DTFT Pairs

X(f) = \sum_{n=-\infty}^\infty x[n] e^{-j2\pi f n} \qquad \text{(analysis)} \\ x[n] = \int_{-\frac{1}{2}}^{\frac{1}{2}} X(f) e^{j2\pi f n} df \qquad \text{(synthesis)}

When discussing the convergence properties of the DTFT, it is useful to define the partial sum:

X_N(f) = \sum_{n=-N}^N x[n] e^{-j2\pi f n}.

$X_N(f), \; N=0, 1, 2, \cdots$ is a sequence of functions.

A signal $x[n]$ is absolutely summable if $\displaystyle \sum_{n=-\infty}^\infty |x[n]| < \infty$ , i.e. the absolutely sum is finite.
The DTFTs of absolutely summable signals are continuous and smooth.
The sequence of partial sums $X_N(f)$ exhibits uniform convergence. Convergence means that given $\epsilon > 0$ , there exists $K$ such that $|X(f) - X_N(f)|<\epsilon$ for all $N \geq K$ . The convergence is uniform in the sense that the same $\epsilon$ holds for all frequencies $f$ .
The DTFT of absolutely summable signals can be computed using the DTFT analysis formula.

A signal $x[n]$ is an energy signal if $\displaystyle \sum_{n=\infty}^\infty |x[n]|^2 < \infty$ , i.e. the energy is finite.
The DTFTs of energy signals can have discontinuities and the partial sum $X_N(f)$ exhibits the Gibbs phenomenon.
The sequence of partial sums $X_N(f)$ converges in the mean square to $X(f)$ . This means that given $\epsilon > 0$ , there exists $K$ such that

\int_{-\frac{1}{2}}^{\frac{1}{2}} \left|X(f) - X_N(f)\right|^2 df < \epsilon

for all $N\geq K$ . Put another way, the energy in the error between $X(f)$ and $X_N(f)$ converges to zero.

The DTFT of energy signals is verified using the DTFT synthesis formula (i.e. the inverse DTFT).

A signal $x[n]$ is a power signal if $\displaystyle \lim_{N\rightarrow \infty} \frac{1}{2N+1} \displaystyle \sum_{n=-\infty}^\infty |x[n]|^2$ , i.e. the power is finite.
Examples of power signals include: $1, u[n], \sin(2\pi f_0 n), \cos(2\pi f_0 n), e^{j2\pi f_0 n}$ .
For example the DTFT of $x[n]=e^{j2\pi f_0 n}$ is defined to be $X(f) = \delta(f-f_0)$ . At $f=f_0$ , the partial sum sequence $X_N(f=f_0)$ does not converge; it diverges.
The DTFT of power signals may be verified using the DTFT synthesis formula (i.e. the inverse DTFT).