I am studying the discrete Fourier transform, and in its most basic definition it is an invertible linear transformation on the complex numbers. From Wikipedia:
The sequence of $N$ complex numbers $x_0, x_1, ..., x_{N-1}$ is transformed into an $N$-periodic sequence of complex numbers:
$$ X_k = \sum_{n=0}^{N-1} x_n * e^{-i2\pi kn/N}, k \in \mathbb Z $$
I believe that I understand what this transformation does (it produces the "power" of the signal at each frequency from $0 Hz$ to $F_s/2 Hz$ (where $F_s$ is the sample frequency), and I can easily perform the required arithmetic. However, I am having some trouble understanding the representation of real samples as complex samples.
A typical signal I might wish to transform can be given by a function $f(t) = a$, where $t$ is some time offset and $a$ is the amplitude of the wave at that particular point in time (say it's in the range $[-1, 1]$ for simplicity's sake).
It is my understanding that I can represent this real sampled signal with $N$ samples as a complex sample with $N/2$ samples. I believe that this can be done with the Hilbert transform?
I'm hoping for some help understanding how a real signal can be transformed into a complex signal with half as many samples, and what those complex samples really represent (it's easy to understand a series of amplitude values, but I don't yet understand what the complex values "mean").
The DFT as you stated it is defined for arbitrary complex-valued sequences $x_n$. If $x_n$ is real-valued then its DFT satisfies the following symmetry condition:
$$X_k=X^*_{N-k},\quad k=0,1,\ldots,N-1$$
This means that, if $N$ is even, the values $X_k$, $k=0,1,\ldots,N/2$ are sufficient to completely represent the sequence $x_n$. So we have $N/2+1$ independent values $X_k$. However, note that if $x_n$ is real-valued, $X_0$ and $X_{N/2}$ are real-valued too, so you have two real-valued and $N/2-1$ complex valued coefficients $X_k$, which in total of course equals exactly $N$ independent (real-valued) numbers to represent the sequence $x_n$, just as expected.
In order to understand what the coefficients $X_k$ mean, it is helpful to look at the inverse transform:
$$x_n=\frac{1}{N}\sum_{k=0}^{N-1}X_ke^{i2\pi kn/N}$$
So $|X_k|$ determines the amplitude and $\arg\{X_k\}$ determines the phase of the contribution to the sequence $x_n$ of the complex exponential with frequency $2\pi k/N$.