After reading this article you will learn about the use of mathematics for the improvement in the methods of communication.
Technical Limitations of Communications:
The rapid developments in Information Technology have primarily been possible because of the introduction of networking which made it possible for organisations to move away from the concept of a sterile “computer centre” (that gave the impression of being highly sophisticated and thereby beyond the reach of the average employee), to putting a computer on the desk of each employee.
This helped in making the computer a more commonplace equipment by driving out the fear of computers from the minds of most employees. This development was largely possible because of the developments in networking. The developments in networking, in turn, were possible because of the improvements in the methods of communication.
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Mathematics for Data Communication:
For a mathematical analysis of transmission, the first necessity is to find out some means whereby, the transmitted signal can be coded and expressed in mathematical terms. In other words, a mathematical model must be formulated for the analysis of data communications. In order to do this, some characteristic of the transmitted signal, whether it is voltage or current, has to be given a mathematical value.
The behaviour of this mathematical value, when represented as a single-valued function of time, can then be analysed. It can be treated as a signal, whose characteristics are periodic over time and depending on the changes of the subject it represents.
By giving it a mathematical basis, we can model the transmission, the subject in question, and also analyse this model. If the function is single-valued, then the analysis becomes much easier. Fortunately, there exists a mathematical method of analysis, known as the Fourier analysis. We shall look at some of the issues in such an analysis.
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Fourier Series:
The 19th century French mathematician Jean-Baptiste Fourier analysed periodic functions and proved that any periodic function can be expressed as a combination of a series of sine- and cosine-based functions. This issue is treated below in an “engineering way”, that is in a way that is understandable in spite of the mathematics that is, occasionally, somewhat difficult to understand.
However, before we come to see how Fourier series is utilised, its mathematics may be of some interest. In particular, if the function f(t) with a time period T is expressed mathematically then its representation, in terms of a series of harmonics, would be
Such a method of describing a periodic function is usually called a Fourier Series of the periodic function. In the above mathematical representation, an and bn are amplitudes of the nth harmonic and f = 1 /T, is the frequency.
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The above equation, therefore, breaks down each periodic function into all its possible harmonics (theoretically infinite, as the symbol ∞ above the summation sign ∑indicates) and expresses the wave shape in the form of the sum of its harmonics. The amplitudes can be obtained by some simple mathematical jugglery.
Multiply both sides of the above equation by a factor of sin (2r k f t) and then integrate them over the interval 0 to T. Thus:
Only one term remains after the integration, viz. an. The bn amplitude, which is associated with the cosine function, disappears as a result of this integration. Similarly, we can obtain the value of bn by multiplying both sides of the above equation by cos(27pkft) and by integrating both sides of the equation, the value of c can be obtained.
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Doing this we get the following values for an, bn and c:
Fast Fourier Transforms:
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If X is a complex-valued Lebesgue integrable function, then the fourier transform to the frequency domain w, is given by the function
for every real number w, when the independent variable t represents time in seconds and the transform variable w represents angular frequency in radians per second.
Fast Fourier Transform—sometimes referred to by its initials FFT—is a discrete Fourier Transform algorithm where, for N points required for a reasonable transform the number of computations required to obtain these N points reduces from 2N2 to 2N log2 N, where log2 N is logarithm to base 2.
Fast Fourier Transform algorithms generally fall into 2 classes:
Decimation in time or decimation in frequency. Although Gauss had described the factorization steps required as early as in 1805, they were not significantly discussed till 1965 by Cooley and Turkey.
However, a discussion on Fast Fourier Transforms becomes somewhat of an academic exercise, since they are likely to be used very rarely in design in data communications. One last mention about Lebesgue functions. These functions are largely used and are important in discrete computations particularly in probability computations.
Bandwidth Limitations:
To consider how Fourier Analysis helps in analysing data transmission, let us consider the wave shape of the binary signal being transmitted.
