# Revision of Lecture 3

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1 Revision of Lecture 3 Modulator/demodulator Basic operations of modulation and demodulation Complex notations for modulation and demodulation Carrier recovery and timing recovery This lecture: bits map symbols MODEM components pulse shaping Tx/Rx filter pair modulator/demodulator bits map symbols equalisation (distorting channel) Recall that to transmit at a rate f s requires at least baseband bandwidth of f s 2 Can you see why do we want to group several bits into a symbol? 40

2 Bits to Symbols The bit stream to be transmitted is serial to parallel multiplexed onto a stream of symbols with q bits per symbol (discrete 2 q levels) Example for q = 2 bits per symbol (4-ary modulation): symbol period T s is twice of bit period T b bit stream time 0 symbol stream z(k) (1,1) (1,0) (0,1) (0,0) time Symbol rate is half of bit rate; symbol stream is then pulse shaped... (what happens to required bandwidth?) 41

3 Mapping to Constellation Pattern It is typical practice to describe a symbol x(k) by a point in constellation diagram, i.e. its in-phase and quadrature components, x i (k) and x q (k) Example for a case of q = 2 bits per symbol (QPSK): From the constellation pattern, the values x i (k) and x q (k) of symbol x(k) are determined There is a one-to-one relationship between symbol set (constellation diagram) and modulation signal set (actually transmitted modulated signal) x q ( k) (0,1) (0,0) (1,1) (1,0) x i ( k) (0,1) (0,0) In the receiver, the constellation point and therefore the transmitted symbol value is determined from the received signal sample ˆx(k) x(k) ^ (1,1) (1,0) 42

4 Phase Shift Keying (PSK) In PSK, carrier phase used to carry symbol information, and modulation signal set: s i (t) = A cos(ω c t + φ i (t)), 0 t T s, 1 i M = 2 q where T s : symbol period, A: constant carrier amplitude, M: number of symbol points in constellation diagram Phase carries symbol information, namely to transmit i-th symbol value (point), signal s(t) = s i (t) is sent, note: s(t) = A cos(ω c t+φ i (t)) = A cos(φ i (t)) }{{} inphse symbol x i (t) cos(ω c t)+ ( A sin(φ i (t))) }{{} sin(ω c t) quadrature symbol x q (t) Recall previously in slide 29, we say transmitted signal is s(t) = x i (t) cos(ω c t) + x q (t)sin(ω c t) 43

5 Binary Phase Shift Keying (BPSK) One bit per symbol, note the mapping from bits to symbols in constellation diagram, where quadrature branch is not used S (t) 1 S (t) 2 Q bit 1 bit 0 I T b T b Modulation signal set s i (t) = A cos(ω c t + φ i ), i = 1, 2 Phase separation: π bit 0 or symbol 1: φ 1 = 0 bit 1 or symbol 2: φ 2 = π 44

6 Quadrature Phase Shift Keying (QPSK) Two bits per symbol with a minimum phase separation of π 2 (0,1) xq( k) A QPSK constellation diagram: (A different one shown in slide 42) Modulation signal set (1,1) (0,0) xi( k) s i (t) = A cos(ω c t + φ i ), 1 i 4 (1,0) S (t) 1 S (t) 2 S (t) S (t) 3 4 T s Ts T s Ts bit (0,0): φ 1 = 0 bit (0,1): φ 2 = π 2 bit (1,1): φ 3 = π bit (1,0): φ 4 = 3π 2 45

7 Amplitude Shift Keying (ASK) Pure ASK: carrier amplitude is used to carry symbol information An example of 4-ASK with constellation diagram and modulation signal set s i (t) = A i cos(ω c t), 1 i 4 Q I T s T T T s s s (1,0): A 1 (1,1): A 2 (0,1): A 3 (0,0): A 4 Note quadrature branch is not used, pure ASK rarely used itself as amplitude can easily be distorted by channel 46

