Digital Modulation Lecture Notes

lecture notes on digital modulation techniques,and what is digital modulation and demodulation what are different digital modulation techniques pdf free download
Dr.LeonBurns Profile Pic
Dr.LeonBurns,New Zealand,Researcher
Published Date:21-07-2017
Your Website URL(Optional)
Comment
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

Advise: Why You Wasting Money in Costly SEO Tools, Use World's Best Free SEO Tool Ubersuggest.