The Fourier transform of a sine function appears as two discrete points, so what is the Fourier transform of a sine function with only one period?

If the Fourier transform of a sine function with only one period is a continuous but aperiodic function. Time domain and frequency domain are two different ways to represent signals. Fourier transform is the mathematical relationship between these two representations. Fourier transform is linear, homogeneous and additive.

phase characteristic

Time-domain migration leads to a constant amplitude but linear phase shift. The time-domain offset of s sampling points changes the phase by 2πfs. As shown in the above figure, a-d represents the change of peak position from 128 to 0, and the corresponding phase shift is shown in the right figure. In this example, the time domain is regarded as a circle. The time domain waveform is symmetrical, so it has a linear phase. The time domain waveform moves to the right and the slope decreases. The time domain waveform moves to the left and the slope increases.

Extended data

Fourier transform does not have displacement symmetry, and time domain displacement cannot cause frequency domain displacement accordingly. Obviously, the displacement of the time domain signal and the sine function also shift correspondingly, and the displacement of the sine function is the change of phase.

If x [n] <; -& gt； Mag X[ f ]？ & phase X[ f], the time-domain displacement result is x [n+s]

If a signal is symmetrical left and right and symmetrical about zero, it is zero phase; If it is not symmetrical about the zero point, it is a linear phase, that is, the phase curve is a straight line. If the signal is asymmetric, it is a nonlinear phase.

The time domain waveform moves to the right, the phase tilt decreases, moves to the left, and the upward tilt gradually increases. The displacement corresponds to the change of slope.

Baidu Encyclopedia-Fourier Transform