Understanding Discrete Fourier Transform (DFT/FFT) Frequency Resolution
The Fast Fourier Transform (FFT) computes the Discrete Fourier Transform of $N$ time-domain samples, converting time-domain data into discrete frequency bins. The fundamental trade-off in spectral analysis states that frequency resolution depends directly on total observation time.
FFT Bin Width ($\Delta f$) and Acquisition Time Formulas
The spacing between adjacent frequency bins ($\Delta f$), also known as the frequency resolution, is defined by the sampling frequency ($f_s$) and the transform length ($N$):
- $$\Delta f = \frac{f_s}{N} = \frac{1}{T_{\text{window}}}$$
Where $T_{\text{window}} = N / f_s$ represents the time-domain acquisition frame duration. Increasing the FFT size ($N$) increases total observation time and produces finer spectral resolution.
Calculating Frequency of Bin Index ($k$)
For a given integer bin index $k \in [0, N-1]$, the exact center frequency ($f_k$) represented by that bin is computed as:
- $$f_k = k \cdot \Delta f = k \cdot \frac{f_s}{N}$$
Real vs. Complex (I/Q) Signal Spectrum Mapping
- Real-Valued Input Data: A real input signal creates a symmetric spectrum mirrored around the Nyquist frequency ($f_s / 2$). Unique positive frequencies span bins $k = 0$ through $k = N/2$.
- Complex Input Data (I/Q Sampling): Complex sampling contains independent In-phase ($I$) and Quadrature ($Q$) channels, expanding the unaliased frequency span to cover the full bandwidth from $-f_s/2$ to $+f_s/2$ (a span equal to $f_s$).