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In designing a two-dimensional digital filter with special consideration of the phase characteristics, particularly the group delay characteristics, a very high-order filter is generally required. This paper extends the coefficients of the filter from real to complex and describes a design technique for comparatively low-order, two-dimensional, circularly symmetric filters with consideration of phase characteristics. First, the method of transforming a two-dimensional input signal into an analytic signal is described and then the design technique of two kinds of complex coefficient filters which use the properties of analytic signals is discussed. In the first kind variable transformation to a 1-D real coefficient FIR filter is used to obtain a 2-D has the merits of permitting independent setting of the sharpness of cutoff characteristics and circular symmetry. In the other kind of complex coefficient filter a recursive digital filter is obtained by using an all-pole filter having constant group delay in the passband of the transfer function and having a 2-D complex coefficient mirror image polynomial. It always then values of the coefficients and the overall group delay is also constants. Moreover, in this paper, design examples based on these methods are given and the validity of our methods is shown. [ABSTRACT FROM AUTHOR]