The Theoretical Basis for Data Communication

The Theoretical Basis for Data Communication: Frequency-domain concept: A sine wave with a frequency of 1 kHz will look like this in the frequency domain: Signal Amplitude, A Frequency, f 1 kHz Fourier Theorem: In the early 19th century, the French mathematician Jean-Baptiste Fourier states that: Any periodic waveform can be expressed as the sum of sine waves with frequencies at integer or harmonic multiples of the fundamental frequency of the waveform and with appropriate maximum amplitudes and phases. A square wave of 1 kHz is composed of a 1 kHz sine wave and its odd multiples (Harmonics) with diminishing amplitudes: A Sin(2p(f1)t) + A/3 sin(2p(3f1)t) + A/5 sin(2p(5f1)t) + A/7 sin(2p(7f1)t) 1 kHz Signal Amplitude, A 7 kHz 5 kHz 3 kHz Frequency, f Reed organ is also called Harmonica, as the reeds are tuned to vibrate at harmonics of the fundamental signal. Addition of Frequency Components: Addition of More Frequency Components: Voice signal: Voice signal is an analog signal with frequency components in the range of 20Hz to 20kHz: Signal Amplitude, A Time, t Voice signal Signal Amplitude, A Frequency, f 1 kHz 3 kHz Spectrum, of a signal is the range of frequencies that it contains. The intelligent voice signal spectrum extends from 300 Hz to 3400 Hz. Bandwidth, of a signal is the width of the spectrum. The bandwidth of the intelligent voice signal is 3100 Hz Telephone equipment allows the voice a bandwidth of 4000Hz, which includes a guard band at top (600Hz) and bottom (300Hz) to prevent interference. This limit on bandwidth is imposed by the telephone and the switching equipment used in the telephone network. Video signal: Video signal is an analog signal with frequency components in the range of 60 Hz to 4 MHz. The bandwidth is almost 4 MHz. With guard bands the standard bandwidth for colour video signaling is 6 MHz. Signal Bandwidth Telephone speech 3.1 kHz Hi-fi stereo 20 kHz AM radio station 10 kHz FM radio station 200 kHz TV station 6 MHz Digital Data Signal: Bit rate, is the number of bits transmitted per second. Baud rate, is the signaling speed. That is the number of times the signal changes its value , e.g., voltage per second. In case of digital transmission through the transmission line similar to telephone line of bandwidth 3.1 kHz the maximum bits can be transmitted is 3.1 kHz x 2 = 6.2 kbps. But in practice we transmit through the telephone lines either tones (1200Hz) of different amplitude or two different tones (1200Hz & 2400Hz) to represent 0 and 1 per unit time of the signal. So, the maximum bit rate is the same as the signal rate, 3.1 kbps. To increase the bit rate different coding scheme has been implemented over time. That is 2, 4, 8 and 16 bits has been transmitted per signal unit. So, the baud rate and the bit rate are no longer the same. Power in Telecommunications and Decibels: Power is the amount of energy per unit time and is unit is Watts. 1 W of electricity is equal to 1 A of electricity flowing through 1 Ohm of resistance. P = I2 R If current is doubled, the power goes up by a factor of 4. In telecommunications, we deal with extremely low power levels such as 10 thousandth of a watt (10 mW). The milliwatt is the unit of measure used for power in telecommunications circuits. Rather than absolute measure of power, we often concerned with the comparison of one power level to another. Such as the output power of an amplifier (RF or AF) is 1000 mW for an input power of 10 mW. We can say this whole this output with one unit, such as the output is 20dB. The relative power of output to input will tell us the gain of the amplifier, Po/PI = 1000mW/10mW = 100. The unit of measure used to compare two power levels is the decibel (dB). A decibel is not an absolute measurement. It is a relative measurement. The decibel level indicates the relationship of one power level to another. The formula for calculating decibel is : dB = 10 log Po/Pi = 10 log 1000mW/10mW = 10 log 100 = 10 x 2 =20 dB tells us the ratio of two power levels, that is it expresses the gain of the system. But some time we want to express the exact output power of a system rather than the gain. In that case, we compare the power levels with a known reference, such as 1mW. So, the output power will be 30 dBm (10 log 1000mW/1mW = 10 x log 1000 = 10 x 3). Notice the addition of a small "m" after the "dB", that says the reference is 1 mW. Decibels referenced to 1 mW provides a convenient measure for measuring the signal levels in telecommunications because most signal levels in the telecommunications system fall within +10 and -10 dBM. i.e., 10 mW to 100 microW. Signal-to-Noise Ratio: The term Signal-to-Noise ratio is used to express in decibels how much higher in level a signal is to the noise on the circuit. In telecommunications, a Signal-to-Noise ratio of 30 dB or more is satisfactory. Signal-to-Noise ratio = signal level in dBm - noise level in dBm So, for a 10 dBm signal (10 mW) the noise level has to be less than -20 dBm (10 microW). Shanon Theorem: Mathematical guidelines have been established to determine the maximum theoretical data transfer over a channel based on the channel's bandwidth and Signal-to-Noise ratio. This is known as the channel capacity (maximum number of bits per sec). One of the most fundamental laws used in telecommunications is Shanon's law. In 1948 Shanon proved that the maximum data rate of a noisy channel whose bandwidth is B Hz, and whose signal-to-noise ratio is S/N, is given by Channel Capacity = B log2 (1+S/N) bps For a bandwidth of 3.1 kHz and a signal to noise ratio of 30 dB (a ratio of 1000/1), the maximum data rate is 31 kbps. Channel Capacity = B log2 (1+S/N) bps = 3100 log2 (1+1000) bps = 3100 log2 (1001) bps = 3100 log2(29.967) bps = 3100 x 9.967 bps = 30,898 bps = 31 kbps Todays Modems reached that limit of data rate on a old telephone line (it is not the wire itself, it includes the circuitry at the local telephone offices).