DFT for discrete Fourier transform

The

DfT is a specialized operation for adapting the Fourier transform of computer analysis, this chapter is the key chapter of the digital signal processing course.
3.7 Spectral analysis using DFT
1. Spectral analysis of continuous signals using DFT
(1) principle
 

(2) Frequency resolution and selection of DFT parameters
Frequency resolution refers to the ability of the algorithm used to separate two very close spectral peaks in a signal.
Set is a band-limited continuous time signal, the highest frequency is FC, according to the time domain sampling theorem, sampling frequency FS>2FC, generally take. 　　　　The
extracts n points on a segment with a time length of TP, obtains a finite long sequence of length n (n), and then has

because FS corresponds to the digital frequency and X (n) as the N-point DFT, the frequency resolution of the number field

At this point, the frequency resolution of the corresponding simulation domain is

above: If you keep the sample number of n constant, to increase the resolution of the Spectrum (f reduction), the sample rate must be reduced, the sample rate reduction will cause spectral analysis range reduction, such as maintaining FS, to increase the resolution can increase the number of sampling points N.
2. Error problems in spectral analysis using DFT
(1) Aliasing
using DFT to approximate the continuous time signal Fourier transform, in order to avoid aliasing distortion, according to the sampling theorem, the sampling frequency is at least twice times the maximum frequency of the signal.
The only way to solve aliasing problems is to ensure that the sampling frequency is high enough.
(2) Truncation effect
when using DFT to process a non-time sequence, the sequence must be truncated. The spectrum of the sequence is the spectrum of the rectangle window function is , then the spectrum of the truncated sequence is


due to the introduction of the spectral of the rectangular window function, the spectrum of the convolution is broadened, called the spectral leakage (truncation effect).
Reduction Method: Select the appropriate shape of the window function, such as Henning window or Hamming window. The
(3) Fence effect
DFT is a spectral interval sampling of a finite length sequence, which is equivalent to observing the spectrum of the original signal through a fence, a phenomenon called the fence effect.

method to reduce the fence effect: 0 at the end.
0 does not add any new information to the original signal and therefore does not increase the frequency resolution. The purpose of zeroing: makes the integer power of the data N 2, so that the fast Fourier transform algorithm (FFT) is used, and the complement 0 can interpolate the original X (k).