5KHz low pass filter. unity at sample 1. Aligned with coeff. 3

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1 Copyright 997, Lavry Egieerig Uderstadig FIR (Fiite Impulse Respose) Filters - A Ituitive Approach by Da Lavry, Lavry Egieerig People less familiar with digital sigal processig, ofte view the theory as "icomprehesible" ad the hardware as "little black boxes". This article is aimed at replacig total mystery with some "feel" ad "commo sese uderstadig". While ot easy readig, a little patiece ad some cocetratio will shed much light regardig the "mechaics" of processig a sigal by simple meas such as multiplicatio ad additio. Itroductio Digital audio sigals are expressed i terms of sample values, equally spaced i time. Filters based o a rather simple "computatioal structure" kow as a FIR, yields impressive results. FIR filter frequecy respose is determied by a set of coefficiets. Bellow are examples of three differet sets of coefficiets: 5KHz low pass filter 5KHz low pass filter 5KHz high pass filter c.5 c c The three examples, each cosistig of 9 coefficiets ( through 8) show three differet coefficiet curves. The filter uses oly the umerical values graphically show as "boxes". The zero coefficiet for the 5KHz low pass has a value. coefficiet 3 is.8, coefficiet is.68 ad so o. Impulse respose The purpose of the coefficiets is to alter the sigal cotet by meas of simple arithmetic. The simplest case to demostrate is the respose of a filter to impulse. A impulse waveform has zero amplitude at all but oe the sample poits. Each sample takes the ozero value sequetially (oe sample at a time). The plots bellow show the propagatio of a impulse from "sample " to "sample " to "sample ". uity at sample uity at sample uity at sample Imp Imp Imp Let us "pass" the impulse sigal through a filter. We use the 9 coefficiet ( through 9) 5KHz low pass show above. Iput sigal sample values are show by circles, ad the coefficiet value as a squares: Aliged with coeff. Aliged with coeff. 3 Aliged with coeff. Imp Imp3 Imp c c c The impulse "travels" from left to right, agaist a "fixed frame" of coefficiets. Each plot shows a differet istat i time, eablig computatio of oe filter output sample. The calculatio is a two step process:. Multiply the values of each pair (samples represeted by circles ad coefficiet represeted by a squares). We get ie ew values kow as products.. Add the ie products. The sum of the products is the value of the correspodig output sample.

2 Copyright 997, Lavry Egieerig Clearly, a impulse iput with amplitude will geerate a output waveform idetical to the shape of the coefficiet plot. The "walkig" impulse has value at all but oe poits at a time. The coefficiets get aliged with the uity impulse, oe at a time, while the rest are aliged with zero's. The sum of the products will reproduce the shape of the coefficiets curve. The FIR output yields a fiite umber of o zero values, thus the ame Fiite Impulse Respose. Real life sigals are more complex, ad the multiplicatios yield may o zero values, but the process of aligmet, sample - coefficiet pair multiplicatio ad fial summatio remais the same. We ext examie the "ext to simplest case". Step respose A step respose iput propagates a "istataeously risig" sigal through the filter. The sigal steps from - to +. The six cosecutive "frames" below show the sigal rise aligmet with coefficiets through 5. The x symbols deotes the products of iput values (circles) ad coefficiet values (expressed by the dotted lie). c V, V. c, c V, V. c, c V, V. c, step aliged with coefficiet step aliged with coefficiet step aliged with coefficiet c V, V. c, c V, 3 V. c, 3 c V, 5 V. c, 5 step aliged with coefficiet step aliged with coefficiet step aliged with coefficiet

3 Copyright 997, Lavry Egieerig The step respose requires very simple arithmetic. All the coefficiets to the left of the step are multiplied by a iput sigal values of - (ivertig the coefficiet curve to the left of the step). The coefficiets to the right of the step are multiplied by, leavig the curve uchaged. Examie samples 3,,5 ad 6 of the right bottom plot. Calculatig output sample value for each "frame" requires summig (accumulatig) all 9 products. Addig the positive ad subtractig the egatives values (show by the x's) yield the output sample value. The "ext" output sample is determied by shiftig the sigal - coefficiet aligmet by oe ad recomputig the sum of the products. The filter used i the above example is a 5KHz low pass (at.khz samplig rate). The plot below shows output samples correspodig to frames. The slower rise ad the overshoot are due to atteuatio of the harmoics above 5KHz. Wider coefficiets "bell shaped" curves would modify the iput step over more output samples, slowig the rise, thus lowerig filter cutoff frequecy. VO, s The frequecy respose of the filter is show bellow. Icomig frequecies below, say KHz are left uatteuated. The atteuatio at KHz is about db.. log M m f m. Let's examie a respose of a 5 coefficiet 5KHz low pass filter. The respose is flat to 5Khz. The atteuatio is db at 6Khz ad db at Khz. A 5 coefficiet filter requires 5 multiplicatio's ad 5 additios for each output sample. Specialized itegrated circuits (digital sigal processors) ca ofte exceed such eormous computatioal requiremet. The the plot shows the value of such a compute egie. Old aalog desig techology does ot yield such results. Other FIR advatages will be discussed later.. log M5 m f m.

