Two Fourier Transformations of discrete Sequences

First, you must know that PI corresponds to 0.5 * FS. The analog frequency can be from negative infinity to positive infinity, but the digital frequency can only take-pi to Pi, but it contains the analog components outside the nycraftsmanship range. The sampling envelope can determine the simulation frequency carried by the sampling sequence.

Review the Fourier series and Fourier transformation of Signals and Systems. Note that the Fourier coefficient is a sample of Fourier transformation. The two dimensions are different. The coefficient is the density-to-frequency integral, while the spectrum is the concept of density. Therefore, the dimension difference between the sampling sequence in the time domain and the original signal is a time multiplier T, while the Fourier series coefficient in the frequency domain is different from the dimension of the continuous spectrum, and the frequency multiplier omigas is used.

A finite bandwidth sequence must have an infinite length in the time domain, while a finite time domain must have unlimited bandwidth in the frequency domain. It is impossible to have signals and sequences with limited time domain length and frequency domain bandwidth. In addition, if the time domain is narrow, the frequency bandwidth is used.

Dtft is time discrete, while the frequency domain is continuous. In Matlab, the special function is [h, w] = freqz (B, A, N ). If n is not specified, 512 is used by default. Here, B is the sequence X, and A is 1. In this way, only 0 ~ Pi, to reflect 0 ~ 2PI must be added with 'whole '.

Note that the dtft of the real sequence has symmetry. If the sequence is not.

Freqz can also be used to calculate the frequency characteristics. B and A are the negative power coefficient vectors of the molecular denominator polynomials, respectively.

DFT is the sampling of dtft in the frequency domain. It seems that this is not accurate. We should say that DFS is a sample of dtft, because they are all periodic. DFT only intercepts a period. Since it is a sample, N must be greater than or equal to the sequence length, otherwise there will be a mixture. MATLAB uses optimized FFT (x, n) calculations. However, the result is 0 ~ 2 * Pi, which can be changed to-pi ~ Through fftshift (XK ~ Pi.

Increase the sequence length to bring DFT closer to dtft. There are two methods: increase the length of the sequence value and increase the order of the input signal. Try to use this method. If the other method is unable to increase, add 0. The latter method does not increase the amount of data information, but can increase the resolution, and does not give a more detailed shape. Note the difference! Strictly speaking, the video resolution is not called Resolution, and the obfuscation is still a mixture.

Two Fourier Transformations of discrete Sequences