FIR and IIR digital filters - IEEE Potentials

Adigitalfilter is a basicbuilding block in any DigitalSignal Processing (DSP) system.The frequency responseof the filter depends on the value of itscoefficients, or taps. Many softwareprograms can compute the values of thecoefficients based on the desired frequencyresponse. These values are typicallyfloating point numbers and theyare represented with a fairly highdegree of precision.However, when a digital filter is implementedin hardware, the designer wantsto represent the coefficients (and alsothe data) with the smallest number ofbits that still gives acceptable resolutionfor the numbers. This is because representinga number with excess bitsincreases the size of the registers, buses,adders, multipliers and other hardwareused to process that signal. The biggersizes result in a chip with a larger diesize, which translates into increasedpower consumption and a higher chipprice. Thus, the bit precisions used torepresent numbers are important in theperformance of real-world signal processingdesigns.FIR digital filtersA Finite Impulse Response (FIR)digital filter is one whose impulseresponse is of finite duration. This canbe stated mathematically ashn ( ) = { 0,n≤τ1 −∞< τ1 < τ2

k = N -1 where N is the number of feedbacktaps in the IIR filter. The right Σdenotes summation from k = 0 to k = M-1 where M is the number of feedforwardtaps.Note that, unlike the FIR filter, theoutput of an IIR filter depends on boththe previous M inputs and the previousN outputs. This feedback mechanism isinherent in any IIR structure. (Feedbackoccurs when a scaled version of the outputis fed back into the input.) It is|H(w)| (dB)Magnitude Of Floating Point FIR Filter0-20-40-60-80-1000 0.1 0.2 0.3 0.4 0.5Digital Frequency (Hz)Fig. 1 Results for floating point FIR and IIR filters|H(w)| (dB)|H(w)| (dB)|H(w)| (dB)Pole-Zero Plot Of Floating Point FIR Filter1.5Imaginary Axis10.50-0.5-1-1.5-1 0 1Real Axisresponsible for the infinite duration ofthe impulse response.How do we show that an IIR filtercan have an infinite duration impulseresponse? The easiest way is to considera simple two-tap IIR filter with the followingdifference equation( ) = ( − ) + ( )y n α y n 1 x n(4)Magnitude Response Of Floating Point IIR Filter0|H(w)| (dB)Magnitude Responses Of Bit Accurate FIR Filters0-50-1000 0.1 0.2 0.3 0.4 0.5Digital Frequency (Hz)0-50Float4 Bit-1000 0.1 0.2 0.3 0.4 0.5Digital Frequency (Hz)0-50Float10 BitFloat16 Bit-1000 0.1 0.2 0.3 0.4 0.5Digital Frequency (Hz)Fig. 2 Results for bit accurate FIR filters-50-100-150-2000 0.1 0.2 0.3 0.4 0.5Digital Frequency (Hz)Pole-Zero Plot Of Floating Point IIR Filter1.5Imaginary Axis10.50-0.5-1-1.5-1 0 1Real AxisPole-Zero Plots Of Bit Accurate FIR FiltersImaginary AxisImaginary AxisImaginary Axis10-110-110-1-1 0 1Real Axis-1 0 1Real Axis-1 0 1Real Axiswhere α is the value of the single feedbacktap a 1 in (3) and b 1 = 1 is the valueof the single feedforward tap. The inputto the filter is δ(n) and the output of thefilter at various times is shown in Table1. Note that although the input to the filterhad only one non-zero value (at timen = 0), the output of the filter is nonzerofor all time 0 ≤ n ≤∞. This is dueto the feedback nature of the IIR filter.From a mathematical standpoint, thisimpulse response takes on non-zero valuesfor an infinite duration. However,the response of this IIR filter when realizedin practice will eventually die out.This happens when the numbersbecome too small to be represented withthe finite precision of the filter.Advantages of IIR filtersIIR filters are useful for high-speeddesigns because they typically require alower number of multiplies compared toFIR filters. IIR filters can also bedesigned to have a frequency responsethat is a discrete version of the frequencyresponse of an analog filter.Unfortunately, IIR filters do not havelinear phase and they can be unstable ifnot designed properly. IIR filters alsoare very sensitive to filter coefficientquantization errors that occur due tousing a finite number of bits to representthe filter coefficients. One way toreduce this sensitivity is to use a cascadeddesign. That is, the IIR filter isimplemented as a series of lower-orderIIR filters as opposed to one high-ordersection. The effect of this implementationwill be shown later in this article.Poles and zerosA detailed discussion of what polesand zeros are is beyond the scope of thisarticle. The interested reader can findout more by looking in any DSP book.(See Read more about it for some possiblesources.) Only a very brief descriptionwill be given here.Suppose that a filter has the followingtransfer functionHz ( ) =( z−z1) ( z−z2) ( z−z3)( z−p1) ( z−p2) ( z−p3)(5)where z n and p n are values on the complexplane (z-domain). The zeros of thefunction are the values of z for whichH(z) equals zero. Hence, the zeros ofthis function are at z = {z 1 , z 2 , z 3 }.Similarly, the poles of the functionare those values of z for which H(z)equals infinity. Thus, the poles of thisfunction are at z = {p 1 , p 2 , and p 3 }.OCTOBER/NOVEMBER 2000 0278-6648/00/$10.00 © 2000 IEEE 29

