# Proving DFT equation

How can I prove the equation $(3)$? I can't understand why there is a $2/N$ in $(3)$. Why he just get the second term in DFT?

Consider a sinusoidal input signal of frequency $\omega$ given by

$$x(t) = \sqrt 2 X \sin\left(\omega t + \phi\right)\tag{1}$$

This signal is conventionally represented by a phasor (a complex number) $\bar X$

$$\bar X = Xe^{j\phi}=X\cos\phi+jX\sin\phi\tag{2}$$

Assuming that $x(t)$ is sampled $N$ times per cycle of the $60\textrm{ Hz}$ waveform to produce the sampled set $\left\{x_k\right\}$

$$x_k = \sqrt 2 X\sin\left(\frac{2\pi}{N}k + \phi\right)\tag{3}$$

The Discrete Fourier Transform of $\left\{x_k\right\}$ contains a fundamental frequency component given by

$$\color{red}{\boxed{\color{black}{\bar X_1= \frac 2N \sum_{k=0}^N x_k e^{-j\frac{2\pi}{N}k}}}}\tag{4}\\$$

• where did this text come from?? 1980 or before? IBM Selectric Typewriters with exchangeable balls? – robert bristow-johnson May 24 '16 at 5:13
• and, in my Electrical Engineering training and experience, Eqs (1) and (2) are incompatible. if $$x(t) = \sqrt{2} X \cos(\omega t + \phi)$$ (note that it's "$\cos()$", not "$\sin()$") then the phasor is $$\bar{X} = X \ e^{j \phi} = X \cos(\phi) + j X \sin(\phi)$$ and $X \ge 0$ is the r.m.s. magnitude of the sinusoid. – robert bristow-johnson May 24 '16 at 5:20
• to answer your question (once the premises are corrected), you first express what $\omega$ is (given that it's a "60 Hz waveform"). then compute what the sampling instances are (values of $t$) given that one cycle of your 60 Hz waveform is sampled with $N$ samples. since this is one cycle going into the DFT, the complex amplitude of the fundamental is in $X_1$ and $X_{N-1}$ of the DFT output (using the crappy subscript notation from pre-1980). and the "2" does not belong. – robert bristow-johnson May 24 '16 at 5:26
• In that time there wasnt a clear phasor representation. – Felipe May 24 '16 at 14:03
• I think there may be a way to get this 2 on equation (4), there is a lot of material using this. Like this one: nptel.ac.in/courses/108101039/download/Lecture-35.pdf – Felipe May 24 '16 at 14:04