Unified Theory: Multirate Transformations

Gianfranco Cariolaro

doi:10.1007/978-0-85729-464-7_7

Unified Signal Theory pp 345-397 | Cite as

Unified Theory: Multirate Transformations

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Abstract

Multirate transformations, better known as multirate systems, operate on discrete signals that have different rates. A single discrete transformation (tf) is characterized by two rates, the input and the output rate, but in a combination of tfs we may find several rates. Typical components of multirate tfs belong to the class of QIL tfs, which include up-samplers, down-samplers, interpolators, decimators and fractional interpolators. Sometimes, exponential modulators, which belong to the broader class of PIL tfs, are considered. The peculiarity of multirate tfs lies in certain properties which allow several alternative architectures with interesting consequences for applications. In this chapter, multirate systems are developed in the signal-domain, where the Unified Signal Theory is particularly useful for the relevance it gives to the domains and consequently to the rates. The signal-domain approach is atypical in the literature where the z-domain-approach is commonly used. Another difference with the literature is in multidimensional multirate systems where we use a representation-independent formulation of lattice operations, whereas other authors make use of heavy matrix manipulations.

Keywords

Orthogonal Frequency Division Multiplex Impulse Response Finite Group Orthogonal Frequency Division Multiplex System Parallel Architecture

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Appendix: Deduction of Parallel Architecture with EMs

To overcome the diagonal form of the PI, we introduce the auxiliary kernel

$$q(t-u,u)=h(t,u) ,\quad (t,u) \in K \times J . $$

(7.76)

Hence, the PI condition (7.44) becomes

$$q(t-u,u+p)=q(t-u,u) ,\quad p \in P, $$

(7.77)

which states that q(v,u) has periodicity P with respect to its second argument. Now the domain of q(v,z) is given by D={(t−u,u)∣t∈K, u∈J}, which defines a nonseparable lattice, in general. But, if K⊃J, the difference t−u belongs to K and D becomes separable as K×J. Then the function q(v,z), v∈K, z∈J has periodicity P in z and, for a fixed v, can be formulated as a signal on I=J/P. The DFT/IDFT of this signal are

$$Q(v,\lambda) =\sum_{z\in J/P} \mathrm{d}(J) \,q(v,z) \mathrm {e}^{-\mathrm{i}2\pi\lambda z},\quad q(v,z) =\sum_{\lambda\in P^\star/J^\star} \mathrm{d}(P^\star)\,Q(v,z) \mathrm{e}^{\mathrm{i}2\pi\lambda z}.$$

Letting g _λ(v)=[1/d(J)]Q(v,λ), the IDFT becomes

$$q(v,z)=\frac{1}{N_J}\sum_{\lambda\in P^\star/J^\star)} g_{\lambda}(v)\mathrm{e}^{\mathrm{i}2\pi\lambda z} .$$

Finally, considering (7.76) we get (7.50) and (7.51). The proof of (7.52) is analogous.

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Unified Theory: Multirate Transformations

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