Thus, a real digital filter maps every real, discrete-time signal to a real, discrete-time signal. A complex filter, on the other hand, may produce a complex output signal even when its input signal is real.
Definition. A real digital filter is defined as any real-valued function of a real signal for each integer .
We may express the input-output relation of a digital filter by the notation
In this book, we are concerned primarily with single-input, single-output (SISO) digital filters. For this reason, the input and output signals of a digital filter are defined as real or complex numbers for each time index (as opposed to vectors). When both the input and output signals are vector-valued, we have what is called a multi-input, multi-out (MIMO) digital filter. We look at MIMO allpass filters in §C.3 and MIMO state-space filter forms in Appendix G, but we will not cover transfer-function analysis of MIMO filters using matrix fraction descriptions [37].