Side-by-side comparison
| Parameter | CTFT | DTFT |
|---|---|---|
| Input signal | Continuous-time signal x(t) | Discrete-time sequence x[n] |
| Analysis formula | X(jω) = ∫_{-∞}^{∞} x(t)·e^{−jωt} dt | X(e^{jω}) = Σ_{n=−∞}^{∞} x[n]·e^{−jωn} |
| Frequency variable | ω in rad/s, continuous and unbounded (−∞ to ∞) | ω̂ in rad/sample, periodic with period 2π |
| Spectrum periodicity | Non-periodic (aperiodic spectrum for aperiodic x(t)) | Always periodic with period 2π — fundamental property |
| Convergence condition | x(t) must be absolutely integrable: ∫|x(t)|dt < ∞ | x[n] must be absolutely summable: Σ|x[n]| < ∞ |
| Inverse transform | x(t) = (1/2π) ∫ X(jω)·e^{jωt} dω | x[n] = (1/2π) ∫_{−π}^{π} X(e^{jω})·e^{jωn} dω |
| Relation to sampling | Applied before ADC | Applied after ADC; spectrum repeats at multiples of 2π |
| Computational use | Analytical tool; computed symbolically | Basis for DFT/FFT; sampled version is DFT |
| Aliasing condition | Not directly applicable | Aliasing occurs if sampling frequency fs < 2fmax |
| Real example | Spectrum of a single 10 µs RF pulse at 2.4 GHz | Frequency response of a 64-tap FIR filter in MATLAB |
Key differences
CTFT operates on continuous signals and produces a continuous, aperiodic spectrum measured in rad/s. DTFT operates on discrete sequences and its spectrum is always periodic with period 2π (in rad/sample) — this periodicity is a direct consequence of sampling. A key relation: if x[n] = x(nT), then X_DTFT(e^{jω}) = (1/T)·Σ X_CTFT(j(ω − 2πk)/T), showing spectral repetition. The DTFT cannot be computed directly on a computer (infinite sum); the DFT is its sampled, finite version.
When to use CTFT
Use the CTFT when working with continuous-time systems — for example, computing the bandwidth of an analog bandpass filter centred at 455 kHz used in an AM receiver's IF stage.
When to use DTFT
Use the DTFT when analysing discrete sequences or FIR/IIR filter responses — for example, plotting the frequency response H(e^{jω}) of a 64-point Hamming-windowed FIR low-pass filter designed for fs = 8 kHz.
Recommendation
For ECE exams, choose CTFT for continuous-time problems and DTFT for anything involving samples or filter coefficients. Know the 2π periodicity of DTFT by heart — it is tested directly. If the problem gives x[n] = δ[n] or a finite sequence, DTFT is the tool.
Exam tip: GATE asks you to find the DTFT of x[n] = a^n·u[n] for |a| < 1 — result is 1/(1 − a·e^{−jω}); also expect questions on the periodicity of DTFT and the condition for its existence.
Interview tip: Interviewers at DSP chip companies ask why DTFT is periodic — explain that sampling in time is multiplication by an impulse train, which causes convolution (repetition) in the frequency domain with period 2π.