# 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.

# FIR Filter Properties

## 2.1 Linear Phase

#### 2.1.1 What is the association between FIR filters and "linear-phase"?

Most FIRs are linear-phase filters; when a linear-phase filter is desired, a FIR is usually used.

#### 2.1.2 What is a linear phase filter?

"Linear Phase" refers to the condition where the phase response of the filter is a linear (straight-line) function of frequency (excluding phase wraps at +/- 180 degrees). This results in the delay through the filter being the same at all frequencies. Therefore, the filter does not cause "phase distortion" or "delay distortion". The lack of phase/delay distortion can be a critical advantage of FIR filters over IIR and analog filters in certain systems, for example, in digital data modems.

#### 2.1.3 What is the condition for linear phase?

FIR filters are usually designed to be linear-phase (but they don't have to be.) A FIR filter is linear-phase if (and only if) its coefficients are symmetrical around the center coefficient, that is, the first coefficient is the same as the last; the second is the same as the next-to-last, etc. (A linear-phase FIR filter having an odd number of coefficients will have a single coefficient in the center which has no mate.)

#### 2.1.4 What is the delay of a linear-phase FIR?

The formula is simple: given a FIR filter which has N taps, the delay is: (N - 1) / (2 * Fs), where Fs is the sampling frequency. So, for example, a 21 tap linear-phase FIR filter operating at a 1 kHz rate has delay: (21 - 1) / (2 * 1 kHz)=10 milliseconds.

#### 2.1.4 What is the alternative to linear phase?

Non-linear phase, of course. ;-) Actually, the most popular alternative is "minimum phase". Minimum-phase filters (which might better be called "minimum delay" filters) have less delay than linear-phase filters with the same amplitude response, at the cost of a non-linear phase characteristic, a.k.a. "phase distortion".

A lowpass FIR filter has its largest-magnitude coefficients in the center of the impulse response. In comparison, the largest-magnitude coefficients of a minimum-phase filter are nearer to the beginning . (See dspGuru's tutorial How To Design Minimum-Phase FIR Filters for more details.)

### 2.2 Frequency Response

#### 2.2.1 What is the Z transform of a FIR filter?

For an N-tap FIR filter with coefficients h(k), whose output is described by:

y(n)=h(0)x(n) + h(1)x(n-1) + h(2)x(n-2) + ... h(N-1)x(n-N-1),

the filter's Z transform is:

H(z)=h(0)z-0 + h(1)z-1 + h(2)z-2 + ... h(N-1)z-(N-1) , or

#### 2.2.2 What is the frequency response formula for a FIR filter?

The variable z in H(z) is a continuous complex variable, and we can describe it as: z=r·ejw, where r is a magnitude and w is the angle of z. If we let r=1, then H(z) around the unit circle becomes the filter's frequency response H(jw). This means that substituting ejw for z in H(z) gives us an expression for the filter's frequency response H(w), which is:

H(jw)=h(0)e-j0w + h(1)e-j1w + h(2)e-j2w + ... h(N-1)e-j(N-1)w , or

Using Euler's identity, e-ja=cos(a) - jsin(a), we can write H(w) in rectangular form as:

H(jw)=h(0)[cos(0w) - jsin(0w)] + h(1)[cos(1w) - jsin(1w)] + ... h(N-1)[cos((N-1)w) - jsin((N-1)w)] , or

#### 2.2.3 Can I calculate the frequency response of a FIR using the Discrete Fourier Transform (DFT)?

Yes. For an N-tap FIR, you can get N evenly-spaced points of the frequency response by doing a DFT on the filter coefficients. However, to get the frequency response of the filter at any arbitrary frequency (that is, at frequencies between the DFT outputs), you will need to use the formula above.

#### 2.2.4 What is the DC gain of a FIR filter?

Consider a DC (zero Hz) input signal consisting of samples which each have value 1.0. After the FIR's delay line had filled with the 1.0 samples, the output would be the sum of the coefficients. Therefore, the gain of a FIR filter at DC is simply the sum of the coefficients.

This intuitive result can be checked against the formula above. If we set w to zero, the cosine term is always 1, and the sine term is always zero, so the frequency response becomes:

#### 2.2.5 How do I scale the gain of a FIR filter?

Simply multiply all coefficients by the scale factor.

### 2.3 Numeric Properties

#### 2.3.1 Are FIR filters inherently stable?

Yes. Since they have no feedback elements, any bounded input results in a bounded output.

#### 2.3.2 What makes the numerical properties of FIR filters "good"?

Again, the key is the lack of feedback. The numeric errors that occur when implementing FIR filters in computer arithmetic occur separately with each calculation; the FIR doesn't "remember" its past numeric errors. In contrast, the feedback aspect of IIR filters can cause numeric errors to compound with each calculation, as numeric errors are fed back.

The practical impact of this is that FIRs can generally be implemented using fewer bits of precision than IIRs. For example, FIRs can usually be implemented with 16 bits, but IIRs generally require 32 bits, or even more.

