Resolution of complement 0 and discrete Fourier transform

The input of a discrete Fourier transform (DFT) is a discrete set of values, and the output is also a discrete set of values. In the case of an input signal, the interval of the adjacent two sampling points is the sampling time TS. In the output signal, the interval of the adjacent two sampling points is the frequency resolution fs/n, where FS is the sampling frequency and its size equals 1/ts,n as the number of sample points for the input signal. This means that the frequency domain resolution of the DfT is not only related to the sampling frequency, but also to the sampling points of the signal. So, if the input signal length is not changed, but the input signal is 0, increase the number of DFT points, at this time the resolution is changed or unchanged?

The answer is that the resolution does not change at this time. From the time domain, it is assumed to distinguish between two signals with a very small frequency difference, intuitively understand, at least to ensure that two signals in the time domain of a complete period of difference, that is, phase difference 2*pi. For example, assuming that the sampling frequency is 1Hz, to distinguish between a sinusoidal signal with a period of 10s and a sinusoidal signal with a period of 11s, the signal must be at least a constant 110s, and two signals can be separated by one cycle, at which point the period of 10s of the signal undergoes a period of 11 cycles. The 11s signal experienced a period of 10. Into the frequency domain, in which case the time domain sample point is 110 and the resolution is 1/110=0.00909, which is exactly equal to two signal frequency difference (1/10-1/11). If the two signals in the time domain do not meet the "difference of a complete period", 0 also can not meet the "difference of a complete period", that is, the resolution does not change. In addition, from the point of view of information theory, it is also very well understood that the input signal 0 does not increase the input signal information, so the resolution will not be changed.

So what is the effect of 0? Since the DfT can be seen as a sampling of DTFT, a complement of 0 only reduces the frequency domain sampling interval. This helps to overcome some of the spectral leaks caused by the fence effect. In other words, the 0 can make the signal more detailed in the frequency domain to observe. If the above requirements of "at least one complete cycle" are not met, even if the DTFT is generally continuous in the frequency domain, it is impossible to distinguish two signals from each other.

So what is the physical mechanism that affects the most fundamental of the DFT resolution? Is the accumulation time of the DfT, and the resolution is the reciprocal of the accumulated time t. This can be easily obtained from the mathematical formula:

fs/n=1/(n*ts) =1/t

For example, if the input signal length is 10s, then regardless of the sampling frequency, of course, the premise is to satisfy the Nyquist theorem, the resolution is 1/10=0.1hz.

Resolution of complement 0 and discrete Fourier transform