DTFT Table

Find common discrete-time Fourier transform pairs and properties in a clear mathematical reference. Compare discrete-time sequences with their frequency-domain expressions, review convergence conditions, and see worked examples using the DTFT.

Table of Common DTFT Pairs

The table lists standard discrete-time sequences, their DTFT expressions, and the conditions under which each pair is valid. Ordinary convergent transforms are distinguished from generalized transforms involving periodic impulses.

Here j² = −1, u[0] = 1, δ[n] is the discrete unit impulse, and δ(ω) is the continuous-frequency Dirac distribution. D(v) = Σ(k∈ℤ) δ(v − 2πk) is a 2π-periodic impulse train. All frequencies are in radians per sample.

24 verified pairs. Lookup highlights a match without hiding any rows.

Impulses and Finite Sequences

Impulses and Finite Sequences — discrete-time Fourier transform pairs
Sequence / pair name	x[n]	X(ω)	Conditions	Notes / Details
Unit impulse	δ[n]	1	All real ω.	Derivation and verification Only the n = 0 term contributes. Verification: Sifting gives e^(−jω·0) = 1. Mathematical reference
Delayed impulse	δ[n − n₀]	e^−jωn₀	Integer n₀.	Derivation and verification A delay produces a negative phase exponent. Verification: Sift the defining sum at n = n₀. Mathematical reference
First impulse difference	δ[n] − δ[n − 1]	1 − e^−jω	Finite support.	Derivation and verification A basic discrete difference filter. Verification: Linearity applied to two impulses. Mathematical reference
Two-point average	(δ[n] + δ[n − 1])/2	(1 + e^−jω)/2	Finite support.	Derivation and verification An averaging filter with unity response at zero frequency. Verification: Sum the two nonzero terms. Mathematical reference
Arbitrary finite sequence	b[n] for 0 ≤ n < N; 0 otherwise	Σ(n=0..N−1) b[n]e^−jωn	Positive integer N; finite b[n].	Derivation and verification A finite polynomial in e^(−jω). Verification: The defining sum terminates. Mathematical reference

Exponential Sequences

Exponential Sequences — discrete-time Fourier transform pairs
Sequence / pair name	x[n]	X(ω)	Conditions	Notes / Details
Causal exponential	aⁿu[n]	1/(1 − ae^−jω)	\|a\| < 1; u[0] = 1.	Derivation and verification An absolutely summable geometric sequence. Verification: Sum the geometric series with ratio ae^(−jω). Mathematical reference
Weighted exponential	(n + 1)aⁿu[n]	1/(1 − ae^−jω)²	\|a\| < 1.	Derivation and verification The convolution of two identical causal exponentials. Verification: Cauchy product has n + 1 equal terms. Mathematical reference
Ramp-weighted exponential	naⁿu[n]	ae^−jω/(1 − ae^−jω)²	\|a\| < 1.	Derivation and verification A weighted geometric series. Verification: Apply a∂/∂a to the geometric sum. Mathematical reference
Higher-order exponential	C(n + r − 1, r − 1)aⁿu[n]	1/(1 − ae^−jω)^r	\|a\| < 1; positive integer r.	Derivation and verification An r-fold convolution; C denotes a binomial coefficient. Verification: The generalized binomial series gives the coefficients. Mathematical reference
Left-sided exponential	aⁿu[−n − 1]	−1/(1 − ae^−jω)	\|a\| > 1.	Derivation and verification A left-sided decaying sequence; n = 0 is excluded. Verification: Set m = −n and sum from m = 1. Mathematical reference
Two-sided exponential	a^\|n\|	(1 − a²)/(1 − 2a cos ω + a²)	Real \|a\| < 1; 0^0 = 1 at n = 0.	Derivation and verification Even and absolutely summable. Verification: Add positive and negative geometric tails and one center term. Mathematical reference
Finite exponential	aⁿ for 0 ≤ n < N; 0 otherwise	(1 − (ae^−jω)ᴺ)/(1 − ae^−jω)	Positive integer N; any finite a; value N if ae^(−jω) = 1.	Derivation and verification Finite support does not require \|a\| < 1. Verification: Finite geometric-series identity, including its limit. Mathematical reference

Rectangular and Symmetric Windows

Rectangular and Symmetric Windows — discrete-time Fourier transform pairs
Sequence / pair name	x[n]	X(ω)	Conditions	Notes / Details
Causal rectangular window	1 for 0 ≤ n ≤ N − 1; 0 otherwise	e^−jω(N−1/2) sin(Nω/2)/sin(ω/2)	Positive integer N; X(2πk) = N by continuous limit.	Derivation and verification A linear phase factor accounts for the window center. Verification: Finite geometric sum; cancellation gives N at each 2πk. Mathematical reference
Symmetric rectangular window	1 for −N ≤ n ≤ N; 0 otherwise	sin((N + 1/2)ω)/sin(ω/2)	Nonnegative integer N; X(2πk) = 2N + 1 by continuous limit.	Derivation and verification A real, even Dirichlet kernel. Verification: Sum 2N + 1 centered complex exponentials. Mathematical reference
Symmetric triangular window	N − \|n\| for \|n\| < N; 0 otherwise	[sin(Nω/2)/sin(ω/2)]²	Positive integer N; X(2πk) = N².	Derivation and verification The unnormalized autocorrelation of an N-point rectangle. Verification: Multiply the rectangle spectrum by its conjugate. Mathematical reference

