Comparison of CTFT, DTFT, and DFT
| Feature | Continuous-Time FT (CTFT) | Discrete-Time FT (DTFT) | Discrete Fourier Transform (DFT) |
|---|---|---|---|
| Input Signal Type | Continuous-time \(x(t)\) | Discrete-time \(x[n]\) | Discrete-time, finite-length \(x[n]\) |
| Output Frequency Type | Continuous frequency \(X(f)\) | Continuous frequency \(X(e^{j\omega})\) | Discrete frequency \(X[k]\) |
| Mathematical Formula | \(X(f) = \int_{-\infty}^{\infty} x(t)e^{-j2\pi ft} \, dt\) | \(X(e^{j\omega}) = \sum_{n=-\infty}^{\infty} x[n] e^{-j\omega n}\) | \(X[k] = \sum_{n=0}^{N-1} x[n] e^{-j \frac{2\pi}{N}kn}\) |
| Inverse Transform | \(x(t) = \int_{-\infty}^{\infty} X(f)e^{j2\pi ft} \, df\) | \(x[n] = \frac{1}{2\pi} \int_{-\pi}^{\pi} X(e^{j\omega}) e^{j\omega n} \, d\omega\) | \(x[n] = \frac{1}{N} \sum_{k=0}^{N-1} X[k] e^{j \frac{2\pi}{N}kn}\) |
| Periodicity (Frequency Domain) | Non-periodic | Periodic with period \(2\pi\) | Periodic with period \(N\) |
| Computation Method | Integral | Infinite summation | Finite summation (numerical via FFT) |
| Spectrum Nature | Continuous, infinite extent | Continuous, periodic | Discrete, finite-length, periodic |
| Applications | Analog signal analysis, circuit theory | Theoretical signal analysis, filter design | Digital signal processing, FFT computations |
| Implementation | Analytical or numerical | Primarily analytical | Digital computation, FFT in hardware/software |
Relationship Between CTFT, DTFT, and DFT
Understanding how sampling in one domain leads to periodicity in the other is crucial in signal processing. This concept ties together the Continuous-Time Fourier Transform (CTFT), the Discrete-Time Fourier Transform (DTFT), and the Discrete Fourier Transform (DFT).
Summary Table
| Step | Time Domain Operation | Frequency Domain Effect | Periodicity Induced In |
|---|---|---|---|
| CTFT â DTFT | Sampling (discretization) | Periodic DTFT (\(2\pi\)-periodic) | Frequency domain |
| DTFT â DFT | Truncation (finite length) | Sampling DTFT at \(N\) points | Time domain |
Illustration
Below is a minimal JavaScript demo illustrating how time-domain sampling causes frequency-domain periodicity, and vice versa.
10 samples
Key Insights
- Sampling in time â Periodicity in frequency (CTFT â DTFT)
- Sampling in frequency (via truncation in time) â Periodicity in time (DTFT â DFT)
- DFT can be viewed as a sampled DTFT, ideal for digital computation.
_Last updated: June 06, 2025