Signal and system 10: discrete Fourier series and Transformation

Like the Fourier series and transformation of continuous signals, the two have the majority of similarities, but the most important difference is that the Fourier coefficient of discrete signals has a period, discrete Signals are split into a complex exponential signal, which is also cyclical. The reason for these phenomena is the difference between E ^ (jkwn) and E ^ (jkwt,

If x (n) = E ^ (jkwn) and X (t) = E ^ (jkwt), x (n) = x (n + n), x (t) = x (t + T ),

However, if X (K) = E ^ (jkwn) and X (K) = x (t) = E ^ (jkwt), then X (K + n) = x (K), while x (t + T) and X (t) are not equal!

This is the reason for the difference among the formulas of the following series, including Fourier series.

For discrete Fourier series =, discrete Fourier transformation is the same as the amount of continuous signals. We can understand the state in another way. The periodic extension in the time domain represents the sampling in the frequency domain, the interval between the period delay and the sampling points in the frequency domain is a reciprocal relationship, that is, the longer the period, the smaller the interval between the sampling points. Therefore, in extreme cases, if the period is infinite, the interval is 0, that is, all samples of the envelope.