Modern communication Theory

Fourier Series
A series proposed by the French mathematician Fourier about the year 1807. The series involves the sines and cosines of whole multiples of a varying angle and is usually written in the following form: y = Ho + A1 sin x + A2 sin 2x + A3 sin 3x + ... B1 cos x + B2 cos 2x + B3 cos 3x + ... By taking a sufficient number of terms the series may be made to represent any periodic function of x
A periodic function is one that repeats itself over time. Fourier proved that any "reasonably behaved" periodic function could be written as a sum of sine and cosine functions. This is important because sine and cosine are easily represented and recreated. The Fourier series allows periodic signals to be sent over a wire

Shannon’s Theorem
The Shannon theorem states that given a noisy channel with information capacity C and information transmitted at a rate R, then if R <> C, an arbitrarily small probability of error is not achievable. So, information cannot be guaranteed to be transmitted reliably across a channel at rates beyond the channel capacity. The theorem does not address the rare situation in which rate and capacity are equal.
Simple schemes such as "send the message 3 times and use at best 2 out of 3 voting scheme if the copies differ" are inefficient error-correction methods, unable to asymptotically guarantee that a block of data can be communicated free of error. Advanced techniques such as Reed-Solomon codes and, more recently, Turbo codes come much closer to reaching the theoretical Shannon limit, but at a cost of high computational complexity. With Turbo codes and the computing power in today's digital signal processors, it is now possible to reach within 1/10 of one decibel of the Shannon limit

Nyquist–Shannon sampling theorem
A signal that is bandlimited is constrained in terms of how fast it can change and therefore how much detail it can convey in between discrete moments of time. The sampling theorem means that the discrete samples are a complete representation of the signal if the bandwidth is less than half the sampling rate, which is referred to as the Nyquist frequency. Frequency components that are at or above the Nyquist frequency are subject to a phenomenon called aliasing, which is undesirable in most applications. The severity of the problem depends on the relative strength of the aliased components.