What are the most important DFT properties?

Linearity, time shift ↔ phase shift, frequency shift ↔ modulation, circular convolution ↔ multiplication, and Parseval's theorem (energy preservation). The convolution property is most practical — it enables fast filtering via FFT.

Why does the DFT give circular convolution instead of linear?

The DFT assumes the signal is periodic with period N. Multiplication of DFT sequences gives circular convolution. To get linear convolution, zero-pad both sequences to length ≥ N₁+N₂−1 before computing the DFT.

How does the DFT relate to the Z-transform?

The DFT samples the Z-transform on the unit circle at N equally spaced points: X[k] = X(z)|_{z=e^{j2πk/N}}. Similarly, the DTFT is the Z-transform evaluated continuously on the unit circle: X(ω) = X(z)|_{z=e^{jω}}.

Dtft Table

Q: What is the difference between DTFT and DFT?

DTFT converts a discrete sequence to a continuous frequency function (theoretical, infinite sum). DFT samples the DTFT at N frequencies (computable from finite data, N-point sum). The FFT is an efficient algorithm for computing the DFT.

Quick Answer

A DFT (Discrete Fourier Transform) property table lists the key transform pairs and properties for discrete-time signals. Essential pairs: δ[n] ↔ 1 (unit impulse has flat spectrum), 1 ↔ 2πδ(ω) (constant has spectrum at DC only), a^n·u[n] ↔ 1/(1−ae^{−jω}) (exponential decay), cos(ω₀n) ↔ π[δ(ω−ω₀)+δ(ω+ω₀)]. Key properties: linearity, time shift ↔ phase shift (x[n−n₀] ↔ e^{−jωn₀}X(ω)), frequency shift (e^{jω₀n}x[n] ↔ X(ω−ω₀)), and convolution ↔ multiplication.

Calculate this instantly on LAPLACE Calculator →

DTFT vs. DFT vs. FFT: Clearing Up the Confusion

The DTFT (Discrete-Time Fourier Transform) converts a discrete-time sequence to a continuous frequency function: X(ω) = Σx[n]e^{−jωn}. It's the theoretical tool for analysis. The DFT (Discrete Fourier Transform) samples the DTFT at N equally spaced frequencies: X[k] = Σx[n]e^{−j2πkn/N}. It's computable from finite data. The FFT (Fast Fourier Transform) is an efficient algorithm for computing the DFT in O(N log N) operations. So: DTFT is the continuous theory, DFT is the discrete computation, and FFT is how you compute the DFT quickly. Tables typically list DTFT pairs since they're more general.

Key Formulas

Essential DTFT/DFT Transform Pairs

Unit impulse δ[n] transforms to 1 (all frequencies present equally). Constant signal 1 transforms to 2πδ(ω) (only DC component). Exponential decay a^n·u[n] (|a|<1) transforms to 1/(1−ae^{−jω}). Unit step u[n] transforms to 1/(1−e^{−jω}) + πδ(ω). Cosine cos(ω₀n) transforms to π[δ(ω−ω₀) + δ(ω+ω₀)] — two spectral lines at ±ω₀. Rectangular window (1 for n=0 to N−1) transforms to e^{−jω(N−1)/2} · sin(Nω/2)/sin(ω/2) — the Dirichlet kernel. These pairs form the foundation for analyzing discrete-time systems.

Compute dtft table Instantly

Get step-by-step solutions with AI-powered explanations. Free for basic computations.

Open Calculator

Key DTFT Properties for Problem Solving

Linearity: αx[n] + βy[n] ↔ αX(ω) + βY(ω). Time shift: x[n−n₀] ↔ e^{−jωn₀}X(ω) — delaying a sequence adds linear phase. Frequency shift: e^{jω₀n}x[n] ↔ X(ω−ω₀) — modulating by a complex exponential shifts the spectrum. Convolution: x[n]*h[n] ↔ X(ω)H(ω) — convolution becomes multiplication. Multiplication: x[n]·y[n] ↔ (1/2π)(X*Y)(ω) — multiplication becomes convolution. Parseval's theorem: Σ|x[n]|² = (1/2π)∫|X(ω)|²dω — energy is preserved across domains.

DFT Properties Specific to Finite Sequences

The N-point DFT has properties that differ from the DTFT due to its periodic/circular nature. Circular shift: x[(n−m) mod N] ↔ e^{−j2πkm/N}X[k]. Circular convolution: (x ⊛ y)[n] ↔ X[k]Y[k] — the DFT naturally computes circular, not linear, convolution. To get linear convolution from the DFT, zero-pad to length ≥ N₁+N₂−1. Duality: the DFT of the DFT (with scaling) gives the time-reversed sequence. Symmetry: for real x[n], X[k] = X*[N−k] — the spectrum has conjugate symmetry.

Using the Table for Z-Transform Problems

The DTFT is a special case of the Z-transform evaluated on the unit circle: X(ω) = X(z)|_{z=e^{jω}}. Therefore, Z-transform pairs directly give DTFT pairs by substituting z = e^{jω}. For example, Z{a^n·u[n]} = 1/(1−az^{−1}), so DTFT{a^n·u[n]} = 1/(1−ae^{−jω}). The ROC (region of convergence) for the Z-transform must include the unit circle for the DTFT to exist. This connection between Z-transform, DTFT, and DFT parallels the relationship between Laplace transform and Fourier transform in continuous time.

Frequently Asked Questions

DTFT converts a discrete sequence to a continuous frequency function (theoretical, infinite sum). DFT samples the DTFT at N frequencies (computable from finite data, N-point sum). The FFT is an efficient algorithm for computing the DFT.

Master Your Engineering Math

Join thousands of students and engineers using LAPLACE Calculator for instant, step-by-step solutions.

Start Calculating Free →