8 Combined ASK / PSK PSK and ASK can be combined. Here is an example of 4-ary or 4-PAM (pulse amplitude modulation) with constellation pattern and transmitted signal s(t): xq( k) (00) (01) (11) (00) (10) (1,1) (1,0) (0,1) (0,0) xi( k) time t 2 amplitude levels and phase shift of π are combined to represent 4-ary symbols Note in M-ary or M-PAM, quadrature component is not used, a more generic scheme of combining PSK/ASK is QAM, which uses both I and Q branches 47

9 Quadrature Amplitude Modulation (QAM) QAM: combines features of PSK and ASK, uses both I and Q components, and is bandwidth very efficient An example of (squared) 16-QAM: xq( k) Note for squared M-QAM, I and Q branches are both M-ary (of previous slide) xi( k) Depending on the channel quality, 64-QAM, 128-QAM, or 256-QAM are possible Why high-order QAM particularly bandwidth efficient? and what is penalty paid? 48

10 Gray Mapping Gray coding: adjacent constellation points only differ in a single bit (minimum Hamming distance) (0000) (0001) xq( k) (0011) (0010) Gray coded (0111) ^ x(k) (0110) (0100) (0101) (0111) (0110) (1100) (1101) (1111) (1110) xi( k) Non gray coded (1111) (0110) (1000) (1001) (1011) (1010) ^ x(k) Symobl (0110) was sent but received sample in neighbor region due to noise If noise or distortions cause misclassification in the receiver, Gray coding can minimise the bit error rate 49

11 Eye Diagram Perfect Channel By now we have discussed all components of MODEM, and we turn to channel again We are looking at stacked 2 symbol period intervals of the demodulated signal ˆx i (t) in a QPSK scheme (ˆx i (t) is BPSK): x i (t) time /symbol periods This is called an eye diagram; ideal sampling of ˆx i (k) will sample the crossing points ˆx i (t) = ±1 clock/timing recovery (τ 0.85T s or t k = kt s T s ) 50

12 Eye Diagram Noisy Channel With channel noise at 10 db SNR, the eye diagram looks different: x i (t) time /symbol periods As long as the sampling points can be clearly determined and the eye is open, ˆx i (k) will correctly resemble x i (k) At higher noise levels, misclassifications can occur if the eye is closed 51

13 Eye Diagram Distorting Channel Non-ideal channel with an impulse response c(t) = δ(t) 1 2 δ(t T s/4), where T s is the symbol period: x i (t) time /symbol periods The eye diagram is distorted; this intersymbol interference together with noise effect will make the eye completely closed, leading to misclassification 52

14 Intersymbol Interference (ISI) Combined impulse response of an ideal pulse shaping filter of regular zero crossings with ideal channel g c (t) = δ(t) and non-ideal channel g c (t) = δ(t) 1 2 δ(t t s/4): txrx filter * channel time / symbol periods For non-ideal channel, the combined Tx-filter channel Rx filter has lost the property of a Nyquist system, no longer has regular zero crossings at symbol spacing 53

15 Dispersive Channel Recall that zero ISI is achieved if combined Tx and Rx filters is a Nyquist system But this is only true if the channel is ideal G Tx (f)g c (f)g Rx (f) = G Tx G Rx (f) If G c (f) is non-ideal, G Tx (f)g c (f)g Rx (f) will not be a Nyquist system; example of a distorting channel: Amplitude Spectrum (db) transmission bandwidth Dispersive channel is caused by: (i) a restricted bandwidth (channel bandwidth is insufficient for the required transmission rate); or (ii) multipath distorting Equalisation is needed for overcoming this channel distortion (next lecture) f 54

16 Summary Mapping bit stream to symbol stream Symbol constellation and modulation signal set Modulation schemes: PSK, ASK, PAM (Q-ary) and QAM Gray code Eye diagram, effects of noise Effects of dispersive channel 55

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