Chapter # 3 Data and Signals

1. Chapter # 3Data and Signals

2. Introduction • One of the major functions of physical layer is to move data in the form of electromagnetic signals across a transmission medium. • Thus, the data must be transformed to electromagnetic signals to be transmitted.

3. Analog and Digital Data • Data can be analog or digital. • The term analog refers to information that is continuous • e.g. analog clock hh:mm:ss • Digital data refers to information that has discrete states. • e.g. digital clock hh:mm • Analog data take on continuous values. • Digital data take on discrete values.

4. Analog and Digital Signals • An analog signal has infinitely many levels of intensity over a period of time. • A digital signal can have only a limited number of defined values.

5. Analog and Digital Signals

6. Periodic and Nonperiodic Signals • Both analog and digital signals can take one of two forms: periodic or nonperiodic • A periodic signal completes a pattern within a measureable time frame. And repeats that pattern over subsequent identical period. • A nonperiodic signal changes without exhibiting a pattern or cycle that repeats over time.

7. Periodic and Nonperiodic Signals • In data communications, we commonly use periodic analog signals ( because they need less bandwidth). • and nonperiodic digital signals ( because they can represent variation in data)

8. Periodic Analog Signals • Periodic analog signals can be classified as simple or composite. • A simple periodic analog signal, a sine wave, cannot be decomposed into simpler signals. • A composite periodic analog signal is composed of multiple sine waves.

9. Sine Waves • The sine wave is the most fundamental form of a periodic analog signal. • A sine wave is represented by three parameters: Peak amplitude, Frequency, and Phase. • Peak amplitude: it is the absolute value of the highest intensity.

10. Sine Waves The figure below show Two signals with the same phase and frequency, but different amplitudes

11. Sine Waves • Frequency: it refers to the number of periods in 1 s. It is formally expressed in Hertz (Hz). • Period is the amount of time, in seconds, a signal needs to complete one cycle (the completion of one full pattern). • Therefore , frequency and period are the inverse of each other.

12. Two signals with the same amplitude and phase, but different frequencies

13. Units of period and frequency

14. Example#3 • The power we use at home has a frequency of 60 Hz. The period of this sine wave can be determined as follows:

15. Example#4 • Express a period of 100 ms in microseconds. • Solution • From Table 3.1 we find the equivalents of 1 ms (1 ms is 10−3 s) and 1 s (1 s is 106μs). We make the following substitutions:.

16. Example#5 • The period of a signal is 100 ms. What is its frequency in kilohertz? • Solution • First we change 100 ms to seconds, and then we calculate the frequency from the period (1 Hz = 10−3 kHz).

17. Notes in Frequency • Frequency is the rate of change with respect to time. • Change in a short span of time means high frequency. • Change over a long span of time means low frequency. • If a signal does not change at all, its frequency is zero. This is because the signal will never change then it will never complete a cycle, thus the frequency is zero. • If a signal changes instantaneously, its frequency is infinite. This is because there is no time it is jump from one level to another in no time. • t=0,  f= 1/0= infinite.

18. Sine Wave: Phase • Phase: • It describes the position of the waveform relative to time 0. • It is measured in degree or radian ( • To look to the phase is in term of shift or offsit: • A sine wave with a phase 0° is not shifted. • A sine wave with a phase 90° is shifted to the left by ¼ cycle. • A sine wave with a phase 180° is shifted to the left by ½ cycle.

19. Wavelength • Wavelength binds the period or frequency of the simple sine wave to the propagation speed of the medium. • Wavelength depends on both the frequency and the medium. • Wavlength = propgation speed X period = progation speed/ frequency

20. Wavelength and period

21. Example#7 • In a vacuum, light is propagated with a speed of 3 X 108 m/s. (that speed is lower in air and cable.) . The frequency of red light is 4 X 1014 • Wavelength is normally measured in micrometers. • Wavelwngth= c/f= (3 X 108 ) / (4 X 1014) • = 0.75 X 10-6 m= 0.75 μm

22. Time and Frequency Domain • A complete sine wave in the time domain can be represented by one single spike in the frequency domain.

23. The time-domain and frequency-domain plots of a sine wave

24. Time and Frequency Domain • The frequency domain is more compact and useful when we are dealing with more than one sine wave. • Example#8, shows three sine waves, each with different amplitude and frequency. All can be represented by three spikes in the frequency domain.

25. Example#8

26. Composite Signals • A single-frequency sine wave is not useful in data communications; we need to send a composite signal, a signal made of many simple sine waves. • e.g.if we use single sine wave to convey a conversation over the telephone. It would just hear a buzz.

