## Introduction

The most common type of filter dealt with in practice is a linear, causal, and time-invariant operator on the vector space consisting of arbitrary real-valued functions of time. Since we are dealing with the space of functions of time, we will use the terms*vector, function,*and

*signal*interchangeably. When time is a continuous variable, the vector space is infinite-dimensional even when time is restricted to a finite interval. Digital filters are simpler in many ways theoretically because finite-time digital signals occupy a finite-dimensional vector space. Furthermore, every linear operator on the space of digital signals may be represented as a matrix.

^{H.1}If the range of time is restricted to N samples then the arbitrary linear operator is an N by N matrix. In the discussion that follows, we will be exclusively concerned with the digital domain. Every linear filter will be representable as a matrix, and every signal will be expressible as a column vector.

Linearity implies the superposition principle which is presently indispensible for a general filter response analysis. The superposition principle states that if a signal is represented as a linear combination of signals , then the response of any linear filter may written as the same linear combination of the signals where . More generally,

*basis functions*. An example of a basis set is the familiar set of sinusoids at all frequencies. The most crucial use of linearity for our purposes is the representation of an arbitrary linear filter as a matrix operator. Causality means that the filter output does not depend on future inputs. This is necessary in analog filters where time is a real entity, but for digital filters causality is highly unnecessary unless the filter must operate in real-time. Requiring a filter to be causal results in a triangular matrix representation. A time-invariant filter is one whose response does not depend on the time of excitation. This allows

*superposition in time*in addition to the superposition of component functions given by linearity. A matrix representing a linear time-invariant filter is Toeplitz (each diagonal is constant). The chief value of time-invariance is that it allows a linear filter to represented by its

*impulse response*which, for digital filters, is the response elicited by the signal . A deeper consequence of superposition in time together with superposition of component signal responses is the fact that every stable linear time invariant filter emits a sinusoid at frequency in response to an input sinusoid at frequency after sufficient time for start-up transients to settle. For this reason sinusoids are called

*eigenfunctions*of linear time-invariant systems. Another way of putting it is that a linear time-invariant filter can only modify a sinusoidal input by a constant scaling of its amplitude and a constant offset in its phase. This is the rationale behind Fourier analysis. The Laplace transform of the impulse response gives the

*transfer function*and the Fourier transform of the impulse response is the

*frequency response*. It is important to note that relaxing time-invariance only prevents us from using superposition in time. Consequently, while we can no longer uniquely characterize a filter in terms of its impulse response, we may still characterize it in terms of its

**basis function response**. This will be developed below for the particular basis functions used in the Discrete Fourier Transform (DFT). These basis functions are defined for the N-dimensional discrete-time signal space as

**Next Section:**

Derivation

**Previous Section:**

State Space Problems