On the Relevancy of the Perfect Reconstruction Property when Minimizing the Mean Square Error in FIR MIMO Filter Systems Are Hjørungnes Signal Processing Laboratory Helsinki University of Technology P. O. Box 3000, FIN-02015 HUT, Finland Tel. +358 9 451 5386, Fax +358 9 452 3614 Email: arehj@wooster.hut.fi Paulo S. R. Diniz Signal Processing Laboratory Universidade Federal do Rio de Janeiro P. O. Box 68504, CEP: 21945-970 RJ, Brazil Tel. +55 21 2562 8211, Fax +55 21 2562 8205 Email: diniz@lps.ufrj.br Abstract— Under the assumption that the transmitter FIR multiple- input multiple-output (MIMO) filter is given and FIR left invertible, conditions are derived for when the minimum minimum mean square error (MSE) FIR MIMO filter system does possess the perfect reconstruc- tion (PR) or zero forcing (ZF) property. In most cases, examples using common models for both source coding and communication systems show that constraining the system to have the PR or ZF property is suboptimal in the minimum MSE sense. I. I NTRODUCTION For a given left invertible transmitter FIR MIMO filter, when is it optimal in the minimum MSE sense to use a system possessing the PR or ZF property? This is the main question answered in this article. The system treated can model for example filter banks used in source coding and communication systems that use linear FIR MIMO filters in both the transmitter and receiver. In source coding systems using analysis and synthesis filter banks, the PR property is often enforced to the system [1], [2], [3], [4]. In communication systems using transmitter and receiver FIR MIMO systems, the filters can be designed under the so-called zero forcing (ZF) constraint [5], [6], [7], [8]. Here, the PR or ZF constraint is questioned, and it is shown under which conditions there is no loss in optimality, in the minimum MSE sense, by enforcing the PR or ZF constraint. It is shown that in almost all source coding and channel models considered, it is suboptimal to enforce the PR or ZF constraint in the system design. The dimension of the vectors in the original time series and the dimension of the input time series to the channel may in general be different, and the system is assumed to be discrete in time. The transmitter and receiver are represented by polyphase matrices. In Figure 1, the matrix and the matrix are causal FIR MIMO filters of order and , respectively, which represent linear time invariant signal processing units, and they can be expressed as: and (1) This means that the matrices and , have dimensions and , respectively. It is assumed that all the vector time series are jointly wide sense stationary (WSS). The PR or ZF property is present if it is possible to recover a delayed version of the vector from , in the absence of noise, i.e., . This is only possible if , because if , then the matrix has normal rank [3] less than or equal to , and then this matrix cannot be left invertible and PR is impossible. Therefore, it is assumed that in this article. In [9], the transform and non-causal infinite order cases were considered for the case where . This article is an extension of Fig. 1. System model. the results found in [9] to the FIR case and to the case where . In [9], it was shown that the condition for PR being optimal in the non-causal infinite order case is: (2) where the operator means complex conjugated transposed, and the condition for the transform case is: (3) It will be shown that, when and non-causal infinite order or transform systems are considered, the conditions proposed in the current article reduces to the results in Equations (2) or (3), respectively. In [9], the transform and the non-causal infinite-order cases were treated for zero block delay through the MIMO filter system. The FIR case has to be treated in a different manner, because in [9], it was assumed that the receiver filter consists of the PR receiver filter followed by a Wiener filter. Then, conditions for when the Wiener polyphase matrix is equal to the identity matrix were given for the transform case and the non-causal infinite-order case. In both these cases, it can be assumed that the Wiener filter has the same order as the first part of the receiver filter, which is either infinite or zero. In the FIR case, this is not possible because, if the first part of the receiver filter is the FIR inverse of the transmitter filter, the following Wiener polyphase filter can only be a memoryless matrix if the order of the total receiver filter is to be kept constant. By using this method, the optimization does not include a search over all polyphase matrices of a given order. Therefore, a different method must be used for the FIR case. The related problems for conditions when the PR solution exists are investigated in [10], [11], and the jointly minimum MSE transmitter and receiver FIR MIMO filters are found in [12]. The rest of this article is organized as follows: In Section II, some of the notation and definitions needed in the article is introduced and in Section III, the equation for the optimal FIR MIMO receiver filter is derived. The conditions for when the PR property is optimal are derived in Section IV. Section V gives examples showing that these conditions for PR being optimal are not in general satisfied for many