Biomechanical Waveform Analysis in theFrequency Domain
Patrick E. Patterson techniques to a wide variety of measurement and analysisLewis Brown tasks. Because of its utility and its increasing availability inDepartment of Industrial Engineering and either software or hardware, the FFT should have evolvedthe Biomedical Engineering Program into a major and commonplace measurement tool for theIowa State University biomechanist. However, general purpose use of the FFT is
still severely limited in medical applications because of theTHE EXPLOSION of medical knowledge in the past 25 years lack of familiarity with Fourier theory. However, the numberUhas created an abundance of individual disciplines within of researchers using frequency analysis is increasing [2, 3, 4].
the general practices of medicine and rehabilitation. Medicine The purpose of this paper is to present Fourier analysis ashas been moving from what was, typically, an empirical art, an alternative to the time domain method of mathematicaltoward a more exact science, by using the tools of engineer- data processing to investigate signals in biomechanical sys-ing and advanced mathematics, not only in research but in tems. This technique is used extensively in the electrical,clinical diagnosis as well. This increase in the use of computer, and mechanical engineering disciplines, but it hasengineering techniques has paralleled the advent of cheaper, not yet found widespread use in either research or diagnosticmore&powerful microcomputers, giving researchers and clini- medicine. A logical basis for using this technique in analyzingcians access to the practical collection and analysis of time biomechanical signals within the human system will beseries data. presented, along with an exemplary application.Time series data is a collection of observations that have
been taken sequentially and may be of two general types- REVIEW OF SINUSOIDScontinuous or discrete. The series is continuous if observa- Many physical processes exhibit sinusoidal behavior.tions are made continuously in time; the series is discrete if Fourier analysis techniques provide researchers with methodsthe observations are made only at specific points in time. A of representing non-sinusoidal processes by a series ofdiscrete time series can obviously be generated from a sinusoidal components. The term "sinusoid" is used for anycontinuous one by sampling at equally-spaced time intervals. response that can be expressed as a sine or cosine function.Time series, as a data type, includes signals such as those A sinusoidal function of time (Fig. 1) can be expressed as:from load cells, accelerometers, and other types of transduc-ers. Some commonly measured time series signals include f(t)=A cos (cwt+±0)EMG, EKG, cinematographic analyses, and pressure measure- wherements. Biomechanical time series data from a human systemmay be converted to a set of numbers by any of a variety of f(t) = instantaneous valuetechniques and then further processed mathematically. A = amplitude (maximum value)
Classically, the nature of the variables observed in biome- w = frequency in radians per secondchanical research has required that they be expressed as t = time in secondsfunctions of time, e.g., position or velocity. Analysis of all 0 = phase anglesuch systems, using time dependent relationships, is referred Because an angle of 2w radians corresponds to one completeto as time domain analysis. For example, the equation: cycle (3600), we define:
d2x dxm ~+a -+kx(t)=f(t)wdt 2 dt f=-= frequency in Hertz (Hz)
2r
might describe a system's behavior in this domain. While this The time for one complete cycle is thereforeequation appears to be very simple, finding its solution canbecome somewhat involved. If there were additional interac- 1tions and components, system equations (actually a set of T=- period in seconds
fsimultaneous differential equations) would result that aretime-consuming, if not impossible, to solve. In some in- The phase angle, 0, may be expressed in either degrees or instances, a more convenient description of the system can be radians (see Fig. 1). It is a measure of the displacementobtained through the process of data transformation. One of difference between the closest positive peak (A) and the y-these approaches is Fourier analysis. axis. The same sinusoid can also be expressed by the sine
For many years, Fourier analysis has existed as an interest- function:ing mathematical approach for obtaining frequency domaininformation, but was generally too difficult to apply in mostpractical cases. The arrival of the digital computer did not a=A sin wt+O+ )create a rush to use the technique-useful Fourier analyses\/were still too time-consuming for widespread use. In the using the trigonometric identity:1960s, J. W. Cooley and J. W. Tukey published "AnAlgorithm for the Machine Calculation of Complex Fourier sin [x+ (r/2) = cos (x)Series [1]."' This algorithm became known as the fast Fouriertransform (FFT). for all values of x.The FFT allows for quick, economical application of Fourier Any sinusoid is thus completely described by its amplitude,
