Fractional biorthogonal partners in channel equalization and signal interpolation

Abstract

Abstract-The concept of biorthogonal partners has been introduced recently by the authors. The work presented here is an extension of some of these results to the case where the upsampling and downsampling ratios are not integers but rational numbers, hence, the name fractional biorthogonal partners. The conditions for the existence of stable and of finite impulse response (FIR) fractional biorthogonal partners are derived. It is also shown that the FIR solutions (when they exist) are not unique. This property is further explored in one of the applications of fractional biorthogonal partners, namely, the fractionally spaced equalization in digital communications. The goal is to construct zero-forcing equalizers (ZFEs) that also combat the channel noise. The performance of these equalizers is assessed through computer simulations. Another application considered is the all-FIR interpolation technique with the minimum amount of oversampling required in the input signal. We also consider the extension of the least squares approximation problem to the setting of fractional biorthogonal partners.

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“…Figure 8a illustrates the fractionally-spaced receiver, while Figure 8b shows the equivalent discrete-time model with fractional sampling. The filters F r j (z) and W r j (z) are known as fractional biorthonormal partners [30] if in the absence of noise, the overall transmission is an exact delay (δ).…”

“…In fact, using well-known multirate noble identities [31], this scalar system can be further expressed via its block transmission equivalent, according to Figure 9. The polynomials E(z) and R(z) correspond to polyphase matrices, which are uniquely determined by polyphase components F r j (z) and W r j (z) respectively, via their order L type-2 and type-1 decompositions (we drop the notation on r j in their factors for simplicity)-see [30]:…”

“…In [30], the authors develop a zero forcing solution for R (z) based on the Smith form of E(z). It is shown that there exists an FIR right vector fractional biorthogonal partner of F i (z) if and only if Q r L >KM , and the greatest common divisor (gcd) of all theK M ×KM minors of E(z) is a delay.…”

“…The additional degree of freedom offered by the resulting receiver is then exploited in order to minimize the noise power at the output. As mentioned in [30], the advantage of the method relies on the fact that neither the input nor the noise autocorrelation matrices are needed when the noise is uncorrelated, in comparison with an MMSE solution. Note that the ZF solution of [30] is not the only one independent of statistics when the noise is uncorrelated; a pure minimum-variance (LS) based receiver also minimizes the output noise power.…”

“…As mentioned in [30], the advantage of the method relies on the fact that neither the input nor the noise autocorrelation matrices are needed when the noise is uncorrelated, in comparison with an MMSE solution. Note that the ZF solution of [30] is not the only one independent of statistics when the noise is uncorrelated; a pure minimum-variance (LS) based receiver also minimizes the output noise power. Also, in the solution proposed in [30], their ZF method is compared to the MMSE solution considering a zero decision delay in the filter design.…”