part transforms the input signal into N filtered and downsampled outputs . The synthesis part recovers the original signal from by upsampling and filtering. This kind of setup is used in many applications such as subband coding, multichannel acquisition, and discrete wavelet transforms. Perfect reconstruction filter banks We can use polyphase representation, so input signal can be represented by a vector of its polyphase components . Denote So we would have , where denotes the j-th polyphase component of the filter . Similarly, for the output signal we would have , where . Also G is a matrix where denotes ith polyphase component of the jth synthesis filter Gj(z). The filter bank has perfect reconstruction if for any input, or equivalently which means that G(z) is a left inverse of H(z). Multidimensional filter design 1-D filter banks have been well developed until today. However, many signals, such as image, video, 3D sound, radar, sonar, are multidimensional, and require the design of multidimensional filter banks. With the fast development of communication technology, signal processing system needs more room to store data during the processing, transmission and reception. In order to reduce the data to be processed, save storage and lower the complexity, multirate sampling techniques were introduced to achieve these goals. Filter banks can be used in various areas, such as image coding, voice coding, radar and so on. Many 1D filter issues were well studied and researchers proposed many 1D filter bank design approaches. But there are still many multidimensional filter bank design problems that need to be solved. Some methods may not well reconstruct the signal, some methods are complex and hard to implement. The simplest approach to design a multi-dimensional filter bank is to cascade 1D filter banks in the form of a tree structure where the decimation matrix is diagonal and data is processed in each dimension separately. Such systems are referred to as separable systems. However, the region of support for the filter banks might not be separable. In that case designing of filter bank gets complex. In most cases we deal with non-separable systems. A filter bank consists of an analysis stage and a synthesis stage. Each stage consists of a set of filters in parallel. The filter bank design is the design of the filters in the analysis and synthesis stages. The analysis filters divide the signal into overlapping or non-overlapping subbands depending on