frequency, and phase. Figure 2b shows three sinusoids. Note period of f(t) and set equal to 2r/IT. The ao coefficient is thethat varying the amplitude (A) changes the vertical displace- DC (zero frequency) term. It is equal to the average value ofment of the waveform, varying the frequency (co) changes the f(t) over one period and is given by:rate of occurrence of the wave-shapes, and varying the phase(0) simply shifts the waveform to the left (phase lead) or right 1 ST(phase lag). ao=- jf(t) dtT oFourier AnalysisThe French mathematician, Francois Marie Charles Fourier The remaining coefficients, a, and bn, are evaluated for n =
(1772-1837), demonstrated that any repeating (periodic) 1, 2, 3, n by:waveform can be described as a sum of sinusoids. The goal ofmany signal processing methods is to decompose a complex 2 rTwaveform into its sinusoidal components. Once these com- an=-j f (t) cos nwot dtponents are known, the physical process can be modeled and °subsequently analyzed in the frequency domain by the Fourier andapproach.Consider the complex waveform shown in Fig. 2a. This 2 T
waveform can be decomposed into three sinusoids, as shown bn=-- f(t) sin nwot dtin Fig. 2b. That is, summing these three sinusoidal compo- T onents results in the original complex waveform as a functionof time. Each of the sinusoidal components is completely Realistically, most processes (waveforms) are acquired bydescribed by its amplitude, frequency, and phase; an alter- periodically sampling and storing the discrete numbers thatnate representation of these components is graphically correspond to the process at the sampling times. In manyshown in Figs. 2c and 2d. Figure 2c depicts sinusoidal cases, investigators desire the Fourier series representationamplitude information, and is commonly called the "ampli- or the spectral content of a nonperiodic sampled waveform.tude spectrum" for the complex waveform. Similarly, Fig. 2d Fourier techniques can still be used in such a case tois called the "phase spectrum" for the complex waveform. decompose the waveform into sinusoidal components. TheseAny complex periodic waveform can be decomposed into components, when summed, produce the same waveform,sinusoids that are completely described by these amplitude but the signal must now appear in a repetitive form. Theand phase spectral plots. period of this resulting Fourier series representation is equalBy examining the amplitude and phase spectrums for a to the total time that the waveform was sampled, called the
complex waveform, a mathematical expression exactly de- "observation time." If the researcher desires a mathematicalscribing that waveform can be written. The expression, model for the waveform, only the first period of this represen-which includes amplitude, frequency, and phase for each tation need be used.sinusoidal component, is called the "Fourier series represen- The discrete Fourier transform (DFT) is an algorithm thattation" of the waveform. Many biomechanical signal analy- transforms a set of time series waveform samples into ases involve obtaining such a representation for an observed corresponding spectral representation. The DFT algorithm isphysical process (e.g., a voltage waveform from a trans- simply a set of equations that accomplishes this transforma-ducer). The general form of the Fourier series is frequently tion. The algorithm requires knowledge of the sampling rate,written as: the number of samples, and the discrete waveform values.
For 2N + 1 discrete samples, the DFT provides the wave-
f(t)=ao+ E (an cos nwot+bn sin nwot)n= 1
The Fourier series for a specific waveform is written by usingsalient features of the waveform to find specific values forthe coefficients in the above series. First, wo is found from the 10.0
- t 12
6.0 3.0 -3COS (6t- /4
2.0 1.0 \ k "A
f(t) A cos (wt +300) -6.0 -3.0 /V5
-10.0 -5 .00.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5
00 3600 T TIME, SEC. TIME, SEC.(2)_ (a) (b)
5 180
2700 go,49 I 0-90
2 2 0 2 T (SEC) 4560 10 2400 36003
0800 5 ~~~~~~~4+(2+,) o-4
Figure 1. Relationship between the displacement of a sinusoidal o 1 2 3 4 5 0 1 2 3 4 5FREQUENCY, Hz ~~~~~FREQUENCY, Hzwaveform and a rotating vector. To reach the ordinate from theFRQEC,H
nearest positive peak of the waveform the rotating vector moves (c) (d)from 0° to 30', the phase angle. Clockwise vector rotation corres- Figure 2. Alternate representations of a complex waveform: (a) theponds to phase lead (0 > 00); counterclockwise vector rotation complex waveform; (b) its three sinusoidal components; {c) itscorresponds to phase lag (0 <0'). amplitude spectrum; and (d) phase spectrum.