Since the signal is binary, it could have the wave shape given in the following figure:
The above wave shape represents an ASCII character represented by 01100011. Fourier analysis of this wave shape will give us some information. The coefficients an, bn and c can be computed to be
The root-mean-square amplitude is √ (an2 + bn2). Of course, the smaller terms have been omitted. A point of interest about these values is the fact that their squares represent the energy transmitted at that frequency. However, in transmission, some energy is invariably lost.
If all the components in the Fourier analysis above are diminished in the same proportion, there would be no problem. The power associated with the transmission is related to the frequency of transmission and therefore, the power loss is also naturally associated with the frequency of transmission. Therefore, the higher frequency harmonics will tend to be reduced more than the lower frequency components.
This will obviously change the wave shape. Usually all frequencies transmitted up to some cut-off frequency fc are transmitted undiminished (or diminished by trivial amounts). Usually this is a physical characteristic of the transmission medium. Suppose that only the first harmonic was being transmitted.
Then what the transmitted wave shape would look like is shown in the following figure:
Similarly, if only the first two harmonics were being transmitted, the wave shape would look like the following figure:
If the first four harmonics are being transmitted then the wave shape would be represented by the following figure:
Finally, let us look at the wave shape of the transmitted wave if the first eight harmonics are transmitted:
It should be obvious from the above figures that the wave shape of the transmitted signal changes as the bandwidth of the carrier increases. The time required to transmit the signal—in the above case a character—depends upon both, the encoding method and the signal transmission speed.
The transmission speed indicates the frequency, or the number of times per second that the signal changes its value. And since the signal is shown as the transmitted voltage, this is the entity whose value changes with time.
The unit used for the speed of transmission of characters is baud. A b baud line, however, does not transmit 6 bits per second, since a signal might convey more than 1 bit at a time. For the sake of argument, if each signal level was used to convey three bits—possible if various voltage levels are used to convey signals—then the baud rate would be one-third of the bit rate.
However, since we are considering only digital signals representing only O’s and l’s, obviously the baud rate is equal to the bit rate. Now given a bit rate of say 6, b bits are being sent per second. Therefore, the time to send 8 bits is 8/6 seconds and the frequency, which is the reciprocal of this rate is, therefore, 6/8 Hz.
In a voice grade telephone, there is a frequency cut off introduced at about 3000 Hz. Therefore, the number of the highest harmonic passed through is 24000/6 (or 3000/b/8). Trying to send 9600 bps on a voice grade line will cause the first two harmonics to be sent, resulting in the wave shape shown in Fig.2.3 above.
Obviously, this will make accurate reception extremely difficult. Beyond 38.4 kbps, by the same token, sending digital signals using single voltage levels becomes impossible, even if the transmission facility is completely noiseless.
However, specialised facilities that use complicated coding schemes by using several voltage levels do exist and can transmit higher frequencies. This brings us to the concept of maximum data rate for a channel.
In 1924, Nyquist proved that there is a finite limit to the maximum data rate for a finite bandwidth noiseless channel and derived an equation to predict this. In 1948, Claude Shannon extended this work to include channels subject to random noise.
If an arbitrary signal has been run through a low-pass filter of bandwidth H then the signal can be reconstructed exactly by taking 2H samples per second. Sampling at higher rates is useless because the higher frequencies will have been filtered out. If the signal consists of N discrete levels, then according to Nyquist’s theorem
Thus, a noiseless 3 kHz channel cannot transmit binary signals in excess of a rate of 6000 bps. However, it is virtually impossible to get an absolutely, noiseless channel.
In normal cases, the amount of noise present is measured by the signal to noise ratio. This ratio is expressed in the form of the logarithm to base 10 of the ratio S/N, where S is the power generating the signal and N that of the noise. This ratio, expressed in decibels, is defined by
Shannon’s modification to Nyquist’s theorem gives the maximum data rate of any channel whose signal to noise ratio is S/N, is given by
For example, a 3-kHz bandwidth with a signal to noise ratio of 30 dB cannot transmit beyond about 30,000 bps irrespective of the number of harmonics transmitted or the sampling interval. It should be remembered that this frequency is the theoretical limit that defines the ultimate achievable target. In actual practice, however, this limit is extremely difficult to achieve or even come close to.