What is SINC Sine Cardinal or Sinus Cardinalis

en.wikipedia.org/wiki/Digital_signal_processing

The SINC function, also known as the Sine Cardinal or Sinus Cardinalis, is a mathematical function that arises frequently in signal processing and related fields. It has two main definitions, each with its own applications:

1. Unnormalized SINC Function (Sa):

This is the historical definition of the SINC function and is often referred to as the sampling function. It's defined for x ≠ 0 as:

Sa(x) = (sin(πx)) / (πx)

Here, π (pi) represents the mathematical constant pi, and x is the independent variable.

Properties of the Unnormalized SINC Function:

• Symmetry: The SINC function is an even function, meaning Sa(-x) = Sa(x) for all x.
• Zeros: The SINC function has zeros at all non-zero integer multiples of π (pi). In other words, Sa(nπ) = 0 for all integer values of n except n = 0.
• Main Lobe: The SINC function has a central lobe centered at x = 0 with a value of 1. This lobe decreases in magnitude as x moves away from 0.
• Side Lobes: The SINC function has additional, smaller lobes on either side of the central lobe. These side lobes decrease in magnitude further away from the central lobe but never reach zero completely.

Applications of the Unnormalized SINC Function:

• Ideal Low-Pass Filter: The unnormalized SINC function can be used to represent the ideal impulse response of a perfect low-pass filter. However, it's not practical to implement in real systems due to its infinite duration.
• Theoretical Foundation: The unnormalized SINC function provides the theoretical basis for understanding the sampling theorem, which states that a band-limited continuous signal can be perfectly reconstructed from its samples if the sampling frequency is greater than twice the highest frequency component in the signal.

2. Normalized SINC Function (sinc):

This is the more commonly used definition in modern signal processing applications. It's defined as:

sinc(x) = { 1 for x = 0
           sin(πx) / (πx) otherwise

This definition incorporates a scaling factor of 1/π compared to the unnormalized SINC function.

Properties of the Normalized SINC Function:

• Inherits many properties from the unnormalized SINC function, including symmetry, zeros at non-zero integer multiples of π, and a central lobe with a value of 1 at x = 0.
• The side lobes of the normalized SINC function decay faster compared to the unnormalized version, making it more desirable for practical applications.

Applications of the Normalized SINC Function:

• Interpolation: The normalized SINC function is used in various interpolation techniques to reconstruct a continuous signal from its discrete samples.
• Windowing: The SINC function is often used as a window function in signal processing to reduce spectral leakage during Fourier transform operations.
• Filter Design: Idealized filter responses can be designed using the SINC function as a starting point, which are then approximated by realizable filter functions.

In Conclusion:

The SINC function, in both its unnormalized and normalized forms, plays a crucial role in various signal processing applications. Its properties of zero crossings, central lobe, and decaying side lobes make it valuable for tasks like interpolation, windowing, and filter design. Understanding the SINC function is essential for anyone working in fields like digital signal processing, communication systems, and image processing.