17.4 : 傅里叶变换的性质 II

The Frequency Shifting property of Fourier Transforms states that a shift in the frequency domain corresponds to a phase shift in the time domain, illuminating the interplay between these two domains.

In radio broadcasting, frequency shifting modulates a carrier signal with the input signal, allowing simultaneous multi-channel transmission.

The concept of Time Differentiation reveals that the Fourier transform of the derivative of a function can be obtained by multiplying its Fourier transform by a factor j-omega.

It's crucial to understand how temporal changes affect signals when applying time-zone-based broadcast delays in audio processing.

The Frequency Differentiation property underscores the interconnectedness of time and frequency domains.

The Duality property highlights the symmetry between the time and frequency domains. If one domain undergoes a Fourier transform, its counterpart reflects that change, albeit with a factor adjustment and a sign reversal in the integral's exponential.

The Convolution property, key in signal processing, asserts that the Fourier transform of two convolved time functions equals their individual Fourier transforms' product, facilitating the combination of multiple signals.