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The first and second antennas then transmit the signals |
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the entire problem into the one discussed in the previous subsection. This principle, with a slight modification, is used in the UMTS downlink for arranging an open loop (without user-BS feedback) transmit diversity [114,115]. To be fair, processing synchronization signals (time-delay measurement) encoded by Alamouti code in a user’s terminal would appear much more complicated as compared to data demodulation. For this reason the transmit diversity mode employed in the synchronization channel is the time-switched coding touched upon in Section 10.3.3 [115]. One more interesting detail is using closedloop diversity in the dedicated (i.e. assigned to a specific user) physical channels. Based on the feedback MS-BS data, the BS knows the current state of the channel linking the BS with the specific user and adjusts the phases of the signals of two transmit antennas to make them sum coherently at the terminal input. The amplitudes of the transmitted signals may also be adjusted to realize the maximal ratio combining in the receive antenna and bring the efficiency of transmit diversity nearer to that of the receive one. The cdma2000 downlink specification includes some similar solutions concerning transmit diversity.
Problems
10.1.Let the first signature a1 be a linear combination of the other signatures. Prove that the linear system (10.12) has no solution, i.e. suppression of MAI automatically removes the useful effect too.
10.2.There is a three-user synchronous DS CDMA system. The signatures are binary of spreading factor N ¼ 3: a1 ¼ ( þ ), a2 ¼ ( þ ), a3 ¼ ( þ ). Find the reference of the first user’s decorrelating receiver, demonstrate elimination of MAI, and evaluate SNR loss of the decorrelating algorithm to the matched filtering.
10.3.Derive the gradient of the function (10.19) and prove that u given by (10.20) is the point of minimum of this function.
10.4.Prove the matrix inverse lemma in the form (10.21).
10.5.Find the reference vector for the MMSE detector for the conditions of Problem 10.2, setting all signature amplitudes and noise variance equal to one. Calculate SINR at the MMSE detector output, compare it with those of the decorrelating detector and matched filtering (Problem 10.2), and explain the results.
10.6.A synchronous CDMA system accommodates 128 users within the spreading factor N ¼ 96. The signatures are columns of the 128th order Hadamard matrix, in which 32 rows are discarded. Find MMSE reference vectors for all users, if their signals have equal intensities.
10.7.The MC-DS-CDMA downlink is realized using three subcarriers. Data is transmitted at each subcarrier by BPSK at the rate 32 kbps with spreading factor 64. The guard frequency interval equals 0:5/D, where D is signature chip duration. How would the potential number of users change if DS-CDMA replaced MC-DS-CDMA?
10.8.Data should be transmitted using QPSK at the rate 2.88 Mbps over a channel whose coherence bandwidth Bc ¼ 50 kHz. Find the minimal number of subcarriers
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necessary for MC transmission. What is the minimum length of DFT in the OFDM scheme, if the overhead due to the guard intervals should not exceed 10%?
10.9.A synchronous MC-CDMA downlink in the OFDM version transmits data
using QPSK |
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the rate 40 kbps |
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channel with |
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max ¼ 10 ms. |
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many users can it |
serve |
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all undistorted |
signatures are |
orthogonal and the overall bandwidth is 5 MHz?
10.10.Would it be reasonable to use zero-forcing combining in MC-CDMA operating on the Rayleigh subchannels?
10.11.Suppose that nR antennas receive in parallel the signal transmitted by a single antenna, intensities of all received signals are the same, as well as of independent Gaussian noises corrupting the signals. How does the Shannon capacity of such a channel differ from that corresponding to a no-diversity case, if the receiver knows the path length differences of all signals?
10.12.Suppose that a transmitter is capable of transmitting data involving nd independent identical diversity branches, total transmitted power being fixed. Suppose that the intensities of all received signals are the same, as well as the independent Gaussian noises corrupting the signals, and the receiver (but not the transmitter!) knows the path length differences of all signals. What is better from the angle of Shannon capacity: to transmit the same or different datastreams over nd branches?
10.13. Prove the upper bound on the complementary error function: Q(x) (1/2) exp ( x2/2) for any x 0.
10.14.Prove convergence of the right-hand side of (10.36) to the upper bound of the error probability for a single non-fading branch: (1/2) exp ( q2/2), when the number of branches grows without limit.
10.15.According to the strict definition [110,112,113] the diversity gain is the minimal
rank among all pairwise differences of distinct space–time codewords, i.e. nT n arrays [uit]. Prove that for the Alamouti code this diversity gain equals 2.
10.16.Find the rate (in data symbols per code symbol) and diversity gain of the space– time code with real symbol codewords [112]:
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10.17.Find the rate (in data symbols per code symbol) and diversity gain of the space– time code with complex symbol codewords [116]:
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(c) Calculate the IDFT of the bit pattern.
(d) Attach a cyclic prefix and plot the OFDM symbol obtained.
(e) Set a random channel delay profile, i.e. integer delays, amplitudes and phases of multiple paths; take the delay spread within 4–6, Rayleigh amplitudes and uniformly distributed over [ , ] phases, all independent of each other.
(f) Calculate and plot the OFDM symbol distorted by the channel.
(g) Discard the prefix and tail (due to a channel delay) samples, and calculate the DFT of the vector obtained.
(h) Calculate the channel transfer function and divide the DFT by it.
(i) Demodulate the samples obtained into bits.
(j) Plot the demodulated bit pattern and compare it with the transmitted one.
10.20. Write a program to analyse the effect of choice of orthogonal MC-CDMA signatures on the peak-factor of multicarrier symbols. Run the program for the Walsh-function signature set and complex signatures being cyclic shifts of the polyphase codes of Section 6.11.2. Explain the discrepancy in peak-factor for these two signature ensembles.
10.21. Write a program demonstrating the gain of Alamouti space–time coding versus the no-diversity system for BPSK data transmission over the Rayleigh channel. Recommended steps:
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Take a stream of Lbs ¼ 104 105 random independent bits. |
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Set their amplitudes independently according to the Rayleigh fading model |
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with the average square equal to one. |
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Add Gaussian noise of a suitable variance to have pre-assigned average bit |
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power SNR q2. |
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Demodulate the observation obtained and calculate the empirical bit error |
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rate. |
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Split the original bit stream into pairs of even and odd bits. |
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Encode every pair of even and odd bits by rule (10.37), forming two new |
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streams of length Lbs corresponding to two antennas. |
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Form two subchannel amplitude vectors, consisting respectively of even and |
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odd elements of the set of item (b); assign to every element of these vectors |
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random independent phase uniformly distributed over the interval [ , ]. |
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Use the first of vectors of item (g) to imitate Rayleigh fading in the first |
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subchannel, assigning its elements to every pair of consecutive bits of the |
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first antenna bit stream as complex amplitudes; do the same for the second |
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antenna bit stream using the second vector of the previous item. |
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Sum the vectors obtained, dividing the result by p2, and add the same noise |
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Demodulate the observation obtained according to rule (10.41), which for |
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(k)Calculate the empirical bit error rate and compare it to that of item (d).
(l)Run all previous steps, varying bit SNR, and build empirical dependences of the bit error rate on SNR for both transmission modes; compare the results with those predicted from Figure 10.9 and explain the discrepancy, if any.