Digital Filtering

Digital filters are fundamental to digital audio processing. While we only have time here for a cursory overview of the essential features of filters, students are encouraged to pursue more advanced courses and references in filter analysis and design.

• In general, we need to manipulate our signals. Even if we only seek to measure and analyze “real world” signals, we still typically need to “process” these signals in order to compensate for measurement system “biases”.

• The processing of signals is called filtering. When applied to discrete-time signals, this processing is called digital filtering.

• Digital filters are defined by their impulse response, $h[n]$, or the filter output given a unit sample impulse input signal. A discrete-time unit impulse signal is defined by:
  $$\delta[n] \ensuremath{\stackrel{\Delta}{=}}\left\{ \begin{array}{ll} 1, & n = 0 \\ 0, & n \neq 0 \end{array} \right.$$

• The filtering operation in the time domain is referred to as convolution, defined as
  $$y[n] = x[n] \ast h[n] = \sum_{m=-\infty}^{\infty} h[m] x[n-m],$$
  where $h$ is the filter impulse response and $x$ is the input signal to the filter.

• Digital filters are often more intuitively understood in terms of their frequency response. That is, how is a sinusoidal signal of a given frequency affected by the filter.

• One way to find the frequency response of a digital filter is by taking the Discrete Fourier Transform of the filter impulse response.

• The frequency response of a filter, which is represented using complex numbers of the form $a + j b$, consists of its magnitude and phase responses. The magnitude response indicates the ratio of a filtered sine wave's output amplitude to its input amplitude (and is given by $\sqrt{a^2 + b^2}$). The phase response describes the phase “offset” or time delay in radians experienced by a sine wave passing through a filter (and is given by $\tan^{-1}(b/a)$).