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## Chapter 3: Data Transmission

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**COE 341: Data & Computer Communications (T061)Dr. Radwan E.**Abdel-Aal Chapter 3: Data Transmission**Remaining Six Chapters:**Chapter 7: Data Link: Flow and Error control, Link management Data Link Chapter 8: Improved utilization: Multiplexing Physical Layer Chapter 6: Data Communication: Synchronization, Error detection and correction Chapter 4: Transmission Media Transmission Medium Chapter 5: Encoding: From data to signals Chapter 3: Signals, their representations, their transmission over media, Resulting impairments**Agenda**• Concepts & Terminology • Signal representation: Time and Frequency domains • Bandwidth and data rate • Decibels and Signal Strength (Appendix 3A ) • Fourier Analysis (Appendix B ) • Analog & Digital Data Transmission • Transmission Impairments • Channel Capacity**Terminology (1)**Transmission system: • Transmitter • Receiver • Transmission Media • Guided media • e.g. twisted pair, coaxial cable, optical fiber • Unguided media • e.g. air, water, vacuum**Terminology (2)**Link Configurations: • Direct link • Nointermediate ‘communication’ devices (these exclude repeaters/amplifiers) Two types: • Point-to-point • Only 2 devices share link • Multi-point • More than two devices share the same link, e.g. Ethernet bus segment A Amplifier C B**Terminology (3)**Transmission Types (ANSI Definitions) • Simplex • In one direction only all the time e.g. Television, Radio broadcasting • Duplex • In both directions • Two types: • Half duplex • Only one direction at a time e.g. Walki-Talki • Full duplex • In both directions at the same time e.g. telephone**Frequency, Spectrum and Bandwidth**• Time domain concepts • Analog signal • Varies in a smooth, continuous way in both time and amplitude • Digital signal • Maintains a constant level for sometime and then changes to another constant level (i.e. amplitude takes only a finite number of discrete levels) • Periodic signal • Same pattern repeated over time • Aperiodic signal • Pattern not repeated over time**T**PeriodicSignals Temporal Period … t t+1T t+2T S (t+nT) = S (t); 0 t T Where: t is time over first period T is the waveform period n is an integer Signal behavior over one period describes behavior at all times**Aperiodic (non periodic) Signals in time**s(t) 1 0 t + X/2 - X/2**T**Sine Wave s(t) = A sin(2ft +) = A sin (F) A • Peak Amplitude (A) • Peak strength of signal, volts • T = Period = time for one repetition (cycle) • Repetition Frequency (f) • Measure of signal variations with time • In Hertz (Hz) (cycles per second) • f = 1/T(sec) = Hz • Angular Frequency (w) w = radians per second = 2 f = 2 /T • Phase () • Relative position in time, radians (how to determine?)**Varying one of the three parameters of a sine wave**carriers(t) = A sin(2ft +) = A sin(wt+F) Can be used to convey information…! M o d u l a t I o n Varying A Varying Varying f**Traveling Sine Waves(t) = A sin (k x - t]** = Angular Frequency = 2 f = 2 / T Spatial Period = Wavelength k = Wave Number = 2 / • For point p on the wave: • Total phase at t = 0: kx - (0) = kx • Total phase at t = t: k(x+ x) - (t) • Same total phase, • kx = k(x+ x) - (t) • k x = t Wave propagation velocity v = x / t v = /k = /T = f x p x Distance, x t = 0 t = t Direction of Wave Travel, Velocity v What is the expression for a wave traveling in the negative x direction? V is constant for a given wave type and medium v = f**Wave Propagation Velocity, v m/s**• Constant for a given wave type (e.g. electromagnetic, seismic, sound, ...) and propagation medium (air, water) • For all types of waves: • v = l f • For a given wave type (given v), higher frequencies correspond to shorter