A filter is a system that is designed to remove some component or modify some characteristic of a signal.

Several different things are introduced, including: - Finite impulse response (FIR) systems: refered as FIR filters, are systems for which each output value is the sum of a finite number of weighted values of the input sequence. - Difference equation: the basis of the input-output structure of the FIR filter as a time-domain computation. - Unit impulse response - Convolution - Linearity and time invariance - Discrete-time systems

Discrete-time systems

A discrete-time system is a computational process for transforming one sequence into another sequence. - \(x[n]\rightarrow \mathcal{T}\{\cdot\}\rightarrow y[n]=\mathcal{T}\{x[n]\}\), where \(x[n]\) is the input signal and \(y[n]\) is the output signal, both of which are discrete-time signals.

The running-average (moving-average) filter

  • Difference equation, e.g., the general, causal, linear and time invariant difference equation: \[\begin{equation} y[n] = \sum\limits_{k=0}^M b_k x[n-k] - \sum\limits_{l=0}^N a_l y[n-l], \end{equation}\label{DE}\] where \(k\) and \(l\) are the "dummy" counting indices for the sum and \(n\) denotes the index of the \(n^{th}\) sample of the output sequence.
  • Causal and noncausal:
    • Causal filter: a filter that uses only the present and past values of the input.
    • Noncausal filter: a filter that uses future values of the input.
  • Causal running averager or backward averager, similarly, we have the centralized running averager and the forward averager.

The general FIR filter

  • The general causal difference equation \[\begin{equation} y[n] = \sum\limits_{k=0}^M b_k x[n-k], \end{equation}\label{FIR}\] where the coefficients \(b_k\) are fixed numbers.
    • \(M\), the order of the FIR filter
    • \(L=M+1\), the number of filter coefficients is the filter length
  • Eq. \(\eqref{FIR}\) could be written as \[\begin{equation} y[n] = \sum\limits_{l=n-M}^n b_{n-l} x[l], \end{equation}\label{FIR_l}\] where \(l=n-k\) showing the FIR is causal using the input \(x[l]\) start from the previous \(M\) samples, i.e. \(l=n-M\), up to the current one \(l=n\)
  • For finite length input signal, i.e., \(x[l]\neq 0\) for \(l\in[0, N-1]\) and a \(M^{th}\)-order FIR filter (of length \(M+1\), i.e., involving \(M+1\) samples), there would be transient component of the output including \(M\) samples running onto and running off session. And the total output length would be \(N+M\text{ (order)}=N+L-1\).

The unit impulse response and convolution

The impulse response provides a complete characterization of the FIR filter. - Three new ideas introduced: - the unit impulse sequence - the unit impulse response - the convolution sum

Unit impulse sequence

  • Unit impulse, or mathematically taken as the Kronecker delta function \[\begin{equation} \delta[n]= \begin{cases} 1\quad n=0\\ 0\quad n\neq 0 \end{cases} \end{equation}\label{deltaFunction}\]
  • Express any sequence interm of delta function \[\begin{equation} x[n]=\sum\limits_k x[k]\delta[n-k] \end{equation}\lab el{x}\]
    • the unit impulse is a sequence
    • \(\mathbf{x}\) is a summation of infinite impulse sequences \(\mathbf{\delta}_k\)

Unit impulse response sequence

  • The output from a filter is called the response to the input.
  • Unit impulse response \(h[n]\) represents the output when the input is the unit impulse \(\delta[n]\).