**Sampling Property of Delta Function**

The delta function, also known as the Dirac delta function, is a mathematical construct used in signal processing and mathematics. It is often represented as δ(t) or δ(t - τ), where τ is a shift parameter.

The sampling property of the delta function states that the area under the product of a function and a shifted impulse is equal to the value of the function located at the unit impulse instant. In other words, it relates the integral of the product of a function and a shifted delta function to the value of the function at the impulse instant.

The sampling property can be mathematically expressed as:

∫ [f(t) δ(t - τ)] dt = f(τ)

where f(t) is a function, δ(t - τ) is a shifted delta function, and f(τ) is the value of the function at the impulse instant τ.

**Explanation**

Let's consider a simple example to understand this property. Suppose we have a function f(t) and a shifted delta function δ(t - τ). The integral of their product over a certain interval is given by:

∫ [f(t) δ(t - τ)] dt

When we integrate the product of f(t) and δ(t - τ), the delta function acts as a sampling operator. It selects the value of f(t) at the impulse instant τ and multiplies it by the function value at that instant.

Since the delta function is nonzero only at t = τ, the integral can be simplified to:

= f(τ) ∫ δ(t - τ) dt

The integral of the delta function δ(t - τ) is equal to 1 when the integration interval includes the impulse instant τ, and 0 otherwise. Therefore, the integral simplifies to:

= f(τ) * 1

= f(τ)

Hence, the area under the product of the function f(t) and the shifted delta function δ(t - τ) is equal to the value of the function at the impulse instant τ.

**Conclusion**

The sampling property of the delta function is a fundamental property that relates the area under the product of a function and a shifted delta function to the value of the function at the impulse instant. This property is widely used in signal processing, especially in applications such as signal reconstruction from samples and convolution operations.