1 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 f c f c  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 f c Spectrum & Bandwidth of BFSK Signals

2 Transmission Bandwidth of BFSK Signals (from Carson’s Rule) B = bandwidth of digital baseband signal B T = transmission bandwidth of BFSK signal B T = 2  f +2B assume 1 st null bandwidth used for digital signal, B - bandwidth for rectangular pulses is given by B = R b - bandwidth of BFSK using rectangular pulse becomes B T = 2(  f + R b ) if RC pulse shaping used, bandwidth reduced to: B T = 2  f +(1+  ) R b Spectrum & Bandwidth of BFSK Signals

3 General FSK signal and orthogonality Two FSK signals, V H (t) and V L (t) are orthogonal if interference between V H (t) and V L (t) will average to 0 during demodulation and integration of received symbol received signal will contain V H (t) and V L (t) demodulation of V H (t) results in (V H (t) + V L (t))V H (t) ? ?

4 = v H (t) v L (t) = then = = and v H (t) v L (t) are orthogonal if Δf sin(4πf c T b ) = -f c (sin(4πΔf T b ) An FSK signal for 0 ≤ t ≤ T b v H (t) = v L (t) = and

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

6 To determine f c and h by substitution 2πf c t + θ(0) += 2πf 2 t+ θ(0)2πf c t + θ(0) -= 2πf 1 t+ θ(0) thus fc=fc= h = T(f 2 – f 1 ) nominal f c = mean of f 1 and f 2 h ≡ f 2 – f 1 normalized by T f 1 =f 2 =yieldsand

7 ‘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 symbol ‘1’  θ(T) - θ(0) = πh symbol ‘0’  θ(T) - θ(0) = -πh θ(T) = θ(0) ± πh At t = T  FSK modulation index = k FSK (similar to FM modulation index) k FSK = peak frequency deviation  F = |f c -f i | =

8 θ(t) - (0) r ads 3πh 2πh πh 0 -πh -2πh -3πh 0 T 2T 3T 4T 5T 6T t  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 Phase Tree

9 Phase Tree is a manifestation of phase continuity – an inherent characteristic of CPFSK 0 ≤ t ≤ Tθ(t) = θ(0) ±  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 θ(t) - (0) 3π 2π π 0 -π -2π -3π 0 T 2T 3T 4T 5T 6T t

10 assume f i given by asfi =fi =n c = fixed integer CPFSK = continuous phase FSK phase continuity during inter-bit switching times s i (t) =0 ≤ t ≤ T for i = 1, 2 = 0 otherwise s i (t) =0 ≤ t ≤ T = 0 otherwise for i = 1, 2

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

12 0 ≤ t ≤ T s 2 (t) = = 0 otherwise = s 1 (t) = 0 ≤ t ≤ T = 0 otherwise =  2 (t)  1 (t) 1 0 BFSK Constellation

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

14 operates in noisy channel without coherent carrier reference pair of matched filters followed by envelope detector - upper path filter matched to f H (binary 1) - lower path filter matched to f L (binary 0) envelope detector output sampled at kT b  compared to threshold P e,BFSK, NC = Average probability of error in non-coherent FSK receiver: r(t) output Decision Circuit +-+-  Envelope Detector Matched Filter f L Envelope Detector TbTb Matched Filter f H Non-coherent Detection of BFSK

15 Non-coherent Quadrature BFSK Detector output Decision Circuit r(t) ++++ (  2/T) cos w H t  (  2/T) sin w H t (.) 2 Z 1 (T) I-channel Q-channel (.) 2 Z 2 (T)  +-+- Z 3 (T) I-channel ++++ (  2/T) cos w L t  (  2/T) sin w L t (.) 2 Q-channel (.) 2 Z 4 (T)

16 Tutorial Derive minimum frequency spacing (f 2 – f 1 ) for  Non-coherent detection (arbitrary phase  ) Coherent detection