# Dft definition and properties

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The discrete Fourier transform (DFT) and its inverse (IDFT) are the primary numerical transforms relating time and frequency in digital signal processing. The DFT has a number of important properties relating time and frequency, including shift, circular convolution, multiplication, time-reversal and conjugation properties, as well as Parseval's theorem equating time and frequency energy.

## Dft

The discrete Fourier transform (DFT) is the primary transform used for numerical computation in digital signal processing. It is very widely used for spectrum analysis , fast convolution , and many other applications. The DFT transforms $N$ discrete-time samples to the same number of discrete frequency samples, and is defined as

$X(k)=\sum_{n=0}^{N-1} x(n)e^{-(i\frac{2\pi nk}{N})}$
The DFT is widely used in part because it can be computed very efficiently using fast Fourier transform (FFT) algorithms.

## Idft

The inverse DFT (IDFT) transforms $N$ discrete-frequency samples to the same number of discrete-time samples. The IDFT has a form very similar to the DFT,

$x(n)=\frac{1}{N}\sum_{k=0}^{N-1} X(k)e^{i\frac{2\pi nk}{N}}$
and can thus also be computed efficiently using FFTs .

## Periodicity

Due to the $N$ -sample periodicity of the complex exponential basis functions $e^{i\frac{2\pi nk}{N}}$ in the DFT and IDFT, the resulting transforms are also periodic with $N$ samples.

$X(k+N)=X(k)$ $x(n)=x(n+N)$

## Circular shift

A shift in time corresponds to a phase shift that is linear in frequency. Because of the periodicity induced by the DFT and IDFT, the shift is circular , or modulo $N$ samples.

$(x((n-m)\mod N), X(k)e^{-(i\frac{2\pi km}{N})})$ The modulus operator $p\mod N$ means the remainder of $p$ when divided by $N$ . For example, $9\mod 5=4$ and $-1\mod 5=4$

## Time reversal

$(x(-n\mod N)=x((N-n)\mod N), X((N-k)\mod N)=X(-k\mod N))$ Note: time-reversal maps $(0, 0)$ , $(1, N-1)$ , $(2, N-2)$ , etc. as illustrated in the figure below.

## Complex conjugate

$(\overline{x(n)}, \overline{X(-k\mod N)})$

## Circular convolution property

Circular convolution is defined as $\doteq ((x(n), h(n)), \sum_{m=0}^{N-1} x(m)x((n-m)\mod N))$

Circular convolution of two discrete-time signals corresponds to multiplication of their DFTs: $((x(n), h(n)), X(k)H(k))$

## Multiplication property

A similar property relates multiplication in time to circular convolution in frequency. $(x(n)h(n), \frac{1}{N}(X(k), H(k)))$

## Parseval's theorem

Parseval's theorem relates the energy of a length- $N$ discrete-time signal (or one period) to the energy of its DFT. $\sum_{n=0}^{N-1} \left|x(n)\right|^{2}=\frac{1}{N}\sum_{k=0}^{N-1} \left|X(k)\right|^{2}$

## Symmetry

The continuous-time Fourier transform , the DTFT , and DFT are all defined as transforms of complex-valueddata to complex-valued spectra. However, in practice signals are often real-valued.The DFT of a real-valued discrete-time signal has a special symmetry, in which the real part of the transform values are DFT even symmetric and the imaginary part is DFT odd symmetric , as illustrated in the equation and figure below.

$x(n)$ real  $X(k)=\overline{X((N-k)\mod N)}$ (This implies $X(0)$ , $X(\frac{N}{2})$ are real-valued.)

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Source:  OpenStax, The dft, fft, and practical spectral analysis. OpenStax CNX. Feb 22, 2007 Download for free at http://cnx.org/content/col10281/1.2
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