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Demodulation, Bit-Error Probability and Diversity Arrangements

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Demodulation, Bit-Error Probability and Diversity Arrangements RADIO SYSTEMS – ETIN15 Lecture no: 6 Demodulation, bit-error probability and diversity arrangements Anders J Johansson, Department of Electrical and Information Technology [email protected] 5 April 2017 1 Contents • Receiver noise calculations [Covered briefly in Chapter 3 of textbook!] • Optimal receiver and bit error probability – Principle of maximum-likelihood receiver – Error probabilities in non-fading channels – Error probabilities in fading channels • Diversity arrangements – The diversity principle – Types of diversity – Spatial (antenna) diversity performance 5 April 2017 2 RECEIVER NOISE 5 April 2017 3 Receiver noise Noise sources The noise situation in a receiver depends on several noise sources Noise picked up Wanted by the antenna signal Output signal Analog Detector with requirement circuits on quality Thermal noise 5 April 2017 4 Receiver noise Equivalent noise source To simplify the situation, we replace all noise sources with a single equivalent noise source. Wanted How do we determine Noise free signal N from the other sources? N Output signal C Analog Detector with requirement circuits on quality Noise free Same “input quality”, signal-to-noise ratio, C/N in the whole chain. 5 April 2017 5 Receiver noise Examples • Thermal noise is caused by random movements of electrons in circuits. It is assumed to be Gaussian and the power is proportional to the temperature of the material, in Kelvin. • Atmospheric noise is caused by electrical activity in the atmosphere, e.g. lightning. This noise is impulsive in its nature and below 20 MHz it is a dominating. • Cosmic noise is caused by radiation from space and the sun is a major contributor. • Artificial (man made) noise can be very strong and, e.g., light switches and ignition systems can produce significant noise well above 100 MHz. 5 April 2017 6 Receiver noise Noise sources The power spectral density of a noise source is usually given in one of the following three ways: This one is 1) Directly [W/Hz]: Ns often given in dB and called 2) Noise temperature [Kelvin]: Ts noise figure. 3) Noise factor [1]: Fs The relation between the three is Ns kT s kF s T0 -23 where k is Boltzmann’s constant (1.38x10 W/Hz) and T0 is the, so called, room temperature of 290 K (17O C). 5 April 2017 7 Receiver noise Noise sources, cont. Antenna example N a Model Noise temperature Noise free of antenna 1600 K antenna Power spectral density of antenna noise is −23 −20 N a=1.38×10 ×1600=2.21×10 W/Hz=−196.6 dB [W/Hz] 5 April 2017 8 Receiver noise System noise The noise factor or noise figure of a system component (with input and output) is: N sys System System Model component component Noise factor F Noise free Due to a definition of noise factor as the ratio of noise powers on the output versus on the input, when a resistor in room temperature (T0=290 K) generates the input noise, the PSD of the equivalent noise source (placed at the input) becomes Nsys k F 1 T0 W/Hz Don’t use dB value! Equivalent noise temperature 5 April 2017 9 Receiver noise Several noise sources A simple example T a System 1 System 2 F F 1 2 Na kT a Noise N1 k F 1 1 T 0 free N N N N2 k F 2 1 T 0 a 1 2 System 1 System 2 Noise Noise free free 5 April 2017 10 Receiver noise Several noise sources, cont. After extraction of the noise sources from each component, we need to move them to one point. When doing this, we must compensate for amplification and attenuation! Amplifier: N NG G G Attenuator: N N/L 1/L 1/L 5 April 2017 11 Receiver noise Pierce’s rule A passive attenuator in room temperature, in this case a transmission line, has a noise factor equal to its attenuation. N f Lf Lf F = L Noise free f f Nf k F f 1 T0 k L f 1 T 0 Remember to convert from dB! 5 April 2017 12 Receiver noise Remember ... Antenna noise is usually given as a noise temperature! Noise factors or noise figures of different system components are determined by their implementation. When adding noise from several sources, remember to convert from the dB-scale noise figures that are usually given, before starting your calculations. A passive attenuator in (room temperature), like a transmission line, has a noise figure/factor equal to its attenuation. 