Sinusoidal and Periodic Sequences

Sinusoidal and Periodic Sequences — discrete-time Fourier transform pairs
Sequence / pair name	x[n]	X(ω)	Conditions	Notes / Details
Constant sequence	1	2π D(ω)	Generalized transform; D(v) = Σ(k∈ℤ) δ(v − 2πk).	Derivation and verification A periodic Dirac impulse train, not a finite-height function. Verification: Inverse sifting over one period gives 1. Mathematical reference
Complex exponential	e^jω₀n	2π D(ω − ω₀)	Real ω₀; generalized transform.	Derivation and verification One spectral line per frequency period. Verification: Frequency shift of the constant; inverse gives e^(jω₀n). Mathematical reference
Cosine	cos(ω₀n)	π[D(ω − ω₀) + D(ω + ω₀)]	Real ω₀; generalized transform.	Derivation and verification Coincident lines add when ω₀ is 0 or π modulo 2π. Verification: Euler expansion into two exponentials gives π per line. Mathematical reference
Sine	sin(ω₀n)	(π/j)[D(ω − ω₀) − D(ω + ω₀)]	Real ω₀; generalized transform.	Derivation and verification Coincident opposite-signed lines cancel. Verification: Euler expansion and the inverse factor 1/(2π) reproduce sine. Mathematical reference
Periodic impulse sequence	Σ(k∈ℤ) δ[n − kN]	(2π/N) Σ(k∈ℤ) δ(ω − 2πk/N)	Positive integer N; generalized transform.	Derivation and verification N equal spectral lines per 2π period. Verification: Inverse finite root-of-unity sum is 1 exactly when N divides n. Mathematical reference

Additional Verified Pairs

Additional Verified Pairs — discrete-time Fourier transform pairs
Sequence / pair name	x[n]	X(ω)	Conditions	Notes / Details
Modulated causal exponential	aⁿe^jω₀nu[n]	1/(1 − ae^{−j(ω−ω₀}))	\|a\| < 1; real ω₀.	Derivation and verification Modulation translates the spectrum. Verification: Replace ω by ω − ω₀ in the geometric series. Mathematical reference
Damped causal cosine	aⁿcos(ω₀n)u[n]	(1 − a cos(ω₀)e^−jω)/(1 − 2a cos(ω₀)e^−jω + a²e^−j2ω)	Real \|a\| < 1; real ω₀.	Derivation and verification A sum of two shifted geometric spectra. Verification: Combine half the two modulated exponential transforms. Mathematical reference
Damped causal sine	aⁿsin(ω₀n)u[n]	a sin(ω₀)e^−jω/(1 − 2a cos(ω₀)e^−jω + a²e^−j2ω)	Real \|a\| < 1; real ω₀.	Derivation and verification A difference of two shifted geometric spectra. Verification: Subtract the modulated exponential transforms and divide by 2j. Mathematical reference
Ideal low-pass sequence	sin(Ωn)/(πn); x[0] = Ω/π	1 for \|ω\| < Ω; 0 for Ω < \|ω\| ≤ π; extend 2π-periodically	0 < Ω < π; L² transform; boundary value 1/2 under symmetric summation.	Derivation and verification Not absolutely summable. Endpoints do not change the inverse integral. Verification: Inverse integrate e^(jωn)/(2π) from −Ω to Ω. Mathematical reference

Discrete-Time Fourier Transform Definition

X (ω) = \sum_{n = - \infty}^{\infty} x [n] e^{- j ω n}

x [n] = \frac{1}{2 π} \int_{- π}^{π} X (ω) e^{j ω n} d ω

The integer n is the sample index. The angular frequency ω is measured in radians per sample, not radians per second. X(ω) repeats with period 2π; the inverse integral may use any complete 2π interval.

DTFT Properties Table

Use x[n] ↔ X(ω) and y[n] ↔ Y(ω) with the definition above. Conjugation is written conj; * denotes convolution in time, not multiplication.

