Definition of a Filter

Definition. A real digital filter ${\cal T}_n$ is defined as any real-valued function of a signal for each integer $n\in{\bf Z}$.

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

$$y(n)={\cal T}_n\{x(\cdot)\} \protect$$ (5.1)

where $x(\cdot)$ denotes the entire input signal, and $y(n)$ is the output signal at time $n$. (We will also refer to $x(\cdot)$ as simply $x$.) The general filter is denoted by ${\cal T}_n\{x\}$, which stands for any transformation from a signal $x$ to a sample value at time $n$. The filter ${\cal T}$ can also be called an operator on the space of signals ${\cal X}$. The operator ${\cal T}$ maps every signal $x\in{\cal X}$ to some new signal $y\in{\cal X}$. (For simplicity, we take ${\cal X}$ to be the space of complex signals whenever ${\cal T}$ is complex.) If ${\cal T}$ is linear, it can be called a linear operator on ${\cal X}$. If, additionally, the signal space ${\cal X}$ consists only of finite-length signals, all $N$ samples long, i.e., ${\cal X}\subset{\bf R}^N$ or ${\cal X}\subset{\bf C}^N$, then every linear filter ${\cal T}$ may be called a linear transformation, which is representable by constant $N\times N$ 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 scalars for each time index $n$ (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 §D.3 and MIMO state-space filter forms in Appendix E, but we will not cover transfer-function analysis of MIMO filters using matrix fraction descriptions [37].