27. Composite Signals • According to Fourier analysis, any composite signal is a combination of simple sine waves with different frequencies, amplitudes, and phases. • A composite signals can be periodic or non periodic. • A periodic composite signal can be decomposed into a series of simple sine waves with discrete frequencies ( with integer values [1,2,3 ,….] ). • A nonperiodic composite signal can be decomposed into a combination of an infinite number of simple sine waves with continuous frequencies. ( with real value)

28. A composite periodic signal

29. Explanation • The previous figure shows a periodic composite signal with frequency f. This type of signal is not typical of those found in data communications. • We can consider it to be three alarm systems, each with a different frequency. • The analysis of this signal can give us a good understanding of how to decompose signals.

30. Decomposition of a composite periodic signal in the time and frequency domains

31. The time and frequency domains of a nonperiodic signal

32. Explanation • The previous figure shows a nonperiodic composite signal. It can be the signal created by a microphone or a telephone set when a word or two is pronounced. • In this case, the composite signal cannot be periodic, because that implies that we are repeating the same word or words with exactly the same tone.

33. Bandwidth • The bandwidth of a composite signal is the difference between the highest and the lowest frequencies contained in that signal. • e.g. if a composite signal contain frequencies between 1000 and 5000, its bandwidth is 5000-1000 = 4000

34. The bandwidth of periodic and non- periodic composite signals

35. Example#1 • If a periodic signal is decomposed into five sine waves with frequencies of 100, 300, 500, 700, and 900 Hz, what is its bandwidth? Draw the spectrum, assuming all components have a maximum amplitude of 10 V. • Solution • Let fh be the highest frequency, fl the lowest frequency, and B the bandwidth. Then • The spectrum has only five spikes, at 100, 300, 500, 700, and 900 Hz (see next figure).

36. Example#1

37. Example#2 • A periodic 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 frequencies of the same amplitude. • Solution • Let fh be the highest frequency, fl the lowest frequency, and B the bandwidth. Then • The spectrum contains all integer frequencies. We show this by a series of spikes (see next Figure).

38. Example#2

39. Example#3 • A nonperiodic composite signal has a bandwidth of 200 kHz, with a middle frequency of 140 kHz and peak amplitude of 20 V. The two extreme frequencies have an amplitude of 0. Draw the frequency domain of the signal. • Solution • The lowest frequency must be at 40 kHz and the highest at 240 kHz. Figure 3.15 shows the frequency domain and the bandwidth.

40. Example#3

41. Example#4 • Another example of a nonperiodic composite signal is the signal received by an old-fashioned analog black-and-white TV. • A TV screen is made up of pixels. If we assume a resolution of 525 × 700, we have 367,500 pixels per screen. If we scan the screen 30 times per second, this is 367,500 × 30 = 11,025,000 pixels per second. • The worst-case scenario is alternating black and white pixels. We can send 2 pixels per cycle. • Therefore, we need 11,025,000 / 2 = 5,512,500 cycles per second, or Hz. The bandwidth needed is 5.5125 MHz.

42. 3-3 DIGITAL SIGNALS In addition to being represented by an analog signal, information can also be represented by a digital signal. For example, a 1 can be encoded as a positive voltage and a 0 as zero voltage. A digital signal can have more than two levels. In this case, we can send more than 1 bit for each level.

43. Figure 3.16 Two digital signals: one with two signal levels and the other with four signal levels

44. Example 3.16 A digital signal has eight levels. How many bits are needed per level? We calculate the number of bits from the formula Each signal level is represented by 3 bits.

45. Example 3.17 • A digital signal has nine levels. How many bits are needed per level? • We calculate the number of bits by using the formula: • Log2 L= number of bits in each level • Log2(9)=3.17bits. • However, this answer is not realistic. The number of bits sent per level needs to be an integer as well as a power of 2. • For this example, 4bits can represent one level.

46. Bit rate and bit interval Most digital signals are nonperiodic, frequency and period are not appropriate. Another terms instead of frequencyisbit rate and instead of period: bit interval(bit duration) Bit rate: number of bits per second bps Bit interval=1/bit rate

47. Example 3.18 • Assume we need to download text documents at the rate of 100 pages per minute. What is the required bit rate of the channel? • Solution • A page is an average of 24 lines with 80 characters in each line. If we assume that one character requires 8bits • The bit rate is: • =100x24x80x8/60 • =25.6Kbps

48. Note A digital signal is a composite analog signal with an infinite bandwidth.

49. Digital Signal as a composite Analog Signal • Fourier analysis can be used to decompose a digital signal • If the digital signal is periodic (rare in data communications), the decomposed signal has a frequency domain representation with an infinite Bandwidth and discrete frequencies. • If it is nonperiodic, the decomposed signal still has infinite Bandwidth, but the frequencies are continuous.

50. Figure 3.17 The time and frequency domains of periodic and nonperiodic digital signals