Bandwidth Estimation of a Signal using Fourier Analysis

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.

Time and Frequency domain representations for an analog electromagnetic signal

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

Fourier analysis states that 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.
Assuming x(t) to be the time domain representation function of a signal and X(f) to be the frequency domain representation function of the same signal, Fourier gave the following formulae  of deriving one from the other.
Fourier transformations between time and frequency domains for an A-periodic signal

Fourier transformations between time and frequency domains for an A-periodic signal

The above formulaes help us to find out the frequency components of a signal as a function of frequency (X(f)), given its time domain function x(t) and vice versa.
For computer communication, wherever analog signalling is used (e.g. ASK, PSK etc.), the above formulae 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
Fourier analysis states that if a signal is periodic, then it can be represented as a weighted sum of sinusoids & cosines consisting of a 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.
The Fourier transformation for periodic signals states that any periodic function, 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:
Fourier transformation for finding out the frequency components of a periodic signal

Fourier transformation for finding out the frequency components of a periodic signal

The above formulae can be used in digital transmission to find out the frequency components of digital signals.
For example, consider the periodic square wave, with period “T” and amplitude “A”, given in the figure below:
A periodic square wave with period "T" and amplitude "A", representing the digital pattern 10101010....

A periodic square wave with period “T” and amplitude “A”, representing the digital pattern 10101010….

Assume that it represents the digital pattern 10101010…… . If we apply the Fourier series for this signal, then we get the following infinite series 
squarewaveexample
The digital square wave is actually a summation of sinusoids of this fundamental frequency and its odd harmonics (f, 3f, 5f, 7f etc.)
Since the summation runs infinitely, the  square wave consists of  infinite number of frequency components and hence its bandwidth is infinity.
But if you consider the amplitude of the kth frequency component (kf), then the peak value of its amplitude (or weightage in the overall sum) is proportional to the reciprocal of k (1/k). Thus as k increases, the peak amplitude decreases exponentially. This means that the weightage of higher harmonics is quite negligible, when compared to the first few components.
Infact, if you sum up the first few sinusoidal components (say upto 5 to 10) and plot the resultant wave, it would fairly resemble the periodic square wave.
So for practical purposes, the bandwidth of the signal can be approximated by those initial frequency components, that have major weightage.
In practise too, for digital transmission, considering the frequency constraints imposed by the standards for different types of transmission, the transmitter only transmits a bandwidth limited signal (not infinite bandwidth), that consists of only those initial harmonics which contribute the major weightage to the overall composition of the signal. The number of harmonics included in the transmission should be such that the receiver is able to reconstruct back the original signal in the presence of channel noise.
For example, if the digital signal to be transmitted 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.
Similarly, if the digital signal 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.
From the above examples, it must be clear that digital signals require higher bandwidth in general, than analog signals, because we need not only the fundamental frequency but also a few higher harmonics to reconstruct the signal properly at the receiver. Also, in general, higher frequency components suffer attenuation and lose signal strength over longer distances. For these reasons, digital transmission is generally preferred for shorter distances.
Thus we see that Fourier series helps us in estimating the bandwidth occupied by both analog and digital signals.

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