Event Detection: Qrs-Complexes in Ecg Signals - microLab

BioMedSigProcAnaExamples of Events: Waves in Ecg’s2Rnoisy ECG signal (random noise)1TECG0PP−1−2QS0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2time in secondsP wave: Duration less than 120 msec with frequencies below10–15 Hz.Pq segment: Lasts about 80 msec; is a “non-event”Qrs wave: Lasts about 70–110 msec, frequencies mostly concentratedin the interval 10–50 HzT wave: Has frequency content similar to that of the P wave.St segment: Duration is usually about 100–120 msec and isnormally flat; is a “non-event”For the various waves in the Ecg signal we have a summaryof the main attributes in [Goe14c, Section 2]. You find a more indepthtreatment in [SL05, Section 6.2.3] and in [Ran02], whereyou find, besides more general information, in Section 4.2.1 adiscussion targeted to event detection.6 EventDet 2 2014

BioMedSigProcAna2 Qrs Detection2.1 General Structure of Qrs DetectorGeneral Requirements on a Qrs Detector• must be able to detect a large number of different Qrsmorphologies to be clinical useful• must be able to follow– sudden changes– gradual changesin prevailing Qrs morphology• must not lock onto certain types of rhythm, but treat nextpossible event as if anything could happen next6 EventDet 3 2014

BioMedSigProcAnaDetector-Critical Types of Noise and Artifacts• noise might be in nature– highly transient– or persistent (e.g. power-line interference)• problem of conditioning Ecg signals w.r.t. noise has beentreated• however, term “noise” has a somewhat different meaningwhen considered form the Qrs-detection viewpoint– P- and T-wave must here be treated as noise– noise might have physiological as well as technicaloriginsWe refer to our accompanying document [Goe14a] for theproblem of conditioning Ecg signals with respect to noise.6 EventDet 4 2014

BioMedSigProcAna5 Hz to 10 Hz. Note that waveform distortion is, in contrast toother types of Ecg-signal filtering, not a critical issue in Qrsdetection; instead, the goal is to improve signal-to-noise ratio toachieve good detector performance.The goal of the nonlinear transformation is to further enhancethe Qrs complex relative to the remaining backgroundnoise. Additionally, it must transform each Qrs complex into asingle positive peak that is better suited to threshold detection.In general, the nonlinear transformation might be memory-less(static)—rectification or squaring are examples—or a more complextransformation involving memory. Note, however, that notall Qrs detectors use a nonlinear transformation in their preprocessingstage.The goal of the block named decision rule is to perform,based on the output of the pre-processor, a test on whether ata certain time a Qrs complex is present or not. The decisionrule might be implemented as simple as an amplitude thresholdprocedure, but it may perform additional tests, for exampleon reasonable durations of certain waveforms, to obtain betterimmunity against various types of noise.Qrs detectors are usually designed to detect heartbeats.They rarely need to produce occurrence times of the Qrs complexeswith high resolution in time. If time resolution is neverthelessimportant, it may be necessary to improve the initialresolution with algorithms that perform time alignment of thedetected beats and thus reduce the problem of smearing whichmight occur when computing the ensemble average of severalbeats (recall the synchronized averaging, which we have discussedin [Goe14a]).6 EventDet 7 2014

BioMedSigProcAnaQrs Detection as an Estimation Problem• theoretical justification of Qrs-detector structure on page 6– investigate statistical models of Ecg signals– use maximum-likelihood estimation techniques to derivedetector structures corresponding to the signalmodel of interest• example model: unknown occurrence-time and amplitude⎧v[n] , 0 ≤ n ≤ θ − 1 ,⎪⎨x[n] = as[n − θ] + v[n] , θ ≤ n ≤ θ + D − 1 ,⎪⎩v[n] , θ + D ≤ n ≤ N − 1 ,whereN ˆ= length of observation intervalv[n] ˆ= noises[n] ˆ= Qrs-complex, known morphology, durationθ ˆ= unknown occurrence time of Qrs-complexa ˆ= unknown amplitude of Qrs-complexSörnmo and Laguna [SL05] discuss in Section 7.4 Qrs detectionin a more theoretically oriented manner. With the proposedsignal model shown above, it is then feasible to estimate the unknownoccurrence time θ and the unknown amplitude a of theQrs-complex. The notion “Qrs detection” suggests, however,that we are faced with a detection problem instead of an estimationproblem. It is indeed true that the problem consideredin [SL05] can likewise be formulated as a detection problem;6 EventDet 8 2014

BioMedSigProcAna2.2 Simplest: Derivative-Based IdeasUsing First- and Second Derivatives• differentiation forms basis of many Qrs detection algorithms• differentiation ˆ= basically a highpass filter– amplifies higher frequencies that are characteristic forthe Qrs complex– attenuates lower frequencies that are characteristicfor P- and T-waves– attenuates low frequencies of baseline wander• however: differentiation also amplifies higher frequenciesdue to noise ❀ additional significant smoothing is required• an algorithm based on first and second derivatives, withsmoothing included– by Balda et. al. in 1977– modified by Ahlstom and Tompkins for high-speedapplications in 1983– additionally modified by Friesen et. al. 1990The references cited in the above slide are [BDD + 77], [AT83],and [FJJ + 90].6 EventDet 10 2014

BioMedSigProcAnaThe Algorithm: Pre-Processing• let x[n] ˆ= input to and y[n] ˆ= output from pre-processingstage of algorithm• use for pre-processing a linear combination of first- andsecond derivative• first derivative: approximated as absolute value of threepointfirst difference∣y 0 [n] = ∣x[n] − x[n − 2] ∣ .• second derivative: approximated as absolute value of threepointfirst difference of three-point first differences∣y 1 [n] = ∣x[n] − 2x[n − 2] + x[n − 4] ∣ .• output y[n] of pre-processing stage as linear combinationy[n] = 1.3 · y 0 [n] + 1.1 · y 1 [n]To understand the rationals behind the given approximationof the first derivative, we start with a first-difference operator:δ 1 [n] ˆ= x[n] − x[n − 1] •—• ∆ 1 (z) =(1 − z −1) X(z) .6 EventDet 11 2014