#### 2.4 Why are FIR filters generally preferred over IIR filters in multirate (decimating and interpolating) systems?

Because only a fraction of the calculations that would be required to implement a decimating or interpolating FIR in a literal way actually needs to be done.

Since FIR filters do not use feedback, only those outputs which are actually going to be used have to be calculated. Therefore, in the case of decimating FIRs (in which only 1 of N outputs will be used), the other N-1 outputs don't have to be calculated. Similarly, for interpolating filters (in which zeroes are inserted between the input samples to raise the sampling rate) you don't actually have to multiply the inserted zeroes with their corresponding FIR coefficients and sum the result; you just omit the multiplication-additions that are associated with the zeroes (because they don't change the result anyway.)

In contrast, since IIR filters use feedback, every input must be used, and every input must be calculated because all inputs and outputs contribute to the feedback in the filter.

#### 2.5 What special types of FIR filters are there?

Aside from "regular" and "extra crispy" there are:

• Boxcar - Boxcar FIR filters are simply filters in which each coefficient is 1.0. Therefore, for an N-tap boxcar, the output is just the sum of the past N samples. Because boxcar FIRs can be implemented using only adders, they are of interest primarily in hardware implementations, where multipliers are expensive to implement.
• Hilbert Transformer - Hilbert Transformers shift the phase of a signal by 90 degrees. They are used primarily for creating the imaginary part of a complex signal, given its real part.
• Differentiator - Differentiators have an amplitude response which is a linear function of frequency. They are not very popular nowadays, but are sometimes used for FM demodulators.
• Lth-Band - Also called "Nyquist" filters, these filters are a special class of filters used primarily in multirate applications. Their key selling point is that one of every L coefficients is zero--a fact which can be exploited to reduce the number of multiply-accumulate operations required to implement the filter. (The famous "half-band" filter is actually an Lth-band filter, with L=2.)
• Raised-Cosine - This is a special kind of filter that is sometimes used for digital data applications. (The frequency response in the passband is a cosine shape which has been "raised" by a constant.) See dspGuru's Raised-Cosine FAQ for more information.
• Lots of others.

# FIR Filter Design

#### 3.1 What are the methods of designing FIR filters?

The three most popular design methods are (in order):

• Parks-McClellan: The Parks-McClellan method (inaccurately called "Remez" by Matlab) is probably the most widely used FIR filter design method. It is an iteration algorithm that accepts filter specifications in terms of passband and stopband frequencies, passband ripple, and stopband attenuation. The fact that you can directly specify all the important filter parameters is what makes this method so popular. The PM method can design not only FIR "filters" but also FIR "differentiators" and FIR "Hilbert transformers".
• Windowing:. In the windowing method, an initial impulse response is derived by taking the Inverse Discrete Fourier Transform (IDFT) of the desired frequency response. Then, the impulse response is refined by applying a data window to it.
• Direct Calculation: The impulse responses of certain types of FIR filters (e.g. Raised Cosine and Windowed Sinc) can be calculated directly from formulas.

#### 3.2 How do I actually design FIR filters?

With a FIR filter design program, of course. ;-) Although it's possible to design FIR filters using manual methods, it is a whole lot easier just to use a FIR filter design program.

#### 3.3 What FIR filter design programs are available?

FIR filter design programs come in three broad categories:
• Filter Design Applications: See dspGuru's Digital Filter Design Software page for a list of filter design programs.
Near and dear to us here at dspGuru is Iowegian's own ScopeFIR product. We believe ScopeFIR offers an excellent combination of professional features, smooth user interface, and affordable price. We sell it for just \$299, with a 30 day trial period. Even if you already use Matlab, ScopeFIR's "point and shoot" capabilities can improve your FIR filter design productivity.
• Math Programs: Matlab and its Free Clones offer built-in FIR filter design functions.
• Source code: One of the best places on the net to find source code to design FIR filters is Charles Poynton's Filter Design Software page.

# FIR Filter Implementation

#### 4.1 What is the basic algorithm for implementing FIR filters?

Structurally, FIR filters consist of just two things: a sample delay line and a set of coefficients. To implement the filter:

1. Put the input sample into the delay line.
2. Multiply each sample in the delay line by the corresponding coefficient and accumulate the result.
3. Shift the delay line by one sample to make room for the next input sample.

#### 4.2 How do I implement FIR filters in C?

There are lots of possibilities, including a few tricks. To illustrate, we have provided a set of FIR filter algorithms implemented in C called "FirAlgs.c" in ScopeFIR's distribution file. FirAlgs.c includes the following functions:

1. fir_basic: Illustrates the basic FIR calculation described above by implementing it very literally.
2. fir_circular: Illustrates how circular buffers are used to implement FIRs.
3. fir_shuffle: Illustrates the "shuffle down" technique used by some of Texas Instruments' processors.
4. fir_split: Splits the FIR calculation into two flat (non-circular) pieces to avoid the use circular buffer logic and shuffling.
5. fir_double_z: Uses a double-sized delay line so that the FIR calculation can be done using a flat buffer.
6. fir_double_h: Similar to fir_double_z, this uses a double-sized coefficient so that the FIR calculation can be done using a flat buffer.

