1 J. G. Proakis, Digital Communications, 5th Edition, McGraw Hill, 2008. 2 Athanasios Papoulis, Probability, Random Variables, and Stochastic Processes, McGrawHill, 1991 (3rd Ed.), 20011 J. G. Proakis, Digital Communications, 5th Edition, McGraw Hill, 2008. 2 Athanasios Papoulis, Probability, Random Variables, and Stochastic Processes, McGrawHill, 1991 (3rd Ed.), 20011 J. G. Proakis, Digital Communications, 5th Edition, McGraw Hill, 2008. 2 Athanasios Papoulis, Probability, Random Variables, and Stochastic Processes, McGrawHill, 1991 (3rd Ed.), 20011 J. G. Proakis, Digital Communications, 5th Edition, McGraw Hill, 2008. 2 Athanasios Papoulis, Probability, Random Variables, and Stochastic Processes, McGrawHill, 1991 (3rd Ed.), 20011 J. G. Proakis, Digital Communications, 5th Edition, McGraw Hill, 2008. 2 Athanasios Papoulis, Probability, Random Variables, and Stochastic Processes, McGrawHill, 1991 (3rd Ed.), 2001
Chapter 5: Signaling on ISI Channels and Equalization Dept of Telecomm Eng Faculty of EEE Comm II 2013 DHT, HCMUT References [1] J G Proakis, Digital Communications, 5th Edition, McGrawHill, 2008 [2] Athanasios Papoulis, Probability, Random Variables, and Stochastic Processes, McGraw-Hill, 1991 (3rd Ed.), 2001 (4th Ed.) Dept of Telecomm Eng Faculty of EEE Comm II 2013 DHT, HCMUT Goals (1) In this chapter, we consider the problem of signal design when the channel is bandlimited to some specified bandwidth of W Hz Under this condition, the channel may be modeled as a linear filter having an equivalent lowpass frequency response C( f ) that is zero for | f | > W The first topic that is treated is the design of the signal pulse g(t) in a linearly modulated signal, represented as that efficiently utilizes the total available channel bandwidth W We shall see that when the channel is ideal for | f| ≤ W, a signal pulse can be designed that allows us to transmit at symbol rates comparable to or exceeding the channel bandwidth W On the other hand, when the channel is not ideal, signal transmission at a symbol rate equal to or exceeding W results in intersymbol interference (ISI) among a number of adjacent symbols Dept of Telecomm Eng Faculty of EEE Comm II 2013 DHT, HCMUT Goals (2) The second topic that we consider is the design of the receiver in the presence of intersymbol interference and AWGN The solution to the ISI problem is to design a receiver that employs a means for compensating or reducing the ISI in the received signal The compensator for the ISI is called an equalizer Dept of Telecomm Eng Faculty of EEE Comm II 2013 DHT, HCMUT Characterization of Band-limited Channel (1) A bandlimited channel such as a telephone channel will be characterized as a linear filter having an equivalent lowpass frequencyresponse characteristic C( f ) Its equivalent lowpass impulse response is denoted by c(t) Then, if a signal of the form: is transmitted over a bandpass telephone channel, the equivalent lowpass received signal is where the integral represents the convolution of c(t) with v(t), and z(t) denotes the additive noise Within the bandwidth of the channel, we may express the frequency response C( f ) as where |C( f )| is the amplitude-response characteristic and θ( f ) is the phase-response characteristic Dept of Telecomm Eng Faculty of EEE Comm II 2013 DHT, HCMUT Characterization of Band-limited Channel (2) Furthermore, the envelope delay characteristic is defined as A channel is said to be nondistorting or ideal if the amplitude response |C(f)| is constant for all | f | ≤ W and θ(f) is a linear function of frequency, i.e., τ (f) is a constant for all | f | ≤ W On the other hand, if |C(f)| is not constant for all | f | ≤ W, we say that the channel distorts the transmitted signal V(f) in amplitude, and, if τ(f) is not constant for all | f | ≤ W, we say that the channel distorts the signal V(f) in delay As a result of the amplitude and delay distortion caused by the nonideal channel frequency-response characteristic C(f), a succession of pulses transmitted through the channel at rates comparable to the bandwidth W are smeared to the point that they are no longer distinguishable as well-defined pulses at the receiving terminal Instead, they overlap, and, thus, we have intersymbol interference Dept of Telecomm Eng Faculty of EEE Comm II 2013 DHT, HCMUT Characterization of Band-limited Channel (3) Example: Effect of channel distortion: (a) channel input; (b) channel output; (c) equalizer output Dept of Telecomm Eng Faculty of EEE Comm II 2013 DHT, HCMUT Digital Transmissions Digital transmission techniques are becoming more popular in all areas of telecommunications Most of the new telecommunication systems are based on digital technology Baseband digital transmission: Digital PAM Line coding Nyquist pulse shaping Digital modulation methods: QAM, PSK, FSK, MSK digital modulations Dept of Telecomm Eng Faculty of EEE Comm II 2013 DHT, HCMUT Bits and Symbols The idea of digital transmission is to transmit bit sequences or, more generally, multilevel symbol sequences using PAM-modulation (pulse amplitude modulation) A multilevel symbol is obtained when several bits are combined into a symbol E.g., if bits are combined into symbol, then the number of bit combinations is 24 = 16 In general, B bits can be represented as M = 2B levels The number of levels or bit combinations depends on the application and on channel requirements, so that levels can be distinct in noisy channel If we combine several bits into one symbol, the symbol rate (or baud rate) is reduced This affects the transmitted signal bandwidth Transmitting bandwidth sets an upper limit to the symbol rate and noise causes