# How to apply Fast Fourier Transform¶

This article demonstrates how to use the Fast Fourier Transform and apply both forward and inverse FFT on complex and real data using the KFR framework.

KFR DFT supports all sizes, and KFR automatically chooses the best algorithm to perform DFT for the given size.

## Quick example¶

### Complex input/output¶

Apply the FFT to the complex input data: See how to pass data to KFR

```
// prepare data (0, 1, 2, ..., 255)
univector<complex<double>, 256> data = cexp(
linspace(0, c_pi<double, 2>, 256) * make_complex(0, 1));
univector<complex<double>, 256> freq;
// do the forward FFT
freq = dft(data);
```

```
data = idft(freq);
```

Scaling is not performed by KFR. To get output in the same scale as input, divide it by `size`

.

### Real input, complex output¶

Frequency data is stored in CCS or Perm format.

The size of the output data is equal to `size/2+1`

for CCS and `size/2`

for Perm format.

For the inverse FFT, you have to prepare frequency data in the CCS or Perm format as well.

For CCS format, you must ensure that freq[0] and freq[N/2] are real numbers to get the correct real result.

```
data = irealdft(freq);
// or to get the properly scaled result:
data = irealdft(freq) / data.size();
```

Note

Real-to-complex and complex-to-real transforms are only available for even sizes. This is caused by the way real DFT is calculated. A pair of real values are interpreted as complex for high performance, so there is a limitation for real DFT size; it must be even. Use complex transform and data conversion instead.

```
const dft_plan<double> dft(N); // N is odd
univector<complex<double>> output(N);
univector<double> input;
univector<u8> temp(dft.temp_size); // temporary buffer
dft.execute(output, univector<complex<double>>(input), temp);
```

## Creating FFT plan¶

The implementation of FFT requires twiddle coefficients to be prepared before actual processing occurs. If FFT will be performed more than once, then it makes sense to keep dft plan and reuse it every time.

## FFT Plan caching¶

If you are using `dft`

, `idft`

, `realdft`

or `irealdft`

functions, all plans will be kept in memory, so the next call to these functions will reuse the saved data.

You can manually get plan from the cache (or create a new if it doesn’t exist in the cache):

```
dft_plan_ptr<T> dft = dft_cache::instance().get(ctype<T>, size);
```

Note

All functions related to the FFT caching are protected with mutex and are thread safe

`ctype`

is a special template variable intended just to pass a type to the function.

`dft_plan_ptr`

is an alias for `std::shared_ptr<const dft_plan<T>>`

To clear all saved plans and free memory, call the clear function:

```
dft_cache::instance().clear();
```

Note

As of version 1.3.0, functions convolve and correlate use these cache internally.

## Examples using plan¶

### Complex-to-complex transform¶

You can create a plan manually without using the cache to get more control over it.

`dft_plan`

class is intended to store all data needed for FFT.

```
template <typename T> // data type, float or double
struct dft_plan
{
dft_plan(size_t size);
void execute(complex<T>* out,
const complex<T>* in,
u8* temp,
bool inverse = false) const;
void execute(univector<complex<T>, N>& out,
const univector<const complex<T>, N>& in,
univector<u8, N2>& temp,
bool inverse = false) const;
...
};
```

`dft_plan`

has constructor that takes FFT size as an argument.

Input and output arguments for calls to the execute function must have types `complex<T>*`

or `univector<complex<T>, N>`

.

You can create a plan and use it as many times as needed. After creating a plan, all access to it is thread-safe, because executing the FFT doesn’t modify its internal data, so you can share this plan across threads in your application without protecting the plan with locks and mutexes.

```
dft_plan<double> plan(1024);
univector<complex<double>, 1024> in;
univector<complex<double>, 1024> out;
// here fill `in` array with our data (samples)
univector<u8> temp(plan.temp_size); // temporary buffer
plan.execute(out, in, temp, false); // direct FFT
// `out` now contains frequencies which have to be processed
plan.execute(in, out, temp, true); // inverse FFT
// `in` now contains processed data (samples)
...
// process new data
plan.execute(out, in, temp, false); // direct FFT
```

### Real-to-complex and complex-to-real transform¶

```
dft_plan_real<double> plan(1024); // dft_plan_real for real transform
univector<double, 1024> in;
univector<complex<double>, 1024> out;
// here fill `in` array with our data (samples)
univector<u8> temp(plan.temp_size); // temporary buffer
plan.execute(out, in, temp); // direct FFT
// `out` now contains frequencies which have to be processed
plan.execute(in, out, temp); // inverse FFT
// `in` now contains processed data (samples)
```

## Multidimensional DFT¶

The multidimensional DFT can be performed the same way as the 1D transform but using the `dft_plan_md`

class.

```
dft_plan_md<double> plan(shape{ 64, 256 });
const std::complex<double>* in = ...; // the number of samples for in and out must be
// the product of sizes (here is 16384)
std::complex<double>* out = ...;
// here fill `in` array with our data (samples)
univector<u8> temp(plan.temp_size); // temporary buffer
plan.execute(out, in, temp, false); // direct FFT
// `out` now contains frequencies which have to be processed
plan.execute(in, out, temp, true); // inverse FFT
// `in` now contains processed data (samples)
...
// process new data
plan.execute(out, in, temp, false); // direct FFT
```

`dft_plan_md`

class also supports passing tensors as `in`

and `out`

.

## Multidimensional Real DFT¶

For multidimensional real-to-complex transforms, the DFT performs a real-to-complex transform for the last axis, followed by a number of complex-to-complex transforms for the other axes. The size of the last axis must be even.

Multidimensional DFT is always performed in CCS format.

Thus, given the size of the input as \((S_0, S_1, ..., S_{n-1})\), the total number of input samples equals \(S_0 \cdot S_1 \cdot ... \cdot S_{n-1}\). The output size will be \((S_0, S_1, ..., \dfrac{S_{n-1}}{2}+1)\).

Use `plan.complex_size(real_size)`

to get the exact size of the complex output for a given real input size.

The KFR FFT implementation supports in-place processing for Multidimensional Real DFT.

Note

For performance reasons, it is advantageous to use a slightly larger output buffer for the complex-to-real transform.
`plan.real_out_size()`

returns the size of the buffer (in real numbers) required for fast processing.
Pass `true`

as a second argument to the `dft_plan_md_real`

constructor to indicate that the output buffer has enough space for fast processing.