BioMedSigProcAnaThis filter has a zero at z = 1—at Dc frequency—and has amagnitude response of 2 sin(ˆω/2). For sufficiently low frequenciesˆω, it approximates the derivative operation as we see from2 sin(ˆω/2) ≈ 2ˆω/2 = ˆω. To obtain a smoothing such that highfrequencynoise is not too much amplified, we post-process thesequence δ 1 [n] by a two-point moving-average filter having transferfunction (1 + z −1 ): 1ỹ 0 [n] ˆ= δ 1 [n] + δ 1 [n − 1]= ( x[n] − x[n − 1] ) + ( x[n − 1] − x[n − 2] )= ( x[n] − x[n − 2] ) .In the z-transformation domain we equivalently haveỸ 0 (z) ===(1 + z −1) ∆ 1 (z)(1 + z −1)( 1 − z −1) X(z)(1 − z −2) X(z) .Clearly, the complete filter (1 − z −2 ) has one zero at z = 1—atfrequency ˆω = 0—, which approximates the derivative, and asecond zero at z = −1—at frequency ˆω = ±π—, which suppresseshigh frequency noise. The magnitude response of the filteris 2 sin(ˆω); for low frequencies we again have 2 sin(ˆω) ≈ 2ˆω,which approximates a derivative with a gain 2.For the approximation of the second derivative consider thesequence ỹ 0 [n] being the output of a three-point first differencefilter: ỹ 0 [n] = x[n] − x[n − 2]. Processing this sequence by1 We denote the output of this processing step by ỹ 0 [n] to distinguish itfrom y 0 [n], which we use in the slide on page 11, and which is the absolutevalue of ỹ 0 [n].6 EventDet 12 2014

BioMedSigProcAnaanother three-point first difference filter, we obtain 2ỹ 1 [n] ˆ= ỹ 0 [n] − ỹ 0 [n − 2]= ( x[n] − x[n − 2] ) − ( x[n − 2] − x[n − 4] )= ( x[n] − 2x[n − 2] + x[n − 4 ) .The Algorithm: Decision Rule• threshold y[n] ˆ= output of pre-processing with a thresholdvalueof 1.0• if y[n 0 ] ≥ 1.0 for a certain n 0 , then compare next 8 samplesto threshold• if 6 or more of these 8 samples pass threshold test, thentake the 8-samples segment as part of a Qrs complex• advantage: additionally to detecting Qrs complexes, thealgorithm produces a pulse that is proportional in widthto the Qrs complex• disadvantage: the method is extremely sensitive to highfrequencynoise2 Again, we denote the output of this processing step by ỹ 1 [n] to distinguishit from y 1 [n], which we use in the slide on page 11, and which is theabsolute value of ỹ 1 [n].6 EventDet 13 2014

BioMedSigProcAna2.3 Aside: “Integer” FiltersMotivation• in real-time operations of digital filters– many filter designs are unsatisfactory– because huge computation-time requirements– surely if implemented in floating-point operations– but often also if implemented in fixed-point operations• integer filters are a special class of filters having only integercoefficients• here more special: coefficients ∈ {0, ±1, ±2}• here even more special: a preparation to the Pan-TompkinsalgorithmWe admit that we are not sure about the generality of thenotion “integer filters.” We feel that the notion “integer filters”used in the signal processing community is more general thanthe same notion often used in the literature on biomedical signalprocessing. Our present discussion treats filters in the morerestricted meaning as given, for example, in [Tom93]—see thereChapter 7 entitled “Integer Filters,” but also see its Chapter 12on Qrs detection.For Qrs detection, for example, it is crucial that the Qrscomplex as well as P- and T-waves remain unchanged. This6 EventDet 14 2014

BioMedSigProcAnademand on preservation of signal forms calls for linear-phasefilters, which must be Fir filters. “Integer” filters as we discussthem here are Fir filters with recursive implementations.The slide on the previous page 14 states that our “integer”filters have only coefficients being 0, ±1, and ±2. This is notthe full truth: The basic blocks that we consider have suchcoefficients, but cascading these basic blocks leads to additionalcoefficient values, see page 27.Simple Example• transfer function• zerosH(z) =N(z z ) = z 6 z − 1 ≡ 01 − z −61 − z −1 + z −2 = z 6 − 1z 4 (z 2 − z + 1)ˆ=N(z)D(z)❀z z = e j·k·2π/6 , k = 0, 1, . . ., 5 ˆ= 6-th roots of unity❝j❝❝.❝1❝❝zeros: 6-th roots of unityNote that the transfer function of the above example definesa filter that has only coefficients ±1.6 EventDet 15 2014

BioMedSigProcAnaSimple Example (cont’d)• poles z p1 = ρe jϕ , z p2 = ρe −jϕ = z ∗ p1 , z pk = 0, k = 3, . . .,6• non-trivial polesz 2 p −ρ}{{}≡ 1˜D(z p ) ˆ= z 2 p − z p + 1 ≡ 0(zp − ρe jϕ) ( z p − ρe −jϕ) ≡ 0(e jϕ + e −jϕ) z p + ρ 2 ≡ 0} {{ } }{{}= 2 cosϕ ≡ 1 ≡ 1} {{ }❀ ϕ = 60 ◦j.60 ◦e jϕ×......1×poles z p1,2 = e ±jϕ , ϕ = 60 ◦6 EventDet 16 2014

BioMedSigProcAnaSimple Example (cont’d)• pole-zero map of complete filter (poles “eat” zeros)❝j❝× 4-fold.❝1❝complete pole-zero map• transfer function (filter is Fir, linear phase)H(z) =1 − z −61 − z −1 + z −2 = 1 + z−1 − z −3 − z −4To see how we obtain the second form—the non-recursiveform—of the above Fir-filter transfer function, just multiplythe terms that correspond to the remaining zeros and poles:(1 − z−1 ) ( 1 + z −1) ( 1 − e j2π/3 z −1)( 1 − e −j2π/3 z −1) .The first pair of terms gives ( 1 − z −2) , the second pair of termsgives—we use the formula of Euler together with the resultcos(120 ◦ ) = −1/2—the polynomial ( 1 + z −1 + z −2) , and wefinally obtain(1 − z−2 ) ( 1 + z −1 + z −2) = 1 + z −1 − z −3 − z −4 .6 EventDet 17 2014

