Fourier Coefficients: A_n & B_n Explained

[dimensionless]

A function $$f(t)$$ can be represented by the expansion

$$f(t) = \frac{1}{2}A_{0} + A_{1}cos(\omega t) + A_{2}cos(2 \omega t) + A_{3}cos(3 \omega t) + ... B_{1}sin(\omega t) + B_{2}sin(2 \omega t) + B_{3}sin(3 \omega t) + ...$$

Do the constants $$A_{n}$$ and $$B_{n}$$ the same thing as the real and imaginary components of the Fourier transform? If so, why is there no imaginary component in the zeroth term?

 

[mathman]

In computing the Fourier transform, the kernel is of the form e^inwt. For A₀, the kernel is simply 1, so there is no imaginary part.

 

[tacman]

The constants A_n and B_n in the Fourier series expansion represent the amplitudes of the cosine and sine terms, respectively. They are not the same as the real and imaginary components of the Fourier transform.

The real and imaginary components of the Fourier transform represent the magnitude and phase of the frequency components in the signal. On the other hand, the A_n and B_n coefficients in the Fourier series expansion represent the amplitude of each cosine and sine term at a specific frequency.

The reason there is no imaginary component in the zeroth term, A_0, is because it represents the average value of the function f(t). Since it is a constant term, it does not have a frequency component and therefore does not have an imaginary component.

Overall, the Fourier series expansion and Fourier transform are different mathematical representations of a function, but they both provide useful information about the frequency components present in a signal.