What Is the Sinc Function and Why Is It Important in Electrical Engineering?

SINC功能及其标准化变体

SINC函数的数学公式，也称为基本正弦函数，如下写：

\[\text{sinc}(x)=\frac{\sin(x)}{x}\]

The function shown above is undefined for x = 0, and consequently we need to define sinc(0) based on the limit as x approaches 0, which is 1. Thus,

\[\text{sinc}(x)=\begin{cases}1 & \text{for } x = 0\\\frac{\sin(x)}{x} & \text{otherwise}\end{cases}\]

In the context of digital signal processing, we often use an alternative form in which the independent variable is multiplied by π:

\ [\ text {sinc} _ \ pi（x）\ equiv \ equiv \ text {sinc}（\ pi x）= \ begin {cases} 1＆\ text {for} for} x = 0 \\\\\\\\\\\\\\\\ frac {\ sin（\ sin（\ sin（\ sin）pi x）} {\ pi x}＆\ text {否则} \ end {cases} \] \]

This second form is called the normalized sinc function, because the definite integral over the entire range of x is equal to 1:

\[\int_{-\infty}^{\infty}\text{sinc}_\pi(x)dx=1\]

我提到标准化版本在数字信号处理中很常见。当我们处理离散数据而不是连续时间变量时，标准化如下表示：

\[\sum_{n=-\infty}^{\infty}\text{sinc}_\pi[n]=1 \]

The following plot shows the shape of the sinc function, and it also conveys the difference created when we multiply the independent variable by π.

The Sinc Function in Signal Processing

TheFourier transformof the sinc function is a rectangle centered on ω = 0. This gives sinc(x) a special place in the realm of signal processing, because a rectangular shape in the frequency domain is the idealized “brick-wall” filter response. In other words, sinc(x) is the impulse response of an ideal low-pass filter.

The use of the sinc function in filtering applications is more apparent in the digital domain. The following diagram illustrates the similarity between the impulse response of aFIR filterand a plot of sinc(x).

The Fourier transform of the sinc function is a rectangle, and the Fourier transform of a rectangular pulse is a sinc function. If we need to shorten a discrete-time signal for the purpose of spectral analysis, we can multiply it by a rectangular window, and this operation is equivalent to卷积the Fourier transform of the signal with a sinc function.

SINC函数也出现在数字到分析转换的分析中。模拟信号的理想化重建是一系列冲动，但是现实生活中的DAC通过将零阶保持在输出样品中，从而产生“楼梯”波形。在频域中，零阶保持会导致输出光谱等于理想化的频谱乘以SINC函数。