# One-Dimensional Fast Fourier Transforms

## One-dimensional FFT's

The computation of an FFT involves some preprocessing steps. When multiple FFT's of the same length need to be computed, it is more efficient to use an Fft<T> object, which caches the results of these calculations.

### Real FFT's

Use the static CreateReal(Int32) method of the Fft<T> class to create an Fft<T> object that can be used to compute one-dimensional real FFT's. The method takes one argument: the length of the signal.

```
int N = 1024;
Fft<double> realFft = Fft<double>.CreateReal(N);
```

The Fft<T>
object has several properties that can be used to fine-tune the transform. The
ForwardScaleFactor
property sets or gets the scale factor used in the forward transform.
The default is *1/N* where *N* is the number of points
in the transform. The
BackwardScaleFactor
property sets or gets the scale factor used in the backward or inverse transform.
The default is *1*. There is also a
InPlace
property, which is not valid for real FFT's.

```
realFft.ForwardScaleFactor = 1.0 / Math.Sqrt(N);
realFft.BackwardScaleFactor = realFft.ForwardScaleFactor;
```

These properties can be changed up to the point where the Fft<T> object is committed. This happens when one of the transform methods is called for the first time. The Committed property indicates whether this is the case.

The actual transform is computed by two methods: ForwardTransform and BackwardTransform. As the name implies, these methods perform the forward and backward (inverse) transform. They have seven overloads, four of which apply to real transforms.

The first overload for ForwardTransform takes one argument: a Vector<T> that specifies the signal to transform. It returns a ComplexConjugateSignalVector<T>, a special type of complex vector that enforces the symmetry properties of real FFT's and does not store any duplicate information. The corresponding BackwardTransform method takes a ComplexConjugateSignalVector<T> and returns a Vector<T>.

```
var r1 = Vector.CreateFromFunction(1000, i => 1.0 / (1 + i));
var c1 = realFft.ForwardTransform(r1);
var r2 = realFft.BackwardTransform(c1);
```

The second pair of overloads is similar to the first, but takes a second parameter that is used to store the result. For the forward transform, the parameters are a Vector<T> and a ComplexConjugateSignalVector<T>. For the backward transform, the parameters are a ComplexConjugateSignalVector<T> and a Vector<T>.

```
var c2 = new ComplexConjugateSignalVector<double>(r1.Length);
realFft.ForwardTransform(r1, c2);
var r3 = Vector.Create<double>(c2.Length);
realFft.BackwardTransform(c2, r3);
```

The third pair of overloads is similar to the second, but allows any vector types for the arguments. The forward transform takes any real and complex Vector<T>. The backward transform takes the same parameter types in reverse order.

The final overload is similar to the third, but allows you to specify whether to return a one-sided or a two-sided transform.
Because of symmetry, only the first *N/2+1* terms of a real FFT are independent. The remaining terms are the complex
conjugate of terms in the first half. A one-sided transform returns only the first N/2+1 terms of the transform,
while a two-sided transform returns the full FFT vector. This fourth overload of
ForwardTransform and
BackwardTransform
takes a third parameter: a
RealFftFormat value with possible values
OneSided and TwoSided. The length of the complex vector argument must equal N/2+1
for a one-sided transform.

```
var c3 = Vector.Create<Complex<double>>(r1.Length / 2 + 1);
realFft.ForwardTransform(r1, c3, RealFftFormat.OneSided);
var r4 = Vector.Create<double>(r1.Length);
realFft.BackwardTransform(c3, r4, RealFftFormat.OneSided);
```

### Complex FFT's

Use the static CreateComplex(Int32) method of the Fft<T> class to create an Fft<T> object that can be used to compute one-dimensional real FFT's. The method takes one argument: the length of the signal.

```
int N = 1024;
Fft<double> complexFft = Fft<double>.CreateComplex(N);
```

Complex FFT's are simpler in that the Fourier transform of a complex signal is always complex also.
This means that the transform can be done in-place. To perform a transform in place, set the
InPlace property to **true**.

The transformation methods, ForwardTransform and BackwardTransform, have three overloads that apply to complex transforms. The first overload takes a single complex Vector<T> and returns a complex Vector<T> that is the forward or backward Fourier transform of its argument.

```
var c1 = Vector.CreateFromFunction(N, i => new Complex<double>(1.0 / (1 + i), i));
var c2 = complexFft.ForwardTransform(c1);
var c3 = complexFft.BackwardTransform(c2);
```

The second pair of overloads take two complex Vector<T> objects. They compute the forward or backward transform of the first argument and return it in the second. The third pair of overloads is more general than the second, taking two complex vectors of any type.

```
var c4 = Vector.Create<Complex<double>>(c1.Length);
complexFft.ForwardTransform(c1, c4);
var c5 = Vector.Create<Complex<double>>(c4.Length);
complexFft.BackwardTransform(c4, c5);
```

Since the Fourier transform of a complex vector is again a complex vector of the same length, it is possible to perform a Fourier transform in-place. The ForwardTransformInPlace and BackwardTransformInPlace methods perform this operation.

```
complexFft.ForwardTransformInPlace(c1);
complexFft.BackwardTransformInPlace(c1);
```