The Discrete-Time Fourier Transform (DTFT) is the primary theoretical tool for understanding the frequency content of a discrete-time (sampled) signal. The DTFT is defined as The inverse DTFT (IDTFT) is defined by an integral formula, because it operates on a continuous-frequency DTFT spectrum:

The DTFT is very useful for theory and analysis, but is not practical for numerically computing a spectrum digitally, because

1. infinite time samples means
   • infinite computation
   • infinite delay
2. The transform is continuous in the discrete-time frequency, ω

For practical computation of the frequency content of real-world signals, the Discrete Fourier Transform (DFT) is used.

The DFT transforms NN samples of a discrete-time signal to the same number of discrete frequency samples, and is defined as The DFT is invertible by the inverse discrete Fourier transform (IDFT): The DFT and IDFT are a self-contained, one-to-one transform pair for a length-NN discrete-time signal. (That is, the DFT is not merely an approximation to the DTFT as discussed next.) However, the DFT is very often used as a practical approximation to the DTFT.

The goal of spectrum analysis is often to determine the frequency content of an analog (continuous-time) signal; very often, as in most modern spectrum analyzers, this is actually accomplished by sampling the analog signal, windowing (truncating) the data, and computing and plotting the magnitude of its DFT. It is thus essential to relate the DFT frequency samples back to the original analog frequency. Assuming that the analog signal is bandlimited and the sampling frequency exceeds twice that limit so that no frequency aliasing occurs, the relationship between the continuous-time Fourier frequency ΩΩ (in radians) and the DTFT frequency ωω imposed by sampling is ω=ΩT ω Ω T where TT is the sampling period. Through the relationship ω k =2πkN ω k 2 k N between the DTFT frequency ωω and the DFT frequency index kk, the correspondence between the DFT frequency index and the original analog frequency can be found: Ω=2πkNT Ω 2 k N T or in terms of analog frequency ff in Hertz (cycles per second rather than radians) f=kNT f k N T for kk in the range kk between 00 and N2 N 2 . It is important to note that k∈ N2+1N-1 k N 2 1 N 1 correspond to negative frequencies due to the periodicity of the DTFT and the DFT.

If more than NN equally spaced frequency samples of a length-NN signal are desired, they can easily be obtained by zero-padding the discrete-time signal and computing a DFT of the longer length. In particular, if LN LN DTFT samples are desired of a length-NN sequence, one can compute the length- LN LN DFT of a length- LN LN zero-padded sequence zn=xnif0≤n≤N-10ifN≤n≤LN-1 z n x n 0 n N 1 0 N n L N 1 X w k =2πkLN=∑n=0N-1xnⅇ-ⅈ2πknLN=∑n=0LN-1znⅇ-ⅈ2πknLN= DFT L N zn X w k 2 k L N n N 1 0 x n 2 k n L N n L N 1 0 z n 2 k n L N DFT L N z n Note that zero-padding interpolates the spectrum. One should always zero-pad (by about at least a factor of 4) when using the DFT to approximate the DTFT to get a clear picture of the DTFT. While performing computations on zeros may at first seem inefficient, using FFT algorithms, which generally expect the same number of input and output samples, actually makes this approach very efficient.

Figure 1 shows the magnitude of the DFT values corresponding to the non-negative frequencies of a real-valued length-64 DFT of a length-64 signal, both in a "stem" format to emphasize the discrete nature of the DFT frequency samples, and as a line plot to emphasize its use as an approximation to the continuous-in-frequency DTFT. From this figure, it appears that the signal has a single dominant frequency component. Zero-padding by a factor of two by appending 64 zero values to the signal and computing a length-128 DFT yields Figure 2. It can now be seen that the signal consists of at least two narrowband frequency components; the gap between them fell between DFT samples in Figure 1, resulting in a misleading picture of the signal's spectral content. This is sometimes called the picket-fence effect, and is a result of insufficient sampling in frequency. While zero-padding by a factor of two has revealed more structure, it is unclear whether the peak magnitudes are reliably rendered, and the jagged linear interpolation in the line graph does not yet reflect the smooth, continuously-differentiable spectrum of the DTFT of a finite-length truncated signal. Errors in the apparent peak magnitude due to insufficient frequency sampling is sometimes referred to as scalloping loss. Zero-padding to four times the length of the signal, as shown in Figure 3, clearly shows the spectral structure and reveals that the magnitude of the two spectral lines are nearly identical. The line graph is still a bit rough and the peak magnitudes and frequencies may not be precisely captured, but the spectral characteristics of the truncated signal are now clear. Zero-padding to a length of 1024, as shown in Figure 4 yields a spectrum that is smooth and continuous to the resolution of the computer screen, and produces a very accurate rendition of the DTFT of the truncated signal. The signal used in this example actually consisted of two pure sinusoids of equal magnitude. The slight difference in magnitude of the two dominant peaks, the breadth of the peaks, and the sinc-like lesser side lobe peaks throughout frequency are artifacts of the truncation, or windowing, process used to practically approximate the DFT. These problems and partial solutions to them are discussed in the following section.