4 Copyright 997, Lavry Egieerig Siusoidal waves ad periodic waveforms The multiply ad accumulate processes ca be doe for ay waveform. To gai further isight, lets view the case of a sie wave filterig from a somewhat differet agle. Impulse ad step respose iputs "walk" a sigle trasitio through the filter, ad each coefficiet is aliged with the trasitio oly oce. Whe "walkig" a sie wave through a filter, each of the coefficiets fids itself lied up with a sie wave (the same sie wave samples move from oe coefficiet tap to aother. The 3 frames below show the sie wave motio. Tap umber receives at first sample time,.77 value at the secod ad for the third. Cotiuig the process will preset tap umber with a periodic set of values, followig the iput sie wave. The same is true for all the taps. The differece betwee taps is oly the "time of arrival" of the sigal (or time delay). first sample time S secod sample time S third sample time S Allow me to express a iterestig fact (proof ca be foud i basic trigoometry): Addig sie waves, all of the same frequecy, produces a sie wave. I other words, while the amplitude ad delay of the sum deped o the amplitude ad delay of the idividual compoets, the wave form of the sum remais a sie wave. Whe a FIR filter adds a umber of delayed sie waves, each compoet has its ow amplitude (each delay tap has its ow multiplyig coefficiet), but the outcome remais a sie wave. Thus a FIR causes o harmoic distortios. The followig plots demostrate how the sum of two equal amplitude.khz waves (sampled at.khz) yield differet outcomes. With sample delay, the sigals are almost i phase, thus the amplitude is almost doubled. With 5 tap delay, the sigals cacel each other ad the output is atteuated to zero. tap delay 3 tap delay 5 tap delay a, a, a, b, b, b, 5 Sum, Sum, Sum, Let us repeat the plots for a 8.8KHz iput sigal, with the same time delays: tap delay 3 tap delay 5 tap delay c, c, c, d, d, d, 5 sum, sum, sum,

5 Copyright 997, Lavry Egieerig Comparig the "filters outputs" at 8.8KHz to the.khz plots show that output amplitudes deped o iput frequecies. The 5 tap delay shows a extreme case (double the amplitude at 8.8KHz ad complete atteuatio at.khz). The above filters are very simple, yet they serve to demostrate the basis for sie wave filterig. Real world FIR's are made of umerous taps ad with coefficiets that are rarely equal to, eablig great cotrol over filter respose. What about periodic waves? All periodic waveforms are, i essece, a sums of basic sie waves compoets (Fourier series). Feedig a FIR with ay wave shape is the same as sedig the idividual compoets through the filter. The FIR filters a complex periodic waveform as if it were operatig o each sie wave compoet separately (idepedetly). Multiplyig a sample by a coefficiet yield the same result as "breakig" a sample ito parts, multiplyig each part by a coefficiet ad summig the products. The cocept is referred to as a "liear system". Filter behavior across the frequecy spectrum is completely defied by it behavior at each iput frequecy (oe at a time). Let us demostrate the workigs of a FIR with a two toe example. The plot shows KHz ad 8KHz toes (at.khz samplig), with dotted lies. The filter iput V (sum of V ad V) is show by a solid lie. V, V, V, Let's filter the first four poits of the dual toe sigal with a tap 3KHz low pass coefficiets. The four frames show a "fixed" coefficiet curve (dotted lies), the sigal (circles) movig through the filter (by oe sample per frame from right to left). The idividual product pairs are marked by x symbols. c V, V. c, aligmet for st output sample c V, V. c, aligmet for d output sample c V, V. c, aligmet for 3rd output sample c V, 3 V. c, 3 aligmet for th output sample

