Discrete-Time Fourier Transform (DTFT)

April 10, 2026 · Luciano Muratore

We have mastered the Discrete Fourier Transform (DFT) for finite signals. Now, we expand our toolkit to handle infinite sequences, both periodic and aperiodic, exploring how the nature of time—whether it is finite, infinite, or periodic—shapes the frequency domain.

Discrete Fourier Series (DFS)

The Discrete Fourier Series (DFS) is the primary tool for the Fourier analysis of discrete-time periodic sequences. Given an NN-periodic sequence where x[n]=x[n+iN]x[n] = x[n+iN] for all integers nn and ii, the DFS maps the signal from CN\mathbb{C}^N back to CN\mathbb{C}^N.

  • DFS Coefficients: X[k]=1N∑n=0N−1x[n]e−j2πNkn,k=0,…,N−1X[k] = \frac{1}{N} \sum_{n=0}^{N-1} x[n] e^{-j\frac{2\pi}{N}kn}, \quad k=0, \dots, N-1.
  • Reconstruction: x[n]=∑k=0N−1X[k]ej2πNkn,n∈Zx[n] = \sum_{k=0}^{N-1} X[k] e^{j\frac{2\pi}{N}kn}, \quad n \in \mathbb{Z}.

Discrete-Time Fourier Transform (DTFT)

For sequences that are aperiodic and of infinite length (x[n]∈l2(Z)x[n] \in l_{2}(\mathbb{Z})), we transition to the DTFT. This transform maps the discrete sequence to a continuous L2L_2-function on the interval [−π,π][-\pi, \pi]:

  • DTFT Analysis: X(ejω)=∑n=−∞∞x[n]e−jωn,ω∈[−π,π]X(e^{j\omega}) = \sum_{n=-\infty}^{\infty} x[n] e^{-j\omega n}, \quad \omega \in [-\pi, \pi].
  • Inverse Relation: x[n]=12π∫−ππX(ejω)ejωndωx[n] = \frac{1}{2\pi} \int_{-\pi}^{\pi} X(e^{j\omega}) e^{j\omega n} d\omega.

DTFT as a Change of Basis

The DTFT effectively rewrites a sequence in terms of frequency components. While the DFS uses a finite basis of NN exponentials {ej(2π/N)kn}\{e^{j(2\pi/N)kn}\}, the DTFT utilizes an uncountable basis {ejωn}ω∈[−π,π]\{e^{j\omega n}\}_{\omega \in [-\pi, \pi]}. These exponentials satisfy distributional orthogonality:

⟨ejωn,ejσn⟩=2πδ(ω−σ)\langle e^{j\omega n}, e^{j\sigma n} \rangle = 2\pi \delta(\omega - \sigma)

Because projections result in Dirac deltas rather than simple numbers, the DTFT requires the use of generalized functions.

DTFT of Simple Signals

Understanding the DTFT of basic signals provides intuition for more complex analysis:

  1. Constant Sequence (x[n]=1x[n]=1): Results in X(ejω)=2πδ(ω)X(e^{j\omega}) = 2\pi \delta(\omega), meaning all energy is concentrated at the zero frequency.
  2. Cosine (cos⁡(ω0n+ϕ)\cos(\omega_0 n + \phi)): Using Euler’s formula, a cosine is decomposed into two complex exponentials, resulting in two impulses at ±ω0\pm\omega_0 in the frequency domain: πejϕδ(ω−ω0)+πe−jϕδ(ω+ω0)\pi e^{j\phi}\delta(\omega - \omega_0) + \pi e^{-j\phi}\delta(\omega + \omega_0)

Relationship Between Periodic and Finite Sequences

The interplay between periodicity in time and discreteness in frequency is central to the DTFT[cite: 234]:

Periodic Extension

If we take a length-NN signal and create an NN-periodic extension x~[n]=x[nmod  N]\tilde{x}[n] = x[n \mod N], its DTFT consists of discrete impulses at the NN roots of unity: X~(ejω)=1N∑k=0N−1X[k]2πδ~(ω−2πNk)\tilde{X}(e^{j\omega}) = \frac{1}{N} \sum_{k=0}^{N-1} X[k] 2\pi \tilde{\delta}(\omega - \frac{2\pi}{N}k)

Finite Support and Interpolation

Alternatively, if we treat the signal as having finite support (zero-padding the rest of the infinite sequence), the DTFT becomes a continuous spectrum. Specifically, the DTFT is a smooth interpolation of the DFT samples using the Dirichlet kernel, Λ(ω)\Lambda(\omega): X‾(ejω)=∑k=0N−1X[k]Λ(ω−2πNk),Λ(ω)=1N∑m=0N−1e−jωm\overline{X}(e^{j\omega}) = \sum_{k=0}^{N-1} X[k] \Lambda(\omega - \frac{2\pi}{N}k), \quad \Lambda(\omega) = \frac{1}{N} \sum_{m=0}^{N-1} e^{-j\omega m}

As the window length NN approaches infinity, these discrete impulses become dense and eventually converge to the continuous DTFT of the aperiodic sequence.