# 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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what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
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da
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Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
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da
Application of nanotechnology in medicine
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I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
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Professor
I think
Professor
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scanning tunneling microscope
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Rafiq
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Rafiq
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Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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