wavelengths and vise versa: Radio: long wave (km), short waves, … light (nm), etc.. • For electromagnetic waves: • In free space, v speed of light in vacuum = c = 3x108 m/sec • Over other guided media v is lower than c**Wavelength, l (meters)**• Is the Spatial period of the wave: i.e. distance between two points in space where the wave has the same total phase • Also: Distance traveled by the wave during one temporal (time) cycle: dT = v T = (l f) T = l**Continuous and Discrete Representations**Availability of the signal over the horizontal axis (Time or Frequency) Continuous: Signal is defined at all points on the horizontal axis Sampling with a train of delta function Discrete: Signal is defined Only at certain points on the horizontal axis**Frequency Domain Concepts**• System response to sine waves is easy to analyze • Signals we deal with in practice are not all sine waves, e.g. Square waves • Can we relate practical waves to sine waves? YES • Fourier analysis shows that any signal can be treated as the sum of many sine wave components having different frequencies, amplitudes, and phases(Fourier Analysis: Appendix B) • This forms the basis for frequency domainanalysis • For a linear system, its response to a complex signal will be the sum of its response to the individual sine wave components of the signal. • Dealing with functions in the frequency domain is simpler than in the time domain**Fundamental**Addition of Twofrequency Components A = 1*(4/) frequency = f + 3rd harmonic A = (1/3)*(4/) frequency = 3f Frequency Spectrum = Approaching a square wave Fourier Series 3 t f Frequency Domain: S(f) vs f Time Domain: s(t) vs t Discrete Function in f Periodic function in t**Asymptotically approaching a square wave by combining the**fundamental + an infinite number of odd harmonics at prescribed amplitudes Topic of a programming assignment What is the highest Harmonic added?**s(t)**1 0 t + X/2 - X/2 time More Frequency Domain Representations: A single square pulse (Aperiodic signal) Fourier Transform frequency 1/X Frequency Domain: S(f) vs f Time Domain: s(t) vs t Continuous Function in f Aperiodic function in t • What happens to the spectrum as the pulse gets broader …-> DC ? • What happens to the spectrum as the pulse gets narrower …-> spike ?**Spectrum & Bandwidth**• Spectrum • range of frequencies contained in a signal • Absolute (theoretical) Bandwidth (BW) • Is the width of spectrum = fmax- fmin • In many situations fmax = • Effective Bandwidth • Often just bandwidth • Narrow band of frequencies containing most of the signal energy • Somewhat arbitrary: what is “most”? • DC Component • Component at zero frequency S(f) 7f 5f f 3f f**Signal with a DC Component**+ _ t + 1V DC Level + t 1V DC Component**= (fmax- fmin)**Bandwidth for these signals:**Received Waveform**Limiting Effect of System Bandwidth 1,3 Better waveforms require larger BW BW = 2f More difficult reception with smaller BW f 3f 1 1,3,5 BW = 4f 5f f 3f 2 Varying System BW 1,3,5,7 BW = 6f 7f 5f f 3f 3 … BW = 1,3,5,7 ,9,… …… 7f 5f f 3f 4 Fourier Series for a Square Wave**System Bandwidth and Achievable Data Rates**• Any transmission system supports only a limitedband of frequencies(bandwidth) for satisfactory transmission • “system” includes: TX, RX, and Transmission medium • For example, this bandwidth is largest for optical fibers and smallest for twisted pair wires. • This limited system bandwidth degrades higher frequency components of the signal transmitted poorer received waveforms more difficult to interpret the signal at the receiver (especially with noise) Data Errors • More degradation occurs when higher data rates are used (signal will have more components at higher frequency ) • This puts a limit on the data rate that can be used with agiven signal to