BioMedSigProcAnaA second, alternative, way to find the second form of thetransfer function in the slide of page 17 is just to carry-out thedivision of the first form: Dividing (1 − z −6 ) by (1 − z −1 + z −2 )gives 1 with remainder (z −1 − z −2 − z −6 ); dividing this remainderby the denominator (1 − z −1 + z −2 ) gives z −1 anda new remainder (−z −3 − z −6 ); again dividing this new remainderby the denominator gives −z −3 and a next remainder(−z −4 + z −5 − z −6 ); finally, dividing this last remainder by thedenominator (1 − z −1 + z −2 ) ends up with −z −4 and a zeroremainder.The two transfer-function formulae on page 17 suggest thatwe could implement the filter as a usual Fir filter by taking thesecond formula, but also in a recursive form by taking the firstformula. The question then is which kind of implementation toselect.To discuss the implementation issue, we take a larger filterof the considered class of filters where the problem becomesevident. So consider the N-point moving average-filter havingthe transfer function forms 3H(z) = 1 N= 1 N(1 + z −1 + z −2 + · · · + z −(N−1)) , (1a)1 − z −N. (1b)1 − z−1 Naming the input sequence to the filter x[n] and the outputsequence from the filter y[n], the form (1a) could be implementedin direct form asy[n] = 1 N()x[n] + x[n − 1] + x[n − 2] + · · · + +x[n − (N − 1)] ,3 We obtain the second form by applying the formula for the sum of ageometric series to the first form.(2)6 EventDet 18 2014

BioMedSigProcAnawhereas the form (1b) could be implemented in direct form asy[n] = y[n − 1] + 1 Nor, in what is called the canonical form, as()x[n] − x[n − N] , (3)w[n] = w[n − 1] + x[n]y[n] = 1 ()w[n] − w[n − N] ;N(4)here in (4) w[n] is a newly introduced interval variable—an internalsignal—of the filter.Comparing the feed-forward implementation (2) to the recursiveimplementations either (3) or (4), we see that the numberof operations needed to compute one output sample favorsthe recursive implementations. This lower computational effortis the reason that many books on biomedical signal processingsuggest the recursive implementations.We note, however, that the recursive implementations mightbecome problems with error accumulation and instability—theimplementations (3) and (4) both have a pole at z = 1, whichis not strictly inside of the unit circle. To get a feeling of whatmight happen, consider the canonical implementation (4) andassume that the input signal is a white-noise signal. 4 The equationw[n] = w[n − 1] + x[n] is an accumulator, and accumulatinga white-noise sequence gives a non-stationary stochasticprocess with a linearly increasing variance, meaning that w[n]must sooner or later become unstable, which is, of course, notacceptable. Similar considerations for the implementation (3)lead to the same conclusions. We do not go into more details,but, instead, refer to [Orf96], where these issues are discussedon pages 402 ff., and which has an Appendix A, pages 728 ff.,discussing discrete-time random signals and their filtering.4 A discrete-time white noise signal is a sequence where any two samplesare uncorrelated.6 EventDet 19 2014

BioMedSigProcAnaHowever, an additional note is in order: The transfer functionform (1b) shows perfect pole/zero canceling and an implementationbased on this form relies on the achievement of thisperfect canceling by the arithmetic used in the implementation.Exact integer arithmetic can achieve it: Both, two’s complementnumber systems and, more general, any of the residuenumber systems 5 have the ability to support error-free arithmetic;in the case of the two’s complement number system, thearithmetic must be performed modulo 2 b , that is, with naturaloverflow as we explain in [Goe14b], and in the case of residuenumber systems, the arithmetic is performed in modulo M.5 The two’s complement number system is a member of the family ofresidue number systems, if overflow is correctly treated; see the main text.6 EventDet 20 2014

BioMedSigProcAnaSimple Example (cont’d)• frequency response of filter∣ H(e jˆω ) ∣ 3.0̸. ˆω−π 0 πH(e jˆω )π/2−π−π/2ˆωπ.6 EventDet 21 2014

BioMedSigProcAnaGeneral: Poles• complex-conjugate pole-pair on unit circlez p1 = e jϕ ,z p2 = e −jϕ = z ∗ p1• thendenominator = ( z − e jϕ) ( z − e −jϕ)= z 2 − ( e jϕ + e −jϕ) z + 1= z 2 − 2 cosϕz + 1• has (integer) coefficients ∈ {0, ±1, ±2} only forϕ cosϕ 2 cosϕ±0 ◦ 1 2±60 ◦ 1/2 1±90 ◦ 0 0±120 ◦ −1/2 −1±180 ◦ −1 −2Note that the constraints on possible pole locations, whichare given in the above slide, lead to corresponding constraintson the choice of sampling rates if we need to process continuoustime(analog) signals by discrete-time filters, 6 and if, therefore,natural (physical) frequencies dictate the useful pole locations;6 Although there are other occasions, we have in biomedical signal processingalways, to our knowledge, the situation where the signals to beprocessed are in the first instance analog signals.6 EventDet 22 2014

BioMedSigProcAnasee our exemplifying discussion on the the bandpass filter of thePan-Tompkins Qrs detection algorithm given on page 42.Also note that we compile, later on page 27 and after a correspondingdiscussion on “general zeros,” the transfer functionsthat can be realized with “integer filters.”General: Poles (cont’d)• possible pole locations for integer polynomial-coefficientsj×× ××2-fold2-fold.×1×××6 EventDet 23 2014

BioMedSigProcAnaGeneral: Zeros• first casenumerator = z m − 1 ,m ˆ= positive integerzeros:❀❀zz m = 1 , z z = e j·k·2π/m , k ∈ {0, m−1}zeros are m-th roots of unityevenly spaced zeros around the unit circlestarting at z = 1• second casenumerator = z m + 1 ,m ˆ= positive integerzeros:❀❀z m z = −1 , z z = e j(π/m+k·2π/m) , k ∈ {0, m−1}zeros are m-th roots of (−1)evenly spaced zeros around the unit circlestarting at z = e jπ/mThe notation k ∈ {0, m − 1} means that k is in the setof integers with given lower and upper limits, that is, k =0, 1, . . .,m − 1.6 EventDet 24 2014

BioMedSigProcAnaExample: Zeros for z m − 1 = z 12 − 1• zeros start at z = 1 for a numerator of type z m − 1❝❝❝❝❝j ❝❝❝❝❝. ❝1❝6 EventDet 25 2014

