## Spectrum of a Windowed Sinusoid

Ideal sinusoids are infinite in duration. In practice, however, we must work with finite-length signals. Therefore, only a finite segment of a sinusoid can be processed at any one time, as if we were looking at the sinusoid through a ``window'' in time. For audio signals, the ear also processes sinusoids in short time windows (much less than 1 second long); thus, audio spectrum analysis is generally carried out using analysis windows comparable to the time-window inherent in hearing. Finally, nothing in nature produces true sinusoids. However, natural oscillations can often be modeled as sinusoidal over some finite time. Thus, it is useful to consider*short-time spectrum analysis*, in which the time-domain signal is analyzed in short time segments (windows). The short-time spectrum then evolves at some rate over time. The shorter our analysis window, the faster the short-time spectrum can change. Very long analysis windows, on the other hand, presuppose a fixed spectral content over the duration of the window.

The easiest windowing operation is simply

*truncating*the sinusoid on the left and right at some finite times. This can be modeled mathematically as a multiplication of the sinusoid by the so-called

*rectangular window*:

(6.12) |

where is the length of the window (assumed odd for simplicity). The windowed sinusoid is then

(6.13) |

However, as we will see, it is often advantageous to

*taper*the window more gracefully to zero on the left and right. Figure 5.1 illustrates the general shape of a more typical window function. Note that it is nonnegative and symmetric about time 0. Such functions are loosely called ``zero-phase'' since their Fourier transforms are real, as shown in §2.3. A more precise adjective is

*zero-centered*for such windows. In some cases, such as when analyzing real time systems, it is appropriate to require our analysis windows be

*causal*. A window is said to be

*causal*if it is zero for all . Figure 5.2 depicts the causal version of the window shown in Fig.5.1. A length zero-centered window can be made causal by shifting (delaying) it in time by half its length . The shift theorem (§2.3.4) tells us this introduces a linear phase term in the frequency domain. That is, the window Fourier transform has gone from being real to being multiplied by .

**Next Section:**

Effect of Windowing

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Spectrum of Sampled Complex Sinusoid