0.12 Linear equalization  (Page 8/17)

This procedure, along with three others that will be discussed in the ensuing sections, is available on the website in the program dae.m . Combining the various approaches makes it easier to compare their behaviors in the examples of "Examples and Observations" .

Complex signals and parameters

The preceding development assumes that the source signal and channel, and therefore the received signal, equalizer, and equalizer output,are all real valued. However, the source signal and channel may bemodeled as complex valued when using modulations such as QAM of [link] . This is explored in some detail in the document A Digital Quadrature Amplitude Modulation Radio , which can be found on the website.The same basic strategy for equalizer design can also be used in the complex case.

Consider a complex delayed source recovery error

where $j=\sqrt{-1}$ . Consider its square,

which is typically complex valued, and potentially real valued and negative when $e_R\approx 0$ . Thus, a sum of $e^2$ is no longer a suitable measure of performance since $\left|e\right|$ might be nonzero but its squared average might be zero.

Instead, consider the product of a complex $e$ with its complex conjugate $e^{*}=e_R-j{e}_I$ ; that is,

In vector form, the summed squared error of interest is $E^{H}E$ (rather than the $E^{T}E$ of [link] ), where the superscript $H$ denotes the operations of both transposition and complex conjugation.Thus, [link] becomes

Note that in implementing this refinement in the M atlab code, the symbol pair .'implements a transpose, while 'alone implements a conjugate transpose.

Fractionally spaced equalization

The preceding development assumes that the sampled input to the equalizer is symbol spaced with thesampling interval equal to the symbol interval of $T$ seconds. Thus, the unit delay in realizing the tapped-delay-line equalizeris $T$ seconds. Sometimes, the inputto the equalizer is oversampled such that the sample interval is shorter than the symbol interval, and theresulting equalizer is said to be fractionally spaced. The same kinds of algorithms and solutions can be usedto calculate the coefficients in fractionally spaced equalizers as are used for $T$ -spaced equalizers. Of course, details of the construction of the matricescorresponding to $\overline{S}$ and $\overline{R}$ will necessarily differ due to the structural differences.The more rapid sampling allows greater latitude in the ordering of the blocks in the receiver. This is discussed at length in Equalization on the website.

Consider the multiuser system shown in [link] . Both users transmit binary $\pm 1$ PAM signals that are independent and equally probable symbols.The signal from the first user is distorted by a frequency selective channel with impulse response

$$h_1\left[k\right]=\delta \left[k\right]+b\delta [k-1].$$

The signal from the second (interfering) user is scaled by $h_2\left[k\right]=c\delta \left[k\right]$ . The difference equation generating the received signal $r$ is

$$r\left[k\right]=s_1\left[k\right]+b{s}_1[k-1]+c{s}_2\left[k\right].$$

The difference equation describing the equalizer input-output behavior is

$$x\left[k\right]=r\left[k\right]+dr[k-1]$$

The decision device is the sign operator.

1. Write out the difference equation relating $s_1\left[k\right]$ and $s_2\left[k\right]$ to $x\left[k\right]$ .
2. Consider the performance function
   $$J_{MSE}=\frac{1}{N}\sum _{k=k_0+1}^{k_0+N}{(s_1\left[k\right]-x\left[k\right])}^2.$$
   For uncorrelated source symbol sequences, $J_{MSE}$ is minimized by minimizing $\sum {\rho }_i^2$ . For $b=0.8$ and $c=0.3$ compute the equalizer setting $d$ that minimizes $J_{MSE}$ .
3. For $b=0.8$ , $c=0.3$ , and $d=0$ , (i.e. equalizer-less reception) does the system exhibit decision errors?
4. For $b=0.8$ , $c=0.3$ , and your $d$ from part (b) does the system exhibit decision errors?Does this system suffer more severe decision errors than the equalizer-less system of part (c)?