The successful transmission of data depends principally on two factors: the quality of the signal being transmitted; and the characteristics of the transmission medium. We must consider the way in which data are transmitted. For example, computers transmit data using digital signals, in a sequence of specified voltage levels. Graphically, they are often represented as a square wave shown in Figure. The horizontal axis represents time and the vertical axis represents the voltage level.

Computers sometimes communicate over telephone lines using analog signals, which are formed by continuously varying voltage levels. These signals are most often represented by their characteristic sine wave shown in Figure.

Analog signals add complexity to data communications. One problem is that digital computers are incompatible with analog transmission media. Because much of the telephone system is analog, and analog is the major medium for computer communications, this incompatibility must be resolved. That is, we need a device that converts a digital signal to an analog one (modulation) and another that converts an analog signal to digital (demodulation). A modem (short form for modulation/ demodulation) does both. This chapter provides an important theoretical foundation for analog signals.
Information can be transmitted on wires by varying some physical properties such as voltage or current. By representing the value of this voltage or current as a single­valued function of time, f(t), we can model the behaviour of the signal and analyze it mathematically.

 FOURIER ANALYSIS
In the early 19th century, a famous mathematician, Jean Baptiste Fourier, developed a theory that any periodic function can be expressed as an infinite series of sums of sine and cosine functions of varying amplitude, frequency, and phase shift.

This series is called Fourier series.
If F(t) is a periodic function with period T, it may be expressed as:

where f = 1/T is called the fundamental frequency. The number of times a signal oscillates per unit time is called its frequency. Its units of measurement are cycle per second or equivalently, Hertz (Hz). an and bnare the sine and cosine amplitudes of the nth harmonies (terms), (amplitude defines the values between which signal oscillates). A decomposition as above is called a Fourier series.
From the Fourier series, the function can be reconstructed, that is, if the period, T, is known and amplitudes are given, the original function of time can be found by performing the sum of Eq. (2.1).
A data signal that has a finite duration (which all of them do) can be handled by just imagining that it repeats the entire pattern over and over forever (i.e. the interval from T to 2T is the same as 0 to T etc.).
The amplitude can be computed for any given F(t) by multiplying both sides of Eq. (2.1) by sin (21ckft) and then integrating from 0 to T. Since

only one term of summation survives: a,t. The b„ summation vanishes completely.
Similarly, by multiplying Eq. (2.1), by cos (2nkft) and integrating between 0 to T, we can derive bn. By integrating both sides of the equation as it stands, c can be found. The results of performing these operations are as follows:

Let us examine the Fourier series that represent bandwidth-limited signals.

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