Definition of a Filter

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

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 ${\cal S}$ . The operator ${\cal T}$ maps every signal $x\in{\cal S}$ to some new signal $y\in{\cal S}$ . (For simplicity, we take ${\cal S}$ 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 ${\cal S}$ . If, additionally, the signal space ${\cal S}$ consists only of finite-length signals, all $N$ samples long, i.e., ${\cal S}\subset\mathbb{R}^N$ or ${\cal S}\subset\mathbb{C}^N$ , then every linear filter ${\cal T}$ may be called a linear transformation, which is representable by a 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 numbers 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 §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].