Understanding the Concept:
The Continuous-Time Fourier Transform (CTFT) maps a time-domain signal \(x(t)\) into a complex frequency-domain function \(X(j\omega)\) via the integral equation:
\[
X(j\omega) = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt
\]
Symmetry properties in the time domain create corresponding constraints on the mathematical properties of \(X(j\omega)\). When a signal is explicitly given as real-valued, meaning \(x(t) = x^*(t)\) (where \(*\) denotes the complex conjugate operation), it imposes a powerful condition known as conjugate symmetry on its frequency spectrum.
Step 1: Apply the complex conjugate operation to the Fourier integral.
Let's take the complex conjugate of both sides of the standard Fourier transform definition equation:
\[
X^*(j\omega) = \left[ \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt \right]^*
\]
Passing the conjugation operator inside the linear integral gives:
\[
X^*(j\omega) = \int_{-\infty}^{\infty} x^*(t) (e^{-j\omega t})^* dt
\]
Since \(x(t)\) is strictly real-valued by definition, we substitute \(x^*(t) = x(t)\). Also, the conjugate of the complex exponential changes the sign of the imaginary component: \((e^{-j\omega t})^* = e^{j\omega t}\).
\[
X^*(j\omega) = \int_{-\infty}^{\infty} x(t) e^{j\omega t} dt \quad \cdots (1)
\]
Step 2: Evaluate the expression at negative frequencies.
Now let's look at the standard expression evaluated at a negative frequency value (\(-\omega\)):
\[
X(-j\omega) = \int_{-\infty}^{\infty} x(t) e^{-j(-\omega)t} dt = \int_{-\infty}^{\infty} x(t) e^{j\omega t} dt \quad \cdots (2)
\]
Step 3: Correlate the equations to prove the symmetry constraint.
Comparing Equation (1) and Equation (2), the right-hand integrals are identical, leading directly to the property:
\[
X^*(j\omega) = X(-j\omega) \quad \text{or equivalently} \quad X(j\omega) = X^*(-j\omega)
\]
This mathematically defines conjugate symmetry (also called Hermitian symmetry).
Step 4: Understand the real and imaginary parts.
Expressing the frequency spectrum in rectangular form, \(X(j\omega) = R(\omega) + jI(\omega)\):
\[
R(\omega) + jI(\omega) = [R(-\omega) + jI(-\omega)]^* = R(-\omega) - jI(-\omega)
\]
Equating real and imaginary components yields:
• \(R(\omega) = R(-\omega)\) (The magnitude/real spectrum has even symmetry).
• \(I(\omega) = -I(-\omega)\) (The phase/imaginary spectrum has odd symmetry).
Because the total complex function combines both an even real part and an odd imaginary part, the complete property is uniquely referred to as conjugate symmetry. Thus, option (C) is the correct choice.