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 C amplifiers (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

Discontinuous Phase FSK Switching between 2 independent oscillators for binary 1 & 0 switch cos w2t cos w1t input data phase jumps sBFSK(t)= vH(t) binary 1 = binary 0 sBFSK(t)= vL(t) results in phase discontinuities discontinuities causes spectral spreading & spurious transmission not suited for tightly designed systems

Continuous Phase FSK sBFSK(t) = 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)

FSK Example 0 1 1 x a0 VCO modulated composite a1 signal 1 cos wct 1 1 0 1 Data FSK Signal 0 1 1 VCO cos wct x 1 a0 a1 modulated composite signal

Spectrum & Bandwidth of BFSK Signals complex envelope of BFSK is nonlinear function of m(t) spectrum evaluation - difficult - performed using actual time averaged measurements PSD of BFSK consists of discrete frequency components at fc fc  nf , 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 from fc

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 = 2f +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 pulse shaping used, bandwidth reduced to: BT = 2f +(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) = = vH(t) vL(t) = then An FSK signal for 0 ≤ t ≤ Tb and vH(t) vL(t) are orthogonal if Δf sin(4πfcTb) = -fc(sin(4πΔf Tb)

CPFSK Modulation elimination of phase discontinuity improves spectral efficiency & noise performance consider binary CPFSK signal defined over the interval 0 ≤ t ≤ T s(t) = θ(t) = phase of 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 0 ≤ t ≤ T

To determine fc and h by substitution 2πfct + θ(0) + = 2πf2 t+ θ(0) 2πfct + θ(0) - = 2πf1t+ θ(0) f1 = f2 = yields and thus fc= h = T(f2 – f1) nominal fc = mean of f1 and f2 h ≡ f2 – f1 normalized by T

FSK modulation index = kFSK (similar to FM modulation index) symbol ‘1’  θ(T) - θ(0) = πh symbol ‘0’  θ(T) - θ(0) = -πh θ(T) = θ(0) ± πh At t = T  ‘1’ sent  increases phase of s(t) by πh ‘0’ sent  decreases phase of s(t) by πh variation of θ(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) kFSK = peak frequency deviation F = |fc-fi | =

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) rads 3πh 2πh πh -πh -2πh -3πh 0 T 2T 3T 4T 5T 6T t

 thus change in phase over T is either π or -π Phase Tree is a manifestation of phase continuity – an inherent characteristic of CPFSK 1 0 0 0 0 1 1 0 ≤ t ≤ T θ(t) = θ(0) ± θ(t) - (0) 3π 2π π -π -2π -3π 0 T 2T 3T 4T 5T 6T t  thus change in phase over T is either π or -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

CPFSK = continuous phase FSK phase continuity during inter-bit switching times si(t) = 0 ≤ t ≤ T = 0 otherwise for i = 1, 2 assume fi given by as fi = nc = fixed integer si(t) = 0 ≤ t ≤ T for i = 1, 2 = 0 otherwise

BFSK constellation: define two coordinates as for i = 1, 2 i(t) = 0 ≤ t ≤ T = 0 otherwise let nc = 2 and T = 1s (1Mbps) then f1 = 3MHz, f2 = 4MHz 1(t) = 0 ≤ t ≤ T = 0 otherwise 2(t) =

BFSK Constellation s1(t) = 0 ≤ t ≤ T = = 0 otherwise 0 ≤ t ≤ T s2(t) = BFSK Constellation s1(t) = 0 ≤ t ≤ T = 0 otherwise = 0 ≤ t ≤ T s2(t) = = 0 otherwise =

Coherent BFSK Detector 2 correlators fed with local coherent reference signals difference in correlator outputs compared with threshold to determine binary value output + - r(t) Decision Circuit sin wct cos wct  Pe,BFSK = Probability of error in coherent FSK receiver given as:

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 r(t) output Decision Circuit + -  Envelope Detector Matched Filter fL Tb fH Average probability of error in non-coherent FSK receiver: Pe,BFSK, NC =