noise requirement, receiver type, and a specified error performance Channel capacity issues**5f**f 3f Data Element = Signal Element Bandwidth and Data Rates Period T = 1/f T/2 Data rate = 1/(T/2) = 2/T bits per sec = 2f B 0 0 1 1 Data B = 4f Given a bandwidth B, Data rate = 2f = B/2 To double the data rate you need to double f: Two ways to do this… 1. Double the bandwidth with same received waveform(same RX conditions & error rate) 2B = 4f’ 2B 1 0 0 1 1 1 0 0 1 0 0 1 1 1 0 0 New bandwidth: 2B, Data rate = 2f’ = 2(2f)= 4f = B f’ 3f’ 5f’ 2. Same bandwidth, B, but tolerate poorer received waveform (needs better receiver, higher S/N ratio, or tolerating more errors in data) 1 B = 2f’ 0 0 0 1 1 1 0 B Bandwidth: B, Data rate = 2f’ = 2(2f) = 4f = B 5f’ 3f’ f’**Bandwidth & Data Rates: Tradeoffs… Compromises**• Increasing the data rate (bps) while keeping BWthe same means working with inferior (poorer) waveforms at the receiver, which may require: • Ensuring higher signal to noise ratio at RX (larger signal relative to noise): • Shorter link distances • Use of more en-route repeaters/amplifiers • Better shielding of cables to reduce noise, etc. • More sensitive (& costly!) receiver • Getting higher bit error rates • Tolerate them? • Add more efficient means for error detection and correction- this also increases overhead!.**Appendix 3A: Decibels and Signal Strength**• The decibel notation (dB) is a logarithmic measure of the ratio between two signal power levels • NdB= number of decibels • P1 = input power level • P2 = output power level • Example: • A signal with power level of 10mW is inserted into a transmission line • Measured power some distance away is 5mW • Power loss in dBs is expressed as NdB =10 log (5/10)=10(-0.3)= -3 dB • - ive dBs: P2 < P1 (Loss), • +ive dBs: P2 > P1 (Gain)**Decibels and Signal Strength**• Decibel notation is a relative, not absolute, measure: • A loss of 3 dB halves the power (could be 100 to 50, 16 to 8, …) • A gain of 3 dB doubles the power (could be 5 to 10, 7.5 to 15, …) • Will see shortly how we can handle absolute levels • Advantage: • The “log” allows replacing: • Multiplication with addition C = A * B Log C = Log A + Log B • and division with subtraction**Amplifier**? 4 mW Gain: 35 dB Transmitted Signal Received Signal Loss: 10 dB Loss: 12 dB Decibels and Signal Strength • Example: Transmission line with an intermediate amplifier • Net power gain over transmission path: + 35 –12 – 10 =+13 dB (+ ive means there is actual net gain) • Received signal power = (4 mW) log10-1(13/10) = 4 x 101.3 mW = 79.8 mW**WK 4**How to represent absolute power levels?Decibel-Watt (dBW) and Decibel-mW (dBm) • As a ratio relative to a fixed reference power level • Value of 1 W is a reference defined as 0 dBW • Value of 1 mW is a reference defined as 0 dBm • Examples: • Power of 1000 W is 30 dBW • –10 dBm represents a power of 0.1 mW Caution!: Must be same units at top and bottom Caution!: Must be same units at top and bottom X dBW = (X + ?) dBm**Decibels and Signal Strength**Everything in terms of dBs and dBm (or dBW) Levels {dBs and dBms} can be added and subtracted. Same for {dBs and dBWs} • Example: Transmission line with an intermediate amplifier ? 4 mW Gain: 35 dB Transmitted Signal Received Signal Loss: 10 dB Loss: 12 dB Amplifier • Net power gain over transmission path: + 35 –12 – 10 =+13 dB (+ ive means actual net gain) TX Signal Power = 4 mW = 10 log (4/1) = 6.02 dBm • RX signal power (dBm) = 6.02 + 13 =19.02 dBm • Check: 19.02 dBm = 10 log (RX signal in mW/1 mW) RX signal = log-1 (19.02/10) = 79.8 mW**Decibels & Voltage ratios**• Power decibels can also be expressed in terms of voltage ratios • Power P = V2/R, assuming same R • Decibel-millivolt (dBmV) is an absolute unit, with 0 dBmV being equivalent to 