BioMedSigProcAnaExample: Zeros for z m + 1 = z 12 + 1• zeros start at z = e jπ/m for a numerator of type z m + 1j❝ ❝❝ ❝❝❝❝.❝ 1❝❝❝❝Note that when the power m is an odd number, the numeratorz m + 1 always places a zero at ±180 ◦ —at ±π—and thecomplete zero pattern appears “left-right flipped” from the zeropattern of the corresponding numerator z m −1. Take the examplem = 3: In the left-hand side of the graphic below we showthe zeros of z 3 − 1, and in its right-hand side we show the zerosof z 3 + 1.❝jj❝. ❝1❝.1❝numerator = z 3 − 1❝numerator = z 3 + 16 EventDet 26 2014

BioMedSigProcAnaGeneral: Transfer Functions• two types of general transfer functionsH 1 (z) =H 2 (z) =• where m, p, q ˆ= integers(1 − z −m ) p(1 − 2 cosϕz −1 + z −2 ) q(1 + z −m ) p(1 − 2 cosϕz −1 + z −2 ) q• and where ϕ ∈ {0 ◦ , ±60 ◦ , ±90 ◦ , ±120 ◦ , 180 ◦ }• note:raising p & q by equal integer-amounts corresponds to cascadingof identical filtersAs an example for cascading, we start with p = q = 1 andobtain from the above H 1 (z)H 11 (z) =Taking next p = q = 2, we obtainH 12 (z) =1 − z −m1 − 2 cosϕz −1 + z −2 .(1 − z −m ) 2(1 − 2 cosϕz −1 + z −2 ) 2 = H 11(z) · H 11 (z) ,which is just the cascade of two identical filters with transferfunction H 11 (z).6 EventDet 27 2014

BioMedSigProcAna2.4 The Pan-Tompkins AlgorithmBlock Diagram of Pan-Tompkins Algorithmbandpassfilterderivativeoperatorsquaringoperatormov-winintegrator• algorithm for Qrs detection in real-time• referring to general structure on page 6– linear filtering:bandpass filter & derivative operator– nonlinear transformation:squaring operator & moving-window (mov-win) integrator– decision rule: not shown• algorithm is intended for Ecg signals that are sampled at200 samples/secThe algorithm for Qrs detection in real time has been developedby Pan and Tompkins [PT85]; it has further been analyzedand described by Hamilton and Tompkins [HT86]. The algorithmrecognizes Qrs complexes by analyzing slope and amplitudeof Ecg signals taking into account the width of appearingEcg waves.6 EventDet 28 2014

BioMedSigProcAnaWe present the algorithm as it is described in [Tom93, Section12.5]; you might also want to consult [Ran02, Section 4.3.2]and/or study the original literature [PT85] and [HT86]. Thedescribed form of the algorithm is targeted to an implementationon a microprocessor using integer programming. In thelaboratory exercises we explore variants that might be useful forhardware implementations.The slide on the previous page 28 states that the describedPan-Tompkins algorithm is intended for discrete-time Ecg signalsthat have been obtained by sampling the true, continuoustimeEcg signal at a rate of 200 samples/sec. This is true becausethe algorithm uses “integer” filters that realize pole- andzero locations that are useful for the intended sampling rate,but become unreasonable for different sampling rates; see ourdiscussion starting on page 42.Because this subsection is quit long, we give an overviewon where you find what. The pre-processing according to theblock diagram shown on page 28 is discussed on pages 30–49;we indicate on page 50 how Pan and Tompkins implement thedecision rule.For the pre-processing stage we treat the first preprocessingblock, the bandpass filter, on pages 30 to 42. Next we discussthe derivative operator on pages 44 to 45. The squaring operatorfollows on page 46, and the final moving-window integrator isdescribed on pages 47 to 49.6 EventDet 29 2014

BioMedSigProcAnaBandpass Filter• isolates the predominant Qrs energy, which is centered inthe 5–15 Hz range• attenuates the low-frequency characteristics– of the P- and T-waves– of baseline wander• attenuates the high frequencies associated with– electromyographic noise– power-line interferences– measurement noise• implementation as– “integer” filter in which poles cancel zeros on unitcircle– cascade of lowpass and highpass filterLp.ˆω−π 0 πHp.. .ˆω−π 0 πBpˆω−π 0 π.6 EventDet 30 2014

BioMedSigProcAnaBandpass Filter: The Lowpass Part• consider the transfer functionH Lp1 (z) = 1 − z−61 − z −1 = 1 + z−1 + z −2 + · · · + z −5• the Dc gain is (ˆω = 0 =⇒ z = e j·0 = 1)H Lp1 (z = 1) = 1 + z −1 + z −2 + · · · + z −5∣ ∣z=1= 6• the pole-zero map and the magnitude response are❝j❝6.0∣ H(e jˆω ) ∣ ∣❝× × .❝5-fold1❝❝. ˆω−π 0 π6 EventDet 31 2014

BioMedSigProcAnaBandpass Filter: The Lowpass Part (cont’d)• the complete Lp filter is the scaled cascade of two H Lp1 (z)• the cascade H Lp1 (z) · H Lp1 (z) = ( H Lp1 (z) ) 2has Dc-gain6 · 6 = 36• for unity Dc-gain it needs a scaling factor 1/36• the used scaling factor is 1/32:– approximates 1/36 to avoid saturation– can be implemented by a right shift of 5 bit-positions• the transfer function of the used lowpass filter then isH Lp (z) ˆ= 1 ( ) 1 − z−6 232 1 − z −1 = 1321 − 2z −6 + z −12 Y (z)1 − 2z −1 ˆ=+ z−2 X(z)6 EventDet 32 2014

BioMedSigProcAnaBandpass Filter: The Lowpass Part (cont’d)• the magnitude response is∣ H(e jˆω ) ∣ 1.0. ˆω−π 0 π• the difference equation for a recursive implementation isy[n] = 2y[n−1] − y[n−2] + 1 ()x[n] − 2x[n−6] + x[n−12]326 EventDet 33 2014

