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zheng89120
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I have a simple question. Is the kernel of a Fourier transform exp(-iwt) or exp(iwt). It feels like my professor sometimes uses one, and sometimes uses the other.
The exponential form, also known as the complex form, is used in Fourier transforms to represent the amplitude and phase information of a signal. It allows for a more concise and intuitive representation of the signal in the frequency domain.
The difference between Exp(-iwt) and Exp(iwt) is the direction of rotation in the complex plane. Exp(-iwt) represents a clockwise rotation, while Exp(iwt) represents a counterclockwise rotation. This difference is important when dealing with phase information in Fourier transforms.
The exponential forms have a direct impact on the frequency domain representation of a signal. Exp(-iwt) shifts the frequency components to the left, while Exp(iwt) shifts them to the right. This shift can be seen in the phase spectrum of the signal.
No, there is no preference for one form over the other in Fourier transforms. Both Exp(-iwt) and Exp(iwt) are equally valid representations and can be used interchangeably. The choice may depend on the specific problem or preference of the user.
The choice of which form to use in a Fourier transform depends on the problem at hand. In some cases, one form may be more convenient or intuitive to use over the other. It is important to understand the properties of both forms and choose the one that best suits the specific problem.