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## Constellation Diagram

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**Constellation Diagram**• Properties of Modulation Scheme can be inferred from the Constellation Diagram: • Bandwidth occupied by the modulation increases as the dimension of the modulated signal increases. • Bandwidth occupied by the modulation decreases as the signal_points per dimension increases (getting more dense). • Probability of bit error is proportional to the distance between the closest points in the constellation. • Euclidean Distance • Bit error decreases as the distance increases (sparse).**Linear Modulation Techniques**• Digital modulation techniques classified as: • Linear • The amplitude of the transmitted signal varies linearly with the modulating digital signal, m(t). • They usually do not have constant envelope. • More spectrally efficient. • Poor power efficiency • Example: QPSK. • Non-linear / Constant Envelope**Constant Envelope Modulation**• constant carrier amplitude - regardless of variations in m(t) • Better immunity to fluctuations due to fading. • Better random noise immunity. • improved power efficiency without degrading occupied spectrum • - use power efficient class Camplifiers (non-linear) • low out of band radiation (-60dB to -70dB) • use limiter-discriminator detection • - simplified receiver design • high immunity against random FM noise & fluctuations • from Rayleigh Fading • larger occupied bandwidth than linear modulation**Constant Envelope Modulation**• Frequency Shift Keying • Minimum Shift Keying • Gaussian Minimum Shift Keying**Frequency Shift Keying (FSK)**• Binary FSK • Frequency of the constant amplitude carrier is changed according to the message state high (1) or low (0) • Discontinuous / Continuous Phase**input data**phase jumps cos w1t switch cos w2t = sBFSK(t)= vH(t) binary 1 binary 0 sBFSK(t)= vL(t) = Discontinuous Phase FSK Switching between 2 independent oscillators for binary 1 & 0 • results in phase discontinuities • discontinuities causes spectral spreading & spurious transmission • not suited for tightly designed systems**Continuous Phase FSK**single carrier that is frequency modulated using m(t) sBFSK(t) = = where (t) = • m(t) = discontinuous bit stream • (t) = continuous phase function proportional to integral of m(t)**x**a0 a1 0 1 VCO Data 1 1 0 1 FSK Signal 0 1 1 modulated composite signal cos wct FSK Example**complex envelope of BFSK is nonlinear function of m(t)**• spectrumevaluation - difficult - performed using actual time • averaged measurements • PSD of BFSK consists of discretefrequency components at • fc • fc nf , n is an integer • PSD decay rate (inversely proportional to spectrum) • PSD decay rate for CP-BFSK • PSD decay rate for non CP-BFSK • f = frequency offset fromfc Spectrum & Bandwidth of BFSK Signals**Spectrum & Bandwidth of BFSK Signals**• Transmission Bandwidth of BFSK Signals (from Carson’s Rule) • B = bandwidth of digital baseband signal • BT = transmission bandwidth of BFSK signal • BT= 2f +2B • assume 1st null bandwidth used for digital signal, B • - bandwidth for rectangular pulses is given by B = Rb • - bandwidth of BFSK using rectangular pulse becomes • BT = 2(f + Rb) • if RC pulseshaping used, bandwidth reduced to: • BT = 2f +(1+) Rb**General FSK signal and orthogonality**• Two FSK signals, VH(t) and VL(t) are orthogonal if ? • interference between VH(t) and VL(t) will average to 0 during • demodulation and integration of received symbol • received signal will contain VH(t) and VL(t) • demodulation of VH(t) results in (VH(t) + VL(t))VH(t) ?**vH(t) =**vL(t) = and then and vH(t)vL(t) = = = = An FSK signal for 0 ≤ t ≤ Tb vH(t)vL(t) are orthogonal if Δf sin(4πfcTb) = -fc(sin(4πΔf Tb)**consider binary CPFSK signal defined over the interval 0**≤ t ≤ T s(t) = • θ(t) = phaseof CPFSK signal • θ(t) is continuous s(t) is continuous at bit switching times • θ(t) increases/decreases linearly with t during T θ(t) = θ(0) ± ‘+’ corresponds to ‘1’ symbol ‘-’ corresponds to ‘0’ symbol h = deviation ratio of CPFSK CPFSK Modulation elimination of phase discontinuity improves spectral efficiency & noise performance 0 ≤ t ≤ T**2πfct +θ(0) +**= 2πf2 t+θ(0) f1 = 2πfct +θ(0) - = 2πf1t+θ(0) fc= f2 = yields and thus h = T(f2 – f1) To determine fc and h by substitution • nominal fc= mean of f1 and f2 • h≡f2 – f1 normalized by T**At t = T**θ(T) = θ(0) ± πh kFSK= symbol‘1’ θ(T) - θ(0) = πh symbol‘0’ θ(T) - θ(0) = -πh • peak frequency deviation F = |fc-fi | = ‘1’ sent increases phase of s(t) by πh ‘0’ sent decreases phase of s(t) by πh • variation ofθ(t)with t follows a path consisting of straight lines • slope of lines represent changes in frequency FSK modulation index =kFSK (similar to FM modulation index)**θ(t) - (0) rads**3πh 2πh πh 0 -πh -2πh -3πh 0 T 2T 3T 4T 5T 6T t Phase Tree • depicted from t = 0 • phase transitions across • interval boundaries of • incoming bit sequence • θ(t) - θ(0) = phase of CPFSK signal is even or odd multiple of πh at even or odd multiples of T**θ(t) - (0)**3π 2π π 0 -π -2π -3π 0 T 2T 3T 4T 5T 6T t θ(t) = θ(0) ± 0 ≤ t ≤ T Phase Tree is a manifestation of phase continuity – an inherent characteristic of CPFSK 1 0 0 0 0 1 1 • thus change in phase over T • is either πor -π • change in phase of π = change in phase of -π • e.g. knowing value of bit i doesn’t help to find the value of bit i+1**fi=**nc = fixed integer assume fi given by as si(t) = 0 ≤ t ≤ T for i = 1, 2 si(t) = 0 ≤ t ≤ T for i = 1, 2 = 0 otherwise = 0 otherwise • CPFSK = continuous phase FSK • phase continuity during inter-bit switching times**1(t) =**2(t) = 0 ≤ t ≤ T 0 ≤ t ≤ T = 0 otherwise = 0 otherwise for i = 1, 2 i(t) = 0 ≤ t ≤ T = 0 otherwise BFSK constellation: define two coordinates as let nc = 2 and T = 1s (1Mbps) then f1= 3MHz,f2 = 4MHz**0 ≤ t ≤ T**s1(t) = = 0 otherwise 0 ≤ t ≤ T = = s2(t) = = 0 otherwise 2(t) 0 1 1(t) BFSK Constellation**cos wct**output + - Decision Circuit r(t) sin wct Probability of error in coherent FSK receiver given as: Pe,BFSK = Coherent BFSK Detector • 2 correlators fed with local coherent reference signals • difference in correlator outputs compared with threshold to • determine binary value**Matched Filter**fH Envelope Detector + - r(t) output Tb Decision Circuit Envelope Detector Matched Filter fL Pe,BFSK, NC = Non-coherent Detection of BFSK • operates in noisy channel without coherent carrier reference • pair of matched filters followed by envelope detector • - upper path filter matched to fH (binary 1) • - lower path filter matched to fL (binary 0) • envelope detector output sampled at kTb compared to threshold Average probability of error in non-coherent FSK receiver: