Diversity Spring 2017 ELE 492 FUNDAMENTALS OF WIRELESS COMMUNICATIONS 1
Diversity A fading channel with an average SNR has worse BER performance as compared to that of an AWGN channel with the same SNR!. A fading channel may collapse randomly with probability p. What if we have two independent channels that convey the same information? Spring 2017 ELE 492 FUNDAMENTALS OF WIRELESS COMMUNICATIONS 2
Diversity Example: The noise power within the RX filter bandwidth is 50 pw, the average received signal power is 1 nw, hence the SNR is 13 db. In an AWGN channel this would yield 10-9 BER for differentially coded FSK. Consider a fading channel where during 90 % of the time the received power is 1.11 nw, and the SNR is thus 13.5 db, (BER = 10-10 ) while for the remainder, received power is zero (BER = 0.5 (why 0.5?)). Average BER is Average RX power is 1nW Assume that we have two antennas experiencing independent fading: Probability that the received power is 0 at both antennas simultaneously is 0.1 x 0.1 = 0.01 1.11 nw at both antennas simultaneously is 0.9 x 0.9 = 0.81, (assume selection diversity, SNR = 1.35 db) 1.11 nw at one antenna and 0 at the other is 0.18 (assume selection diversity, SNR = 1.35 db) Average BER is Approx. square of the BER for a single antenna! Spring 2017 ELE 492 FUNDAMENTALS OF WIRELESS COMMUNICATIONS 3
Diversity Example (cont. d): If there are three antennas, The probability that the signal power is 0 at all antennas is 0.1 3, Average BER is Approx. the third power of the BER for a single-antenna system. General rule is: If the BER of a single antenna system, i.e. one receive channel is BER oc With N r diversity antennas, we obtain a bit error probability Spring 2017 ELE 492 FUNDAMENTALS OF WIRELESS COMMUNICATIONS 4
Correlation Coefficient Diversity is most efficient when the different transmission channels carry independent fading copies of the same signal. That is, the joint pdf of field strength for all channels is equal to the product of the marginal pdf.s, i.e. Otherwise, any correlation between the fading of the channels will decrease the effectiveness of diversity. Correlation coefficient characterizes the correlation between signals on diversity branches. A commonly used metric is the correlation coefficient of signal envelopes x and y If x and y are independent E{x y} =E{x} E{y}, hence ρ xy = 0 Signals are often said to be «effectively decorrelated» if ρ is below a certain threshold (typically 0.5 or 0.7). Spring 2017 ELE 492 FUNDAMENTALS OF WIRELESS COMMUNICATIONS 5
Microdiversity Diversity is a powerful technique to combat small-scale fading. Spatial diversity: several antenna elements (TX and/or RX) separated in space. Temporal diversity: transmission of the transmit signal at different times. Frequency diversity: transmission of the signal on different frequencies. Angular diversity: multiple antennas (with or without spatial separation) with different antenna patterns. Polarization diversity: multiple antennas with different polarizations (e.g. vertical and horizontal) Consider the correlation coefficient of two signals that have a temporal separation τ and a frequency separation of f 1 f 2, then the correlation coefficient is (assumptions: WSSUS valid, no LOS, exponential PDP, isotropic distribution of incident power, omnidirectional antennas) For a moving RX, temporal separation can be converted into spatial separation they are mathematically equivalent Above correlation coefficient can be applied to spatial, temporal and frequency diversity. Spring 2017 ELE 492 FUNDAMENTALS OF WIRELESS COMMUNICATIONS 6
Microdiversity Example: Compute the correlation coefficient of two frequencies with separation (i) 30 khz, (ii) 200 khz, (iii) 5 MHz, in the «typical urban» environment as defined in COST 207 channel model. (no temporal or spatial separation) No temporal separation J 0 (0) = 1 rms delay spread of the typical urban environment in COST 207 model is Correlation coefficient is IS 136 TDMA GSM WCDMA Spring 2017 ELE 492 FUNDAMENTALS OF WIRELESS COMMUNICATIONS 7
Spatial Diversity Transmitted signal is received at several antenna elements, and received signals are processed jointly. Correlation among the antenna elements has an elementary role, Large correlation between signals at antenna elements decreases the effectiveness of diversity. Establish a relation between antenna spacing and the correlation coefficient. There may be different scenarios for antenna positioning: 1. MS in cellular and cordless systems: It is assumed that waves are incident from all directions at the MS Points of constructive and destructive interference of MPCs are spaced approximately λ/4 apart (verify from the corr. coef. eqn.) For 900/1800 MHz (GSM) min. antenna spacing should be about 8/4 cm, For 2400 MHz (WiFi) min. antenna spacing should be about 3 cm, 2. BS in cordless systems and WLANs: The angular distribution of incident radiation at indoor BSs is assumed to be uniform the same rules apply as for MSs. Spring 2017 ELE 492 FUNDAMENTALS OF WIRELESS COMMUNICATIONS 8
Spatial Diversity BS in cellular systems: BS antenna is on a pole/mast high above (~30m) IOs in the channel are generally concentrated around MS, not the BS. Since all waves from a BS are incident from one direction, the correlation coefficient is much higher. The antenna spacing required to obtain sufficient decorrelation increases. Angular spread ~ S ϕ Spring 2017 ELE 492 FUNDAMENTALS OF WIRELESS COMMUNICATIONS 9
Temporal Diversity If the channel is time varying, the signals received at different times are uncorrelated. can be used to generate diversity. If the channel is static (e.g. WLAN), there is no temporal variation correlation coef. ρ=1 temporal diversity is useless. For sufficient decorrelation, the temporal distance must be at least 1/(2ν max ) (ν max : max. Doppler shift) Temporal diversity can be realized in different ways: 1. Repetition coding: The signal is repeated several times, where the repetition intervals are long enough to achieve decorrelation. 2. Automatic Repeat request (ARQ): Transmission is repeated if RX cannot receive the information correctly (ACK/NACK is used) 3. Combination of interleaving and coding: Forward error correction with interleaving. Spring 2017 ELE 492 FUNDAMENTALS OF WIRELESS COMMUNICATIONS 10
Frequency Diversity The same signal is transmitted from two or more different frequencies. If these frequencies are separated apart more than the coherence bandwidth of the channel, then their fading is independent. (Requires frequency selective channel, no diversity for flat channels!) Can be used to generate diversity. For an exponential PDP, the correlation coefficient was given as (if there is no temporal variation in the channel) Great waste of spectrum! A solution is to spread the signal over a wider band small parts of information are conveyed by different freq. components. Compressing the information in time (TDMA) Code Division Multiple Access (CDMA) Multicarrier CDMA and coded Orthogonal Frequency Division Multiplexing/Multiple Access (OFDM/OFDMA) Loss is gained back by hosting multiple users in the same band! Spring 2017 ELE 492 FUNDAMENTALS OF WIRELESS COMMUNICATIONS 11
Angle Diversity (Pattern diversity) Fading is created by destructive accumulation of MPCs at the receiver. If some of the waves are attenuated/eliminated, location of the fading dip changes. Two co-located antennas with different antenna patterns see different weighted MPCs, hence each antenna experiences different fading. Spring 2017 ELE 492 FUNDAMENTALS OF WIRELESS COMMUNICATIONS 12
Polarization Diversity Horizontally and vertically polarized MPCs propagate differently and separately in the medium (remember TE-TM wave comparison) Even if the TX transmits in a single polarization, both polarizations arrive at the RX due to the effect of the channel. The average received signal strength in the two diversity branches may differ 3-20 db as compared to the other one. Fading of the two received signals are independent. Can be used to generate diversity. Spring 2017 ELE 492 FUNDAMENTALS OF WIRELESS COMMUNICATIONS 13
