1. 1 Chapter 3 Physical Layer: Digital Transmission Fundamentals

2. Objective Given the characteristics of the communication channel and system resources, what is required to achieve the required performance? r Performance: probability of transmission error (bit error rate) r Resource: signal power (energy), bandwidth 2

3. Digital Representation of Information 3 Stream: ex. Voice A continuous-time varying signal Two steps: sampling, quantizing

4. Digital Representation of Information 4 Stream: ex. Voice Sampling: Highest frequency W=4k Hz: Min. sample rate: 2W=8 kHz

5. Why Digital Communication? 5 The resistance to signal distortion r Distortion: unexpected change of signal during transmission due to imperfect channel r Analog: cannot resist distortion, degraded performance, quality r Digital: can correct the signal as long as the distortion is not severe enough to change the signal polarity (e.g., from 0 to 1) Many others: suitable for long-distance transmission, handling many types of services, various protocol (error correction, data encryption, etc.),…

6. Block diagram of digital transmissions 6 source (digital) source encoder channel encoder modulator r eliminate redundancy in the data, send same information in fewer bits r compression ratio: #bits (original file)/#bits (compressed file) r lossless compression: Zip (in windows) r lossy compression: JPEG (compression ratio of about 15), MPEG (e.g., 300Mbps to 6Mbps)

7. Block diagram of digital transmissions 7 source (digital) source encoder channel encoder modulator r Transforming signals to improve communications performance by increasing the robustness against channel impairments (noise, interference, fading, …) r add redundancy: map binary source output sequences of k into binary channel input sequences of n (n>k) compression

8. Block diagram of digital transmissions 8 source (digital) source encoder channel encoder modulator compression add redundancy r Converts the electrical signals (from the source) into a form that is suitable for transmission over the physical media m DSL modem, Cable modem, wireless modulator

9. Block diagram of digital transmissions 9 source (digital) source encoder channel encoder modulator demodulator channel decoder source decoder destination repair the impaired signal internet compression add redundancy decompression frequency moving Video/audio/ text

10. r Service: transmitting raw bits (0/1) over wire/physical link between two systems Layer 1: Physical Layer 10 Data Link PHY Receiver Communication channel Data Link PHY Transmitter Transmitter: r Converts information into signal/waveform suitable for transmission Receiver: r Converts received signal into form suitable for delivery to user

11. Communication Channel Data Link PHY Receiver Communication channel Data Link PHY Transmitter Transmitted Signal Received Signal r Twisted pair, Coaxial cable, Radio, Light in optical fiber, etc. r Wireless communication: propagates through the medium as electromagnetic waves Twisted pair Coaxial cable Electromagnetic waves 11

12. Low-pass and Bandpass Channels 12 r Low-pass channel: low frequency signal can pass the channel while high frequency signal is blocked r Bandpass channel: pass the signal with a frequency range [f1,f2] f0f0

13. Attenuation r Attention of a signal is defined as the reduction or loss in signal power as it is transferred over a channel m Cause: signal energy absorbed by media or object m Computed in dB as Channel 13

14. r The channel capacity is the maximum rate at which bits can be reliably transmitted over a channel r Channel noise affects the reliability m Thermal electronic noise (due to the vibrations of electrons) is inevitable 14 Channel Capacity and Noise

15. 15 Physical Layer r Physical Layer: Implements a digital communication link that delivers bits. Suppose that: The duration of Si(t) is T, where T is called the symbol duration and 1/T is called the symbol rate/data rate. r Objective: Given the characteristics of the communication channel, what system resources are needed in order to achieve the required performance? m Performance: probability of transmission error (bit error) m Resource: signal power (energy), bandwidth Figure ： The Stochastic Process

16. 16 Additive White Gaussian Noise (AWGN) r AWGN Channel m The channel has infinite bandwidth.

