Physical Meaning of 1DFT

Here we only consider the FT of continuous and periodic signals. The result here should be easily generalized to other cases. The Fourier expansion of a 1D periodic signal

$$x_T(t)=\sum_{n=-\infty}^{\infty} X[n]e^{j2\pi nf_0t}$$

represents the signal as a weighted sum of complex exponential functions. Here $X[n]$ is the weight of the nth term (the Fourier coefficient) which is in general a complex number

$$X[n]=X_r[n]+jX_j[n]=A_n e^{j\Phi_n} =\frac{1}{T}\int_T x_T(t)e^{-j2\pi nf_0t} dt$$

where

$$\left\{ \begin{array}{l} \vert X[n]\vert=\sqrt{X_r[n]^2+X_j[... ... \angle{X[n]}=tan^{-1} [ X_j[n]/X_r[n] ] \end{array} \right.$$

We assume the signal $x_T(t)=x_T^*(t)$ is real, then $X[-n]=X^*[n]$, and we can rewrite the Fourier expansion of $x(t)$ above as

$$x(t) = \sum_{n=-\infty}^{\infty} X[n]e^{j2\pi nf_0t} = X[0]+\sum_{n=1}^{\infty}[ X[n]e^{j2\pi nf_0t}+X[-n]e^{-j2\pi nf_0t}]$$

$$= X[0]+\sum_{n=1}^{\infty}[ X[n]e^{j2\pi nf_0t}+X^*[n]e^{-j2\pi nf_... ...j\angle X[n]}e^{j2\pi nf_0t}+\vert X[n]\vert e^{-j\angle X[n]}e^{-j2\pi nf_0t}]$$

$$= X[0]+\sum_{n=1}^{\infty} \vert X[n]\vert [e^{j(2\pi nf_0+\angle X... ...] )}] = X[0]+\sum_{n=1}^{\infty} 2\vert X[n]\vert\;cos(2\pi nf_0t+\angle X[n] )$$

where $X[0]=\frac{1}{T}\int_T x(t) dt$ is the average or DC component of the signal. Now we see that any real signal $x(t)$ can be represented as a weighted sum of infinite number of sinusoids of different frequencies $nf_0$ with different amplitudes $2X[n]$ and phases $\angle X[n]$.