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r 2 2 t sin 2pfc t ; 0 T in Software Printer USS Code 128 in Software r 2 2 t sin 2pfc t ; 0 T

r 2 2 t sin 2pfc t ; 0 T generate, create uss code 128 none for software projects EAN-8 The signal space barcode code 128 for None representation is plotted in Figure 6.5. p The error rate for this situation was treated in 5.

The minimum distance is 2E and there are two neighbors at dmin. Consequently s ! r  E 2Eb 2Q P e % 2Q : N0 N0. Digital communications In-phase bits dt T 1(t ). 1(t ). Quadrature bits dt T 2(t ). Modulator 2(t ). Demodulator Figure 6.6 Modulator and demodulator for 4-QAM. Figure 6.7 Decision regions for 16-QAM. This is as effic Code 128 for None ient in terms of Eb/N0 as BPSK or PAM, with twice the bits per symbol. It uses the same bandwidth, and is therefore twice as spectrally efficient. The modulator and demodulator are shown in Figure 6.

6. In the modulator, the input streams of 1s and 0s are converted to 1s and 1s. In the demodulator, the integrator outputs are sampled every T seconds, and then a threshold detector decides the value is 1 if the result is positive or 0 otherwise.

M-QAM generally uses larger PAM constellations on each of the quadrature carriers. For 16-QAM, draw the optimal decision regions in signal space and a receiver that would implement these decisions, then compute the approximate error probability. Solution There remain only two orthogonal functions.

The decision boundaries are composed of the right bisectors of the lines joining neighbors. Due to the symmetry of the signal set, the boundaries are as depicted in Figure 6.7 The demodulator is exactly as depicted in Figure 6.

6, except that four-level uniform slicers replace the two-level slicers (quantizers). Let the coordinates of the lowest-energy signal in the first quadrant be (a, a). Then the minimum distance is 2a, and the average number of neighbors at this distance is [4(4) 8(3) 4(2)]/16 3.

The average energy E may be computed using the fourfold symmetry and similarity for the two coordinates as [4(a)2 4(3a)2]/4=10a2. Alternatively one could have doubled the average energy of 4-PAM. In either case, dmin is related to Eb by dmin 2a 2 r r r E 4Eb 8Eb 2 10 10 5.

6.2 Communication over the Gaussian channel so that s Software Code 128 Code Set A ! 4Eb P e 3Q : 5N0 Gray coding, which labels symbols so that the nearest symbols differ in exactly one bit, exists for all QAM configurations, and thus the bit error rate performance as a function of Eb/N0 is the same for 16-QAM as for 4-PAM. This is achieved with twice the spectral efficiency..

Signal sets can also be constructed using orthogonal frequencies. In BFSK two carriers separated in frequency by 1/T are used, with the bit conveyed depending upon the frequency. A two-branch detector is required.

The squared distance between two orthogonal signals with energy E is half that of two antipodal signals with energy E, and thus BFSK performs 3 dB worse with respect to Eb/N0 than BPSK. In contrast to other M-ary modulations, however, M-FSK actually has better performance as M increases, with s ! r  E Eb log2 M P e % M 1 Q M 1 Q : (6:5) N0 N0 The price of this performance increase is a linear increase in the required bandwidth, in contrast to the phase/amplitude modulations for which bandwidth is independent of M. In practice, a better trade in bandwidth and Eb/N0 performance is obtained through the use of channel codes and a modulation such as PSK, making these the methods of choice for deep-space communication.

FSK finds application mainly as a non-coherent scheme in low-cost applications such as paging. Channel coding Channel coding provides a means of exchanging redundancy for improved Eb/N0 performance. This redundancy can be in the form of sending an increased number of symbols (the traditional trade), essentially a bandwidth cost, or in the form of expanding the number of signals beyond M to convey log2M bits of information ( coded modulation ).

Channel coding consists of several components. The first is the encoding which for every k input bits produces n output bits. The code rate is then R k/n, with k < n.

The second is estimation of channel parameters so that the decoder can properly weight the decision variables required in the decoder. In hard-decision decoding, this takes the form of simply making decisions. In soft-decision decoding, the decision variables that might be sent to a slicer are instead preserved for operation by the decoder.

The decoder, acting upon a sequence of decision variables, then produces the ML estimate. Because the decoder acts upon a sequence of variables that have some structure imposed via the memory in the channel encoder, the dmin is improved compared with treating uncoded symbols one by one. The degree to which the error probability is improved with respect to Eb/N0 is termed the coding gain.

Note that it is the energy per information bit that is important, excluding the redundancy required by the code. Thus if an M-point constellation is used in conjunction with a code of rate R, Eb E/Rlog2M. In this way simply repeating each information symbol.

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