17. Signal + Noise Ratio 1-17 signal noise signal + noise signal noise signal + noise High SNR Low SNR virtually error-free error-prone

18. 18 Bandwidth Limited Channel r Bandwidth Limited Channel r Inter-symbol Interference (ISI): In band-limited channels, the channel introduces inter-symbol interference, since the transmitted “signal” pulse is “stretched”. r Question: How to remove the ISI effect? T 3T/2 T 5T/2 =

19. 19 Random Signal r Random Process m A random process is a random variable indexed by t. m A random process is a rule for assigned a real-valued time function (or signal) to each outcome S i of an experiment E. For example, take the transmission of two bits: Transmission of Two Bits

20. 20 Random Signal r A random process x(t,s) is m A family of deterministic functions, where t and S are variables m A random variable at t=t 0, X(t 0,S), where S is a variable m A single time function given that S is fixed where t is a time variable: X(t,S 2 ) = X 2 (t) m A real number if both t and S are fixed r Covariance and correlation functions are:

21. 21 Model of Physical Layer Communications r Model r For a random process X(t), with the correlation function defined above, if the correlation function satisfies the following conditions, then the random process is called Wide Sense Stationary (WSS). Block Diagram of Physical Layer

22. 22 Model of Physical Layer Communications r Properties of the Autocorrelation Functions m Proof: m Proof:, R X (τ) is an even function.

23. 23 Model of Physical Layer Communications r Properties of the Autocorrelation Functions (Cont.) m Proof: R X (0) is the average power of the random process.

24. 24 Model of Physical Layer Communications r Wiener-Khintchine (W-K) Relation m The autocorrelation function and the power spectral density are a Fourier Transform pair.

25. 25 White Gaussian Noise r Conditions for White Gaussian Noise: m r Properties m

26. 26 White Gaussian Noise r Properties m Proof:

27. 27 White Gaussian Noise r Properties m Proof (cont.) :

28. 28 Model of Physical Layer Communications r Model i. ii. AWGN Channel Block Diagram of Physical Layer

29. 29 Model of Physical Layer Communications

30. 30 Signal to Noise Ratio r Signal to Noise Ratio It has been proved that in order to maximize the signal-to- noise ratio, the design must follow: h d (t) is matched to the time revered and delayed version of S(t). This implies that the optimum detector is a Matched Filter. This applies to AWGN channel models.

31. 31 Examples 1. 2.

32. 32 Signal Energy r For S 1 (t) = S(t) and S 2 (t) = -S(t) where The noise component is: Signal energy E S is given by: Which is not constant, it is NOT White Gaussian Noise, it is simply Gaussian Noise.

33. 33 Noise Energy r First, review Parseval’s Theorem: (From the Parseval’s Theorem) Then:

34. 34 Average Signal r Calculation of Average Signal Value: Therefore :

35. 35 Decision Threshold r Decision Threshold Here the sampled values: r(T) is a Gaussian random variable. The following notation denotes that r(T) is a Gaussian random variable with a (mean, variance). Let μ = r(T), then: If, then bit “1” was sent If, then bit “0” was sent

36. 36 Error Events An error occurs in the following cases: Since and

37. Shannon Channel Capacity r The maximum reliable transmission rate over an ideal channel with bandwidth W Hz, with Gaussian distributed noise, and with SNR is r Reliable means that the error rate can be made arbitrarily small by using proper coding 37

38. Example r Consider a telephone line with the frequencies from 300Hz to 3400Hz and with a SNR of 35dB. What is the Shannon channel capacity? 38 Since Therefore, bps

39. 39 Physical Layer (general signal) r Model Block Diagram of Physical Layer Transmitter ： bit -> symbol Channels: AWGN, the simplest channel Receiver: demodulator/detector, sampling, decision device

40. Blog Diagram: Transmitter (1/2) Additive White Gaussian Noise (AWGN) channel r Binary signaling: two signals with symbol duration T m “0” → s 0 (t) with probability P(b=0)=0.5 m “1” → s 1 (t) with probability P(b=1)=0.5 r The average signal energy is Receiver 40

41. Blog Diagram: Transmitter (2/2) Additive White Gaussian Noise (AWGN) channel r AWGN channel model: m Channel response: m Fourier transform (Channel of infinite bandwidth) r Received signal convolution 41

42. Block diagram: the Optimal Receiver r How to detect what ‘bit’ has been transmitted from the received signal r(t)? r What is the probability of detection error? r Objective: m Design a detection and decision devices to minimize 42

43. Detector (matched filter) 43 r In order to minimize, the impulse response of the detector is matched to the time-reversed version of input signal

44. r Output of the detection: r The decision variable 44 Signal componentNoise component

