17.9 : 离散傅里叶变换

Consider a vibration sensor that continuously captures data in the form of a continuous time-dependent signal.

However, in reality, the sensor can only record a finite number of vibrations as discrete data points at specific time intervals.

Here, the Discrete Fourier Transform, DFT, can be used to analyze the frequency components of the vibrations.

The DFT decomposes any signal into a sum of simple sine and cosine waves, for which the frequency, amplitude, and phase can be measured.

In the DFT amplitude spectrum, the signal obtained from the sensor can be represented as the bar graph corresponding to the four sine waves.

The bar height, after normalization, is the amplitude of that corresponding sine wave signal in the time domain.

Mathematically, the DFT is represented as a finite summation of the product of the time-domain signal with a complex exponent dependent on frequency k.

If X as a function of k is large for a certain k, it indicates that the vibration signal has strong frequency components at those frequencies.