When plotting the pole-zero plot of afunction, a circle (o) is used to denotethe location of a zero, and a cross (x) isused to denote a pole.ImplementationImplementing a digital filter in practicetypically involves using software todetermine the filter coefficients basedMagnitude Responses Of Bit Accurate IIR Filters0|H(w)| (dB)|H(w)| (dB)|H(w)| (dB)-50-100-150-2000 0.1 0.2 0.3 0.4 0.5Digital Frequency (Hz)0-50-100-150|H(w)| (dB)|H(w)| (dB)|H(w)| (dB)-50-100-1500Magnitude Responses Of BitAccurate Cascaded IIR FiltersFloat10 Bit-2000 0.1 0.2 0.3 0.4 0.5Digital Frequency (Hz)0-50-100-150Float15 Bit-2000 0.1 0.2 0.3 0.4 0.5Digital Frequency (Hz)0-50-100-150Float10 Bit-2000 0.1 0.2 0.3 0.4 0.5Digital Frequency (Hz)0-50-100-150Float15 BitFloat20 Bit-2000 0.1 0.2 0.3 0.4 0.5Digital Frequency (Hz)Fig. 3 Results for bit accurate IIR filtersFloat20 Bit-2000 0.1 0.2 0.3 0.4 0.5Digital Frequency (Hz)Fig. 4 Results for bit accurate cascaded IIR filterson various user-defined parameters. Thefilter design software usually computesand displays the filter coefficients witha high degree of precision. If the digitalfilter can be implemented using thatsame degree of precision, then the filterwill behave as predicted by the filterdesign software.In practice, only a finite number ofPole-Zero Plots Of Bit Accurate IIR FiltersImaginary AxisImaginary AxisImaginary Axis10-110-110-1Pole-Zero Plots Of BitAccurate Cascaded IIR FiltersImaginary AxisImaginary AxisImaginary Axis1 0 1Real Axis-1 0 1Real Axis-1 0 1Real Axis10-110-110-1-1 0 1Real Axis-1 0 1Real Axis-1 0 1Real Axisbits can be used to represent the digitalfilter coefficients. This reduction ineach coefficient’s precision causes thefrequency response of the filter to differfrom the “ideal” response due to coefficientquantization errors.When using B coeff bits to representthe filter coefficients, the total numberof possible values that the filter coefficientscan take on is 2 Bcoeff . Thus,instead of having an infinite range ofvalues for the coefficients, they areinstead constrained to one of the 2 Bcoefflevels. The location of the poles andzeros of the filter are also quantized.This is because they depend on thevalue of the filter coefficients.The quantization of the pole and zerolocations will typically move the polesand zeros of the filter to locations thatare different from the “ideal” setting.This can have drastic effects on the performanceof the filter.For example, suppose a pole is to belocated at a distance of 0.999 from theorigin. However, the quantization ofthat pole’s location can cause the poleto be moved to a location that is at a distanceof 1 from the origin (sinceQ{0.999} = 1 for certain bit precisionswhere Q{.} denotes quantization). Sincefilters with poles on the unit circle areunstable, quantization can cause stablefilter designs to become unstable whenactually implemented. Because of theeffects of filter coefficient quantization,the number of bits to assign to the filtercoefficients (and for the data as well)must be carefully chosen.Fixed point arithmeticFixed point arithmetic is a typical formatused to implement digital filters onboth DSP processors and in VLSI implementations.(Another popular format isfloating point arithmetic in which numbersare represented using two parts:mantissa and exponent.) A very popularform of fixed point arithmetic is thetwo’s complement fixed point format.In this format, a B-bit number’s mostsignificant bit (MSB) represents the signof the number. The lower B-1 bits representthe magnitude. Using this format, aB-bit number can represent signed numbersin the range from -2 B-1 to 2 B-1 -1.In two’s complement arithmetic, thenegative of a binary number is formed byinverting each bit of the number. We thenadd a 1 to the least significant bit (LSB).Table 2 shows all the numbers that can berepresented with 3 bits using the two’scomplement fixed point format.30 IEEE POTENTIALS

Simulation resultsTwo ninth-order lowpass filters weredesigned to demonstrate the effects ofusing finite bit precisions to representdigital filter coefficients. The first filteris a FIR filter, and the second one is anIIR filter. Both were designed by placingthe poles and zeros to get a filterwith a lowpass response.Figure 1 shows the frequencyresponse and the pole-zero plots forthese filters. These filters are referred toas the floating point filters: The coefficientsfor these filters are represented infloating point format with the full precisionof the computer. The frequencyresponses shown in Fig. 1 represent thedesired response. The other figures showhow using a reduced precision adverselyaffects the filter’s response. These filtersare referred to as bit accurate.The coefficients for these filters arerepresented using the two’s complementfixed point format with a finite numberof bits. Three different bit precisions areused for each bit accurate filter to showthe performance for different precisions.Figure 2 shows the results for the bitaccurate FIR filters. The three