45. r Mean of Y(T) 45

46. r Variance of Y(T) 46 Normal distribution the mean and variance

47. Hypothesis testing r Given bit “0” was sent r Given bit “1” was sent μ0μ0 47

48. Decision Rule z μ1μ1 μ0μ0 α r Assuming r Threshold m If, then, i.e., is regarded to be sent 48

49. Bit Error (1/3) z μ1μ1 μ0μ0 α r Error event: an error occur in the following cases: m When (i.e., ), but was sent 49

50. Bit Error (2/3) z μ1μ1 μ0μ0 α r Bit Error Probability 50 anddue to symmetrywhere Assuming that

51. Bit Error (3/3) z μ1μ1 μ0μ0 α r Bit Error Probability 51 Since

52. Signal to Noise Ratio (SNR) 52 : average amplitude of the received signal where

53. Signal Power 53 (assuming that μ 1 > 0 ≥ μ 0 ) where ( )

54. Noise Power 54 (due to )

55. Performance metrics r SNR: r Probability of error: 55 ( )

56. What is Line Coding? r Mapping of binary information sequence into the digital signal that enters the channel m Ex. “1” maps to +A square pulse; “0” to –A pulse r Baseband modulated signal has form where is the pulse of duration (the bit time) with amplitude m e.g, a k = A for a “1” bit and 0 for a “0” bit) m The data rate is 1/T b 101 0 11001 Unipolar NRZ 56

57. Line coding examples NRZ-inverted (differential encoding) 101 0 11001 Unipolar NRZ Bipolar encoding Manchester encoding Differential Manchester encoding Polar NRZ 57

58. 58 The Band-Limited Model In the case of the band-limited model the channel does not have infinite bandwidth and thus, the time-domain is not an impulse at t=0. As such, the convolution causes an Inter- Symbol Interference (ISI). For example:, if the impulse response is: And the following data is transmitted:

59. 59 The Band-Limited Model The result is the linear superposition of the impulse responses. The individual responses and the net summation are shown below. In order to accomplish this, the Modular is split into two parts: the pre-coder and the pulse-shaping filter. The pulse shaping filter generates a unit pulse only. The inputs to the pre-coder are the “1” and “0” bits. The output is “d” and “-d”. The pre-coder transforms {a n } to a desired form {b n } for baseband transmission.

60. 60 Pulse Shapes Observations: 1.The duration of the transmitted pulse is ‘stretched’ through a channel with memory. 2.The pulse stretching is referred to as “time dispersion”, and the channel is called a “time dispersion channel.” 3.Time dispersion causes overlaps between adjacent symbols at the output of the channel. The overlaps are called Inter- Symbol Interference (ISI). 4.ISI can’t be suppressed by simply increasing the signal energy.

61. 61 ISI Free Samples The above figure represents the band-limited model for the physical layer. From this, the following equations are obtained: The output of the convolution is: Let, then: where μ is a scaling factor such that c(0) = 1.

62. 62 ISI Free Samples μC(f) or μc(t) is the transfer function of the effective channel. The sampled r(t) value at ti = iT is: with

63. 63 Time Domain Condition for ISI Free Samples or equivalently: To achieve ISI free samples:

64. 64 Frequency Domain Condition Consider sampling c(t) at t k = kT, k=0,±1, ±2, ±3,.. where R = 1/T is the bit rate. For ISI free samples:

65. 65 Ideal Nyquist Channel B 0 is the effective channel single-side bandwidth. Here the ISI Free Sample Condition gives the following: where Practical Limitations c(t) decreases slowly as |t| increases 1.an ideal Low-pass filter is not physically realizable 2.2.

66. 66 ISI Free Samples

67. 67 Raised Cosine Spectrum To overcome the limitation, a particular pulse spectrum that has desirable spectral properties and has been readily used in practice is the raised cosine spectrum. Here, taking the inverse Fourier Transform gives: is called the roll-off factor. is the single-sided bandwidth. Advantages: C(f) is practically realizable c(t) decreases proportionally to Disadvantages: Lower efficiency of frequency spectrum.

68. 68 Raised Cosine Spectrum

69. Summary of Chapter 3 Given the characteristics of the communication channel (AWGN, Bandlimited) and system resources (signal power, bandwidth), we can calculate the probability of transmission error (bit error rate), i.e., the required performance. 69