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Chi-Cheng Lin, Winona State University CS412 Introduction to Computer Networking & Telecommunication Theoretical Basis of Data Communication

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2 Topics l Analog/Digital Signals l Time and Frequency Domains l Bandwidth and Channel Capacity l Data Communication Measurements

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3 Signals l Information must be transformed into electromagnetic signals to be transmitted l Signal forms Analog or digital Periodic or aperiodic

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4 Analog/Digital Signals l Analog signal Continuous waveform Can have a infinite number of values in a range l Digital signal Discrete Can have only a limited number of values E.g., 0 or 1

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5 Figure 3.1 Comparison of analog and digital signals

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6 Periodic/Aperiodic Signals l Periodical signal Contains continuously repeated pattern Period (T): amount of time needed for the pattern to complete l Aperiodical signal Contains no repetitive signals

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7 Analog Signals l Simple analog signal Sine wave 3 characteristics 1. Peak amplitude (A) 2. Frequency (f) 3. Phase ( ) l Composite analog signal Composed of multiple sine waves

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8 Figure 3.2 A sine wave

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9 Figure 3.3 Amplitude t s(t): instantaneous amplitude

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10 Characteristics of Analog Signal l Peak amplitude: highest intensity l Frequency (f) Number of cycles/rate of change per second Measured in Hertz (Hz), KHz, MHz, GHz, … Period (T): amount of time it takes to complete one cycle f = 1/T l Phase: position of the waveform relative to time 0

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11 Figure 3.4 Period and frequency

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12 Table 3.1 Units of periods and frequencies UnitEquivalentUnitEquivalent Seconds (s)1 shertz (Hz)1 Hz Milliseconds (ms)10 –3 skilohertz (KHz)10 3 Hz Microseconds (ms)10 –6 smegahertz (MHz)10 6 Hz Nanoseconds (ns)10 –9 sgigahertz (GHz)10 9 Hz Picoseconds (ps)10 –12 sterahertz (THz)10 12 Hz

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13 Figure 3.5 Relationships between different phases

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14 Figure 3.6 Sine wave examples

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15 Figure 3.6 Sine wave examples (continued)

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16 Figure 3.6 Sine wave examples (continued)

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17 Characteristics of Analog Signal l Changes in the three characteristics provides the basis for telecommunication Used by modems (later …)

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18 Time Vs. Frequency Domain l The sine waves shown previously are plotted in its time domain. l An analog signal is best represented in the frequency domain.

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19 Figure 3.7 Time and frequency domains

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20 Composite Signals l A composite signal can be decomposed into component sine waves - harmonics l The decomposition is performed by Fourier Analysis

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Figure 4-13 WCB/McGraw-Hill The McGraw-Hill Companies, Inc., 1998 Signal with DC Component

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22 Figure Square wave and the first three harmonics

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23 Figure 3.11 Frequency spectrum comparison

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24 Frequency Spectrum and Bandwidth l Frequency spectrum Collection of all component frequencies it contains l Bandwidth Width of frequency spectrum

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25 Figure 3.13 Bandwidth

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26 Example 3 If a periodic signal is decomposed into five sine waves with frequencies of 100, 300, 500, 700, and 900 Hz, what is the bandwidth? Draw the spectrum, assuming all components have a maximum amplitude of 10 V. Solution B = f h f l = 900 100 = 800 Hz The spectrum has only five spikes, at 100, 300, 500, 700, and 900 (see Figure 13.4 )

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27 Figure 3.14 Example 3

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28 Example 4 A signal has a bandwidth of 20 Hz. The highest frequency is 60 Hz. What is the lowest frequency? Draw the spectrum if the signal contains all integral frequencies of the same amplitude. Solution B = f h f l 20 = 60 f l f l = 60 20 = 40 Hz

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29 Figure 3.15 Example 4

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30 Example 5 A signal has a spectrum with frequencies between 1000 and 2000 Hz (bandwidth of 1000 Hz). A medium can pass frequencies from 3000 to 4000 Hz (a bandwidth of 1000 Hz). Can this signal faithfully pass through this medium? Solution The answer is definitely no. Although the signal can have the same bandwidth (1000 Hz), the range does not overlap. The medium can only pass the frequencies between 3000 and 4000 Hz; the signal is totally lost.