#### 4.3 How do I implement FIR filters in assembly?

FIR assembly algorithms are quite processor-specific, but the most common system uses a circular buffer mechanism provided by the DSP processor. The basic steps are:

1. Configure the circular buffer; load the coefficient and delay-line pointers. Then, for each input sample:
2. Store the incoming data in the delay line; increment the delay-line pointer.
3. Clear the multiplier-accumulator.
4. Loop over all coefficients/delays; accumulate the values obtained by multiplying the coefficients by the delayed samples.
5. Round or truncate the result as the FIR output.

Alternatively, a "shuffle down" method is used in Texas Instruments' older fixed-point processors to implement circular buffers. The processor literally moves each sample delay values by one slot during each multiply-accumulate (via the "MACD" instruction).

Each DSP microprocessor manufacturer provides example FIR assembly code in its data books or its application handbooks, so be sure to look at those before you "reinvent the circular buffer".

#### 4.4 How do I test my FIR implementation?

Here are a few methods:

• Impulse Test: A very simple and effective test is to put an impulse into it (which is just a "1" sample followed by at lest N - 1 zeroes.) You can also put in an "impulse train", with the "1" samples spaced at least N samples apart. If all the coefficients of the filter come out in the proper order, there is a good chance your filter is working correctly. (You might want to test with non-linear phase coefficients so you can see the order they come out.) We recommend you do this test whenever you write a new FIR filter routine.
• Step Test: Input N or more "1" samples. The output after N samples, should be the sum (DC gain) of the FIR filter.
• Sine Test: Input a sine wave at one or more frequencies and see if the output sine has the expected amplitude.
• Swept FM Test: From Eric Jacobsen: "My favorite test after an impulse train is to take two identical instances of the filter under test, use them as I and Q filters and put a complex FM linear sweep through them from DC to Fs/2. You can do an FFT on the result and see the complete frequency response of the filter, make sure the phase is nice and continuous everywhere, and match the response to what you'd expect from the coefficient set, the precision, etc."

#### 4.5 What tricks are useful in implementing FIR filters?

FIR tricks center on two things 1) not calculating things that don't need to be calculated, and 2) "faking" circular buffers in software.

#### 4.5.1 How do I skip needless calculations?

First, if your filter has zero-valued coefficients, you don't actually have to calculate those taps; you can leave them out. A common case of this is "half-band" filter, which have the property that every-other coefficient is zero.

Second, if your filter is "symmetric" (linear phase), you can "pre-add" the samples which will be multiplied by the same coefficient value, prior to doing the multiply. Since this technique essentially trades an add for a multiply, it isn't really useful in DSP microprocessors which can do a multiply in a single instruction cycle. However, it is useful in ASIC implementations (in which addition is usually much less expensive than multiplication); also, some newer DSP processors now offer special hardware and instructions to make use of this trick.

#### 4.5.2 How do I fake circular buffers in software?

When hardware support for circular buffers isn't available, you have to "fake" them. Also, since ANSI C has no construct to describe circular buffers, most C compilers can't generate code to use them, even if the target processor has them.

You can always implement a circular buffer by duplicating the logic of a circular buffer in software (and many have), but the overhead can be prohibitive; the circular-fake might take several instructions to implement, compared to just a single instruction to do the multiply-accumulate operation. Therefore you need to fake it.

Here are several basic techniques to fake circular buffers:

1. Split the calculation: You can split any FIR calculation into its "pre-wrap" and "post-wrap" parts. By splitting the calculation into these two parts, you essentially can do the circular logic only once, rather than once per tap. (See fir_double_z in FirAlgs.c above.)
2. Duplicate the delay line: For a FIR with N taps, use a delay line of size 2N. Copy each sample to its proper location, as well as at location-plus-N. Therefore, the FIR calculation's MAC loop can be done on a flat buffer of N points, starting anywhere within the first set of N points. The second set of N delayed samples provides the "wrap around" comparable to a true circular buffer. (See fir_double_z in FirAlgs.c above.)
3. Duplicate the coefficients: This is similar to the above, except that the duplication occurs in terms of the coefficients, not the delay line. Compared to the previous method, this has a calculation advantage of not having to store each incoming sample twice, and it also has a memory advantage when the same coefficient set will be used on multiple delay lines. (See fir_double_h in FirAlgs.c above.)
4. Use block processing: In block processing, you use a delay line which is a multiple of the number of taps. You therefore only have to move the data once per block to implement the delay-line mechanism. When the block size becomes "large", the overhead of a moving the delay line once per block becomes negligible.