Digital Modulation Lecture Notes

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Digital Modulation 1 Lecture Notes Ingmar Land and Bernard H. Fleury Navigation and Communications (NavCom) Department of Electronic Systems Aalborg University, DK Version: February 5, 20072 1 The Basic Constituents of a Digi- tal Communication System PSfrag replacements Block diagram of a digital communication system: Binary Information Source Digital Transmitter U X(t) k Information Source Vector Waveform Source Encoder Encoder Modulator 1=T b 1=T Waveform Channel Binary Information Sink Digital Receiver 1=T 1=T b Information Source Vector Waveform Sink Decoder Decoder Demodulator Y (t) U k  Time duration of one binary symbol (bit)U : T k b  Time duration of one waveformX(t): T  Waveform channel may introduce - distortion, - interference, - noise Goal: Signal of information source and signal at information sink should be as similar as possible according to some measure. Land, Fleury: Digital Modulation 1 NavCom3 Some properties:  Information source: may be analog or digital  Source encoder: may perform sampling, quantization and compression; generates one binary symbol (bit) U 2f0; 1g k per time intervalT b  Waveform modulator: generates one waveformx(t) per time intervalT  Vector decoder: generates one binary symbol (bit) U 2 k f0; 1g (estimates ofU ) per time intervalT k b  Source decoder: reconstructs the original source signal Land, Fleury: Digital Modulation 1 NavCom4 1.1 Discrete Information Sources A discrete memoryless source (DMS) is a source that generates a sequence U ;U ;::: of independent and identically distributed 1 2 (i.i.d.) discrete random variables, called an i.i.d. sequence. PSfrag replacements U ;U ;::: 1 2 DMS Properties of the output sequence: discrete U ;U ;:::2fa ;a ;:::;a g. 1 2 1 2 Q memoryless U ;U ;::: are statistically independent. 1 2 stationary U ;U ;::: are identically distributed. 1 2 Notice: The symbol alphabetfa ;a ;:::;a g has cardinality Q. 1 2 Q The value ofQ may be in nite, the elements of the alphabet only have to be countable. Example: Discrete sources (i) Output of the PC keyboard, SMS (usually not memoryless). (ii) Compressed le (near memoryless). (iii) The numbers drawn on a roulette table in a casino (ought to be memoryless, but may not. . . ). 3 Land, Fleury: Digital Modulation 1 NavCom5 A DMS is statistically completely described by the probabilities p (a ) = Pr(U =a ) U q q of the symbolsa ,q = 1; 2;:::;Q. Notice that q (i) p (a ) 0 for allq = 1; 2;:::;Q, and U q Q X (ii) p (a ) = 1. U q q=1 As the sequence is i.i.d., we have Pr(U =a ) = Pr(U =a ). i q j q For convenience, we may writep (a) shortly asp(a). U Binary Symmetric Source A binary memoryless source is a DMS with a binary symbol al- phabet. Remark: Commonly used binary symbol alphabets are F :=f0; 1g and 2 B :=f1; +1g. In the following, we will use F . 2 A binary symmetric source (BSS) is a binary memoryless source with equiprobable symbols, 1 p (0) =p (1) = : U U 2 Example: Binary Symmetric Source All length-K sequences of a BSS have the same probability K   Y K 1 p (u u :::u ) = p (u ) = : U U :::U 1 2 K U i 1 2 K 2 i=1 3 Land, Fleury: Digital Modulation 1 NavCom6 1.2 Considered System Model This course focuses on the transmitter and the receiver. Therefore, we replace the information source by a BSS Information Source U ;U ;::: 1 2 PSfrag replacements Source