BioMedSigProcAnaBandpass Filter: The Highpass Part• implemented by a delay-complementary transfer functionhighpass = delay allpass − lowpassdelayallpass+−32-pointmoving average• delay of M-point moving averager is (M − 1)/2• for M = 32: half-sample delay 31/2 ❀ 16 is usedYou might want to read more about complementary transferfunctions; [Mit98, pp. 245–249] discusses not only delay-complementarytransfer functions, but also more general allpasscomplementaryfilters, power-complementary filters and magnitude-complementaryfilters. Alternatively, you might use thenewer edition of that book; [Mit06, pp. 391–396] additionallydiscusses doubly-complementary transfer functions, which are atthe same time allpass-complementary and power-complementary.Concerning the delay of a moving-average Fir filter we havethe following: First consider a pure delay systemy[n] = x[n − n 0 ] ,6 EventDet 34 2014

BioMedSigProcAnawhich delays each sample of the input sequences by n 0 samples.The transfer function of this system is H(z) = z −n0 , and itsfrequency response becomes H(ˆω) = e −jˆωn0 .Next consider an M-point moving-average filter. Its transferfunction isH(z) = 1 M(1 + z −1 + . . . + z −(M−1)) = 1 1 − z −MM 1 − z −1 .Its frequency response can be expressed asH(ˆω) ˆ= H(z = e jˆω ) = 1 M1 − e −jˆωM1 − e −jˆω= 1 e ( −jˆωM/2 e jˆωM/2 − e −jˆωM/2)M e ( −jˆω/2 e jˆω/2 − e −jˆω/2)( ) 1 sin(ˆωM/2)=e −jˆω(M−1)/2 ,M sin(ˆω/2)} {{ }Dirichlet functionwhere we have used Euler’s formula, and where the indicatedDirichlet function is purely real. We see then that H(ˆω) is theproduct of two functions, the Dirichlet function and e −jˆω(M−1)/2 ,with the latter indicating a shift of (M −1)/2 samples. Clearly,if the filter length M is an odd integer, (M − 1)/2 becomes aninteger, and the meaning of delaying by an integer number ofsamples is clear. In M is, however, an even integer, (M − 1)/2is not an integer, and the delay needs a special interpretation;[MSY98, pp. 192–194] gives such an interpretation by applyinga discrete-time moving average filter to process continuous-timesignals.6 EventDet 35 2014

BioMedSigProcAnaBandpass Filter: The Highpass Part (cont’d)• the 32-point moving average filter has transfer functionH MA (z) = 1 (1 + z −1 + . . . + z −31) = 1 1 − z −323232 1 − z −1 .• the pole-zero map and the magnitude response are❝ ❝❝❝❝❝ ❝ ❝❝❝❝❝ ❝❝j ❝∣❝H(e❝jˆω ) ∣ ❝❝1.0❝❝❝× × . ❝31-fold❝❝ 1❝❝❝ ❝ ❝ ❝ ❝. ˆω−π 0 πWe note that the phase response of the moving-average transferfunction H MA (z) is linear—it is just a symmetric Fir filter,and its recursive realization does, of course, neither change themagnitude- nor the phase response.6 EventDet 36 2014

BioMedSigProcAnaBandpass Filter: The Highpass Part (cont’d)• the highpass transfer function becomesH Hp (z) = z −16 − H MA (z)= z −16 − 1 1 − z −3232 1 − z −1 .• the pole-zero map and the magnitude response are❝❝❝❝❝❝❝❝❝❝❝❝❝❝j❝❝❝❝❝❝❝× × . ❝31-fold❝ 1❝❝❝ ❝❝❝ ❝2-foldzero1.0∣∣H(e jˆω ) ∣ ∣..ˆω−π 0 π.Note that the phase response of the considered highpass filteris not exactly linear but merely “almost linear.” This is becausethe involved moving-average filter is a 32-point moving averagefilter—the length is selected as a power-of-two such that neededmultiplication by 1/32 can be realized by just bit shifting—and32 being an even integer gives a half-point delay of (32 −1)/2 =15.5 The delay allpass must, however, of course have an integerdelay, which has been selected to be 16.6 EventDet 37 2014

BioMedSigProcAnaIf we would use instead of the 32-point a 33-point movingaveragefilter, we would obtain a highpass filter with exactlylinear phase response, but we would loose the possibility to realizethe needed multiplication by mere bit shifting. To illustratethese arguments, we consider the situation for shorter lengthsof the involved moving average filter.First consider a 5-point moving-average filter. Its delay is(M − 1)/2 = (5 − 1)/2 = 2, and we use a delay allpass having adelay of 2. The transfer function of the realized highpass filterbecomesH Hp5 (z) = z −2 − H MA5 (z)= z −2 − 1 (1 + z −1 + z −2 + z −3 + z −4)5= − 1 (1 + z −1 − 4z −2 + z −3 + z −4) .5Clearly, H Hp5 (z) realizes a symmetric Fir filter which has apurely linear phase.Next consider a 4-point moving-average filter. Its delay is(M − 1)/2 = (4 − 1)/2 = 1.5, and we must use a delay allpasshaving either a delay of 1 or of 2. Consider first the delay 2.The transfer function of the realized highpass filter becomesH Hp4,2 (z) = z −2 − H MA4 (z)= z −2 − 1 (1 + z −1 + z −2 + z −3)4= − 1 (1 + z −1 − 3z −2 + z −3) .4Obviously, H Hp4,2 (z) is not a symmetric Fir filter and we cannotexpect that it has a linear phase. If we consider to realizethe delay allpass with the delay 1, the transfer function of the6 EventDet 38 2014

BioMedSigProcAnarealized highpass filter becomesH Hp4,1 (z) = z −1 − H MA4 (z)= z −1 − 1 (1 + z −1 + z −2 + z −3)4= − 1 (1 − 3z −1 + z −2 + z −3) .4Again, H Hp4,1 (z) is not a symmetric Fir filter and we cannotexpect that it has a linear phase.6 EventDet 39 2014

BioMedSigProcAnaBandpass Filter: The Highpass Part (cont’d)• denote byx[n] : the input signal to the highpass filtery[n] : the output signal of the highpass filtery 1 [n] : the output signal of the moving-average filter part• then: the difference equation for the moving-average filterisy 1 [n] = 1 ()y 1 [n − 1] + x[n] − x[n − 32]32• then: the difference equation for the highpass filter isy[n] = x[n − 16] − y 1 [n]The difference equations given above are nearly the differenceequations given in [Tom93, p. 250] and [Ran02, pp. 187–188]with a slight difference: These two references compute accordingto (note that the following two equations stem unaltered fromthe articles)ỹ 1 [n] = ỹ 1 [n − 1] + x[n] − x[n − 32]) , (5)y[n] = x[n − 16] − 1 ()ỹ 1 [n − 1] + x[n] − x[n − 32] , (6)32that is, they have the moving-average filter not scaled and do thescaling only when subtracting the moving-average filter output6 EventDet 40 2014

