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Digital Signal Processing with Applications in Medicine

Paulo S. R. Diniz***, David M. Simpson**, A. De Stefano**, and Ronaldo C. Gismondi*.

* School of Medicine, State University of Rio de Janeiro, Brazil

** Institute of Sound and Vibration Research, University of Southampton, UK

*** Program of Electrical Engineering, COPPE/EE/Federal University of Rio de Janeiro, Brazil

Corresponding Author:

Paulo S. R. Diniz

Prog. de Engenharia Eltrica e Depto. de Eletrnica

COPPE/EE/Federal University of Rio de Janeiro,Caixa Postal 68504, Rio de Janeiro, 21945970

Brazil

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1. Introduction

Most natural phenomena occur continuously in time, such as the variations of temperature of the

human body, forces exerted by muscles, or electrical potentials generated on the surface of the

scalp. These are analogue signals, being able to take on any value (though usually limited to a finite

range). They are also continuous in time, i.e. at all instants in time is their value available. However,analogue, continuous-time signals are not suitable for processing on the now ubiquitous computer-

type processors (or other digital machines), which are built to deal with sequential computations

involving numbers. These require digital signals, which are formed by sampling the original

analogue data.

The theory of sampling was developed in the early 20 th century by Nyquist [1] and others, and

revolutionized signal processing and analysis [2]-[6] especially from the 1960s onwards, when the

appropriate computer technology became widely available. The rapid development of high-speed

digital integrated circuit technology in the last three decades has made digital signal processing the

technique of choice for many most applications, including multimedia, speech analysis and

synthesis, mobile radio, sonar, radar, seismology and biomedical engineering. Digital signalprocessing presents many advantages over analog approaches: digital machines are flexible,

reliable, easily reproduced and relatively cheap. As a consequence, many signal processing tasks

originally performed in the analog domain are now routinely implemented in the digital domain,

and others can only feasibly be implemented in digital form. In most cases, a digital signal

processing systems is implemented using software on a general-purpose digital computer or digital

signal processor (DSP). Alternatively, application specific hardware usually in the form of an

integrated circuit can also be employed.

In this chapter we will discuss a number of fundamental principles and basic tools for digital signal

processing. These will be illustrated with examples from medical applications, where signal

analysis has been widely applied in patient monitoring, diagnosis and prognosis, as well asphysiological investigation and in some therapeutic settings (e.g. muscle and sensory stimulation,

hearing aids).

We will first discuss the principles of sampling. Here it is demonstrated that certain signals (those

limited in the range of frequencies they contain), can be fully represented by a sequence of samples.

These digital signals coincide with the original analog (and continuous-time) signals at predefined

time instants. By interpolation, the continuous-time signal can be recovered (without error!) from

the sequence of samples.

Next we will discuss digital filters, which are one of the main tools employed in signal processing,

as they suppress certain frequency bands, and enhance others. This is particularly useful in reducingnoise and other sources of interference, which are an almost constant problem in medical

applications. In these signals the most common sources of noise are electromagnetic interference at

50/60 Hz (and in higher frequency ranges), and contamination by other, unwanted physiological

signals (e.g. electrical signals from muscles may obscure signals of neural origin). Patient motion

often results in short-lasting artifacts in recorded data, and filters can again be used to mitigate

these. Digital filters can also be employed to describe the relationship between physiological

signals, such as that between blood pressure and blood flow. As such, they are also useful in

characterizing physiological systems.

In Fourier analysis, a signal is split into constituent sinusoidal oscillations (we will assume that

readers are familiar with the continuous time Fourier transform), and the discrete Fourier transform

is an extension developed for the analysis of digital signals. For the current work in particular, it

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provides a very convenient tool for specifying and describing digital filters, and it also provides the

basis for the sampling theorem.

Many signals, including most from a biomedical origin, can be classified as random: repeated

recordings result in signals that are all different from each other, but share the same statistical

characteristics. The power spectrum reflects some of the most interesting of these characteristics.

The interpretation and estimation of the power spectral density will be addressed in the final part ofthis chapter.

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2. Digital Signal Processing of Continuous-Time Signals

A typical digital signal processing system includes the following subsystems, as illustrated in Fig 1.

