This post describes the important role of Fourier analysis in data communication.

In general, electromagnetic signals can be represented either as a function of time (t) or as a function of frequency (f), as shown in the diagram below.

But the frequency perspective of a signal plays a much more significant role in communication than the time perspective. This is because the whole electromagnetic spectrum is split into different frequency ranges and each of these frequency ranges are used for different applications like broadcast (radio, TV), data communication etc.

Within data communication itself, different media use different frequency spectrum. For e.g. in wired media, while copper and coaxial cables use the spectrum upto 100 MHz (10ˆ8Hz), fiber optical communication uses electromagnetic signals of a much higher frequency range (10ˆ15Hz). Similarly in wireless communication, while 802.11b and 802.11g use 2.4Ghz, 802.11a uses 5Ghz.

**The beauty lies in the fact that all these signals spanning different frequency spectrum, can coexist simultaneously in the time domain, thereby enabling us to simultaneously use different forms of communication and entertainment. That means that at any instant of time, you would find electromagnetic signals of a wide range of frequencies around us.**

Thus it is extremelly important to understand the frequency spectrum of an electromagnetic signal. It is here that Fourier series comes in handy, as it helps us to decompose a signal into its frequency components, thereby enabling us to estimate the bandwidth occupied by any electromagnetic signal.

**Decomposition of Electromagnetic signals in the frequency domain**

Any electromagnetic signal, whether it is analog or digital, is generally composed of a range of different frequencies, with each frequency component having a specific weightage in the overall value of the signal, at any instant of time. Given the time domain representation of a signal, **Fourier Analysis helps us in finding the different frequency components of the signal, **along with their respective weightages**. **Fouries analysis **also helps in the reverse process**, namely, if the frequency components of an electromagnetic signal are known, along with their weights, then it enables us to get the time domain representation of the signal.

**Fourier Analysis for an A-periodic signal**

**any electromagnetic signal can be represented as a weighted sum of sinusoids and cosines of various frequencies**. In simple terms, this means that any signal (whether periodic or not periodic) can be constructed by adding a series of sines and cosines of different frequencies.

**can be used to find out the frequency components (and hence the bandwidth) of the analog signal that is to be transmitted**.

**Fourier Analysis for a Periodic signal**

**fundamental frequency (f) and its harmonics**(2f, 3f, 4f etc.)

**alone**. The main difference between the A-periodic and periodic case is that the A-periodic signal typically has frequency components of varying values (not necessarily harmonics of a fundamental frequency), whereas a periodic signal

**only has frequency components that are multiples of a single fundamental frequency**.

*g*(

*t*), with period

*T*can be constructed by summing a (possibly infinite) number of sines and cosines, of a fundamental frequency (f = 1/T) and its harmonics. The actual formulae is given below:

**The digital square wave is actually a summation of sinusoids of this fundamental frequency and its odd harmonics (f, 3f, 5f, 7f etc.)**

**the square wave consists of infinite number of frequency components and hence its bandwidth is infinity.**

**This means that the weightage of higher harmonics is quite negligible, when compared to the first few components.**

**t would fairly resemble the periodic square wave**.

**is approximated by the fundamental frequency f and the first 2 harmonics**, namely, 3f and 5f,

**then the bandwidth occupied by the signal is (5f -f) = 4f**.

**is approximated by the fundamental frequency f and only one additional harmonic,**namely 3f,

**then the bandwidth occupied by the signal is (3f – f) = 2f**.