Encoder Binary Decoder U ;U ;::: 1 2 BSS Sink and the information sink by a binary (information) sink. PSfrag replacements Information Source U ;U ;::: 1 2 BSS Sink Decoder Encoder Binary U ;U ;::: U ;U ;::: 1 2 1 2 Sink Land, Fleury: Digital Modulation 1 NavCom7 Some Remarks (i) There is no loss of generality resulting from these substitutions. Indeed it can be demonstrated within Shannon's Information The- ory that an ecient source encoder converts the output of an in- formation source into a random binary independent and uniformly distributed (i.u.d.) sequence. Thus, the output of a perfect source encoder looks like the output of a BSS. (ii) Our main concern is the design of communication systems for reliable transmission of the output symbols of a BSS. We will not address the various methods for ecient source encoding. Land, Fleury: Digital Modulation 1 NavComPSfrag replacements 8 Digital communication system considered in this course: X(t) U Digital BSS Transmitter Waveform bit rate 1=T rate 1=T b Channel Binary Digital Sink Receiver Y (t) U  Source: U = U ;:::;U ,U 2f0; 1g 1 K i  Transmitter: X(t) =x(t; u)  Sink: U = U ;:::;U ,U 2f0; 1g 1 K i  Bit rate: 1=T b  Rate (of waveforms): 1=T = 1=(KT ) b  Bit error probability K X 1 P = Pr(U =6 U ) b k k K k=1 Objective: Design of ecient digital communication systems ( small bit error probability and Eciency means high bit rate 1=T . b Land, Fleury: Digital Modulation 1 NavCom9 Constraints and limitations:  limited power  limited bandwidth  impairments (distortion, interference, noise) of the channel Design goals:  \good" waveforms  low-complexity transmitters  low-complexity receivers Land, Fleury: Digital Modulation 1 NavCom10 1.3 The Digital Transmitter PSfrag replacements 1.3.1 Waveform Look-up Table PSfrag replacements PSfrag replacements Example: 4PPM (Pulse-Position Modulation) PSfrag replacements Set of four di erent waveforms, S =fs (t);s (t);s (t);s (t)g: 1 2 3 4 s (t) 1 s (t) s (t) s (t) s (t) 1 2 3 4 s (t) s (t) 1 2 A A A A s (t) s (t) s (t) 1 2 3 0 t 0 t 0 t 0 t 0 0 0 0 T T T T 2 Each waveformX(t)2 S is addressed by vector U ;U 2f0; 1g : 1 2 U ;U 7 X(t): 1 2 The mapping may be implemented by a waveform look-up table. PSfrag replacements 4PPM Transmitter 007s (t) 1 U ;U X(t) 1 2 017s (t) 2 PSfrag replacements 107s (t) 3 117s (t) 4 Example: 00011100 is transmitted as x(t) A 0 t 0 T 2T 3T 4T 3 Land, Fleury: Digital Modulation 1 NavCom11 PSfrag replacements For an arbitrary digital transmitter, we have the following: Digital Transmitter U ;U ;:::;U X(t) 1 2 K Waveform Look-up Table Input Binary vectors of lengthK from the input set U: K U ;U ;:::;U 2 U :=f0; 1g : 1 2 K The \duration" of one binary symbolU isT . k b Output Waveforms of durationT from the output set S: x(t)2 S :=fs (t);s (t);:::;s (t)g: 1 2 M Waveform durationT means that form = 1; 2;:::;M, s (t) = 0 fort2= 0;T: m The look-up table maps each input vector u ;:::;u 2 U to one 1 K waveform x(t)2 S. Thus the digital transmitter may de ned by a mapping U S with u ;:::;u 7x(t): 1 K The mapping is one-to-one and onto such that K M = 2 : Relation between signaling interval (waveform duration)T and bit interval (\bit duration")T : b T =KT : b Land, Fleury: Digital Modulation 1 NavCom12 1.3.2 Waveform Synthesis The set of waveforms, S :=fs (t);s (t);:::;s (t)g; 1 2 M spans a vector space. Applying the Gram-Schmidt orthogonaliza- tion procedure, we