1. 1 Dr. Essam Sourour Alexandria University, Faculty of Engineering, Dept. Of Electrical Engineering Introduction to Fading Channels, part 2

2. 2 Local reflections cause multipath Each path has a random gain, with random magnitude and random phase Each gain is represented in baseband as Receiver, and/or reflectors, may be moving Small Scale Fading

3. 3 Assume a group of paths with small relative delay Net effect is one path with random gain and phase According to Central Limit Theorem, the net gain Re j  is complex Gaussian with zero mean The envelope R is Rayleigh distributed and the phase  is uniform [0, 2  ] Small Scale Rayleigh Fading

4. 4 The net gain is the sum of all closely delayed paths: Each of g real and g imag is the sum of many independent random variables Hence g real and g imag are independent and Gaussian with zero mean and variance σ 2 each Fading gain g = g real + j g imag is complex Gaussian with zero mean and variance 2σ 2 (sum of two variances) Rayleigh Fading

5. Rayleigh Distribution From probability theory we know: Received amplitude follows Rayleigh distribution Received power follows Exponential distribution Received phase follows Uniform distribution 5

6. Amplitude, Rayleigh Distribution 6

7. Power, Exponential Distribution 7

8. Fading of 16 QAM signal Signal has a higher probability of being week For example, to receive the 16-QAM signal we must estimate and compensate for the amplitude and phase 8 No fadingFaded signals with random amplitude and phase

9. Effect of Mobility Fading gain changes with time g(t)=g real (t) + j g imag (t) Fading change rate depends on the maximum Doppler frequency Coherence time << 1/f D Example: f c =1GHz, v=100 km/h gives: f D = 92.6 Hz, Coherence time << 10.8 ms 9

10. Fading Example for R 10

11. Statistical Properties 1 Complex fading gain g(t) The two parts g real (t) and g imag (t) are zero mean The two parts g real (t) and g imag (t) are statistically independent 11

12. Statistical Properties 2 Fading gain is correlated over time Usually Jakes model is used in mobile comm. Autocorrelation function given by J o () is the Bessel function of order zero A g (  t) indicates how much the gain is correlated with itself after delay  t Power spectral density of fading is the FT of the autocorrelation function 12

13. Statistical Properties 3 Usually the fading gain is normalized to unity power, i.e,  2 =1/2 13 fD tfD t f/f D

14. Rician Fading Channel If the channel also includes a LOS component we get Rician fading Fading gain is now g’ real is Gaussian with mean S and variance  2 The envelope R is Rician distributed (see Proakis chapter 2)

15. Rician Fading Channel The channel amplitude R is Rician I 0 = modified Bessel function of order zero Now, when s(t) is transmitted Power ratio K=LOS/faded=S 2 /(2  2 ) If K=0 we are back to Rayleigh fading As K increases, more power to LOS 15

16. Rician PDF 16 K=1, 2, 3

17. 17 Channel may consists of groups of delays (echoes) Each group is composed of many closely delayed paths Maximum Delay Spread: Delay between first and last Typically few microseconds outdoor and less than hundred of nanoseconds indoor Channel with large delay spread is an FIR filter: Large Delays Effect

18. Power Delay Profile Power of the multipath decay as delay increases according to power delay profile Each path g l has a variance Example, exponential profile Example, uniform profile Typically, fading is normalized Mean delay spread: RMS delay spread 18

19. 19 Time and Delay Picture Channel may have many “resolvable” paths Each path at a certain delay Each path changes with time, t, and has its delay,  Autocorrelation function: Scattering function: twice Fourier Transform of the Autocorrelation function, over  t and 

20. 20 Simulating Classical Fading Model Jakes model

21. 21 Simulating Classical Fading Model Assume a mobile station in the middle of 4N reflectors Reflections with equal amplitude but different Doppler Doppler from path with incident angle  n is f n =f M cos(  n ), f M is the maximum Doppler Reflectors have different propagation delay around the circle

22. 22 Classical Fading Model After some mathematical manipulations, the gain of the path h k (t):

23. 23 Classical Fading Model With L resolvable multipath, the channel model is given by The gains v l select the desire delay profile They are normalize the total channel power to 1

24. 24 Walch Codes of length 16 1111111111111111 11 1 1 1 1 1 1 11 11 11 11 1 11 11 11 1 1111 1111 1 1 1 11 1 1 1 11 1111 11 1 1 11 1 1 11 11111111 1 1 1 1 1 1 1 1 11 11 11 11 1 11 1 11 11 1111 1111 1 1 1 1 1 11 1 11 11 1111 1 1 11 11 1 1 n k