BioMedSigProcAnafrom the delayed input; we do not understand why they seemto recompute ỹ 1 [n] in (6).The reference [HT86] computes differently according toy[n] = y[n − 1]− 1 1x[n] + x[n − 16] − x[n − 17] +32 32 x[n − 32] . (7)We obtain that difference equation by reformulating the transferfunction from page 37 asH Hp (z) = z −16 − 1 1 − z −3232 1 − z −1( ) 1 − z−1z −16 − 1 32=1 − z −1(1 − z−32 )= − 132 + z−16 − z −17 + 1 32 z−321 − z −1 .Finally, the original paper [PT85] uses a completely differenthighpass filter. We suspect, however, that the formula giventhere might be in error; indeed, [HT86] state in their introductorysection that they use “the processor developed by Pan andTompkins” with slight modifications.6 EventDet 41 2014

..BioMedSigProcAnaBandpass Filter: The Complete Filterthe magnitude response is• linear∣ H(e jˆω ) ∣ ∣1.0. ˆω−π 0 π• in [dB]∣ H(e jˆω ) ∣ ∣dB. .0dB−60dB.. . .. . . ˆω−π 0 πIf we analyze the complete bandpass filter in detail, we findthat the filter has a middle frequency 7 of ˆω mid = 0.2520 rad, aleft −3 dB frequency of ˆω left = 0.1539 rad, and a right −3 dB7 With a slight abuse of the notion, we call that frequency where themagnitude is maximum the “middle” frequency.6 EventDet 42 2014

BioMedSigProcAnafrequency of ˆω right = 0.3707 rad. If the signal to be processedstems from a continuous-time Ecg signal which has been sampledat a rate of f s = 200 samples/sec, these characteristicstranslate 8 to left, middle, and right natural frequencies being{4.9, 8.0, 11.8}Hz. In the light of the frequency characteristicsof the Qrs complex of Ecg signals as we have compiled onpage 30—the dominant Qrs energy is contained in the 5–15 Hzfrequency range—, the realized filter characteristics are reasonable.Note, however, that if the input signal to the Qrs detectorstems from a continuous-time Ecg signal sampled at arate of f s = 1000 samples/sec, for example, the discrete-timefilter characteristics translate to left, middle, and right naturalfrequencies being {24.5, 40, 59}Hz. Obviously, in that case thebandpass filter is not useful.8 Recall that the discrete- and continuous-time frequencies are relatedby ˆω ˆ= ω · T s = 2πf/f s, where ω is the continuous-time frequency inrad/sec, T s ˆ= 1/f s is the sampling time-interval in sec with f s being thecorresponding sampling frequency in Hz, and f is the natural (continuoustime)frequency in Hz.6 EventDet 43 2014

BioMedSigProcAnaThe Derivative Operator• a five-point derivative operator(H dif (z) = 0.1 2 + z −1 − z −3 − 2z −4) .• the true gain 0.1 is approximated by 1/8• let x[n] ˆ= input, y[n] ˆ= output• the difference equation then becomesy[n] = 1 8()2x[n] + x[n − 1] − x[n − 3] − 2x[n − 4] .Note that the defined derivative operator is an Fir filter withan impulse response of length 5; this is the reason that we call ita “five-point derivative operator” although it uses only 4 termsfor the computations—imagine that the fifth term x[n − 2] ismultiplied with a coefficient being zero.6 EventDet 44 2014

BioMedSigProcAnaThe Derivative Operator (cont’d)the magnitude response is• linear∣ H(e jˆω ) ∣ ∣0.7. ˆω−π 0 π• in [dB] versus log-frequency∣∣ H(e jˆω )[dB]∣0−20−40.−6010 −3 10 −2 10 −1 10 0log 10 (ˆω)We note from the magnitude plots of the bandpass filter onpage 42 that essentially only signal component-frequencies ˆωbelow 1 rad/sample arrive at the derivative operator. For thesefrequencies the proposed operator well acts as a differentiator,as the second of the above plots—magnitude in [dB] versus logfrequency—clearlyreveals: We have in this double logarithmicplot a slope of +20 dB/decade in the relevant frequency interval.6 EventDet 45 2014

BioMedSigProcAnaThe Squaring Operator• non-linear operation (as opposed to all other processingsteps)• but static operation (no memory)• let x[n] ˆ= input, y[n] ˆ= outputthen: squaring operation is y[n] =(x[n]) 2.• makes output-signal samples positive• amplifies output of derivative operator non-linearly– emphasizes higher frequencies in Ecg signal– higher frequencies are mainly due to Qrs complex– smaller differences arising from P- and T-waves aresuppressedWe note that in microprocessor implementations of the Pan-Tompkins algorithm the output of the squaring-operation stageshould be hard-limited to a certain level corresponding to thenumber of bits used to represent the data type of the signals.In hardware implementations, however, we do not need a singledata type for all signals in the chain of operations, and thedecision whether and how to limit the squaring-operator outputmust be taken based on other considerations.6 EventDet 46 2014

BioMedSigProcAnaThe Moving-Window Integrator• slope of the R-wave in a Qrs complex is not a guaranteedmeasure to detect a Qrs event• many abnormal Qrs complexes that have– large amplitude– large duration– but not very steep slopesare not detected using only information about slope of R-wave• ❀ need to extract more information from an Ecg signalto detect a Qrs complex• moving-window integration extracts additional features6 EventDet 47 2014