Applying the DTFT multiplication property X ω k ̂=∑n=-∞∞xnwnⅇ-ⅈ ω k n=12πX ω k *W ω k X ω k n ∞ ∞ x n w n ω k n 1 2 X ω k W ω k we find that the DFT of the windowed (truncated) signal produces samples not of the true (desired) DTFT spectrum Xω X ω , but of a smoothed verson Xω*Wω X ω W ω . We want this to resemble Xω X ω as closely as possible, so Wω W ω should be as close to an impulse as possible. The window wn w n need not be a simple truncation (or rectangle, or boxcar) window; other shapes can also be used as long as they limit the sequence to at most NN consecutive nonzero samples. All good windows are impulse-like, and represent various tradeoffs between three criteria:

1. main lobe width: (limits resolution of closely-spaced peaks of equal height)
2. height of first sidelobe: (limits ability to see a small peak near a big peak)
3. slope of sidelobe drop-off: (limits ability to see small peaks further away from a big peak)

Many different window functions have been developed for truncating and shaping a length-NN signal segment for spectral analysis. The simple truncation window has a periodic sinc DTFT, as shown in Figure 5. It has the narrowest main-lobe width, 2πN 2 N at the -3 dB level and 4πN 4 N between the two zeros surrounding the main lobe, of the common window functions, but also the largest side-lobe peak, at about -13 dB. The side-lobes also taper off relatively slowly.

The Hann window (sometimes also called the hanning window), illustrated in Figure 6, takes the form wn=0.5-0.5cos2πnN-1 w n 0.5 0.5 2 n N 1 for nn between 00 and N-1 N 1 . It has a main-lobe width (about 3πN 3 N at the -3 dB level and 8πN 8 N between the two zeros surrounding the main lobe) considerably larger than the rectangular window, but the largest side-lobe peak is much lower, at about -31.5 dB. The side-lobes also taper off much faster. For a given length, this window is worse than the boxcar window at separating closely-spaced spectral components of similar magnitude, but better for identifying smaller-magnitude components at a greater distance from the larger components.

The Hamming window, illustrated in Figure 7, has a form similar to the Hann window but with slightly different constants: wn=0.538-0.462cos2πnN-1 w n 0.538 0.462 2 n N 1 for nn between 00 and N-1 N 1 . Since it is composed of the same Fourier series harmonics as the Hann window, it has a similar main-lobe width (a bit less than 3πN 3 N at the -3 dB level and 8πN 8 N between the two zeros surrounding the main lobe), but the largest side-lobe peak is much lower, at about -42.5 dB. However, the side-lobes also taper off much more slowly than with the Hann window. For a given length, the Hamming window is better than the Hann (and of course the boxcar) windows at separating a small component relatively near to a large component, but worse than the Hann for identifying very small components at considerable frequency separation. Due to their shape and form, the Hann and Hamming windows are also known as raised-cosine windows.

The main-lobe width of a window is an inverse function of the window-length N N; for any type of window, a longer window will always provide better resolution.

Many other windows exist that make various other tradeoffs between main-lobe width, height of largest side-lobe, and side-lobe rolloff rate. The Kaiser window family, based on a modified Bessel function, has an adjustable parameter that allows the user to tune the tradeoff over a continuous range. The Kaiser window has near-optimal time-frequency resolution and is widely used. A list of many different windows can be found here.