# Spectrum analysis using the discrete fourier transform

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The discrete Fourier transform (DFT) maps a finite number of discrete time-domain samples to the same number of discrete Fourier-domain samples. Being practical to compute, it is the primary transform applied to real-world sampled data in digital signal processing.The DFT has special relationships with the discrete-time Fourier transform and the continuous-time Fourier transform that let it be used as a practical approximation of them through truncation and windowing of an infinite-length signal. Different window functions make various tradeoffs in the spectral distortions andartifacts introduced by DFT-based spectrum analysis.

## Discrete-time fourier transform

The Discrete-Time Fourier Transform (DTFT) is the primary theoretical tool for understanding the frequency content of a discrete-time (sampled) signal.The DTFT is defined as

$X(\omega )=\sum_{n=()}$ x n ω n
The inverse DTFT (IDTFT) is defined by an integral formula, because it operates on a continuous-frequency DTFT spectrum:
$x(n)=\frac{1}{2\pi }\int_{-\pi }^{\pi } X(k)e^{i\omega n}\,d \omega$

The DTFT is very useful for theory and analysis, but is not practical for numerically computing a spectrum digitally, because

• infinite time samples means
• infinite computation
• infinite delay
• The transform is continuous in the discrete-time frequency, ω

For practical computation of the frequency content of real-world signals, the Discrete Fourier Transform (DFT) is used.

## Discrete fourier transform

The DFT transforms $N$ samples of a discrete-time signal to the same number of discrete frequency samples, and is defined as

$X(k)=\sum_{n=0}^{N-1} x(n)e^{-\left(\frac{i\times 2\pi nk}{N}\right)}$
The DFT is invertible by the inverse discrete Fourier transform (IDFT):
$x(n)=\frac{1}{N}\sum_{k=0}^{N-1} X(k)e^{i\frac{2\pi nk}{N}}$
The DFT and IDFT are a self-contained, one-to-one transform pair for alength- $N$ discrete-time signal. (That is, the DFT is not merely an approximation to the DTFT as discussed next.) However, the DFT is very often used as a practical approximation to the DTFT .

## Dft and discrete fourier series

The DFT gives the discrete-time Fourierseries coefficients of a periodic sequence ( $x(n)=x(n+N)$ ) of period $N$ samples, or

$X(\omega )=\frac{2\pi }{N}\sum X(k)\delta (\omega -\frac{2\pi k}{N})$
as can easily be confirmed by computing the inverse DTFT of the corresponding line spectrum:

$x(n)=\frac{1}{2\pi }\int_{-\pi }^{\pi } \frac{2\pi }{N}\sum X(k)\delta (\omega -\frac{2\pi k}{N})e^{i\omega n}\,d \omega =\frac{1}{N}\sum_{k=0}^{N-1} X(k)e^{i\frac{2\pi nk}{N}}=\mathrm{IDFT}(X(k))=x(n)$

The DFT can thus be used to exactly compute the relative values of the $N$ line spectral components of the DTFT of any periodic discrete-time sequence with an integer-length period.

## Dft and dtft of finite-length data

When a discrete-time sequence happens to equal zero for all samples except for those between $0$ and $N-1$ , the infinite sum in the DTFT equation becomes the same as the finite sum from $0$ to $N-1$ in the DFT equation. By matching the arguments in the exponential terms, we observe that theDFT values exactly equal the DTFT for specific DTFT frequencies ${\omega }_{k}=\frac{2\pi k}{N}$ . That is, the DFT computes exact samples of the DTFT at $N$ equally spaced frequencies ${\omega }_{k}=\frac{2\pi k}{N}$ , or

$X({\omega }_{k}=\frac{2\pi k}{N})=\sum_{n=-\infty }^{\infty } x(n)e^{-(i{\omega }_{k}n)}=\sum_{n=0}^{N-1} x(n)e^{-\left(\frac{i\times 2\pi nk}{N}\right)}=X(k)$

## Dft as a dtft approximation

In most cases, the signal is neither exactly periodic nor truly of finite length; in such cases, the DFT of a finite block of $N$ consecutive discrete-time samples does not exactly equal samples of the DTFT at specific frequencies. Instead, the DFT gives frequency samples of a windowed (truncated) DTFT $\stackrel{^}{X}({\omega }_{k}=\frac{2\pi k}{N})=\sum_{n=0}^{N-1} x(n)e^{-(i{\omega }_{k}n)}=\sum_{n=-\infty }^{\infty } x(n)w(n)e^{-(i{\omega }_{k}n)}=X(k)$ where $w(n)=\begin{cases}1 & \text{if 0\le n< N}\\ 0 & \text{if \text{else}}\end{cases}$ Once again, $X(k)$ exactly equals $X({\omega }_{k})$ a DTFT frequency sample only when $\forall n, n\notin \left[0 , N-1\right]\colon x(n)=0$

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