can nd a set of orthonormal functions S :=f (t); (t);:::; (t)g withDM 1 2 D such that the space spanned by S contains the space spanned by S. Hence, each waveforms (t) can be represented by a linear combi- m nation of the orthonormal functions: D X s (t) = s  (t); m m;i i i=1 m = 1; 2;:::;M. Each signal s (t) can thus be geometrically m represented by theD-dimensional vector T D s = s ;s ;:::;s 2 R m m;1 m;2 m;D with respect to the set S . Further details are given in Appendix A. Land, Fleury: Digital Modulation 1 NavCom13 1.3.3 Canonical Decomposition The digital transmitter may be represented by a waveform look-up table: u = u ;:::;u 7 x(t); 1 K where u2 U andx(t)2 S. From the previous section, we know how to synthesize the wave- forms (t) from s (see also Appendix A) with respect to a set of m m basis functions S :=f (t); (t);:::; (t)g: 1 2 D Making use of this method, we can split the waveform look-up table into a vector look-up table and a waveform synthesizer: u = u ;:::;u 7 x = x ;x ;:::;x 7 x(t); 1 K 1 2 D where K u2 U =f0; 1g ; D x2 X =fs ; s ;:::; s g R ; 1 2 D PSfrag replacements x(t)2 S =fs (t);s (t);:::;s (t)g: 1 2 M This splitting procedure leads to the sought canonical decomposi- tion of a digital transmitter: Vector Waveform Encoder Modulator X 1 Vector Look-up Table U ;:::;U X(t) (t) 1 K 1 0::: 07 s 1 X D 1::: 17 s M (t) D Land, Fleury: Digital Modulation 1 NavComPSfrag replacements PSfrag replacements 14 PSfrag replacements p 2= T p PSfrag replacements Example: 4PPM PSfrag replacements 2= T p Vector Encoder 2= T (t) The four orthonormal basis functions are 1 Vector (t) 1 (t) (t) (t) (t) 1 2 3 4 p (t) (t) 1 2 2= T Waveform Modulator (t) (t) 2 3 0 0 0 0 t t t t 0 T 0 T 0 T 0 T The canonical decomposition of the receiver is the following, where p p E =A T=2. s Vector Waveform Encoder Modulator X 1 Vector Look-up Table (t) 1 X 2 p T 007 E ; 0; 0; 0 s U ;U 1 2 p (t) T 2 X(t) 017 0; E ; 0; 0 s X 3 p T 107 0; 0; E ; 0 s p (t) T 3 117 0; 0; 0; E s X 4 (t) 4 Remarks: - The signal energy of each basis function is equal to one: R (t)dt = 1 (due to their construction). d - The basis functions are orthonormal (due to their construction). p - The value E is chosen such that the synthesized signals have s the same energy as the original signals, namely E (compare s Example 3). 3 Land, Fleury: Digital Modulation 1 NavCom15 1.4 The Additive White Gaussian-Noise Channel PSfrag replacements W(t) X(t) Y (t) The additive white Gaussian noise (AWGN) channel-model is widely used in communications. The transmitted signal X(t) is superimposed by a stochastic noise signal W(t), such that the transmitted signal reads Y (t) =X(t) +W(t): The stochastic process W(t) is a stationary process with the following properties: (i) W(t) is a Gaussian process, i.e., for each time t, the samples v = w(t) are Gaussian distributed with zero mean 2 and variance : 2  1 v p (v) =p exp : V 2 2 2 2 (ii) W(t) has a at power spectrum with heightN =2: 0 N 0 S (f) = W 2 (Therefore it is called \white".) (iii) W(t) has the autocorrelation function N 0 R () = (): W 2 Land, Fleury: Digital Modulation 1 NavCom16 Remarks 1. Autocorrelation function and power spectrum:    R () = E W(t)W(t +) ; S (f) =F R () : W W W 2. For Gaussian processes, weak stationarity implies strong sta- tionarity. 3. An WGN is an idealized process without physical reality:  The process is so \wild" that its realizations are not ordinary functions of time.  