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Maretha Prinsloo and Paul Barrett
Clearly both schemes fulfill your constraints. Question: which is the configuration which minimizes the overall mean square error between the original signal and the digitized signal? Show why. As a guideline, note that the MSE will be composed of two independent parts: the one introduced by the quantizers and, for the second scheme, the one which is introduced by the lowpass filter before the downsampler. For the quantizer error, you can assume that the downsampled process still remains a uniform, i.i.d. process.
and this is the factor by which we need to upsample; this can be achieved with a combination of a 25times upsampler followed by a 4times downsampler as in Figure where _{}() is a lowpass filter with cutoff frequency 25 and gain _{} = 425. At the receiver the chain is inverted, with an upsampler by four, followed by a lowpass filter with cutoff frequency 4 and gain _{} = 254 followed by a 25times downsampler.
(20101109)  ISBN13: 9786133506787
Timing recovery is the ensemble of strategies which are put in place to recover the synchronism between transmitter and receiver at the level of discretetime samples. This synchronism, which was one of the assumptions of backtoback operation, is lost in realworld situations because of propagation delays and because of slight hardware differences between devices. The D/A and A/D, being physically separate, run on independent clocks which may exhibit small frequency differences and a slow drift. The purpose of timing recovery is to offset such hardware discrepancies in the discretetime domain.
This decisiondirected feedback method is almost always able to “lock” the constellation in place; due to the fourfold symmetry of regular square constellations, however, there is no guarantee that the final orientation of the locked pattern be the same as the original. This difficulty is overcome by a mapping technique called ; in differential encoding the first two bits of each symbol actually encode the of the symbol with respect to the previous one, while the remaining bits indicate the actual point within the quadrant. In so doing, the encoded symbol sequence becomes independent of the constellation’s absolute orientation.
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where we have assumed that the shaper () is an ideal lowpass. As a general rule, . Moreover, if we follow the normal processing order, we can equivalently say that a symbol sequence generated at symbols per second gives rise to a modulated signal whose positive passband is The effective bandwidth _{} depends on the modulation scheme and, especially, on the frequency leakage introduced by the shaper.
The of a communication system is the number of which can be transmitted in one second. Considering that the interpolator works at _{} samples per second and that, because of upsampling, there are exactly samples per symbol in the signal [], the baud rate of the system is
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where the multiplicity of the three types of terms as well as the relative coefficients are dependent (in a nontrivial way) on the original filter coefficients. This generates a parallel structure of filters, whose outputs are summed together. The first branch corresponds to the first sum and it is an FIR filter; a further set of branches are associated to each term in the second sum, each one of them a first order IIR; the last set of branches is a collection of second order sections, one for each term of the third sum.
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where _{} is the number of real zeros, _{} is the number of complexconjugate zeros and _{} + 2_{} = (and, equivalently, for the poles, _{} + 2_{} = ). From this representation of the transfer function we can obtain an alternative structure for a filter; recall that if we apply a series of filters in sequence, the overall transfer function is the product of the single transfer functions. Working backwards, we can interpret () as the cascade of smaller sections. The resulting structure is called a and it is particularly important for IIR filters, as we will see later.
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As a consequence, the transfer function can be factored into the product of first and secondorder terms in which the coefficients are all strictly real; namely:
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where, of course, the coefficients correspond to the nonzero values of the impulse response [], i.e. _{} = []. Using the constitutive elements outlined above, we can immediately draw a block diagram of an FIR filter as in Figure . In practice, however, additions are distributed as shown in Figure ; this kind of implementation is called a . Further, adhoc optimizations for FIR structures can be obtained in the the case of symmetric and antisymmetric linear phase filters; these are considered in the exercises.
(20100917)  ISBN13: 9786130629427
where the _{} are the 1 (complex) roots of the numerator polynomial and the _{} are the 1 (complex) roots of the denominator polynomial. Since the coefficients of the CCDE are assumed to be real, complex roots for both polynomials always appear in complexconjugate pairs. A pair of firstorder terms with complexconjugate roots can be combined into a secondorder term with real coefficients: