {
                *val_long = *val;
            }
            for (val_long, val) in rustfft_check
                .iter_mut()
                .rev()
                .take(length / 2)
                .zip(indata.iter().skip(1))
            {
                *val_long = val.conj();
            }
            let mut fft_planner = FftPlanner::<f64>::new();
            let fft = fft_planner.plan_fft_inverse(length);

            c2r.process(&mut indata, &mut out_a).unwrap();
            fft.process(&mut rustfft_check);

            let check_real = rustfft_check.iter().map(|val| val.re).collect::<Vec<f64>>();
            let maxdiff = compare_scalars(&out_a, &check_real);
            assert!(
                maxdiff < 1.0e-9,
                "Length: {}, too large error: {}",
                length,
                maxdiff
            );
        }
    }
```

```rust
    #[test]
    fn complex_to_real_32() {
        for length in 1..1000 {
            let mut real_planner = RealFftPlanner::<f32>::new();
            let c2r = real_planner.plan_fft_inverse(length);
            let mut out_a = c2r.make_output_vec();
            let mut indata = c2r.make_input_vec();
            let mut rustfft_check: Vec<Complex<f32>> = vec![Complex::zero(); length];
            let mut rng = rand::rng();
            for val in indata.iter_mut() {
                *val = Complex::new(rng.random::<f32>(), rng.random::<f32>());
            }
            indata[0].im = 0.0;
            if length % 2 == 0 {
                indata[length / 2].im = 0.0;
            }
            for (val_long, val) in rustfft_check
                .iter_mut()
                .take(c2r.complex_len())
                .zip(indata.iter())
            {
                *val_long = *val;
            }
```

--------------------------------

### RealToComplexEven::process

Source: https://docs.rs/realfft/3.5.0/realfft/struct.RealToComplexEven.html

Transforms a signal of N real-valued samples into a complex spectrum.

```APIDOC
## fn process(&self, input: &mut [T], output: &mut [Complex<T>]) -> Result<(), FftError>

### Description
Transform a signal of N real-valued samples, storing the resulting complex spectrum in the N/2+1 element long output slice. The input buffer is used as scratch space, so the contents of input should be considered garbage after calling.
```

--------------------------------

### Complex FFT output of real-valued input

Source: https://docs.rs/realfft/3.5.0/realfft

The resulting complex spectrum from a standard complex FFT on a real-valued input.

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

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

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

--------------------------------

### Real-to-complex FFT output

Source: https://docs.rs/realfft/3.5.0/realfft

The compact output format produced by the RealFFT library for real-valued inputs.

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

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

--------------------------------

### ComplexToRealEven Struct Definition

Source: https://docs.rs/realfft/3.5.0/realfft/struct.ComplexToRealEven.html

The definition of the ComplexToRealEven struct.

```rust
pub struct ComplexToRealEven<T> { /* private fields */ }
```

--------------------------------

### Compute twiddle factors

Source: https://docs.rs/realfft/3.5.0/src/realfft/lib.rs.html

Internal helper function to calculate complex twiddle factors for FFT algorithms.

```rust
fn compute_twiddle<T: FftNum>(index: usize, fft_len: usize) -> Complex<T> {
    let constant = -2f64 * std::f64::consts::PI / fft_len as f64;
    let angle = constant * index as f64;
    Complex {
        re: T::from_f64(angle.cos()).unwrap(),
        im: T::from_f64(angle.sin()).unwrap(),
    }
}
```

--------------------------------

### FftNum Trait Implementation

Source: https://docs.rs/realfft/3.5.0/realfft/trait.FftNum.html

The blanket implementation of FftNum for types satisfying the required trait bounds.

```rust
impl<T> FftNum for T
where T: Copy + FromPrimitive + Signed + Sync + Send + Debug + 'static,
```

--------------------------------

### Validate ComplexToReal Error Handling for Odd Lengths

Source: https://docs.rs/realfft/3.5.0/src/realfft/lib.rs.html

Tests that the inverse FFT correctly identifies invalid input data at the first point for odd-length sequences.

```rust
#[test]
    fn complex_to_real_errors_odd() {
        let length = 101;
        let mut real_planner = RealFftPlanner::<f64>::new();
        let c2r = real_planner.plan_fft_inverse(length);
        let mut out_a = c2r.make_output_vec();
        let mut indata = c2r.make_input_vec();
        let mut rng = rand::rng();

        // Make some valid data
        for val in indata.iter_mut() {
            *val = Complex::new(rng.random::<f64>(), rng.random::<f64>());
        }
        indata[0].im = 0.0;
        // this should be ok
        assert!(c2r.process(&mut indata, &mut out_a).is_ok());

        // Make some invalid data, first point invalid
        for val in indata.iter_mut() {
            *val = Complex::new(rng.random::<f64>(), rng.random::<f64>());
        }
        let res = c2r.process(&mut indata, &mut out_a);
        assert!(res.is_err());
        assert!(matches!(res, Err(FftError::InputValues(true, false))));
    }
```

--------------------------------

### Validate ComplexToReal Error Handling for Even Lengths

Source: https://docs.rs/realfft/3.5.0/src/realfft/lib.rs.html

Tests that the inverse FFT correctly identifies invalid input data at the first and last points for even-length sequences.

```rust
#[test]
    fn complex_to_real_errors_even() {
        let length = 100;
        let mut real_planner = RealFftPlanner::<f64>::new();
        let c2r = real_planner.plan_fft_inverse(length);
        let mut out_a = c2r.make_output_vec();
        let mut indata = c2r.make_input_vec();
        let mut rng = rand::rng();

        // Make some valid data
        for val in indata.iter_mut() {
            *val = Complex::new(rng.random::<f64>(), rng.random::<f64>());
        }
        indata[0].im = 0.0;
        indata[50].im = 0.0;
        // this should be ok
        assert!(c2r.process(&mut indata, &mut out_a).is_ok());

        // Make some invalid data, first point invalid
        for val in indata.iter_mut() {
            *val = Complex::new(rng.random::<f64>(), rng.random::<f64>());
        }
        indata[50].im = 0.0;
        let res = c2r.process(&mut indata, &mut out_a);
        assert!(res.is_err());
        assert!(matches!(res, Err(FftError::InputValues(true, false))));

        // Make some invalid data, last point invalid
        for val in indata.iter_mut() {
            *val = Complex::new(rng.random::<f64>(), rng.random::<f64>());
        }
        indata[0].im = 0.0;
        let res = c2r.process(&mut indata, &mut out_a);
        assert!(res.is_err());
        assert!(matches!(res, Err(FftError::InputValues(false, true))));
    }
```

--------------------------------

### Define FftError Enum

Source: https://docs.rs/realfft/3.5.0/realfft/enum.FftError.html

The structure of the FftError enum, detailing variants for input, output, and scratch buffer size errors, as well as input value validation errors.

```rust
pub enum FftError {
    InputBuffer(usize, usize),
    OutputBuffer(usize, usize),
    ScratchBuffer(usize, usize),
    InputValues(bool, bool),
}
```