Finite Impulse Response Digital Filters

In §5.1 we defined the general difference equation for IIR filters, and a couple of second-order examples were diagrammed in Fig.5.1. In this section, we take a more detailed look at the special case of Finite Impulse Response (FIR) digital filters. In addition to introducing various terminology and practical considerations associated with FIR filters, we'll look at a preview of transfer-function analysis (Chapter 6) for this simple special case.

Figure: The general, causal, length $ N=M+1$, finite-impulse-response (FIR) digital filter. For FIR filters, direct-form I and direct-form II are the same (see Chapter 9).
\begin{figure}\input fig/fir.pstex_t
\end{figure}

Figure 5.5 gives the signal flow graph for a general causal FIR filter Such a filter is also called a transversal filter, or a tapped delay line. The implementation shown is classified as a direct-form implementation.

FIR impulse response

The impulse response $ h(n)$ is obtained at the output when the input signal is the impulse signal $ \delta = [1,0,0,0,\ldots]$5.6). If the $ k$th tap is denoted $ b_k$, then it is obvious from Fig.5.5 that the impulse response signal is given by

$\displaystyle h(n)\isdef \left\{\begin{array}{ll} 0, & n<0 \\ [5pt] b_n, & 0\leq n\leq M \\ [5pt] 0, & n> M \\ \end{array} \right. \protect$ (6.6)

In other words, the impulse response simply consists of the tap coefficients, prepended and appended by zeros.


Convolution Representation of FIR Filters

Notice that the output of the $ k$th delay element in Fig.5.5 is $ x(n-k)$, $ k=0,1,2,\ldots,M$, where $ x(n)$ is the input signal amplitude at time $ n$. The output signal $ y(n)$ is therefore

$\displaystyle y(n)$ $\displaystyle =$ $\displaystyle b_0 x(n) + b_1 x(n-1) + b_2 x(n-2) + \cdots + b_M x(n-M)$  
  $\displaystyle =$ $\displaystyle \sum_{m=0}^M b_m x(n-m)$  
  $\displaystyle =$ $\displaystyle \sum_{m=-\infty}^{\infty} h(m) x(n-m)$  
  $\displaystyle \isdef$ $\displaystyle (h\ast x)(n)$ (6.7)

where we have used the convolution operator ``$ \ast $'' to denote the convolution of $ h$ and $ x$, as defined in Eq.$ \,$(5.4). An FIR filter thus operates by convolving the input signal $ x$ with the filter's impulse response $ h$.


The ``Finite'' in FIR

From Eq.$ \,$(5.7), we can see that the impulse response becomes zero after time $ M=N-1$. Therefore, a tapped delay line (Fig.5.5) can only implement finite-duration impulse responses in the sense that the non-zero portion of the impulse response must be finite. This is what is meant by the term finite impulse response (FIR). We may say that the impulse response has finite support [52].


Causal FIR Filters

From Eq.$ \,$(5.6), we see also that the impulse response $ h(n)$ is always zero for $ n < 0$. Recall from §5.3 that any LTI filter having a zero impulse response prior to time 0 is said to be causal. Thus, a tapped delay line such as that depicted in Fig.5.5 can only implement causal FIR filters. In software, on the other hand, we may easily implement non-causal FIR filters as well, based simply on the definition of convolution.


FIR Transfer Function

The transfer function of an FIR filter is given by the z transform of its impulse response. This is true for any LTI filter, as discussed in Chapter 6. For FIR filters in particular, we have, from Eq.$ \,$(5.6),

$\displaystyle H(z) \isdef \sum_{n=-\infty}^{\infty} h_n z^{-n} = \sum_{n=0}^M b_n z^{-n} \protect$ (6.8)

Thus, the transfer function of every length $ N=M+1$ FIR filter is an $ M$th-order polynomial in $ z^{-1}$.


FIR Order

The order of a filter is defined as the order of its transfer function, as discussed in Chapter 6. For FIR filters, this is just the order of the transfer-function polynomial. Thus, from Equation (5.8), the order of the general, causal, length $ N=M+1$ FIR filter is $ M$ (provided $ b_M\neq 0$).

Note from Fig.5.5 that the order $ M$ is also the total number of delay elements in the filter. This is typical of practical digital filter implementations. When the number of delay elements in the implementation (Fig.5.5) is equal to the filter order, the filter implementation is said to be canonical with respect to delay. It is not possible to implement a given transfer function in fewer delays than the transfer function order, but it is possible (and sometimes even desirable) to have extra delays.


FIR Software Implementations

In matlab, an efficient FIR filter is implemented by calling

        outputsignal = filter(B,1,inputsignal);
where

$\displaystyle \texttt{B} = [b_0, b_1, \ldots, b_M].
$

It is relatively efficient because filter is a built-in function (compiled C code in most matlab implementations). However, for FIR filters longer than a hundred or so taps, FFT convolution should be used for maximum speed. In Octave and the Matlab Signal Processing Toolbox, fftfilt implements FIR filters using FFT convolution (say ``help fftfilt'').

Figure 5.6 lists a second-order FIR filter implementation in the C programming language.

Figure 5.6: C code for implementing a length 3 FIR filter.

 
  typedef double *pp;  // pointer to array of length NTICK
  typedef double word; // signal and coefficient data type

  typedef struct _fir3Vars {
      pp outputAout;
      pp inputAinp;
      word b0;
      word b1;
      word b2;
      word s1;
      word s2;
  } fir3Vars;

  void fir3(fir3Vars *a)
  {
      int i;
      word input;
      for (i=0; i<NTICK; i++) {
          input = a->inputAinp[i];
          a->outputAout[i] = a->b0 * input
                +  a->b1 * a->s1  +  a->b2 * a->s2;
          a->s2 = a->s1;
          a->s1 = input;
      }
  }


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Convolution Representation