BioMedSigProcAnaThe Moving-Window Integrator (cont’d)• let x[n] ˆ= input, y[n] ˆ= output of moving-window integrator• then, difference equation isy[n] = 1 N()x[n] + x[n − 1] + · · · + x[n − (N − 1)]• is an N-point moving average filter• parameter N ˆ= window-length must be chosen carefully– window-length N approximately the same as widestpossible Qrs complex– usually N is chosen experimentally– for Ecg signals sampled at a rate of 200 samples/sec,a value of N ≈ 30 is reasonableThe given difference equation of the moving-average filtercan, of course, also be written in recursive form: The transferfunction isY (z)X(z) = 1 (1 + z −1 + · · · + z −(N−1)) = 1 1 − z −NNN 1 − z −1 ,and the difference equation in recursive form then readsy[n] = y[n − 1] + 1 N()x[n] − x[n − N] .6 EventDet 48 2014

BioMedSigProcAnaWe also note that it might not be necessary to compute thetrue average, that is, we could leave away the factor 1/N to simplifythe computation, because the subsequent processing steprealizing the “decision rule” can take account of the resultingsignal gain in that it works with adjusted threshold values. Ifwe nevertheless like to have the normalization—the division bythe number N of samples—it is clearly best to choose N = 32,which is in the order of the proposed 30, because the divisionthen becomes a mere bit shift.Concerning the selection of the window size N, the argumentationis as follows: If the window size of the moving-averagefilter is too large—we have a too wide window—, then the integrationwaveform y[n] tends to merge Qrs complexes and subsequentT-waves together. If, however, the window size is toosmall, a single Qrs complex might produce several peaks at theintegrator-stage output. For the considered Ecg signals sampledat a rate of 200 samples/sec, a choice of N = 30 has 29intervals between first and last sample in the window and correspondsto an real-world time interval length of 145 msec. Correspondingly,we obtain for a choice of N = 31 a real-world timeinterval length of 150 msec, and for N = 32 one of 155 msec.6 EventDet 49 2014

BioMedSigProcAnaDecision Rule• adaptive thresholding– algorithm adapts to changes in the Ecg signal– by: computing running estimates of signal and noisepeaks– a peak is said to be detected whenever the final outputchanges direction within a specified interval• adjusting average heartbeat-rate estimates– R-R interval: R-wave to R-wave interval ˆ= heartbeatinterval– works with two averages– 1. average: over 8 most recent beats– 2. average: over 8 most recent beats that fall intocertain limits– reason (idea): to be able to adapt to quickly changingor irregular heartbeat rates• for details: see original literatureThe original original literature is [PT85] and [HT86]; wesupply these papers as pdf-copies along with the laboratoryexercises. Alternatively, you might want to use the textbooks[Tom93] or [Ran02].6 EventDet 50 2014

BioMedSigProcAnaSampling-Rate Conversion• note: the Pan-Tompkins algorithm is designed for Ecgsignals sampled with f s = 200 samples/sec• if we have an Ecg signal with a different sampling rate,say f s = 1000 samples/sec, we need a pre-processing• pre-processing: sample-rate conversion• in fixed-point implementations: sample-rate conversion oftenperformed with Cic filtersThe acronym Cic stands for cascaded integrator-comb filters,being a special form of multirate decimation and interpolationfilters. We have the original paper [Hog81], but you might alsowant to consult [Don00], [MB07, Section 5.3], [Xil08], or thecorresponding items in the Matlab manuals. 9We note that although the Cic decimation-structures haveas first stages integrators—see our simple example on page 52which has just one integrator—there is no problem with overflows,if the used data type is two’s complement with naturaloverflow computations, because then the computations are modulo2 b , and final overall results staying in the representable rangeare correctly computed even if intermediate results have overflowed;see our discussion in [Goe14b].9 In the Matlab help window just search for Cic.6 EventDet 51 2014

BioMedSigProcAnaSampling-Rate Conversion: Cic-Filter Example• to do: realize down-sampling by a factor 5• needed: lowpass filter for limiting to ˆω p = π/5• as example use: 10-point moving-sum filterH(z) =(1 + z −1 + · · · + z −9) = 1 − z−101 − z −1• with subsequent down-sampling1 − z −101 − z −1 ↓ 511 − z −1 1 − z −10 ↓ 5• with Cic-filter structure11 − z −1 ↓ 5 1 − z −2We here call a moving-average filter that comes “withoutaveraging” a moving-sum filter. The moving-average filter correspondingto the moving-sum filter introduced above isH(z) = 1 (1 + z −1 + · · · + z −9) = 1 1 − z −101010 1 − z −1 ,6 EventDet 52 2014

BioMedSigProcAnathat is, the moving-average filter has, as compared to the movingsumfilter, an additional scaling factor 1/10—the computationof the average.Reconsider the Cic-filter decimator structure in the graphicson page 52. We see that the first processing block is anaccumulator—we may likewise term it an integrator if we prefercontinuous-time (Ct) system parlance. Obviously, that accumulatorcontributes a not strictly-stable pole at z = 1 to thesystem, and the result might be register overflow in the integratorstage. 10 However, such a register overflow will have noconsequence if we ensure that the following two conditions hold:(i) The filter is implemented in two’s complement arithmetic; 11(ii) The range of the used number system is equal to or exceedsthe maximum magnitude expected at the output of the completeCic filter.Corresponding to the operation of decimation, there is thecomplementary operation of interpolation: Here, an incomingsignal is first upsampled to a higher rate by inserting zero samplesand subsequently lowpass filtered for interpolation. In theinterpolation-operation too, Cic-filter structures can be used,but the sequence of comb sections and integrator sections is reversed:Corresponding to the example on page 52 we would havethe comb block (1 − z −2 ), followed by an upsampler, followedby the integrator. In this situation of the Cic-filter interpolator,the input signal to the overall filter is preconditioned bythe comb section—the comb section suppresses a potential Dcterm—so that overflow will not occur in the integrator stages.10 If the input signal to the integrator contains a non-zero Dc component,the register will overflow sooner or later independently of how small thatDc component might be.11 The filter can likewise be implemented by any other number system thathas “wrap-around” between the most positive and most negative numbers—the number system must have modulo arithmetic.6 EventDet 53 2014

