How it works
The DTFT is defined as X(e^jω) = Σ x[n]e^(−jωn) summed over all integers n, where ω is normalised frequency in radians/sample (range −π to π). The inverse DTFT recovers x[n] = (1/2π)∫X(e^jω)e^(jωn)dω over [−π, π]. Because discrete-time signals are indexed in samples, the frequency axis is periodic with period 2π — frequencies ω and ω + 2π are identical. For the causal exponential x[n] = aⁿu[n] with |a| < 1, the DTFT is X(e^jω) = 1/(1 − ae^(−jω)), converging for |a| < 1.
Key points to remember
Key DTFT properties: linearity, time shift x[n−n₀] ↔ e^(−jωn₀)X(e^jω), frequency shift x[n]e^(jω₀n) ↔ X(e^j(ω−ω₀)), and convolution x[n]*h[n] ↔ X(e^jω)H(e^jω). The DTFT always produces a continuous and periodic spectrum in ω. The DFT is a sampled version of the DTFT — taking N equally spaced samples over [0, 2π) gives the N-point DFT. Parseval's theorem for DTFT: Σ|x[n]|² = (1/2π)∫|X(e^jω)|² dω. The unit impulse δ[n] has DTFT equal to 1 for all ω — the flattest possible spectrum.
Exam tip
The examiner always asks you to find the DTFT of aⁿu[n] and sketch the magnitude spectrum — include the 2π-periodic repetition in your sketch because leaving it out costs marks even when the formula is correct.