SEPTEMBER 1987 IEEE ENGINEERING IN MEDICINE AND BIOLOGY MAGAZINE 13
form's DC value (zero frequency component), and N ampli- Signal averaging techniques are also used to minimize thetudes and N phases for N sinusoidal components of the effects of noise, enhancing the signal-to-noise ratio. If thewaveform. Intuitively, supplying more samples gives more desired signal waveform is periodic and the noise is random,spectral information (resolution), because the longer a proc- averaging many waveforms can increase the signal-to-noiseess is observed (sampled), the more information is gained. ratio. This method is particularly useful when the noise isDFT algorithms are inefficient, and in terms of computer additive and its amplitude is Gaussian distributed with a zero
time, expensive, because many of the intermediate computa- mean (i.e., Gaussian white noise). In such cases, averagingtions are repeated. More efficient algorithms called "fast the frequency domain DFT/FFT spectrums reduces the noiseFourier transforms" (FFTs) have been developed. The FFT effects. Combinations of filtering and averaging are com-algorithms require the same input information as the DFTs monly used to obtain the "best" separation of signal andexcept that the number of discrete samples must be 2k (k = noise.any integer). FFTs provide exactly the same spectral informa- Another cause of misinterpretation of signal analysis arisestion as the DFTs. For analysis of 1,024 discrete samples, a when only a portion of a complete waveform is sampled.DFT requires more than one million complex multiplications- Obviously, if a researcher observes a truncated version of thean FFT requires about 5,000. DFT and FFT algorithms can be true signal, the resulting interpretation of the sampled wave-programmed many ways, but the results are the same. form will be incorrect. The misinterpretation is greater whenAmplitude and phase spectrums which describe the sinusoidswhich compose a sampled waveform are produced.To prevent misinterpreting the DFT/FFT results, some 4.0
precautions must be observed. When sampling a continuoussignal, information may be lost because no information is 2 A0transmitted between the sample points. The higher the .0sampling frequency, the greater the information retained. Tocorrectly identify a sinusoid with Fourier techniques, sample lthe waveform at a rate greater than twice per period. If DFT/ F 0.0FFT techniques are to be used to decompose a waveform into Xits sinusoidal components, sample the waveform at a ratethat is at least twice the highest frequency component of the -2.0 -
waveform. This frequency is known as the waveform's"Nyquist frequency." The effect of disregarding this "rule"and sampling at lower rates will result in loss of high -4.0 I_Ifrequency components (see Fig. 3). (a)An additional problem which must be addressed in signal 2_0
analysis is that of noise (any undesired signal). For biome- 2.0chanical applications where waveforms are obtained fromskin electrodes (EMG, EKG, etc.), the noise amplitude canexceed the amplitude of the desired signal. Many techniques 1.0 _can be used to minimize the effects of such noise. If the noiseis additive and its range of frequency components differs Dfrom the signal's range, the combined waveforms may be F 0.0filtered to remove the noise from the signal. However, if the C
noise and desired signal share part of the frequency spec- <trum, complete removal of the noise becomes impossible -1.0 -
without distorting the desired signal. Other methods maythen need to be considered.
-2.0 _(b)
4.05.0
3.0 2.0 -
1.0 ~-0.0
a 1.0-2.0-
-3.0
04.005101.0.0 0.5 1.0 1.5 TIME, SEC.
TIME, SEC. (c)
Figure 3. An example of too low a sampling frequency: a plot of two Figure 4. An example of windowing: (a) a continuous sinusoidalsinusoids which have the same amplitude at the periodic sampling waveform; (bI window to be applied; and (c) resulting windowedintervals shown. waveform.
14 IEEE ENGINEERING IN MEDICINE AND BIOLOGY MAGAZINE SEPTEMBER 1987
the endpoints of the observed waveform do not match (i.e., TABLE Ithe first and last numbers in the time series do not coincide). Summary of DFT spectral results.In an attempt to resolve this problem, many researchersartificially reduce the amplitude of the endpoint values in the Frequency (Hz) Amplitude (N) Phase (Deg)time series so that they match. This is accomplished bymultiplying the discrete numbers in the entire time series by afunction which is unity over most of the observation time, but 0 582smoothly decreases to zero at the endpoints. This function is 1.38 185 178called a "window" for obvious reasons, and the multiplica- 2.76 334 -166tion process is termed "windowing" (see Fig. 4). The 4.14 79 - 154resulting modified time series will have matching endpoints(equal to zero), but the spectral information from the timeseries has been modified. Researchers must be careful wheninterpreting the modified results. Windowing is normally notneeded for transient signals, which themselves decay to zeroif they are totally contained within the observation timeframe. experiment design. A priori, the investigator knows that in aWhile this discussion of experimental signal analysis is "normal" gait, 99% of the power contained in the forceplate
intended to serve only as a brief introduction, it does address waveform will occur at frequencies below 1 5 Hz [7]. Thissome of the major aspects that researchers must consider for knowledge is used to select the sampling rate of the datavalid experimental results. For more detailed treatment of acquisition system that will obtain the discrete samples of theexperimental and theoretical considerations, read Lynn [5] forceplate waveform. To prevent errors, the sampling rateand Jackson [6]. must be greater than twice the highest suspected frequencyAn Analysis Example component of the process; 30 Hz. A higher sampling rate ofA biomechanical experiment will serve as an example of 300 Hz is chosen to provide good resolution of the waveform.
how to correctly apply spectral analysis. The experiment The experiment proceeds with the subject in normal gait,consists of analyzing a typical vertical forceplate waveform and the data acquisition system sampling the forceplatefrom the foot of a subject during normal gait. waveform for one cycle (-0.7 s). Approximately 210 sam-
Before proceeding, the investigator uses any a priori ples are acquired and stored. An interpolated plot of theseknowledge of the biomechanical process to aid in the discrete samples is shown in Fig. 5a.
