DFT of discrete Fourier transform

The

DfT is a specialized operation for the analysis of Fourier transforms in a computer, and this chapter is a key section 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 DFT parameter selection
The frequency resolution refers to the ability of the algorithm to separate the two spectral peaks that are close to the signal.
Set is a limited continuous time signal, the highest frequency FC, according to the time domain sampling theorem, sampling frequency FS>2FC, generally taken. 　　　　
to extract n points on a segment of TP for a length of N, and to get a finite long sequence x (n) with an amount of "
" because FS corresponds to the digital frequency , to X (n) as the N-point DFT, the frequency resolution of the numeric field

At this point, the corresponding analog domain has a frequency resolution of

Upper-Class Description: If you keep the sampling point n unchanged, to improve the resolution of the Spectrum (f reduction), the sampling rate must be reduced, the reduction of sampling rate will cause the spectral analysis range to be reduced; If you maintain FS unchanged, you can increase the number of sampling points to increase the resolution N.
2. Error problem of spectral analysis using DFT
(1) Aliasing
using DFT to approximate the Fourier transform of continuous time signal, in order to avoid aliasing distortion, the sampling frequency is at least twice times higher than the maximum frequency of the signal according to the requirement of sampling theorem. The only way to
solve the aliasing problem is to ensure that the sampling frequency is high enough.
(2) truncation effect
The sequence must be truncated when using DFT to process a non time series. Set the spectrum of the sequence, the spectrum of the rectangular window function is , then the spectrum of the truncated sequence is


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

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