plots onthe left show the frequency response forthe floating point filter in blue and forthe bit accurate filters in red. The bitprecisions for each bit accurate filter areshown on the plot’s legend.The three plots on the right show thepole-zero plots for the various FIR filters.Again, the blue corresponds to thelocations for the floating point filter andthe red corresponds to the locations forthe bit accurate filters.At 16 bits, the bit accurate responsematches that of the floating point filter.When the precision drops to 10 bits, thelocation of the zeros changes. The frequencyresponse is altered slightly.Using only 4 bits for the filter coefficientshas a drastic effect on the locationof the zeros. (Some are moved intodifferent quadrants and even outside ofthe unit circle.) The frequency responseflattens out considerably.The results for the IIR filters areshown in Fig. 3. As the bottom plotshows, 20 bits gives a frequencyresponse that matches the floating pointfilter’s response. At 15 bits there is achange in the position of some of thepoles. The frequency response is shiftedupward. Using 10 bits leads to a slightchange in the position of a few zerosand significant changes in the positionsof the poles. The frequency response isTable 1 Output of example IIR filterfor an input of δ(n)Discrete time Filter input Filter outputinstance x(n) y(n)0 1 11 0 α2 0 α 23 0 α 3n 0 α nshifted upward even more.Note that the bit accurate FIR filter’sresponse matches the floating pointresponse when using 16 bits. Also, thereis only a slight degradation in the performancewhen using 10 bits. However,for the IIR filter, 20 bits matches thefloating point response, but droppingthe precision to 15 bits causes the frequencyresponse to change.These plots demonstrate that IIR filtersare particularly susceptible to finitebit precision effects. This is due to thefeedback nature of the IIR structure. Itcauses the filter quantization effects tomagnify. One way to reduce this sensitivityis to implement the IIR filter as acascaded series of lower-order filtersinstead of one higher-order filter.For the plots shown in Fig. 3, the IIRfilter was implemented as one ninthorderfilter. Figure 4 shows the resultsfrom implementing the same filter (samezero and pole locations) as a cascade offour second-order filters and one firstorderfilter. The bit accurate frequencyresponses match the response of thefloating point filter at precisions of 20and 15 bits (contrast this to the plots inFig. 3). Dropping the precision to 10 bitsshows a slight movement in the locationof some poles and zeros. But the effecton the frequency response is negligible.From these results, we see that thecascaded implementation requires alower bit precision when compared withthe precision needed for implementingthe IIR filter as a single ninth-order filter.The differences in the results shownin Figs. 3 and 4 highlight the effect thatimplementing a filter with differentstructures can have on its performance.ConclusionsThe simulation results presentedshow how finite bit precisions can affectthe performance of a digital filter. IIRfilters were shown to be even more susceptibleto finite bit precision effectsthan FIR filters. However, these effectsTable 2 Two’s complement fixedpoint format for 3 bit numbersDecimal format Fixed point format0 0001 0012 0103 011-4 100-3 101-2 110-1 111can be reduced using the IIR filter witha cascaded structure.AcknowledgmentsThe author would like to thank Dr.Tom “Fear & Loathing” Endres and Dr.Samir Hulyalkar (Sarnoff Digital Communications)for first getting the authorinterested in the wonderful world offinite bit precisions. Thanks also toTom Krauss (Purdue University) fornumerous discussions on finite bit precisions,cascaded filters, and the financialimplications of the Year 2000 bug.Read more about it• Ifeachor, E., and Jervis, B., DigitalSignal Processing: A PracticalApproach, Addison-Wesley, 1995.• Litwin, L., Endres, T., Hulyalkar,S., and Zoltowski, M., “The Effects OfFinite Bit Precision For A Fixed PointVLSI Implementation Of The ConstantModulus Algorithm,” InternationalConference on Acoustics, Speech, andSignal Processing, Phoenix, AZ, Vol. 4,pp. 2013-2016, March 1999.• Lyons, R., Understanding DigitalSignal Processing, Addison-Wesley, ’97.• Proakis, J., and Manolakis, D., DigitalSignal Processing: Principles, Algorithms,and Applications, Prentice-Hall, ’96.About the authorLouis Litwin is a Member of the TechnicalStaff in the Corporate Researchdepartment at Thomson Multimedia wherehe works on wireless digital home networkingtechnology. Mr. Litwin receivedhis M.S. degree in Electrical Engineeringfrom Purdue University in ’99 and his B.S.degree in Electrical Engineering with distinctionfrom Drexel University in ’97. Hewas named by Eta Kappa Nu as the AltonB. Zerby and Carl T. Koerner OutstandingElectrical Engineering Student for ’97. Hisinterests include digital signal processingand digital VLSI design. He often annoysthe IRS by computing his income taxesusing only 3 bits of precision.OCTOBER/NOVEMBER 2000 31