Signal Constellation, Geometric Interpretation of Signals
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The document discusses the geometric representation of signals and the concept of constellation diagrams in digital communication. It focuses on Quadrature Phase Shift Keying (QPSK), detailing its types, advantages in bandwidth efficiency, and the elimination of traditional analog components through the use of FPGA in modulators. The references provided at the end suggest sources for further exploration of orthonormal bases and signal constellations.
Signal Constellation, Geometric Interpretation of Signals
- 1. Signal Constellation, Geometric Interpretation of Signals Group: 9 Ayan Das 11500320073 Arnab Paul 11500320075 Arnab Chatterjee 11500320076 Arijit Dhali 11500320078 Digital Communication & Stochastic Process EC 503
- 2. Contents 1. Introduction 2. Geometrical Representation of signal 3. Constellation diagram 4. QPSK and its Constellation Diagram 5. Types of QPSK 6. Conclusion 7. References
- 3. Introduction The essence of geometric representation of signals is to represent any set of M energy signals{si(t)} as linear combinations of N orthogonal basics functions, where N<=M. That part is to say, given a set of real- valued energy signals s1(t),s2(t),….,sM(t), each of duration T second, we write Si(t) =σ𝑗=1 𝑁 𝑠ijɸj(t), ቊ 0 ≤ 𝑡 ≤ 𝑇 𝑖 = 1,2, … , 𝑀 Where the coefficients of the expansion are defined by: Sij = 0 𝑇 𝑠i(t)ɸj(t) dt, ቊ 𝑖 = 1,2, … . , 𝑀 𝑗 = 1,2, … . , 𝑁
- 4. Geometrical Representation of Signals Derivation of geometrical representation of signals. Combination of two orthonormal basis functions Φ1(t) and Φ2(t). Φ1(t) and Φ2(t) are orthonormal if: - 0 𝑇𝑏 𝛷1(𝑡)𝛷2(𝑡) 𝑑𝑡 = 0 0 𝑇𝑏 𝛷1 2 𝑡 = 0 𝑇𝑏 𝛷2 2 𝑡 = 1 (Normalised to have unit energy). The representations are: - 𝑠1 𝑡 = 𝑠11𝛷1 𝑡 + 𝑠12𝛷2 𝑡 , 𝑠2 𝑡 = 𝑠21𝛷1 𝑡 + 𝑠22𝛷2 𝑡 . where 𝑠𝑖𝑗 = 0 𝑇𝑏 𝑠𝑖 𝑡 𝛷𝑗 𝑡 𝑑𝑡, 𝑖, 𝑗 𝜖 {1,2} 0 𝑇𝑏 𝑠𝑖 𝑡 𝛷𝑗 𝑡 𝑑𝑡, is the projection of 𝑠𝑖 𝑡 onto 𝛷𝑗 𝑡 .
- 5. Geometrical Representation of Signals Basis Vectors: The set of basis vectors {e1, e2, …,en} of a space are chosen such that: Should be complete or span the vector space: any vector can be expressed as a linear combination of these vectors. Each basis vector should be orthogonal to all others • Each basis vector should be normalized. • A set of basis vectors satisfying these properties is also said to be a complete orthonormal basis • In an n-dim space, we can have at most n basis vectors Signal Space: If a signal can be represented by n-tuple, then it can be treated in much the same way as a n-dim vector. Let φ1(t), φ2(t),…., φn(t) be n signals Consider a signal x(t) and suppose that If every signal can be written as above ⇒ ~ ~ basis functions and we have a n-dim signal space Orthonormal Basis: In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of vectors. The image of the standard basis under a rotation or reflection (or any orthogonal transformation) is also orthonormal, and every orthonormal basis for Rn arises in this fashion.
- 6. Constellation Diagram It is the graphical representation of the complex envelope of each possible state. • The x-axis represents the in-phase component and the y-axis the quadrature component of the complex envelope. • The distance between signals on a constellation diagram relates to how different the modulation waveforms are and how easily a receiver can differentiate between them. it(t) qt(t) -i0 +i0 +q0 -q0 I I it(t) = ± i0 cos(2𝜋fot) qt(t) = ± q0 cos(2𝜋fot)
- 7. QPSK & Constellation Diagram Quadrature Phase Shift Keying (QPSK) can be interpreted as two independent BPSK systems (one on the I-channel and one on Q), and thus the same performance but twice the bandwidth efficiency. Carrier Phases: {0 , π 2 , π , 3π 2 } Carrier Phases: { π 4 , 3π 4 , 5π 4 , 7π 4 } Q Q I I Quadrature Phase Shift Keying has twice the bandwidth efficiency of BPSK since 2 bits are transmitted in a single modulation symbol.
- 8. Types of QPSK Conventional QPSK has transitions through zero (i.e., 1800 phase transition). Linear amplifiers are required highly. In Offset QPSK, the phase transitions are limited to 900, the transitions on the I and Q channels are staggered. In /4 QPSK the set of constellation points are toggled each symbol, so transitions through zero cannot occur. This scheme produces the lowest envelope variations. All QPSK schemes require linear power amplifiers. π 4 QPSK Offset QPSK Conventional QPSK Q Q Q I I I
- 9. Conclusion In this modulator, analog components like local oscillator and mixer are completely eliminated which are frequency and temperature sensitive . Here all the functions are performed by single FPGA. So the limitations of modulator are completely removed from satellite communication.
- 10. References https://www.testandmeasurementtips.com/what-is-orthogonal- part-3-signal-constellations-faq/ https://www.rcet.org.in/uploads/files/LectureNotes/ece/S6/Digital%2 0communication/UNIT-4.pdf https://en.wikipedia.org/wiki/Orthonormal_basis https://www.slideshare.net/gongadi/design-and-implementation-of- qpsk-modulator-using-digital-subcarrier-18370723 https://dsp.stackexchange.com/questions/74713/geometric- representation-of-a-signal-using-basis-functions http://gn.dronacharya.info/EEEDept/Downloads/subjectinfo/Digital_C ommunication/Unit-1/Lecture-6.pdf