X b s t w t t dt b E ( ) t dt

Transcription

1 Consider the following correlator receiver architecture: T dt X si () t S xt () * () t Wt () T dt X Suppose s (t) is sent, then * () t t T T T X s t w t t dt E t t dt w t dt E W t t T T T X s t w t t dt E ( ) t dt w t dt W 76

2 Likewise, if s t is sent, then X W and X E W. Suppose all symols are transmitted with equal proaility, then the decision regions in the -dimensional oservation space are symmetric (see next figure). s R E X s X R E Oservation space and decision regions M if the random oservation vector T Decision strategy: choose X X X in Rand choose Mif X falls in R. Equivalently, choose M if X X or choose M if X X. falls 77

3 Alternatively, consider the following modification of the correlator receiver: T dt X si () t S xt () * () t S Y Decision ˆk Wt () T dt X threshold = * () t Thus, if we choose Y X X as the decision variale, then a decision strategy ased on a single variale can e implemented, namely, select M if Y >, otherwise, select M. Since oth X and X are Gaussian, then Y is also Gaussian. Now, M M E E W EW E E EW E + E M M EW E E W E E E Y M E X X M E X M E X M E E W E W E Y M E X M E X M E W E E W 78

4 E Y - E Y M M = E E + W - W - E M = E W - W M * * = E W - W = E W - E W W - E W W + E W N N o o = = No and * * E Y E Y M M N o, ecause Y G E, N o E W W E W W. y E y E N o No Y M e Y M e πno πno f y m and f y m Thus, the average BER is given y P P Y and M sent P Y and M sent e P Y M sent P M sent P Y M sent P M sent. 79

5 P Msent P Msent, then Pe PY M sent PY M sent y E E u N o No PY M sent e dy e du πn π If o E E N π N o o v e dv Q, y E u N E o PY M sent e dy E e du Q πn o N π o N o E E and Pe Q =Q. N o N o 8

6 Note that coherent FSK has the same performance as OOK. Generation of coherent inary FSK mt k On-off level encoder Inverter mt t + + si t t An implementation of inary FSK 8

7 The aseand power spectrum of coherent inary FSK is given y S f δ f δ f 8E cos E FSK, B T T T π 4T f πt f Linear and db plots of the PSD of inary FSK, T s, are shown in the next two figures f in Hz PSD of inary FSK in Watts/Hz 8

8 - - - db f in Hz PSD of inary FSK in db/hz 83

9 Differential Phase-Shift Keying (DPSK) It is the noncoherent version of PSK. Thus, it can e noncoherently demodulated at the receiver. Briefly, to send, we phase advance the current signal waveform y π radians, and to send a we leave the phase of the current signal waveform unchanged. To achieve this, we must process the input it stream at the logical level y performing a simple pre-coding operation. Suppose a is transmitted in the interval [, T ], then the symol is either descried y s t E cos π f c t, t T T E cos π f c t, T t T T when a is transmitted in t T,T 84

10 or s t E cos π f c t, t T T E E cosπ fct π cosπ fct, T t T T T when a is transmitted in t T,T. Let us now find out how s (t) and s (t) are related to each other. cos ct T T E T E s t s tdt cos ctdt cosct cosct πdt T T T T E T cos t cos t dt dt T c c T s t s t in the interval,t,t s and Es E. 85

11 Generation of DPSK: The input it sequence is modified y an encoder as follows: dk k dk In lock diagram form, the DPSK modulator is implemented as shown in the next figure. k d k T delay d k amplitude level shifter (±B) B E cos π f t T c sdpsk t DPSK modulator Example of DPSK encoding for the sequence. In order for the pre-coding to work properly, the encoder has to e initialized with a desired initial state (since the pre-coder contains a memory element). 86

12 k k d k d π π k Transmitted phase The following plot shows the DPSK modulated waveform that is transmitted when the it rate is R ps and the carrier frequency is R Hz, assuming an initial condition of dk "" t/t DPSK waveform representation of it sequence 87

13 Consider the following noncoherent receiver with a phase error : x di T dt X I xt cos t θ c sin t θ c x dq T dt X Q T delay T delay + + Y Decision threshold = ˆ k A DPSK noncoherent demodulator 88

14 With the phase error θ at the local oscillator, the output of the down converter is E given y (assuming no noise and A ). T A xdi A cosc t θcosct cos ct θ cosθ A xdq A sinct θcosct sin ct θ sinθ, when a is transmitted in [, T ], and A xdi A cosct θcosct cosct θ cosθ A xdq A sinct θcosct sinct θ sinθ when a is transmitted in [, T ], 89

15 At the output of the integrator, assuming θ is constant for t nt, ( n ) T, we get AT AT XI cos, XQ sinθ, AT AT Thus, X I, XQ is either cosθ, sinθ or AT AT cosθ, sin θ. XI X I Let X X e the oservation at t T and X Q X e the oservation Q at t T. Then X and X point in the same direction if their inner product is positive, otherwise, they point in different directions and the transmitted it in interval T, T is different than that transmitted in interval, T. Mathematically, "" T X X X I XI X Q XQ Y 9 ""

16 Let s now otain the performance of the receiver in the presence of AWGN. Suppose t, T, then for a given θ the conditional error s tis transmitted, proaility is descried y T e P, s P X X s t sent, θ T T X x t cos t θ dt Acos t W t cos t θ dt I c c c AT T cosθ W, W W t cos t θ dt I I c T T X x t sin t θ dt Acos t W t sin t θ dt Q c c c AT T sinθ W, W W t sin t θ dt Q Q c T AT T XI xtcosct θdt cosθ W I, WI Wtcosct θdt T T T AT T XQ xtsinct θdt sinθ W Q, WQ Wtsinct θdt, T T 9

