Gram-Schmidt procedure and constellations
- 2. INTRODUCTION ❑ The Gram-Schmidt orthogonalization procedure is a straightforward way by which an appropriate set of orthonormal functions can be obtained from any given signal set. ❑ PASSBAND SYSTEM Passband transmission shifts the signal to be transmitted in frequency to a higher frequency and then transmits it
- 3. Gram-Schmidt procedure First orthonomal waveform:- Second orthonomal waveform:- Substiution:-
- 4. Gram-Schmidt procedure with constant k: Expressing as linear combination:- ; N <= M
- 5. Gram-Schmidt procedure By, sk (t) expression we can say that, (or) N-dimensional signal space with coordinates {sk i, i=1,2,,…N}. Any signal can be represented geometrically as a point in the signal space spanned by the orthonormal functions {fn (t)}.
- 6. EXAMPLE signals {si(t), i=1,2,3,4} and its corresponding signals
- 7. EXAMPLE We see that C12 =0 s2 (t) and f2 (t) are orthogonal Consequently, s4 (t) is a linear combination of f1 (t) and f3 (t) and and, hence, f4 (t) = 0 .
- 8. CONSTELLATIONS A constellation diagram is a representation of a signal modulated by a digital modulation scheme such as quadrature amplitude modulation or phase-shift keying. The distance of a point from the origin represents a measure of the amplitude or power of the signal.
- 9. PROPERTIES MODULATION SCHEME: ► Bandwidth occupied by the modulation , dimension of the modulated signal ► Bandwidth occupied by the modulation , signal_points per dimension ► Probability of bit error is proportional to the distance between the closest points in the constellations
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