## Definition of a Filter

Definition. A real digital filter is defined as any real-valued function of a real signal for each integer .
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.

We may express the input-output relation of a digital filter by the notation

 (5.1)

where denotes the entire input signal, and is the output signal at time . (We will also refer to as simply .) The general filter is denoted by , which stands for any transformation from a signal to a sample value at time . The filter can also be called an operator on the space of signals . The operator maps every signal to some new signal . (For simplicity, we take to be the space of complex signals whenever is complex.) If is linear, it can be called a linear operator on . If, additionally, the signal space consists only of finite-length signals, all samples long, i.e., or , then every linear filter may be called a linear transformation, which is representable by constant matrix.

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].

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Examples of Digital Filters
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Definition of a Signal