Discrete-time Fourier transform ($X(\nu )={\sum }_n{x}_n{e}^{-2\pi in\nu }$)

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 isFor 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 $1/(2\pi )$.^[ref]

Because $e^{-2\pi in}=1$, $X(\nu +1)=X(\nu )$. 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. ThenAt zero frequency its value is one; at $\nu =1/4$ its magnitude is $1/\sqrt{2}$ 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.^[ref]

For a finite record $x_0,\dots ,x_{N-1}$ extended by zero, the unnormalized discrete Fourier transform gives samples of the same continuous function: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 $n\ge 0$, zero otherwise, and $|a|<1$, the geometric sum givesAt $a=1$, 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.