Introduction:
The Fourier Transform is a mathematical tool that decomposes a function into its constituent frequencies. It is widely used in signal processing, image processing, and many other fields. A conjugate symmetric function is a special kind of function that exhibits certain symmetries. In this response, we will explain why the Fourier Transform of a conjugate symmetric function is always conjugate symmetric.

Explanation:
To understand why the Fourier Transform of a conjugate symmetric function is always conjugate symmetric, let's start by defining what a conjugate symmetric function is.

A function f(t) is said to be conjugate symmetric if it satisfies the following condition:

f(t) = f*(t)

where f*(t) represents the complex conjugate of f(t).

Now, let's consider the Fourier Transform of f(t), denoted as F(ω), where ω is the frequency variable. The Fourier Transform of f(t) is given by the following equation:

F(ω) = ∫[f(t)e^(-jωt)]dt

where j is the imaginary unit.

Proof:
To prove that the Fourier Transform of a conjugate symmetric function is always conjugate symmetric, we need to show that:

F(ω) = F*(-ω)

Let's substitute f(t) = f*(-t) in the Fourier Transform equation:

F(ω) = ∫[f*(-t)e^(-jωt)]dt

Now, let's change the variable of integration from t to -t:

F(ω) = ∫[f*(t)e^(jωt)]dt

Notice that we have f*(t) and e^(jωt), which are the complex conjugates of f*(-t) and e^(-jωt), respectively.

Using the property that the complex conjugate of a product is equal to the product of the complex conjugates, we can write:

F(ω) = [∫f*(-t)dt][∫e^(jωt)dt]

The first integral on the right-hand side is simply the Fourier Transform of f*(-t), denoted as F*(-ω). The second integral represents the inverse Fourier Transform of e^(jωt), which is a complex sinusoidal function.

Therefore, we can conclude that:

F(ω) = F*(-ω)

which means that the Fourier Transform of a conjugate symmetric function is always conjugate symmetric.

Conclusion:
In summary, the Fourier Transform of a conjugate symmetric function is always conjugate symmetric. This property is useful in many applications, such as the analysis of real-valued signals and the simplification of certain mathematical operations. Understanding the symmetry properties of functions and their Fourier Transforms is essential in signal processing and related fields.