Its power is in nite. However, a WGN is a useful approximation of a noise with a at power spectrum in the bandwidthB used by a commu- nication system: S (f) noise PSfrag replacements N =2 0 f f f 0 0 B B 4. In satellite communications,W(t) is the thermal noise of the receiver front-end. In this case, N =2 is proportional to the 0 squared temperature. Land, Fleury: Digital Modulation 1 NavCom17 1.5 The Digital Receiver Consider a transmission system using the waveforms n o S = s (t);s (t);:::;s (t) 1 2 M with s (t) = 0 for t2= 0;T, m = 1; 2;:::;M, i.e., with dura- m tionT. Assume transmission over an AWGN channel, such that y(t) =x(t) +w(t); x(t)2 S. 1.5.1 Bank of Correlators The transmitted waveform may be recovered from the received waveformy(t) by correlatingy(t) with all possible waveforms: PSfrag replacements c =hy(t);s (t)i; m m m = 1; 2;:::;M. Based on the correlations c ;:::;c , the wave- 1 M forms (t) with the highest correlation is chosen and the correspond- ing (estimated) source vector u = u ;:::;u is output. 1 K c 1 R T (:)dt 0 estimation u ;:::;u 1 K ofx (t) s (t) 1 y(t) and table c R M T look-up (:)dt 0 s (t) M Disadvantage: High complexity. Land, Fleury: Digital Modulation 1 NavCom18 1.5.2 Canonical Decomposition Consider a set of orthonormal functions n o S = (t); (t);:::; (t) 1 2 D obtained by applying the Gram-Schmidt procedure to s (t);s (t);:::;s (t). Then, 1 2 M D X s (t) = s  (t); m m;d d d=1 m = 1; 2;:::;M. The vector T s = s ;s ;:::;s m m;1 m;2 m;D entirely determiness (t) with respect to the orthonormal set S . m The set S spans the vector space D n o X D S := s(t) = s (t) : s ;s ;:::;s 2 R : i i 1 2 D i=1 Basic Idea  The received waveformy(t) may contain components outside ofS . However, only the components insideS are relevant, asx(t)2S .  Determine the vector representation y of the components ofy(t) that are insideS .  This vector y is sucient for estimating the transmitted waveform. Land, Fleury: Digital Modulation 1 NavCom19 Canonical decomposition of the optimal receiver for the AWGN channel Appendix A describes two ways to compute the vector represen- tation of a waveform. Accordingly, the demodulator may be im- plemented in two ways, leading to the following two receiver struc- tures. PSfrag replacements Correlator-based Demodulator and Vector Decoder R Y T 1 (:)dt 0 U ;:::;U Y (t) Vector 1 K (t) 1 Decoder R Y T D (:)dt 0 (t) D PSfrag replacements Matched- lter based Demodulator and Vector De- coder Y MF 1 (Tt) 1 Y (t) U ;:::;U Vector 1 K Decoder Y MF D (Tt) D T The optimality of the receiver structures is shown in the following sections. Land, Fleury: Digital Modulation 1 NavCom20 1.5.3 Analysis of the correlator outputs Assume that the waveforms (t) is transmitted, i.e., m x(t) =s (t): m The received waveform is thus y(t) =x(t) +w(t) =s (t) +w(t): m The correlator outputs are T Z y = y(t) (t)dt d d 0 T Z i  = s (t) +w(t) (t)dt m d 0 T T Z Z = s (t) (t)dt + w(t) (t)dt m d d 0 0 z z s w m;d d =s +w ; m;d d d = 1; 2;:::;D. Using the notation T y = y ;:::;y ; 1 D T Z T w = w ;:::;w ; w = w(t) (t)dt; 1 D d d 0 we can recast the correlator outputs in the compact form y = s + w: m The waveform channel betweenx(t) =s (t) andy(t) is thus trans- m formed into a vector channel between x = s and y. m Land, Fleury: Digital Modulation 1 NavCom

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