Discrete-Time Fourier Transform (DTFT)

Core Idea

The DTFT X(e^jω) = Σ x[n]e^-jωn is the Fourier transform of a discrete-time signal, relating discrete-time signals to periodic continuous frequency responses. Unlike the Z-transform (valid on the unit circle), the DTFT is defined only for |z|=1.

Explainer

You already know two Fourier tools: the continuous-time Fourier transform (CTFT), which decomposes a continuous signal x(t) into complex exponentials e^jΩt over all real frequencies Ω, and the Z-transform, which maps a discrete sequence x[n] to a function X(z) of a complex variable z. The DTFT bridges these — it is the Fourier transform for discrete-time signals, and it turns out to be the Z-transform evaluated on a specific contour in the z-plane.

Start from the Z-transform: X(z) = Σ x[n]z^-n. This sum converges for values of z within the region of convergence (ROC). Now restrict z to the unit circle: z = e^jω, so |z| = 1 and the angle ω is the normalized digital frequency in radians. Substituting gives X(e^jω) = Σ x[n]e^-jωn — the DTFT. The signal x[n] at each sample n is being multiplied by a phasor of frequency ω and summed, exactly analogous to the CTFT integral. The output X(e^jω) is a continuous function of ω that completely describes the frequency content of the discrete sequence. The crucial constraint is that the unit circle must lie inside the Z-transform ROC; for a stable system (poles inside unit circle), this is always satisfied.

The most important structural property of the DTFT is its 2π-periodicity: X(e^(j(ω+2π)}) = X(e^jω) for all ω. This is not an approximation — it is exact and fundamental. Digital frequency ω is only meaningful modulo 2π, because multiplying a discrete sequence by e^(j(ω+2π)n) = e^jωne^j2πn = e^jωn·1ⁿ = e^jωn. Frequencies that differ by 2π are indistinguishable to a discrete-time system. This is the frequency-domain signature of the aliasing you already understand from sampling: when you sample a continuous signal at rate f_s, all continuous frequencies Ω and Ω + k·f_s map to the same digital frequency ω = Ω/f_s·2π. The DTFT's periodicity reflects that aliasing is baked into the discrete-time representation.

In practice, the DTFT tells you the frequency response of a discrete-time filter. If h[n] is the impulse response of a digital filter and you compute H(e^jω), you get the complex gain the filter applies at each digital frequency ω — magnitude |H(e^jω)| is the amplitude response, and angle ∠H(e^jω) is the phase shift. Designing a digital filter is equivalent to shaping H(e^jω) to pass desired frequencies and attenuate others. The DTFT is not directly computable for infinite-length sequences (the sum has infinitely many terms), which is why the DFT (its finite, sampled version) and the FFT algorithm exist — but the DTFT is the theoretical foundation. Every digital filter specification, every frequency response plot, and every aliasing argument ultimately rests on the DTFT's continuous periodic frequency-domain representation of discrete-time signals.