Signal Constellation

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<p>Modulation Principles Almost all communication systems transmit digital data using a sinusoidal carrier waveform. Electromagnetic signals propagate well Choice of carrier frequency allows placement of signal in arbitrary part of spectrum Physical system implements modulation by: Processing digital information at baseband Pulse shaping and filtering of digital waveform Baseband signal is mixed with signal from oscillator RF signal is filtered, amplified and coupled with antennaRepresentation of Modulation Signals We can modify amplitude, phase or frequency. Amplitude Shift Keying (ASK) or On/Off Keying (OOK): Frequency Shift Keying (FSK): Phase Shift Keying (PSK):( )1 2 0 0 A f tccos , ( ) ( )1 2 0 21 0 A f t A f t cos , cos ( )( ) ( )1 20 2 2 + = A f tA f t A f tcc ccoscos cos Representation of Bandpass SignalsBandpass signals (signals with small bandwidth compared to carrier frequency) can be represented in any of three standard formats: Quadrature Notationwhere x(t) and y(t) are real-valued baseband signals called the in-phase and quadrature components of s(t)( ) ( )s t x t f t y t f tc c( ) ( ) cos ( ) sin = 2 2 Representation of Bandpass Signals (continued) Complex Envelope Notationwhere is the complex envelope of s(t). Magnitude and Phasewhere is the magnitude of s(t),and is the phase of s(t).( )| | ( )| |s t x t jy t e s t ej f tl j f tc c( ) Re ( ) ( ) Re = + = 2 2 ( ) s tl( ) ( )s t a t f t tc( ) ( ) cos = + 2 a t x t y t ( ) ( ) ( ) = +2 2( ) tan( )( )t y tx t= </p> <p>1Key Ideas from I/Q Representation of Signals We can represent bandpass signals independent of carrier frequency. The idea of quadrature sets up a coordinate system for looking at common modulation types. The coordinate system is sometimes called a signal constellation diagram.Example of Signal Constellation Diagram: BPSKx t ( )y t ( )X X{ }x t y t ( ) , ( ) = 1 0Example of Signal Constellation Diagram: QPSK{ } { }x t y t ( ) , ( ) 1 1x t ( )y t ( )XXXXExample of Signal Constellation Diagram: QAM{ } { }x t y t ( ) , , , , ( ) , , , + + + + 3 1 1 3 3 1 1 3y t ( )XXXX XX XXXX XXXX XXInterpretation of Signal Constellation Diagram Axis are labeled with x(t) and y(t) Possible signals are plotted as points Signal power is proportional to distance from origin Probability of mistaking one signal for another is related to the distance between signal points Decisions are made on the received signal based on the distance to signal points in constellation New Way of Viewing Modulation The I/Q representation of modulation is very convenient for some modulation types. We will examine an even more general way of looking at modulation using signal spaces. By choosing an appropriate set of axis for our signal constellation, we will be able to: Design modulation types which have desirable properties Construct optimal receivers for a given type of modulation Analyze the performance of modulation types using very general techniques.Vector Spaces An n-dimensional vector consists of n scalar components The norm (length) of a vector is given by: The inner product of two vectors and is given by:| |v = v v vn 1 2, , , { }v v vn 1 2, , , vv = = viin21| |v1 11 12 1= v v v n, , , | |v2 21 22 2= v v v n, , , v v1 2 1 21 = = v vi iinBasis Vectors A vector may be expressed as a linear combination of its basis vectors :where Think of the basis vectors as a coordinate system (x-y-z... axis) for describing the vector What makes a good choice of coordinate system?v{ }e e e1 2, , , nv e = = vi iin1vi i= e vvA Complete Orthornomal Basis The set of basis vectors should be complete or span the vector space . Any vector can be expressed as for some Each basis vector should be orthogonal to all others: Each basis vector should be normalized: A set of basis vectors which satisfies these three properties is said to be a complete orthonormal basis.{ }e e e1 2, , , nnv e = = vi iin1{ }vie ei j i j = 0,ei i = 1,Signal SpacesSignals can be treated in much the same way as vectors. The norm of a signal is given by: The inner product of signals and is: Signals can be represented as the sum of basis functions:| |x t t a b ( ), , ( ) ( ) x t x t dt Eabx= |\</p> <p> |.| =21 2 /x t x t x t x t dtab1 2 1 2( ), ( ) ( ) ( ) = x t1( ) x t2( )x t x f tk kin( ) ( ), = =1x x t f tk k= ( ), ( )Basis Functions for a Signal Set One of M signals is transmitted: The functions ( ) form a complete orthonormal basis for the signal set if Any signal can be described by a linear combination: The basis functions are orthogonal to each other: The basis functions are normalized:{ }s t s tM 1( ), , ( ) { }f t f tK 1( ), , ( ) K M s t s f t i Mi i k kkK( ) ( ), , ,,= ==11 f t f t dt i ji jab( ) ( ) , = 0( ) f t dt kkab21 = ,Example of Signal SpaceConsider the following signal signal set:-1+1ts t1( )-1+1ts t3( )-1+1ts t2( )-1+1ts t4( )1 2121 21 2Example of Signal Space (continued) We can express each of the signals in terms of the following basis functions: Therefore the basis is complete-1+1tf t1( )1 2-1+1tf t2( )1 2s t f t f t1 1 21 1 ( ) ( ) ( ) = + s t f t f t2 1 21 1 ( ) ( ) ( ) = s t f t f t3 1 21 1 ( ) ( ) ( ) = + s t f t f t4 1 21 1 ( ) ( ) ( ) = Example of Signal Space (continued) The basis is orthogonal: The basis is normalized:f t f t dt1 20 ( ) ( ) =f t dt f t dt12221 ( ) ( ) = =Signal Constellation for Example Weve seen this signal constellation beforeXXXXf t2( )f t1( )s t1( )s t2( )s t3( )s t4( )Notes on Signal Spaces Two entirely different signal sets can have the same geometric representation. The underlying geometry will determine the performance and the receiver structure for a signal set. In both of these cases we were fortunate enough to guess the correct basis functions. Is there a general method to find a complete orthonormal basis for an arbitrary signal set? Gram-Schmidt Procedure</p>