Analog vs Digital PSD and FFT Usage
1. Analog PSD vs Digital PSD
- Analog PSD:
- Describes the continuous-time signal's power distribution over frequency.
- Not directly measurable on a computer; mostly used in theory, RF systems, or analog filter design.
- Calculated using continuous-time Fourier Transform (CTFT):
\[ S_x(f) = \int_{-\infty}^{\infty} R_x(\tau) e^{-j 2 \pi f \tau} d\tau \]
- Digital PSD:
- Describes the discrete-time or sampled signal's power distribution.
- Computed from samples \(x[n]\) using DFT/FFT or discrete-time Fourier Transform (DTFT).
- Examples include: Classical periodogram, Welch, Bartlett, Blackman-Tukey.
Rule of thumb:
- Sampled signals → use digital PSD with FFT/DFT.
- Continuous signals → use analytical/analog PSD (CTFT).
2. Where FFT is used
- FFT is a fast algorithm to compute the DFT, approximating the DTFT.
- Used in digital PSD estimation whenever you have discrete samples.
- Methods using FFT:
- Classical periodogram
- Windowed periodogram
- Bartlett (segment DFTs)
- Welch
- Blackman-Tukey (FFT of windowed autocorrelation)
- Important: FFT produces discrete frequency bins, not a continuous function.
3. Where Fourier (continuous) is used
- Continuous-time Fourier Transform (CTFT) is used:
- To derive theoretical PSD formulas.
- For analog signal analysis in communications, control, or RF systems.
- To relate autocorrelation to PSD in theory:
\[ S_x(f) = \int R_x(\tau) e^{-j2\pi f\tau} d\tau \]
- Discrete-Time Fourier Transform (DTFT) bridges theory and digital signals:
- Continuous in frequency, but in practice approximated by DFT/FFT at discrete points.
Summary Table
| Signal Type | PSD Method | Transform Used | Practical Tool |
|---|---|---|---|
| Continuous-time (analog) | Theoretical PSD | CTFT | Analytical formula |
| Discrete-time (sampled) | Classical/Windowed/Bartlett/Welch/Blackman-Tukey | DFT / FFT | FFT in software (MATLAB, Python, etc.) |
| Autocorrelation-based | Correlogram / Blackman-Tukey | DTFT / FFT | FFT of windowed autocorrelation |
If you have data in the computer, always think FFT / discrete PSD. Continuous Fourier Transform is mainly for derivation or analog theory.