A/D Converter transforms the analogue input signal into a digital signal. It does so by

acquiring samples from the signal at (normally) equally spaced time intervals and converting thelevel of these samples into a numeric representation that can be used by a digital signal processing

system. In accordance with the sampling theorem, a low-pass (anti-alias) filter is usually required

prior to A/D conversion.

Digital Signal Processing - the digital signal processing system (DSPS) performs arithmeticoperations on the input sequence. In a typical application, the desired signal features are enhanced

in output signal, and unwanted components (such as noise and artifact) are suppressed. Further

analysis of the output signal may be carried out in order to extract information. In a medical

application, this could for example be to determine the average heart-rate, the peak blood pressure,

or an index of muscle activity. If an output signal is required (e.g. in a hearing aid), the next two

steps are carried out.

D/A Converter - converts the DSPS output into analog samples that are equally spaced in time.

Lowpass Filter - converts the analog samples into a continuous-time signal. This step isequivalent to an interpolation operation between the discrete analogue samples produced in the

previous step.

Figure 1. Architecture of a complete DSPS used to process an analogue signal.

EXAMPLE 1.A hearing aid

A hearing aid may be used to illustrate the operation of a DSPS: the sound is picked up

by the microphone, converted into an electric signal, and then digitized. The digital signal

is then filtered to selectively amplify those frequency bands in which the patient shows

the most severe hearing loss. Further processes may also be applied, including amplitude

compression, in which the system gain is reduced when the amplitude exceeds some pre-

defined threshold values, in order to avoid excessive loudness to the ear. Finally the

processed digital signal is converted back to analog form in the digital-to-analog (D/A)

converter, and delivered to the ear via the earphone.

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2.7 2.75 2.8 2.85 2.9 2.95 3-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

time [s]

a.u.

Figure 2. A short segment of speech signal, that may form the input to a hearing aid. The

segment displayed corresponds to the sound /um/.

2.7 2.75 2.8 2.85 2.9 2.95 3-6

-4

-2

0

2

4

6

time [s]

a.u.

Figure 3. The output of the digital filter (digital signal). The frequencies around 1500 Hz,where the patient showed the greatest loss in hearing, have been most amplified. Further

details on the filter used may be found in an example below.

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0 2 4 6 8

-100

-50

0

50

100

time [s]

a.u.

input signaloutput signal

Figure 4. In a further stage of signal processing in hearing aids, the amplitude-range

may be compressed. Thus the gain of the system is progressively reduced when the

sound-volume exceeds a specified level, as is the case near the end of this recording. A

small delay in adapting the gain to the signal-levels may also be noted, as the average of

the most recent amplitudes drives the gain control.

In signal analysis the periodic signals play a major role since they serve as basic signals from

which many other signals can be constructed and analyzed. Lets take as example the sinusoid

sin , that is a periodic signalsincesin( = sin . There are periodic signalsthat are harmonically related, meaning that they consist of a set of periodic signals whose

fundamental frequencies are all multiples of a single positive frequency . For example, theperiodic signalssin k , for any integer k, is called the k-th harmonic of sin . Also insignal analysis it is often performed a linear transformation of the signal from the time domain

to the frequency domain and vice versa, depending on the domain in which either the relevant

information is exposed in a clearer way or the mathematical manipulations are simpler. In the

particular case of periodic signals, they have quite compact representation in the frequency

domain where only the fundamental frequency information and their harmonics are required,whereas in the time domain they are represented by a continuous function of time.

t0 t)20 + t0

0

t0 t0

0

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3. Sampling of Continuous-Time Signals

The sampling theorem states that a band-limited continuous-time signal, , whose frequency

content has no energy beyond the frequency can be exactly recovered from sample values, ,

taken at the time instants t , provided the sampling frequency is larger than twice

. The sampling rate is called the Nyquist rate. The original continuous-time signal can berecovered from the sampled signal by the interpolation formula

)(tx

T/

cf )(nx

nT=

c

fs 1=

cf f2)(nx

[ ]

=

=

n s

s

nTt

nTtsinnxtx

2)(

2)()()(

(1)

withT

fss

2

2 == . Directly applying the above interpolation formula for the recovery of a

continuous-time signal from its samples is however not feasible, since this involves the summation

of an infinite number of terms, and requires future as well as past samples of the signal . The

digital-to-analog (D/A) converter and low-pass filter of Fig. 1 approximate this equation in anefficient manner.

