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Conventional linear phase finite impulse response digital filters have real-valued coefficients forming a symmetric or antisymmetric pulse response. Such filters have a frequency response with an even magnitude about 0 Hz. Odd-length filters with complex coefficients forming a complex conjugate symmetric pulse response also have a linear phase response but can have a magnitude response which is not even about 0 Hz. These filters have applications in single-sideband modulation systems and in processing in-phase and quadrature signals in sonar and radar. Only minor modifications are required in two FIR filter design computer programs (the Parks and McClellan Remez exchange algorithm and the Leon and McCallig constrained ripple algorithm computer programs) to allow the design of linear phase filters with complex coefficients.