= Discrete-time Fourier transform
{title2=$X(\nu)=\sum_nx_ne^{-2\pi i n\nu}$}
= DTFT
{c}
{synonym}
The <discrete-time Fourier transform> represents a sequence indexed by integer sample number using a continuous, periodic <frequency> variable. In cycles per sample, the transform pair is
$$
X(\nu)=\sum_{n\in\mathbb Z}x_n e^{-2\pi i n\nu},\qquad
x_n=\int_{-1/2}^{1/2}X(\nu)e^{2\pi i n\nu}\,d\nu.
$$
For an absolutely summable sequence, the sum converges uniformly to a continuous function, and the inverse formula follows by integrating the exponentials term by term. Their integral over one <frequency> period is one when the integer exponent is zero and zero otherwise. The primary text gives the same pair in angular <frequency> $\omega=2\pi\nu$, where the inverse integral carries a factor $1/(2\pi)$.<Spectral Audio Signal Processing by Julius O. Smith III>[Discrete Time Fourier Transform (DTFT), equations (3.2)–(3.3)]{ref}
Because $e^{-2\pi i n}=1$, $X(\nu+1)=X(\nu)$. Discrete time therefore does not imply discrete <frequency>: the <DTFT> is defined at every <frequency> in one period. With sampling <frequency> $f_s$, a physical <frequency> $f$ corresponds to $\nu=f/f_s$; <frequencies> separated by integer multiples of $f_s$ give identical sampled complex exponentials. This is the basic <frequency> ambiguity behind <aliasing>.
An original two-sample example takes $x_0=x_1=1/2$ and all other entries zero. Then
$$
X(\nu)=\tfrac12(1+e^{-2\pi i\nu})
=e^{-\pi i\nu}\cos(\pi\nu).
$$
At zero <frequency> its value is one; at $\nu=1/4$ its magnitude is $1/\sqrt2$ with phase $-\pi/4$; at $\nu=1/2$ it vanishes. As the <impulse response> of a two-tap moving average, it preserves a constant input and removes an alternating-sign input. SciPy’s <frequency>-response definition evaluates precisely such sums when its denominator is one; its <frequency> units depend on the fs parameter.<SciPy 1.16.2 documentation>[scipy.signal.freqz, defining transfer-function sum and Parameters: fs]{ref}
For a finite record $x_0,\ldots,x_{N-1}$ extended by zero, the unnormalized <discrete Fourier transform> gives samples of the same continuous function:
$$
X_k=\sum_{n=0}^{N-1}x_n e^{-2\pi i nk/N}=X(k/N).
$$
<Zero padding> the record to length $M>N$ instead evaluates $X(k/M)$ on a denser grid. It does not change $X(\nu)$ or restore unobserved time samples. The <rectangular window> and <Hann window> demonstrate how the selected finite record shapes that continuous response.
Convergence assumptions matter. For $x_n=a^n$ when $n\ge0$, zero otherwise, and $|a|<1$, the geometric sum gives
$$
X(\nu)=\frac1{1-ae^{-2\pi i\nu}}.
$$
At $a=1$, that ordinary convergent-sum argument fails. Square-summable sequences admit an $L^2$ interpretation, while perpetual tones require a generalized interpretation; neither should be treated as an absolutely convergent version of the same series without stating the interpretation.
A useful check of numerical transform conventions is to evaluate a finite sum at arbitrary <frequencies>, compare its DFT-grid values with the FFT output, and repeat after zero-padding. This tests the exponent sign, normalization and <frequency> units independently. In a physical acquisition, one must also state sampling rate, record duration and the response before sampling; dense numerical <frequency> points alone do not validate those experimental inputs.
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