Question

Let x (t) be a periodic signal of period T. Prove that the Fourier coefficients can be computed by integrating over any continuous time-interval of T seconds duration, i.e., \

x ₙ = 1/T ∫ᵀ₀ dt x(t) e ⁻ᶦ²πⁿᵗ/ᵀ
= 1/T ∫ᵀ⁰ ⁺ ᵀₜ₀ dt x (t)e ⁻ᶦ²πⁿᵗ/ᵀ ∀T₀

Answer 1

The Fourier coefficients of a periodic signal can be computed by integrating over any continuous time-interval of T seconds duration.

Step-by-step explanation:

The Fourier coefficients of a periodic signal can be computed by integrating over any continuous time-interval of T seconds duration. This means that the Fourier coefficients can be found by integrating the product of the signal x(t) and the complex exponential function e^(-i2πnt/T) over one period T.

To prove this, we can separate the variables t and x and integrate from an initial time t=0 to an arbitrary time. By performing this integration, we can arrive at the equation for the Fourier coefficients.

The equation for the Fourier coefficients is given by x_n = (1/T) ∫(0 to T) x(t) e^(-i2πnt/T) dt, where x_n is the nth Fourier coefficient.

Answer 2

To prove that the Fourier coefficients ��xn can be computed by integrating over any continuous time-interval of �T seconds duration, let's use the definition of the Fourier coefficients:

��=1�∫0��(�)�−�2���/���xn=T1∫