BioMedSigProcAnaThe summary of economical advantages of Cic-filters is,1. Cic-filters need no multipliers;2. consequently, Cic-filters need no storage for filter coefficients;3. Cic-filters reduce internal signal-sample storage, as comparedto equivalent implementations with uniform Fir filters,by integrating at the high sampling rate and combfiltering at the low sampling rate;4. Cic-filters have a very “regular” structure consisting oftwo basic building blocks;5. Cic-filters can be implemented with little external controlor complicated internal timing;6. Cic-filters allow to use the same filter design for a widerange of rate-change factors, with only the addition of ascaling circuit and minimum changes to the filter timing.Problems encountered with Cic-filters include: (i) Registerwidths might become large if large sampling-rate factors areto be implemented; (ii) The frequency response can only beinfluenced by three integer parameters.To understand this second statement, we must generalizethe example on page 52: Instead of using there the 10-pointmoving-sum filter for lowpass filtering prior to downsamplingby a factor of 5, we could start with only a 5-point movingsum filter. Correspondingly, the comb section would become(1 − z −1 ), and in general, the comb section is (1 − z −M ), whereM is named the differential delay. 12 Next, instead of startingjust with the 10-point moving-sum filter as done on page 52, wecould use a cascade of N such moving sum filters, leading then12 Of course we could also start with a 15-point, or a 20-point, or . ..,moving-sum filter, leading to M = 3, M = 4, ..., but in practice thedifferential delay is usually held to M = 1,2.6 EventDet 54 2014

BioMedSigProcAnato a Cic-filter structure corresponding to that on page 52 whichwould first have N integrator stages, next the downsampler bythe factor 5, and finally N comb stages each of the form (1 −z −2 ). Finally, of course, the downsampling factor could be somegeneral positive integer R, and the transfer function of the Cicfilterreferenced to the high sampling rate would become( ) (1 − z−RM N RM−1) NH(z) =(1 − z −1 ) N = z −k .∑k=0From the above transfer-function formula we now understandthat the frequency response of the Cic-filter depends only on thethree integer parameters {R, M, N}, that is, on the samplingratechange factor R, the differential delay M, and the numberof cascaded stages N.6 EventDet 55 2014

BioMedSigProcAnaReferences[AT83][AT85]M. L. Ahlstrom and W. J. Tompkins. Automatedhigh-speed analysis of holter tapes with microcomputers.IEEE Trans. Biomed. Eng., 30(10):651–657,October 1983.M. L. Ahlstrom and W. J. Tompkins. Digital filtersfor real-time Ecg signal processing using microprocessors.IEEE Trans. Biomed. Eng., 32(9):708–713,September 1985.[BDD + 77] R. A. Balda, G. Diller, E. Deardorff, J. Doue, andP. Hsieh. The Hp Ecg analysis program. In J. H.van Bemmel and J. L. Willems, editors, Trends inComputer-Processed Electrocardiograms, pages 197–205. North Holland, Amsterdam, 1977.[Don00] Matthew P. Donadio. Cic filter introduction.On the Web: For Free Publication by Iowegian,July 2000. m.p.donadio@ieee.org, http://www.-dspguru.com/info/tutor/cic.htm.[FJJ + 90] G. M. Friesen, T. C. Jannett, M. A. Jadallah, S. L.Yates, S. R. Quint, and H. T. Nagel. A comparisonof the noise sensitivity of nine Qrs detection algorithms.IEEE Trans. Biomed. Eng., 37(1):85–97,January 1990.[Goe14a] Josef Goette. Biomedical Signal Processing andAnalysis—Filtering for Removing Artifacts. BernUniversity of Applied Sciences, Script at the BfhtiBiel/Bienne, HuCE-microLab, February 2014.6 EventDet 56 2014

BioMedSigProcAna[Goe14b] Josef Goette. Biomedical Signal Processing andAnalysis—On Fixed-Point Filter Realizations. BernUniversity of Applied Sciences, Script at the Bfh-tiBiel/Bienne, HuCE-microLab, February 2014.[Goe14c] Josef Goette. Biomedical Signal Processing andAnalysis—Review of Various Biomedical Signals.Bern University of Applied Sciences, Script at theBfh-ti Biel/Bienne, HuCE-microLab, February2014.[Hog81]Eugene B. Hogenauer. An economical class of digitalfilters for decimation and interpolation. IEEE Trans.Acoust., Speech, Signal Processing, 29(2):155–162,April 1981.[HT86] P. S. Hamilton and W. J. Tompkins. Quantitativeinvestigation of Qrs detection rules usingthe Mit/Bih arrhythmia database. IEEE Trans.Biomed. Eng., 33(12):1157–1165, December 1986.[Los08][MB07][Mit98][Mit06]Ricardo A. Losada. Digital filters with Matlab.Technical report, The MathWorks, Inc., May 2008.Uwe Meyer-Baese. Digital Signal Processing withField Programmable Gate Arrays. Springer Verlag,Berlin Heidelberg New York, 3rd edition, 2007.Sanjit K. Mitra. Digital Signal Processing: A ComputerBased Approach. Mc Graw Hill, New York,1998. Bfh-ti Biel/Bienne Library 621.391 MITRA.Sanjit K. Mitra. Digital Signal Processing: A ComputerBased Approach. Mc Graw Hill, New York, 3rdedition, 2006. Bfh-ti Biel/Bienne Library 621.391MITRA.6 EventDet 57 2014

BioMedSigProcAna[MSY98][Orf96]James H. McClellan, Ronald W. Schafer, andMark A. Yoder. Dsp First: A Multimedia Approach.Prentice-Hall Inc., 1998.Sophocles J. Orfanidis. Introduction to Signal Processing.Prentice-Hall Inc., Upper Saddle River, NewJersey, 1996. Bfh-ti Biel/Bienne Library 621.391ORFAN.[PT85] J. Pan and W. J. Tompkins. A real-time Qrsdetection algorithm. IEEE Trans. Biomed. Eng.,32(3):230–236, March 1985.[Ran02][SL05][Tom93][Xil08]Rangaraj M. Rangayyan. Biomedical Signal Analysis:A Case-Study Approach. IEEE Press, New York,2002. Bfh-ti Biel/Bienne Library 57.08 RANGA.Leif Sörnmo and Pablo Laguna. Bioelectrical SignalProcessing in Cardiac and Neurological Applications.Elsevier Academic Press, Amsterdam, 2005. Bfh-tiBiel/Bienne Library 616 SOERN.Willis J. Tompkins. Biomedical Digital Signal Processing.Prentice-Hall Inc., Englewood Cliffs, N.J.,1993. Eth Bib: +521 458.Xilinx. Cic Compiler v1.1. On the Web, March2008. http://www.xilinx.com.6 EventDet 58 2014