5 April 2017 13 Receiver noise A final example Let’s consider two (incomplete) receiver chains with equal gain from point A to B: T a 1 Lf G1 G2 A B Would there be any reason to choose one F1 F2 over the other? Ta Let’s calculate the equivalent noise 2 Lf at point A for both! G1 G2 A B F1 F2 5 April 2017 14 Receiver noise A final example 1 2 Ta Ta Lf Lf G1 G2 G1 G2 A B A B F1 F2 F1 F2 Equivalent noise sources at point A for the two cases would have the power spectral densities: NkTkF1 L 1 / GF 1 LGT / 1 0a 1 f 1 2 f 1 0 2 NkTkL1 F 1 LF 1 LGT / 0a f 1 f 2 f 1 0 Two of the noise contributions are equal and two are larger in (2), which makes (1) a better arrangement. This is why we want a low-noise amplifier (LNA) close to the antenna. 5 April 2017 15 Receiver noise Noise power We have discussed noise in terms of power spectral density N0 [W/Hz]. For a certain receiver bandwidth B [Hz], we can calculate the equivalent noise power: N =B×N 0 [W] N ∣dB=B∣dBN 0∣dB [dBW] This is the version we will use in our link budget. 5 April 2017 16 Receiver noise The link budget ”POWER” [dB] PTX Lf, TX G a, TX Lp Ga, RX Lf, RX C F [dB] is the noise figure of the equivalent C / N noise source at the reference point and N B [dBHz] the system bandwidth . B The receiver noise calculations N 0 F show up here. Noise reference level = kT0 = -204 dB[W/Hz] Transmitter Receiver In this version the reference point is here 5 April 2017 17 OPTIMAL RECEIVER AND BIT ERROR PROBABILITY 5 April 2017 18 Optimal receiver What do we mean by optimal? Every receiver is optimal according to some criterion! We would like to use optimal in the sense that we achieve a minimal probability of error. In all calculations, we will assume that the noise is white and Gaussian – unless otherwise stated. 5 April 2017 19 Optimal receiver Transmitted and received signal Transmitted signals Channel Received (noisy) signals s1(t) r(t) 1: t n(t) t s(t) r(t) s0(t) r(t) 0: t t 5 April 2017 20 Optimal receiver A first “intuitive” approach “Look” at the received signal and compare it to the possible received noise free signals. Select the one with the best “fit”. Assume that the following Comparing it to the two possible signal is received: noise free received signals: r(t), s1(t) r(t) 1: This seems to be t the best “fit”. We assume that “0” t was the r(t), s2(t) transmitted bit. 0: t 5 April 2017 21 Optimal receiver Let’s make it more measurable To be able to better measure the “fit” we look at the energy of the residual (difference) between received and the possible noise free signals: s1(t) - r(t) r(t), s1(t) 2 1: e1= ∣s1 t −r t ∣ dt t t ∫ s0(t) - r(t) r(t), s0(t) 2 0: e 0=∫∣s0 t −r t ∣ dt t t This residual energy is much smaller. We assume that “0” was transmitted. 5 April 2017 22 Optimal receiver The AWGN channel The additive white Gaussian noise (AWGN) channel n t s t r t s t n t s t - transmitted signal In our digital transmission system, the transmitted - channel attenuation signal s(t) would be one of, n t - white Gaussian noise let’s say M, different alternatives s0(t), s1(t), ... , sM-1(t). r t - received signal 5 April 2017 23 Optimal receiver The AWGN channel, cont. It can be shown that finding the minimal residual energy (as we did before) is the optimal way of deciding which of s0(t), s1(t), ... , sM-1(t) was transmitted over the AWGN channel (if they are equally probable). For a received r(t), the residual energy ei for each possible transmitted alternative si(t) is calculated as 2 * ei=∫∣r t− si t∣ dt=∫ r t− si t r t− si t dt 2 * * 2 2 =∫∣r t∣ dt−2 Re { ∫ r t si t dt }∣∣ ∫∣si t∣ dt Same for all i Same for all i, if the transmitted The residual energy is minimized by signals are of maximizing this part of the expression. equal energy. 5 April 2017 24 Optimal receiver The AWGN channel, cont. The central part of the comparison of different signal alternatives is a correlation, that can be implemented as a correlator: r t ∫ T s * * si t The real part of the output from or a matched filter either of these is sampled at t = Ts r t * si T s t * where Ts is the symbol time (duration). 5 April 2017 25 Optimal receiver Antipodal signals In antipodal signaling, the alternatives (for “0” and “1”) are s0 t t s1 t t This means that we only need ONE correlation in the receiver for simplicity: If the real part r t at T=T is ∫ s T s >0 decide “0” * <0 decide “1” * t 5 April 2017 26 Optimal receiver Orthogonal signals In binary orthogonal signaling, with equal energy alternatives s0(t) and s1(t) (for “0” and “1”) we require the property: * 〈 so t, s1t〉=∫ s0t s1t dt=0 The approach here is to use two correlators: ∫ T s Compare real * * part at t=T s s0 t r t and decide in favor of the ∫ larger.
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