21. Orthonormal Representation of Signals

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1 1. Orthonormal Representation of Signals Introduction An analogue communication system is designed for the transmission of information in analogue form. he source information is in analogue form. In practice, the communication channel is an analogue channel. At the sending end of the channel, the analogue information (the modulating signal) is used to modulate a carrier. he modulated signal is now suitable for transmission over the channel. he operation performed on the analogue signal is called analogue modulation. Many analogue communication systems are still in wide use today. hese include AM, FM, and PM systems. he growth of digital computers has created the need for digital systems. Broadly speaking, there are two kinds of digital systems. hese are : 1. Digital communication system - designed for the transmission of information in digital form. he source information may be an analogue or digital source. An analogue source can be converted into digital form by analogue-to-digital conversion (e.g., pulse-code modulation and delta modulation).. Data communication system - designed for the transmission of information in digital form. he source information is already in digital form. In a data/digital communication system, the digital information is used to modulate a carrier. he operation performed with the digital signal is called digital modulation. At the receiving end, a process of demodulation is used to recover the original signal. able 1.1 lists a number of digital modem (modulation/demodulation) techniques. able 1.1 Some digital modem techniques. We are here concerned with the principles of digital modulation techniques with coherent (with carrier recovery) demodulation/detection of a digital signal in the presence of additive white Gaussian noise (AWGN). In general, the digital signal may be transmitted directly (transmission at baseband) or as a modulated-carrier signal (transmission at radio frequency). In both transmission cases, the concept of signal space can be used to represent a set of signals in terms of a set of orthonormal functions. he Gram-Schmidt orthogonal procedure is therefore discussed. We shall make calculations of error probability for various digital modems based on matched-filter detection and signal space concepts. Orthonormal Series Representation of Signals It is often convenient to represent a set of signals (functions) in terms of a set of 1.1

2 orthonormal basis functions which is both orthogonal and normalised. All possible linear combinations of the orthonormal basis functions form a linear space known as a signal space (function-space coordinate system). he coordinate axes in the signal space are the basis functions u 1 (t), u (t),..., u n (t). Any signal formed from the basis functions can be represented as a point in the signal space and the conventional vector theory applies. he graphical representation of the signals is called a phasor or a signal constellation diagram and the coefficients that represent signal s i (t) can be written by a vector S i = [s i1 s i... s in ]. (1.1) Example 1.1 Gram-Schmidt Orthogonal Process Figure 1.1 -dimensional signal space. Any set of finite energy signals (functions) can be represented by a set of orthonormal basis functions. hese basis functions u 1 (t), u (t),..., u n (t) are derived from the signals s 1 (t), s (t),..., s m (t) where n s i (t) = s ij u j (t) (1.) j =1 and the mathematical definition of an orthonormal set over the interval (, ) is u j (t) u k (t) dt = 1 if j = k if j k (1.3) he method by which the basis set is generated is called the Gram-Schmidt process and is as follows. Let the signal set be {s i (t)}, i = 1,,..., m, and the orthonormal basis set be {u j (t)}, j = 1,,..., n for m < n. In most cases, it is convenient to let m = n. 1. Set s ij = except s 11 in equation (1.). We then have [s 1 (t)] dt = s 11 u 1 (t) u 1 (t) dt 1.

3 { [s 1 (t)] dt } 1/ = s 11 (1.4) and we can find u 1 (t) = s 1 (t)/s 11. (1.5). Set s ij = except s 1 and s in equation (1.). We then have s (t) = s 1 u 1 (t) + s u (t) (1.6) s (t) u 1 (t) dt = s 1 [u 1 (t)] dt + s u (t) u 1 (t) dt s (t) u 1 (t) dt = s 1 + (1.7) and we can now evaluate s. Equation (1.6) can be rewritten as s (t) - s 1 u 1 (t) = s u (t). (1.8) Squaring and integrating, we have [s (t) - s 1 u 1 (t)] dt = s [u (t)] dt { [s (t) - s 1 u 1 (t)] dt } 1/ = s (1.9) and we can use equation (1.6) to find u (t) = [s (t) - s 1 u 1 (t)]/s (1.1) 3. he process is continued in the same manner until we have found m orthonormal basis functions u 1 (t), u (t),..., u m (t). Given m linearly independent signals s 1 (t), s (t),..., s m (t), the process will generate m < n) orthonormal basis functions. If the process had been started with a different signal initially (e.g., s 3 (t)), the basis functions would have been different. 1.3

4 Summary 1. Given s i (t) for i = 1,,..., m, the i-th signal is s i (t) = s i1 u 1 (t) + s i u (t) s i(j = i) u j (t) (1.11). At the (j = i)-th step, u 1 (t), u (t),..., u j-1 (t) are known. We then find s i1, s i,..., s ij, where s i1 = s i (t) u 1 (t) dt s i = s i (t) u (t) dt s ij ={ [s i (t) - s i1 u 1 (t) s ij-1 u j-1 (t)] dt} 1/ (1.1a) (1.1b) : (1.1j) 3. Find u (j = i) (t) from equation (1.11). Example 1. Given s 1 (t) = for < t <, elsewhere = and s (t) = 4 for < t < /, elsewhere =, find s 11, s 1, and s. s 11 = { [s 1 (t)] dt } 1/ = { dt } 1/ =. u 1 (t) = s 1 (t)/s 11 = / 4. / s 1 = s (t)u 1 (t) dt = 4[s 1 (t)/s 11 ] dt =. s = { [s (t) - s 1 u 1 (t)] dt } 1/ 1.4

5 = { [4 - (/ 4 )] dt } 1/ =. Reference s 1 (t) : S 1 = [s 11 s 1 ] = [ ] s (t) : S = [s 1 s ] = [ ]. [1] aub, H. and Schilling, D. L., Principles of Communication Systems, /e, McGraw-Hill,

6 Modem MASK OOK MPSK DE-MPSK DMPSK OQPSK/ SQPSK SI-OQPSK M-ary FSK MSK/FFSK DMSK GMSK FM Multi-h FM CPFSK SFSK M-QAM SM-QAM Description M-ary amplitude-shift keying On-off keying M-ary phase-shift keying Differentially encoded, coherent MPSK Differential MPSK (no carrier recovery) Offset quaternary PSK/Staggered QPSK wo-symbol interval OQPSK MFSK Minimum-shift keying/fast FSK Differential MSK Gaussian MSK amed frequency modulation Multi-index; correlative; duobinary FM Continuous-phase FSK Sinusoidal FSK M-point quadrature amplitude modulation Superimposed M-point QAM able 1.1 Some digital modem techniques. 1.6

7 u - axis [ s 11 s 1 ] u (t) s (t) u (t) 1 s (t) 1 [ s 1 s ] u - axis 1 Figure 1.1 -dimensional signal space. 1.7

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