25. 25 Fading References 1.Classical Model: W. C. Jakes, editor, Microwave Mobile Communications, New York, Wiley 1974 2.Modifications: P. Dent, G. E. Bottomley, and T. Croft, “Jakes fading model revisited,” Electronic letters, vol. 29, pp. 1162-1163, June 1993 3.Good reference: Chapter on Fading channels in “Digital Communications” by Bernard Sklar

26. 26 Small Scale Fading FastSlow Small Scale Fading Non-selective Selective Fast: Channel changes within symbol. T c <T s Slow: Channel constant during more than symbol time. Tc>Ts Selective: Delay Spread > symbol time T s Non-Selective: Delay Spread < symbol time Ts Speed and Selectivity are independent issues Effects on Signal

27. Definitions Coherence time = 1/max doppler = 1/f D Coherence bandwidth = 1/max delay spread Slow fading: Symbol time < coherence time Non-selective fading: Signal bandwidth < coherence bandwidth Fast fading and selective fading are the opposite 27

28. 28 Fast Fading: –Due to high speed –High distortion to the received signal Slow Fading: –Terminal may fall in a fading null for long time –Worse performance Effects on Signal, cont.

29. 29 Selective Fading: –Due to high Delay Spread w.r.t symbol duration –Channel is random FIR filter Non-Selective Fading: –Delay Spread << symbol duration –Channel is one tap Effects on Signal, cont.

30. 30 Receiver Antenna Diversity Transmitter Antenna Diversity Transmitter and Receiver Antenna Diversity (MIMO Systems) Rake Receiver Channel Equalization Channel Coding Fading Counter Measures

31. 31 Receiver may have two or more antennas Two main types: –Antenna Selection: Select stronger antenna signal. Best for slow, non-selective fading –Antenna Combining: Optimally combine signal of antennas (MRC) More complexity & better performance Receiver Antenna Diversity

32. 32 Maximal Ratio Combining

33. 33 Two antennas are used in Tx Two successive symbols are pre-coded as shown Need two orthogonal sources for two channels estimation Transmit Diversity

34. 34 Same as Tx Diversity, but with two Rx’s We have 4 channels, h 0, h 1, h 2 and h 3 Each receiver combines as before The two receivers are then combined Tx & Rx Diversity (MIMO)

35. 35 Used for Direct Sequence Spread Spectrum Systems Multipath diversity = multipath is advantageous One finger (correlator) per path Each finger synchronized to one path Finger outputs combined (MRC) Rake Receiver

36. 36 Need to estimate channel gain for each path Rake Receiver performs Maximal Ratio Combining Number of fingers = number of paths (ideally) Small inter-path interference Rake Receiver, Cont.

37. 37 Equalizers attempt to compensate for channel fading effects Linear Equalizer: FIR filter with adaptive tap weights Adaptation to minimize some criteria Most famous: Least Mean Square (LMS) Other criteria: Recursive Least Squares, Kalman Filter, etc. Channel Equalization

38. 38 LMS: w j (n+1)=w j (n) –  e*(n) y j (n) Linear Equalizer

39. 39 Summary Fading Types: –Large Scale: Distance + Shadowing –Small Scale: Fast or Slow & Flat or Selective Counter Measures: –Diversity Types –Rake –Equalization

40. throughput Throughput is the number of messages successfully delivered per unit time. Throughput is controlled by available bandwidth, as well as the available signal- to-noise ratio and hardware limitations. For example, in Ethernet the maximum frame size 1526 bytes (maximum 1500 byte payload + 8 byte preamble + 14 byte header + 4 Byte trailer). An additional minimum interframe gap corresponding to 12 byte is inserted after each frame. This corresponds to a maximum channel utilization of 1526/(1526+12)100% = 99.22%,. The maximum throughput is 1500/(1526+12) = 97.5 Mbit/s exclusive of Ethernet protocol overhead (preamble is used to synchronize data )

41. latency Latency is a measure of time delay experienced in a system, the precise definition of which depends on the system and the time being measured In networking, the amount of time it takes a packet to travel from source to destination. Together, latency and bandwidth define the speed and capacity of a network. packetbandwidthnetwork (3) In VoIP terminology, latency refers to a delay in packet delivery. VoIP latency is a service issue that is usually based on physical distance, hops, or voice to data conversion.VoIPpacket hops

42. interworking Interworking means that LTE technology must support legacy equipments designed to operate with legacy standards such as 3gpp

43. Co-existance Co-existence in the same geographical area and colocation with GERAN/UTRAN shall be ensured.