17 where W I,W Q,W I, and W Q are uncorrelated, zero-mean Gaussian r.v. s with variances NT / 4 they are independent. Hence, AT AT AT AT cosθ WI cosθ WI sin θ WQ sin θ WQ P e θ,s P s t,θ Let us simplify things a it y introducing a new variale that is equal to the dot product of the two vectors, i.e. T Y X X XI XI XQ XQ XI XI XQ XQ 4 AT cosθ W W AT sinθ W W W W W W 4 I I Q Q I I Q Q With this definition, the event {Y < } is equivalent to the event AT cosθ W W AT sinθ W W I I WQ W Q W I I Q WQ 9

18 WI WI WI W W I Q WQ WQ WQ Let W, W, W 3, W 4, NT then Wi G,, i,,3,4. 8 Let AT AT 3 4 R cosθ W sinθ W, R W W, then Ris a Ricean r.v. and R is a Rayleigh r.v. (this will e proved when we consider non-coherent inary FSK detection), and the conditional proaility of a inary error, given that we know oth and s is given y P s P R R s t sent, θ P R R s t sent, θ e f r,r s,θ dr dr f r r, s,θ f r s,θ dr dr R R S, R R,S, R S, f r f r s,θ dr dr f r f r s,θ dr dr R R S, R R S, r 93

19 r r r r B B fr r e and f R, r s,θ e I r, N T AT where and B. Clearly, fr S, r s,θ fr S r s, independent 8 of θ, which implies that P e s P e s P s = BER s, or Explicitly, S BER s f r dr f r s dr If S r R R r r r r B B e dr e I r dr r σ P S i x r r B B -e e I r dr, x e B r / r σ B A T E B B B e N N 4 4 re I r dr e e e e σ, i,,3,4 4 N, then if we average out over the 4 symols, r E BER e. 94

20 The following plot shows the BER performance of DPSK in AWGN. DPSK BER performance in AWGN 95

21 Noncoherent Detection of Binary FSK Let the transmitted signal e descried y E cos i t, t T, i, si t T, elsewhere where ω and ω are such that s t s t Let. cos t t, t T, i,. Then s t E t, i,., elsewhere i i T i i 96

22 Consider the following noncoherent receiver: * t T T dt dt X Y + + L Xt ˆ* t * t T T dt dt X Y + + L ˆ* t Noncoherent receiver for orthogonal FSK 97

23 ˆ π t cos t sin t, t T, i,. Where i i i T T Assuming an AWGN channel, the received signal e descried y then E xt cosit θ Wt, t T, θ U π, π, T E E x t cosθ cosit sin θ sinit W t, i, T T ˆ i i E cosθ t E sin θ t W t, i, 98

24 Consider the L path: T * X x t t dt T ˆ * E cosθ i t E sin θ i t W t t dt E cosθ W, if i, θ is constant in t,t W, if i T * Y x t t dt T ˆ ˆ * ˆ E cosθ i t E sin θ i t W t t dt E ˆ sinθ W, if i, θ is constant in t,t Ŵ, if i 99

25 Likewise, for the L path X Y E cosθ W, i W, i E ˆ sinθ W, i Ŵ, i Now, Li Xi + Y i, i,. Both ˆ N W i and Wi G,, i,. Since the receiver is symmetric, Pchoosing M M sent = Pchoosing M M sent. Hence, we need to compute only one of them. that L L Suppose M was sent in t T, then an error is made if Wt is such. In this case, L M E cos W E sin Wˆ L M W Wˆ.

26 But, L Msent has a Rayleigh distriution, i.e. its pdf is descried y f L M m N, otherwise N e, and Now, P error M sent P L L L,M m f m d. L M P L L L,M f m d d and N N N l L M e e e N L m sent X Y. N N For a given θ, X G E cosθ, and Y G E sinθ,. x E cos θ E sinθ N y Thus, fx,y M, x,y m,θ e, πn ecause X and Y are independent of each other.

27 Let N N Y X R cos and Y R sin, tan. Then X x, y f r, m f R, M X,Y x, y m J, M r N x rcos x N rsin where J N N x y cos sin x y r r N N r r sin r cos x y N r Hence, N N r cos E cosθ r sin E sin θ N N f R, M r, m e r πn r cos θ N N r E E r E NE r cos θ r N N r e e π π

28 and E r r π E r cos θ N N f R M r m f R, M r, m d d. π π π e e π But, the last integral is independent of θ, ecause it is periodic regardless of the value of θ. Hence, π π E N E r cosθ π r cos N e d e d. Now, order zero. π π is the modified Bessel function of the first kind of π x cos I x e d π r R M e E N E f r m r I r, r R is a Ricean distriuted r.v.. N 3

29 Clearly, N N N r rcos rsin. Thus, r N L M e R M P error M sent f m d f r m e dr ax But, e E E N r N r e I r dr E N r E e e a r I r dr. N 4a x I x dx e. Hence, E E N E N 4 N P error M e. e e. 4

30 , E N then BER P e. Clearly, the BER performance of noncoherent FSK in If oth symols are transmitted with equal proaility, i.e. PM PM AWGN is the same as that of DPSK. 5