FIR Filter Basics
1.1 What are "FIR filters"?
FIR filters are one of two primary types of digital filters used in Digital Signal Processing (DSP) applications, the other type being IIR.
1.2 What does "FIR" mean?
"FIR" means "Finite Impulse Response". If you put in an impulse, that is, a single "1" sample followed by many "0" samples, zeroes will come out after the "1" sample has made its way through the delay line of the filter.
1.3 Why is the impulse response "finite"?
In the common case, the impulse response is finite because there is no feedback in the FIR. A lack of feedback guarantees that the impulse response will be finite. Therefore, the term "finite impulse response" is nearly synonymous with "no feedback".
However, if feedback is employed yet the impulse response is finite, the filter still is a FIR. An example is the moving average filter, in which the Nth prior sample is subtracted (fed back) each time a new sample comes in. This filter has a finite impulse response even though it uses feedback: after N samples of an impulse, the output will always be zero.
1.4 How do I pronounce "FIR"?
Some people say the letters F-I-R; other people pronounce as if it were a type of tree. We prefer the tree. (The difference is whether you talk about an F-I-R filter or a FIR filter.)
1.5 What is the alternative to FIR filters?
DSP filters can also be "Infinite Impulse Response" (IIR). (See dspGuru's IIR FAQ.) IIR filters use feedback, so when you input an impulse the output theoretically rings indefinitely.
1.6 How do FIR filters compare to IIR filters?
Each has advantages and disadvantages. Overall, though, the advantages of FIR filters outweigh the disadvantages, so they are used much more than IIRs.
1.6.1 What are the advantages of FIR Filters (compared to IIR filters)?
Compared to IIR filters, FIR filters offer the following advantages:
- They can easily be designed to be "linear phase" (and usually are). Put simply, linear-phase filters delay the input signal but don’t distort its phase.
- They are simple to implement. On most DSP microprocessors, the FIR calculation can be done by looping a single instruction.
- They are suited to multi-rate applications. By multi-rate, we mean either "decimation" (reducing the sampling rate), "interpolation" (increasing the sampling rate), or both. Whether decimating or interpolating, the use of FIR filters allows some of the calculations to be omitted, thus providing an important computational efficiency. In contrast, if IIR filters are used, each output must be individually calculated, even if it that output will discarded (so the feedback will be incorporated into the filter).
- They have desireable numeric properties. In practice, all DSP filters must be implemented using finite-precision arithmetic, that is, a limited number of bits. The use of finite-precision arithmetic in IIR filters can cause significant problems due to the use of feedback, but FIR filters without feedback can usually be implemented using fewer bits, and the designer has fewer practical problems to solve related to non-ideal arithmetic.
- They can be implemented using fractional arithmetic. Unlike IIR filters, it is always possible to implement a FIR filter using coefficients with magnitude of less than 1.0. (The overall gain of the FIR filter can be adjusted at its output, if desired.) This is an important consideration when using fixed-point DSP's, because it makes the implementation much simpler.
1.6.2 What are the disadvantages of FIR Filters (compared to IIR filters)?
Compared to IIR filters, FIR filters sometimes have the disadvantage that they require more memory and/or calculation to achieve a given filter response characteristic. Also, certain responses are not practical to implement with FIR filters.
1.7 What terms are used in describing FIR filters?
- Impulse Response - The "impulse response" of a FIR filter is actually just the set of FIR coefficients. (If you put an "impluse" into a FIR filter which consists of a "1" sample followed by many "0" samples, the output of the filter will be the set of coefficients, as the 1 sample moves past each coefficient in turn to form the output.)
- Tap - A FIR "tap" is simply a coefficient/delay pair. The number of FIR taps, (often designated as "N") is an indication of 1) the amount of memory required to implement the filter, 2) the number of calculations required, and 3) the amount of "filtering" the filter can do; in effect, more taps means more stopband attenuation, less ripple, narrower filters, etc.
- Multiply-Accumulate (MAC) - In a FIR context, a "MAC" is the operation of multiplying a coefficient by the corresponding delayed data sample and accumulating the result. FIRs usually require one MAC per tap. Most DSP microprocessors implement the MAC operation in a single instruction cycle.
- Transition Band - The band of frequencies between passband and stopband edges. The narrower the transition band, the more taps are required to implement the filter. (A "small" transition band results in a "sharp" filter.)
- Delay Line - The set of memory elements that implement the "Z^-1" delay elements of the FIR calculation.
- Circular Buffer - A special buffer which is "circular" because incrementing at the end causes it to wrap around to the beginning, or because decrementing from the beginning causes it to wrap around to the end. Circular buffers are often provided by DSP microprocessors to implement the "movement" of the samples through the FIR delay-line without having to literally move the data in memory. When a new sample is added to the buffer, it automatically replaces the oldest one.