1mV. Note that this is still a power ratio… But expressed in voltages Caution!: Must be same units at top and bottom**Appendix B: Fourier Analysis**Signals in Time Aperiodic Periodic … Discrete Continuous Discrete Continuous DFS FS FT DFT Use Fourier Series Use Fourier Transform FS : Fourier Series DFS : Discrete Fourier Series FT : Fourier Transform DFT : Discrete Fourier Transform**Fourier Series for periodic continuous signals**• Any periodic signal x(t) of period T and repetition frequency f0 (f0 = 1/T) can be represented as an infinite sum of sinusoids of different frequencies and amplitudes – its Fourier Series. Expressed in Two forms: • 1. The sine/cosine form: Frequencies are multiples of the fundamental frequency f0 f0 = fundamental frequency = 1/T Where: DC Component = f(n) Two components at each frequency All integrals over one period only If A0 is not 0, x(t) has a DC component = f’(n)**Fourier Series: 2. The Amplitude-Phase form:**• Previous form had two components at each frequency (sine, cosine i.e. in quadrature) : An, Bn coefficients • The equivalent Amplitude-Phase representation has only one component at each frequency: Cn, qn • Derived from the previous form using trigonometry: cos (a) cos (b) - sin (a) sin (b) = cos [a +b] Now one component at each frequency nf0 Now components have different amplitudes, frequencies, and phases The C’s and ’s are obtained from the previous A’s and B’s using these equations. They are functions of n**1**-3/2 -1 -1/2 1/2 1 3/2 2 -1 T Fourier Series Example x(t) Note: (1) x(– t)=x(t) x(t) is an even function (2) f0 = 1 / T = ½ Hz Note: A0 by definition is 2 x the DC content**1**-3/2 -1 -1/2 1/2 1 3/2 2 -1 T Contd… = 0 for n even = (4/n) sin (n/2) for n odd f0 =1/2 a function of n only Replacing t by –t in the first integral sin(-2pnf t)= - sin(2pnf t) Since x(– t)=x(t) as x(t) is an even function, then Bn = 0 for all n**Contd…**f0 = ½, so 2 f0 = A0 = 0, Bn = 0 for all n, An = 0 for n even: 2, 4, … = (4/n) sin (n/2) for n odd: 1, 3, … Original x(t) is an even function! Amplitudes, n odd Cosine is an even function 3rd Harmonic 2 p 3 (1/2) t**Another Example**Previous Example x1(t) 1 -2 -1 1 2 -1 T Note that x1(-t)= -x1(t) x(t) is an odd function Also, x1(t)=x(t-1/2) This waveform is the previous waveform shifted right by 1/2**Another Example, Contd…**Sine is an odd function As given before on slide 23. Because:**Fourier Transform**• For aperiodic (non-periodic) signal in time, the spectrum consists of a continuum of frequencies (not discrete components) • This spectrum is defined by the Fourier Transform • For a signal x(t) and a corresponding spectrum X(f), the following relations hold Imaginary nf0 f Inverse FT (from frequency to time ) Forward FT (from time to frequency) Real • X(f) is always complex (Has both real & Imaginary parts), even for x(t) real.**(Continuous**in Frequency) (non-periodic in time) Sinc function Sinc2 function**Fourier Transform Example**x(t) A Sin (x) / x i.e. “sinc” function**A**= A t f 1/ Fourier Transform Example, contd. Sin (x) / x “sinc” function Lim x0 (sin x)/x = (cos x)x=0/1 =1 Study the effect of the pulse width **The narrower a function is**in one domain, the wider its transform is in the other domain The Extreme Cases**Power Spectral Density (PSD) & Bandwidth**• Absolute bandwidth of any time-limited signal is infinite • But luckily, most of the signal power will be concentrated in a finite band of lower frequencies • Power spectral density (PSD) describes the distribution of the power content of a signal as a function of frequency • Effective bandwidth is the width of the spectrum portion containing most of the total signal power • We estimate the total signal power in the time domain

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