DSP First - Chapter 5 - FIR Filters

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 $\delta[n]$.