Next, the 210 discrete numbers of the time series are readinto a DFT algorithm, which computes the magnitude andphase angle of the sinusoidal components of the waveform.
1000 The results show a zero frequency (DC) component and threeprominent sinusoidal components. These components are
800- n / \ described by the DFT output results summarized in Table I.If desired, the investigator can then model the original
forceplate waveform by using the DFT spectral results. A600 suitable mathematical description of the waveform using the
DFT results would be
° 400 f(t)=582+185 cos (27x1.38t+1780)200 + 334 cos (2ir2.76t- 166°)
+ 79 cos (27r4.14t- 154°) Newtons0
(a) A plot of this waveform is shown in Fig. 5b. The similarityof these results and the actual forceplate waveform of Fig. 5a
1000 verify the accuracy of the spectral analysis and the resultinginterpretation-an important preliminary step in the analysis.
800-CONCLUSIONSA signal can be described either as a function of time or as a
z 600 - characterization of its frequency content. Both representa-L: / \tions have advantages; leaving the choice of which to use° 400 _ / \ dependent upon the system being investigated, how the
results will be used, and most frequently, the investigator'sfamiliarity with the representations.
200 Once obtained, spectral information such as that found inthe above example may be useful for comparing changes inwaveforms due to certain pathologies. For instance, certain
0 100 200 300 400 500 600 700 800 pathological conditions such as cerebral palsy or arthritis mayTIME, mS show distinct changes in the spectral content of gait wave-
(b) ~~~~~~formsthat may not be readily detected in the time domain.(b) ~~~~~~Analysis of waveforms in the frequency domain may provideFigure 5. Vertical forceplate waveforms: (a) interpolated plot or researchers with methods of detecting the presence or onsetoriginal sampled waveform and (bI plot of waveform's mathematical of pathological conditions once a template of "normal"model obtained from DFT spectral results. spectral waveforms has been established.
SEPTEMBER 1987 IEEE ENGINEERING IN MEDICINE AND BIOLOGY MAGAZINE 15
REFERENCES _* Patrick E. Patterson received the B.S. degree1. Cooley JW and Tukey JW: An algorithm for the machine calculation of in physical education from Springfield College,
complex Fourier series. Math. Computations, 19:297-301, 1965.2. Schneider E and Chao EY: Fourier analysis of ground reaction forces in Springfield. MA in 1972. the M.S. in adapted
normals and patients with knee joint disease. Biomechanics, 16181:591-601, and developmental physical education from1983. Cleveland State University, Ohio in 1978, and
3. Voloshin AS and Wosk J: Shock absorption of meniscectomized and the Ph.D. in industrial engineering (ergonom-painful knees: a comparative in vivo study. Biomedical Engineering, 5:157- ics) f160, 1983. ics) from Texas A & M University, College
4. Doemiand HH and Jacobs RR, Spence J, Roberts FG: Assessment of Station, in 1984. His research interests in-fracture healing by spectral analysis. Medical Engineering and Technology, clude injury prevention and rehabilitation engi-10(4):180-187, 1986. neering. Dr. Patterson is presently Assistant
5. Lynn PA: An lntroduction to the Analysis and Processing of Signals, 2nd P e i t DEd. The MacMillan Press Ltd., London, 1982.PrfsointeDptm tofIdtiaEg-
6. Jackson LB: Digital Filters and Signal Processing. Kluwer Academic neering and the Biomedical Engineering Pro-Publishers, Hingham, Mass., 1986. gram at Iowa State University, Ames, IA 50011.
7. Antonsson EK and Mann RW: The frequency content of gait. Biomechan-ics, 18(11:39-47, 1985. Lewis F. Brown received the B.S. degree in
electrical engineering from South DakotaState University, Brookings, in 1984 and theM.S., in electrical engineering from Iowa StateUniversity, Ames, in 1986. He is presently aResearch Assistant in the Biomedical Engi-neering Program at Iowa State University,
__ ,~, where his current research interests are elec-tromechanical modeling of ultrasonic trans-ducers, ultrasonic tissue characterization, andEKG signaldeetoanprcsig
1 6 IEEE ENGINEERING IN MEDICINE AND BIOLOGY MAGAZINE SEPTEMBER 1987