errors Thus, bit/symbol rates and error probability play important roles in digital transmission (similar to bandwidth and S/N in analog transmission) Dept of Telecomm Eng Faculty of EEE Comm II 2013 DHT, HCMUT Digital PAM Signals (1) Digital message representation at baseband takes a form of an amplitude modulated pulse (PAM) train Digital PAM-signal is transmitted over the continuous-time channel as the following waveform: Here p(t) is a basic pulse waveform whose amplitude is scaled by the transmitted symbol ak It’s important that adjacent pulses not interfere with each other in the reception Ideally, this happens when the following condition holds: This condition ensures that we can recover the message by sampling x(t) periodically at t = KD, where K = ± 1, ± 2, … since: Dept of Telecomm Eng Faculty of EEE 10 Comm II 2013 DHT, HCMUT Smart Antennas (10) The general case for max SIR: It shows one desired signal arriving from the angle θ0 and N interferers arriving from angles θ1, , θN The signal and the interferers are received by an array of M elements with M potential weights Each received signal at element m also includes additive Gaussian noise Time is represented by the kth time sample Dept of Telecomm Eng Faculty of EEE 99 Comm II 2013 DHT, HCMUT Smart Antennas (11) The array output y can be given in the following form: where Dept of Telecomm Eng Faculty of EEE 100 Comm II 2013 DHT, HCMUT Smart Antennas (12) Therefore, the array output can be re-writen as where The (SIR) is defined as the ratio of the desired signal power divided by the undesired signal power where Dept of Telecomm Eng Faculty of EEE 101 Comm II 2013 DHT, HCMUT Smart Antennas (13) The final optimized weight for max SIR is where See example 8.2, [3]: The M = 3-element array with spacing d = 5λ has a noise variance σ2n = 001, a desired received signal arriving at θ = 30 , and two interferers arriving at angles θ1 = −30◦ and θ2 = 45◦ Assume that the signal and interferer amplitudes are constant Dept of Telecomm Eng Faculty of EEE 102 ◦ Comm II 2013 DHT, HCMUT Smart Antennas (14) Dept of Telecomm Eng Faculty of EEE 103 Comm II 2013 DHT, HCMUT Smart Antennas (15) Minimum variance (or minimum variance distortionless response): The goal of the minimum variance (MV) method is to minimize the array output noise variance The weighted array output is given by In order to ensure a distortionless response, we must also add the constraint that Finally, the minimum variance optimum weights can be obtained as where is the correlation matrix of unwanted signals and noise Dept of Telecomm Eng Faculty of EEE 104 Comm II 2013 DHT, HCMUT Smart Antennas (16) See example 8.5, [3] Dept of Telecomm Eng Faculty of EEE 105 Comm II 2013 DHT, HCMUT Smart Antennas (17) Adaptive Beamforming If the desired arrival angles change with time, it is necessary to devise an optimization scheme that operates on-the-fly so as to keep recalculating the optimum array weights The receiver signal processing algorithm then must allow for the continuous adaptation to an ever-changing electromagnetic environment The adaptive algorithm takes the fixed beamforming process one step further and allows for the calculation of continuously updated weights Least mean squares (LMS): The updated weight vector according to LMS is where Dept of Telecomm Eng Faculty of EEE 106 Comm II 2013 DHT, HCMUT Smart Antennas (18) Adaptive beamforming scheme: Dept of Telecomm Eng Faculty of EEE 107 Comm II 2013 DHT, HCMUT Smart Antennas (19) The convergence of the LMS algorithm is directly proportional to the step-size parameter μ If the step-size is too small, the convergence is slow If the convergence is slower than the changing angles of arrival, it is possible that the adaptive array cannot acquire the signal of interest fast enough to track the changing signal If the step-size is too large, the LMS algorithm will overshoot the optimum weights of interest If attempted convergence is too fast, the weights will oscillate about the optimum weights but will not accurately track the solution desired It is therefore imperative to choose a step-size in a range that insures convergence It can be shown that stability is insured provided if where λmax is the largest eigenvalue of the input correlation matrix Rxx Dept of Telecomm Eng Faculty of EEE 108 Comm II 2013 DHT, HCMUT Smart Antennas (20) See example 8.6 [3] Dept of Telecomm Eng Faculty of EEE 109 Comm II 2013 DHT, HCMUT Smart Antennas (21) Magnitude weights (example 8.6): Dept of Telecomm Eng Faculty of EEE 110 Comm II 2013 DHT, HCMUT Smart Antennas (22) Mean square error (example 8.6): Dept of Telecomm Eng Faculty of EEE 111 Comm II 2013 DHT, HCMUT Smart Antennas (23) In the example 8.6, the LMS algorithm did not converge until after 70 iterations 70 iterations corresponded to more than half of the period of the waveform of interest Dept of Telecomm Eng Faculty of EEE 112 Comm II 2013 DHT, HCMUT Smart Antennas (24) Array factor (example 8.6): Dept of Telecomm Eng Faculty of EEE 113 Comm II 2013 DHT, HCMUT ... Transmission Limitations (1) Digital baseband transmission model: The signal-plus-noise -and- interference waveform: stands for pulse shape with where td is transmission delay and transmission distortion... to spread out which would increase the ISI Consequently, the fundamental limitations of digital transmission is the relationship between ISI, bandwidth and signaling rate Dept of Telecomm Eng Faculty... management and spectrum shaping To remove the variation of DC-component in AC-coupled systems To avoid synchronization problems when the transmitted symbol train consists of long sequences with constant