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31 Digital Signals l 0s and 1s l Bit interval and bit rate Bit interval: time required to send 1 bit Bit rate: #bit intervals in one second

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32 Example 6 A digital signal has a bit rate of 2000 bps. What is the duration of each bit (bit interval) Solution The bit interval is the inverse of the bit rate. Bit interval = 1/ 2000 s = s = x 10 6 s = 500 s

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33 Digital Signal - Decomposition l A digital signal can be decomposed into an infinite number of simple sine waves (harmonics), each with a different amplitude, frequency, and phase A digital signal is a composite signal with an infinite bandwidth. A digital signal is a composite signal with an infinite bandwidth. l Significant spectrum Components required to reconstruct the digital signal

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Figure 4-20 WCB/McGraw-Hill The McGraw-Hill Companies, Inc., 1998 Harmonics of a Digital Signal

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35 Bandwidth-Limited Signals l (a) A binary signal and its root-mean- square Fourier amplitudes.

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36 Bandwidth-Limited Signals (2) l (b) – (e) Successive approximations to the original signal.

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Figure 4-21 WCB/McGraw-Hill The McGraw-Hill Companies, Inc., 1998 Exact and Significant Spectrums

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38 Channel Capacity l Channel capacity Max. bit rate a transmission medium can transfer l Nyquist theorem C = 2H log 2 V where C: channel capacity (bit per second) H: bandwidth (Hz) V: signal levels (2 for binary) C is proportional to H Significant bandwidth puts a limit on channel capacity

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39 Figure 3.18 Digital versus analog To transmit 6bps, we need a bandwidth = = 3Hz

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40 Channel Capacity l Nyquist theorem is for noiseless (error- free) channels. l Shannon Capacity C = H log 2 (1 + S/N) where C: (noisy) channel capacity (bps) H: bandwidth (Hz) S/N: signal-to-noise ratio dB = 10 log 10 S/N l In practice, we have to apply both for determining the channel capacity.

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41 Example 7 Consider a noiseless channel with a bandwidth of 3000 Hz transmitting a signal with two signal levels. The maximum bit rate can be calculated as Bit Rate = 2 3000 log 2 2 = 6000 bps Example 8 Consider the same noiseless channel, transmitting a signal with four signal levels (for each level, we send two bits). The maximum bit rate can be calculated as: Bit Rate = 2 x 3000 x log 2 4 = 12,000 bps Bit Rate = 2 x 3000 x log 2 4 = 12,000 bps

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42 Example 9 Consider an extremely noisy channel in which the value of the signal-to-noise ratio is almost zero. In other words, the noise is so strong that the signal is faint. For this channel the capacity is calculated as C = B log 2 (1 + S/N) = B log 2 (1 + 0) = B log 2 (1) = B 0 = 0

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43 Example 10 We can calculate the theoretical highest bit rate of a regular telephone line. A telephone line normally has a bandwidth of 3000 Hz (300 Hz to 3300 Hz). The signal- to-noise ratio is usually 35dB, i.e., For this channel the capacity is calculated as C = B log 2 (1 + S/N) = 3000 log 2 ( ) = 3000 log 2 (3163) C = 3000 = 34,860 bps

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44 Example 11 We have a channel with a 1 MHz bandwidth. The S/N for this channel is 63; what is the appropriate bit rate and signal level? Solution C = B log 2 (1 + S/N) = 10 6 log 2 (1 + 63) = 10 6 log 2 (64) = 6 Mbps Then we use the Nyquist formula to find the number of signal levels. 4 Mbps = 2 1 MHz log 2 L L = 4 First, we use the Shannon formula to find our upper limit.

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45 Data Communication Measurements l Throughput How fast data can pass through an entity l Propagation speed Depends on medium and signal frequency l Propagation time (propagation delay) Time required for one bit to travel from one point to another l Wavelength Propagation speed = wavelength X frequency

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46 Figure 3.25 Throughput

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47 Figure 3.26 Propagation time

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48 Figure 3.27 Wavelength

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