Macrodiversity The aformentioned diversity methods are a remedy for small-scale fading. Large-scale fading shadowing Polarization, frequency diversity do not work for shadowing. Spatial/temporal diversity may work but the distance to get out of shadowing is ~10-100 λs. May take too long. We may transmit the same signal from BSs at different locations. Forms independently fading (large-scale) paths, If one of the paths is in shadowing, other(s) may observe a stronger path. On-frequency repeaters: receives the signal and retransmits an amplified version of it. Simple but introduces latency to the signal Simulcast: Same signal is transmitted simultaneously from different BSs. Difficult to synchronize the BSs. Signal flow between BSs should be very high speed (fiberoptic links). Spring 2017 ELE 492 FUNDAMENTALS OF WIRELESS COMMUNICATIONS 14
Combination/Selection of Signals How can we exploit diversity? Although there are several ways to generate diversity, the mathematical method is similar for all. Let us consider spatial diversity. Multiple antennas at the RX. Two ways to exploit diversity: Selection diversity: the «best» signal copy is selected and processed, while other copies are discarded. Combination diversity: all copies of the signal are combined and the combined signal is decoded. Since all available information is used combination diversity performs better. RX structure of selection diversity is simpler. In selection diversity, the strength of the signal at each antenna can be measured at the RF stage. Only one RF-chain is adequate. In combination diversity, all signals have to be downconverted and processed in the baseband. Keep in mind!: There are two types of gain due to multiple antennas: Diversity gain: signal level increases due to avoiding fading dips Beamforming gain: combiner performs an averaging over the noise at different antennas. Even if the signal levels at all antenna elements are identical, combiner output SNR is higher than the SNR at a single-antenna element. (E{noise} = 0.) Spring 2017 ELE 492 FUNDAMENTALS OF WIRELESS COMMUNICATIONS 15
Selection Diversity RSSI (Received-Signal-Strength-Indication) Driven Diversity RX selects the signal with the largest instantaneous power (e.g. RSSI) and process this signal only. Requires N r antenna elements, N r RSSI sensors (?), N r 1 multiplexer, 1 RF chain. Can track the change in RSSI in fast fading channels. Pros and cons 1. If the BER is determined by noise only, selection diversity based on RSSI is the best among other selection diversity techniques Maximizes SNR 2. If there is co-channel interference (signals of other users in the same freq. band (FDMA)/time slot (TDMA)), it may be counted for signal by the RSSI sensor. An antenna with low SINR (Signal-to-Interference plus Noise Ratio) can be selected. 3. If the channel is frequency selective, there is no guarantee that the RSSI gives the best choice. (It only gives the average power received from the channel) What is the SNR at the selector output? For Rayleigh fading channels, the pdf of the n th diversity branch is ( is the mean branch SNR, assumed to be the same for all antennas) Spring 2017 ELE 492 FUNDAMENTALS OF WIRELESS COMMUNICATIONS 16
Selection Diversity cdf of the nth branch is: When RX selects the signal with highest SNR, the probability that the SNR of the chosen branch is lower than a threshold γ is the product of the probabilities that all branches are below that level: Example: Compute the probability that the output power of a selection diversity system is 5 db lower than the mean power of each branch, when using N r = 1, 2, 4 antenna elements. Spring 2017 ELE 492 FUNDAMENTALS OF WIRELESS COMMUNICATIONS 17
Selection Diversity Example: If N r = 2 and the mean powers in the branches are 1.5 and 0.5 (unequal mean powers), what is the probability that the output power of the selector is 5 db below than? Diversity is less efficient when the average branch powers are different. Spring 2017 ELE 492 FUNDAMENTALS OF WIRELESS COMMUNICATIONS 18
Selection Diversity BER (Bit-Error-Rate) Driven Diversity A known sequence (training sequence) is transmitted regularly. RX measures the BER of each branch by using this sequence and selects the one with lowest BER. Pros and cons: 1. RX needs either N r RF chains and demodulators (expensive solution), or the training sequence is repeated Nr times and RX evaluates each antenna sequentially by a single RF chain and demodulator (waste of time slot) 2. Second method may not work correctly if the channel is fast time varying, 3. Correctness of the BER measurement depends on the length of the training sequence. Longer sequence gives better estimates of BER. 4. If done correctly, this method gives a better measure of the link quality rather than RSSI. Spring 2017 ELE 492 FUNDAMENTALS OF WIRELESS COMMUNICATIONS 19
Switched Diversity Selection diversity may be expensive (several RF chains + demodulators) or have low spectral efficiency (require several slots for training sequence). A solution to these issues can be switched diversity. Determine a target QoS (Quality of Service) (such as BER, SNR, etc.) As long as the current branch satisfies the QoS, continue with this branch (even if the other branch has a better channel) If the quality of the current branch decreases to a level lower than the target QoS, switch to the other link (for systems with two diversity braches) or search for a better link. Performance of this technique is worse than that of selection diversity. Complexity is lower as compared to selection diversity. Spring 2017 ELE 492 FUNDAMENTALS OF WIRELESS COMMUNICATIONS 20
Combining Diversity Although simple, selection diversity wastes signal energy by discarding (N r -1) copies of the received signal. They may also be accumulated to increase SNR. Combining diversity aligns all received signals and add them together. (Coherent addition) Each signal is multiplied by a complex weight (amplitude + phase correction) and then added up. Phase correction causes the signal amplitudes to add up coherently, while, on the other hand, noise is added incoherently signal powers add up, noise filters out There are two approaches for amplitude weighting, MRC (Maximum Ratio Combining) weighs all signal copies by their amplitude. Maximizes SNR EGC (Equal Gain Combining): all amplitude weights are the same. Spring 2017 ELE 492 FUNDAMENTALS OF WIRELESS COMMUNICATIONS 21
Maximum Ratio Combining Assume a slow fading and frequency flat (narrowband) channel with AWGN: α n : instantaneous gain of the n th diversity branch. Multiply the received signal by, then SNR n is We want to maximize SNR n wrt w n, Keep the denominator constant, i.e. w n 2 : constant (= 1). Use the Cauchy-Schwartz Inequality for the numerator. Satisfied with equality if w n = c α n for any constant c. That is, SNR is maximized. Spring 2017 ELE 492 FUNDAMENTALS OF WIRELESS COMMUNICATIONS 22
Maximum Ratio Combining Remember that RX multiplies each branch by All signals are phase-corrected (due to conjugation, α n* ) and scaled by their amplitude α n. The output SNR of the diversity combiner is the sum of the branch SNRs: For Rayleigh fading channels, if the mean SNR of all branches are the same, e.g., then - # of diversity branches determine the slope - Slopes are the same for both selection diversity and also MRC. - Difference between the curves is due to the N r - 1 signal copies ignored by selection diversity. Spring 2017 ELE 492 FUNDAMENTALS OF WIRELESS COMMUNICATIONS 23
Equal Gain Combiner Only phase alignment is carried out, signal amplitudes are not scaled. It can be shown that the SNR of the combiner output is If all diversity branches experience independent and identical Rayleigh fading, the mean SNR at the EGC output is ( is the mean SNR of all branches) For equal SNR branches, and large N r, the difference between MRC and EGC is. Difference is larger for unequal branch SNRs. Spring 2017 ELE 492 FUNDAMENTALS OF WIRELESS COMMUNICATIONS 24
Error Probability in Fading Channels with Diversity Reception Average symbol (bit) error rate can be found by using What is the performance of BPSK in Rayleigh fading channels with N r diversity branches with MRC? 1. Instantaneous SER for BPSK, γ: symbol (bit) SNR 2. pdf of SNR with MRC for Rayleigh fading channels is 3. Substitute into the integral to get 4. For large values of γ Spring 2017 ELE 492 FUNDAMENTALS OF WIRELESS COMMUNICATIONS 25
Error Probability in Fading Channels with Diversity Reception Diversity gain: determines the slope of decay in BER-SNR plots. Diversity order: SNR required: 1 10 db/decade 2 5 db/decade 3 3.3 db/decade 4 2.5 db/decade 5 2 db/decade Spring 2017 ELE 492 FUNDAMENTALS OF WIRELESS COMMUNICATIONS 26