If a signal is sampled at a given sampling rate, the following equation defines the time interval between the samples, or the sampling interval.
where Δt is the sampling interval and fs is the sampling rate in samples per second (S/s).
The sampling interval is the smallest frequency that the system can resolve through the DFT or related routines.
The following equation defines the DFT. The equation results in X[k], which is the frequency-domain representation of the sample signal.
where x[i] is the time-domain representation of the sample signal and N is the total number of samples. Both the time domain x and the frequency domain X have a total of N samples.
Similar to the time spacing of Δt between the samples of x in the time domain, you have a frequency spacing, or frequency resolution, between the components of X in the frequency domain, as the following equation defines.
where Δf is the frequency resolution, fs is the sampling rate, N is the number of samples, Δt is the sampling interval, and NΔt is the total acquisition time.
To improve the frequency resolution, that is, to decrease Δf, you must increase N and keep fs constant or decrease fs and keep N constant. Both approaches are equivalent to increasing NΔt, which is the time duration of the acquired samples.