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
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
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
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
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
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
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..9.8.7.6.5.4.3.. - -.8 -.6 -.4 -...4.6.8 f in Hz PSD of inary FSK in Watts/Hz 8
- - - db - - - 3 - -.8 -.6 -.4 -...4.6.8 f in Hz PSD of inary FSK in db/hz 83
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
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
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
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 ""..5 -.5-3 4 5 6 7 8 9 t/t DPSK waveform representation of it sequence 87
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
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
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 ""
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
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
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
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
The following plot shows the BER performance of DPSK in AWGN. DPSK BER performance in AWGN 95
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
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
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
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
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.
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.
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
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
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
, 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