)(nx

When a continuous-time signal is sampled with a sampling frequency smaller than the

Nyquist rate, distinct frequency regions of will be mixed, causing an undesirable and

irremovable distortion in the digital signal (and the continuous-time signal recovered from the

samples), known as aliasing. This issue will be further discussed.

)(tx sf

)(tx

The sampling process can be considered as multiplying (in the time-domain) the continuous-time

signal by a train of unitary impulses at time intervals of T: . The resulting

signal is , which consists of a sequence of samples of . Fig. 5-b illustrates thefrequency representation of the original signal before sampling. Fig. 5-c shows the spectrum of

the impulse train . According to the convolution theorem [6], the Fourier transform of the

sampled signal is given by the convolution, in the frequency domain, of the Fourier transform

of the impulse train and the Fourier transform of the original signal before sampling. This leads to

the following result, illustrated in Figs. 5- b and c:

)(tx

x

)()( iTttxi =

)(tx)()()( txtxt i=

)(txi

)(tx

)(tx

)(121

)(* ljjXTT

lj

TjX

TeX s

l

a

l

j

=

=

+=

+= , (2)

where is the Fourier transform of the sampled signal, is the Fourier

transform pair of the original analogue signal, and l is any integer. From the above equation and

Fig. 6 it can be inferred that the spectrum of the sampled signal will be repeated at a frequency

interval of

)(* jeX )()( txeX j

T

2s = .

This periodic repetition of the spectrum is illustrated in Fig. 6. The continuous-time signal is shown

in Fig. 6-a; Figs. 6-b and 6-c show the sampled signal spectra for the cases 2sc > and

2sc < , respectively. It can be concluded that in the latter case the original spectrum is

preserved in the frequency range and . Figure 6-b however shows that when the sampledsignal has frequency components above

c c2s (half the sampling frequency), the repeated spectra

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overlap and result in distortion at frequencies around 2s . This is aliasing, and undesirable in most

cases, as a frequency component (say ) originally above1 2s will be transferred to a frequency

an equal distance below 2s1.

th )(

To avoid aliasing, clearly the maximum frequency of the continuous-time signal must be known.

This is usually achieved by applying a low-pass filter (the so-called anti-alias filter) prior tosampling. This filter eliminates frequency-components above its cut-off frequency, which may then

be taken as . In practice the filter only attenuates high frequency components to a level at whichthey become insignificant (see example below). Due to the imperfections of the anti-alias filter and

more generally the smooth decay of signal spectra, usually sampling rates of 35 times (whichis only a nominal value) are chosen.

c

c

The recovery of the continuous-time signal spectrum from the sampled signal sampling can be

accomplished, in theory, by using an analog filter (filters will be discussed in more detail below)

with a flat frequency response, that retains only the components between 2s and 2s of the

spectra shown in Fig. 6-c, and that sets the spectrum outside the range 2s to 2s to zero. Thefrequency and impulse responses of the analog filter with these characteristics are shown in Fig. 7.

The impulse response of this filter is

t

tsinT lp

)(= (3)

1

More rigorously, will be transferred to a frequency (which is a negative frequency). However, for realsignals (i.e. those without an imaginary component) spectra are symmetric (as illustrated in the figure), and will bealiased to .

1 s1

1

1s

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(a) Signal sampling corresponds to time-domain multiplication.

((b) Spectrum of the continuous-time signal )tx .

)(tx(c) Spectrum of the impulse train .i

Figure 5. Sampling of continuous-time signals.

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(a) Spectrum of the continuous-time signal )(tx .

)(* tx(b) Spectrum of sampled signal for 2sc > . The spectrum of the sampled signal is given by

the sum of the overlapping spectra.

)(* tx f(c) Spectrum of sampled signal or 2sc < .

Figure 6. The effect of sampling on the spectrum.

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Figure 7. Ideal lowpass filter.

a means analog frequency.

lp

means cutoff frequency of a lowpass filter.

1F means inverse Fourier transform.)( ajH frequency response of an ideal lowpass filter.