6 Copyright 997, Lavry Egieerig Accumulatig the x vales for each of frames (four of which are show above) yield the output samples (circles i the plot bellow).the dotted lie is the iput. Note the high frequecy atteuatio: VO V, Phase liearity A importat FIR advatage is phase liearity. Most filters (aalog ad digital) itroduce differet amout of delay for differet frequecy compoets of the sigals. The FIR keeps the delay for all frequecy compoets the same. Let us examie the outcome of varyig the delay of frequecy compoets. We start by approximatig a KHz triagle shape wave by addig, 3, 5 ad 7KHz compoets of appropriate amplitude ad phase relatioships. Notice that the four sie wave compoets peak together ad add up to maximize the triagle wave peak poit (the circles shows the sum): t t3 t5 t7 ts If we ivert the third harmoic (delay it by half a cycle time) it's cotributio at "peak time" is iappropriate as show bellow. The harmoics o loger "peak together" ad the outcome is far from triagle. t ti3 t5 t7 tis

7 Copyright 997, Lavry Egieerig While somewhat crude, the example serves to demostrate that wave shape retetio requires costat time delay at all frequecies. How importat is the shape of the wave? The aswer depeds o the applicatio. A frequecy depedet delay is aalogous to havig differet distace from soud source to the ear for differet frequecy compoets. The impact is ot very oticeable with a moophoic soud source, but rather dramatic for stereo or higher umber of chaels. Phase oliearities (frequecy depedet delay) is cosidered "eemy umber oe of proper soud field imagig" by may recordig egieers. FIR's do ot itroduce such problems. The coefficiets curve: Most filter desigers use a "ready made software". Others (icludig the author of this article) ofte go back to mathematics ad digital sigal processig literature. Uderstadig the relatioships betwee coefficiets ad filter performace requires some math, yet some aspects remai ituitive: DC characteristics: whe feedig a FIR filter a DC sigal, all filter taps receive a costat value. Let us assume a DC value of. The sample - coefficiet product at each tap equal the coefficiet value (times ). Each output sample becomes the sum of the coefficiets. Settig the coefficiets sum to makes a filter with uity gai at DC. Geerally speakig, oe ca scale the coefficiets to adjust gai. Let us restrict all further commets to uity gai filters (gai scalig is rather straight forward). All low pass ad otch FIR filters must pass DC thus their coefficiets sum is. Blockig DC (high pass, bad pass or other DC blockig filters) requires a coefficiet sum of. Relative value of the coefficiets: large value coefficiets carry more weight i the costructio of output samples. Small value coefficiet serve to "fie tue" the filter respose. db atteuatio require fie details of part per millio. The umber of required coefficiet: (or the width of the coefficiets curve): A "all pass" filter cosists of oe tap (each sample value is multiplied by ad is set to the output). Oe iput sample is isufficiet for ay other filter respose. Filterig requires examiig a umber of samples aroud the poit of iterest. Icreasig the umber of samples (filter taps) yield more accurate iformatio. A very low frequecy filter receives a slowly chagig sigal, thus requirig the desiger "to view" sigal behavior may samples away. The three low pass filters bellow cosists of coefficiets ( to ). The KHz filter low pass relays heavily o the "ceter" coefficiets. "tap umber 5" cotributes almost / of the output value. The values away from ceter approach zero value fast. The Hz filter curve is sigificatly wider i shape. The largest coefficiet cotributes less the.5% of the fial outcome, ad taps further from ceter play a sigificat role. KHz low pass coeffi..6 KHz low pass coeffi..6 Hz low pass coeffi..6 LPK LPK LP Closig remarks: Itegrated circuits for digital sigal processig (DSP) are a specialized breed of computatioal egies desiged, for the most part, to simultaeously move sampled data from tap to tap, while computig very large umbers of multiplicatios ad additios. Processig a sigle.khz chael with a 5 tap filter requires 88. millio computatios per secod. Despite may clever schemes for icreased computatioal efficiecy, a compromise betwee desired respose ad the umber of taps is ot ucommo. The tradeoff is betwee atteuatio, flat respose, ripple i the passbad (ad atteuatio regio), trasitio bad(s) ad more. Other compromises have to do with computatioal accuracy. The umber of digits (bits) available for both coefficiets ad iput sigals may plays a major role i filter quality. Each sample ad each coefficiet is dealt with to a fiite hardware depedet accuracy. The accumulated errors may take a toll o the fial output result. At times, addig taps may cause more harm the good. The filter desiger should take all such factors ito cosideratio. Yet, the FIR offers a viable solutio over much of the frequecy rage. Shapig the frequecy respose at very low frequecies or ear Nyquist (half the samplig frequecy) ofte requires prohibitive amout of compute power, ad use of aother filter type: the IIR. Aalog filters ad IIR's yields similar frequecy ad phase characteristics, yet aalog relays o compoet values (such as resistors ad capacitor). FIR ad IIR filters use arithmetic.