13 DTFT properties with assumptions
Property	Time domain	Frequency domain	Assumptions / explanation
Linearity	Ax[n] + By[n]	AX(ω) + BY(ω)	For transforms in a common valid interpretation.
Time shifting	x[n − n₀]	e^−jωn₀X(ω)	Integer n₀; reindex the bilateral sum.
Frequency shifting	e^jω₀nx[n]	X(ω − ω₀)	Real ω₀; frequency taken modulo 2π.
Time reversal	x[−n]	X(−ω)	Reverse the integer summation index.
Conjugation	conj(x[n])	conj(X(−ω))	Conjugate both signal and kernel.
Convolution in time	Σ(k∈ℤ) x[k]y[n − k]	X(ω)Y(ω)	For example, both sequences absolutely summable; no 2π factor.
Multiplication in time	x[n]y[n]	(1/(2π))∫(−π..π) X(θ)Y(ω − θ)dθ	Periodic convolution; absolute summability suffices.
First difference	x[n] − x[n − 1]	(1 − e^−jω)X(ω)	Subtract the delayed sequence.
Frequency differentiation	nx[n]	j dX(ω)/dω	Σ \|nx[n]\| < ∞ is a sufficient termwise-differentiation condition.
Parseval’s identity	Σ(n∈ℤ) \|x[n]\|²	(1/(2π))∫(−π..π) \|X(ω)\|²dω	Square-summable sequences; equality of energy.
2π periodicity	x[n]	X(ω + 2π) = X(ω)	Integer n makes e^(−j2πn) = 1.
Real-signal symmetry	Real x[n]	X(−ω) = conj(X(ω))	Real part even; imaginary part odd. Phase only where X ≠ 0.
Even real sequence	Real x[n] = x[−n]	Real, even X(ω)	Within a valid convergence or L² interpretation.

How to Use the DTFT Table

Start with the discrete-time sequence x[n] and identify the matching row. Read its transform X(ω) and check the conditions before applying a property such as shifting or convolution. For an inverse lookup, start with the frequency-domain expression and find the corresponding sequence.

Why Is the DTFT Periodic?

The DTFT of a discrete-time sequence repeats every 2π radians per sample. The periodicity follows from the integer-valued time index n: multiplying the transform kernel by e^(−j2πn) does not change its value. Therefore X(ω + 2π) = X(ω).

DTFT vs. DFT vs. Fourier Transform

The DTFT describes the frequency content of a discrete-time sequence using a continuous, periodic frequency variable. The DFT evaluates a finite sequence at a discrete set of frequency bins. The continuous-time Fourier transform operates on a continuous-time function. These transforms are related, but their definitions, frequency variables, and scaling conventions are not interchangeable.

DTFT and the Z-Transform

For a sequence with a bilateral Z-transform, the DTFT can be obtained by evaluating the Z-transform on the unit circle when that circle lies in the region of convergence. If the unit circle is excluded, the ordinary DTFT may not exist, even though the Z-transform has a valid expression elsewhere.

Convergence and Generalized Transforms

Absolute summability, Σ |x[n]| < ∞, guarantees an ordinary continuous DTFT. Square-summable sequences have an L² interpretation. Pure constant and sinusoidal sequences instead use periodic Dirac impulse distributions: each impulse has a specified weight, not a finite height.

No principal-value unit-step pair is included here. An ordinary geometric formula must not be extended to |a| = 1 without a separate justified distributional interpretation. For inverse sifting at a period boundary, choose an interval with the line in its interior or use consistent periodic endpoint handling.

Worked DTFT Examples

Example 1: Unit Impulse

For x[n] = δ[n], only n = 0 contributes to the DTFT sum. Therefore X(ω) = 1 at every frequency.

Example 2: Delayed Unit Impulse

For x[n] = δ[n−3], only n = 3 contributes, giving X(ω) = e^(−j3ω). A delay changes phase without changing spectral magnitude.

Example 3: Causal Exponential

For x[n] = (1/2)ⁿu[n], the DTFT is a convergent geometric series. Because |1/2| < 1, the transform is X(ω) = 1/(1 − (1/2)e^(−jω)).

Frequently Asked Questions

What does DTFT stand for?

DTFT stands for Discrete-Time Fourier Transform. It converts a discrete-time sequence into a frequency-domain representation with a continuous, periodic frequency variable.

What is the DTFT formula?

The forward transform is X(ω) = Σ x[n]e^(−jωn), summed over all integer n. The inverse uses a 1/(2π) integral over any complete 2π frequency interval.

Why is the DTFT periodic?

The transform repeats every 2π radians per sample because e^(−j2πn) = 1 for every integer n.

What is the difference between DTFT and DFT?

The DTFT has a continuous frequency variable, while the DFT evaluates a finite-length sequence at a finite set of frequency bins. They are related but are not the same transform.

Does every discrete-time sequence have a DTFT?

No. An ordinary DTFT requires suitable convergence or an appropriate function-space interpretation. Some sequences have transforms only in a generalized distributional sense.

Can I use a Z-transform table to find a DTFT?

Sometimes. If the Z-transform’s region of convergence includes the unit circle, substituting z = e^(jω) gives the DTFT.

Why do some DTFT entries contain Dirac impulses?

Periodic sequences such as pure complex exponentials have line spectra. Their frequency-domain representations are periodic impulse distributions rather than ordinary smooth functions.

What is the inverse DTFT?

The inverse DTFT reconstructs x[n] by integrating X(ω)e^(jωn) over one complete frequency period and dividing by 2π.

Sources and Methodology

This reference uses angular frequency in radians per sample, a negative exponential forward kernel, and inverse scaling 1/(2π). Each pair states a derivation and verification note. Absolutely summable examples are checked against direct sums; periodic impulses are verified by inverse sifting and Fourier-series coefficients rather than finite spike approximations.

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