For a real x(t), the Fourier or Laplace transform X(s) typically exhibits conjugate reciprocal pairs unless x(t) is of a specific form. When x(t) is an exponential function like e^{at}, X(s) can have a real pole at s = a, indicating that not all transforms yield conjugate pairs. In contrast, sinusoidal x(t) results in complex conjugate poles in X(s). This distinction highlights the relationship between the nature of x(t) and the characteristics of its transform X(s). Understanding these properties is crucial for analyzing signal behavior in the frequency domain.