)(th impulse response of an ideal lowpass filter.

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The parameter is the cutoff frequency of the designed lowpass filter. The frequency is

chosen to ensure that all the desired frequencies, and no others, are included - i.e.,

lp lp

2lpc


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0 20 40 60 80 10010

-8

10-6

10-4

10-2

100

frequency (Hz)

normalizedu

nits

Figure 9. The power spectrum of this signal shows that most of the power is concentrated

in frequencies below approximately 20 Hz, with peaks at the heart-rate (about 1.4 Hz)

and multiples of this frequency (the harmonics). In addition there is a sharp peak at

mains frequency (50Hz).

0 5 10 15 20 25 30 35

10-6

10-4

10

-2

100

frequency (Hz)

normalizedunits

Figure 10. The power spectrum of this signal, sampled at 67 Hz. Without the anti-alias

filter (dotted lines), the 50 Hz noise is aliased, and appears as a very large peak at 17 Hz

(67-50 Hz). With the anti-alias filter (solid line), this peak is very much reduced (though

not eliminated, due to the imperfection of the filter), and can be considered insignificant.

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4. Digital Signal Processing Systems

Once the signal has been digitized, it may be desirable to calculate signal parameters that reflect the

signal characteristics. The most commonly used are the mean value, the power, the peak-to-peak

amplitude, and the signal-to-noise ratio. These will now be defined, and their significance and use

will be illustrated on some biomedical examples.

The mean value of a signal is defined as

N

nx

m

N

n

==

1

0

)(

(4)

This represents the value around which the signal fluctuates, and is also known as the signal's 'DC

value'. The fluctuating part (i.e. the residual when the mean is subtracted) is known as the 'AC

component' of the signal. The peak-to-peak amplitude of the signal is given by the range from the

minimum to the maximum value. This is also sometimes known as the dynamic range of the signal.

EXAMPLE 3. Arterial blood pressure

The arterial blood pressure signal shows the fluctuations of the pressure in an artery

during the cardiac cycle. This increases when the heart contracts, and decreases as the

blood drains away while the heart relaxes. There is also a 'notch' (the 'dichrotic notch'),

associated with the closure of cardiac valves, as the pressure decreases.

0 1 2 3 480

100

120

140

160

180

time [s]

mmHg

blood pressuremean value

Figure 11. A segment of arterial blood pressure recorded in an adult subject, showing 6

heart-beats. The mean value of this signal is 124 mmHg. The peak-to-peak amplitude is

74 mmHg.

The mean blood pressure gives the average value during the recorded period. The

fluctuations around the mean constitute the AC component of the signal. In many

applications (though not for the case of blood pressure), only the AC component is of

clinical interest.

The power of a signal is defined by

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N

nx

P

N

nAV

==

1

0

2 )(

(5)

and thus represents its mean-square value. The square root of PAV is the root-mean-square (rms)

value of the signal, and gives a measure of the mean amplitude, which takes both the DC and the

AC component into account. However, in many cases it is only the power (or rms value) of the AC

component that is of interest, and this corresponds to the variance (or standard deviation) of the

signal. The power is in squared units (e.g. mmHg2, or V2), and therefore rather difficult tointerpret; the rms value is more convenient since it has easier interpretation and it is directly related

to the measure data unit. The mean absolute value (or magnitude) of a signal is also often used, but

the power lends itself more readily to further statistical analyses (see section on random signals,

below).

EXAMPLE 4.The power and standard deviation in the electromyogram (EMG)

The electromyogram is the electrical signal recorded from muscles. During muscle

contraction, the signal amplitude increases, and is used extensively in studies of neuro-

muscular disease.

0 1 2 3 4

-2

-1.5

-1

-0.5

0

0.5

1

1.5

time [s]

a.u

.

Figure 12. The EMG signal recorded from a quadriceps muscle during two cycles ofhuman gait, showing how the muscle is periodically activated. The mean value of the

signal is zero. The standard deviation (rms value) of the signal during the shaded periods

are 3.5894, 0.1983 and 4.1880, respectively, reflecting the change in signal amplitude.

The powers (variance) for the same periods are 12.8836, 0.0393 and 17.5396,

respectively. The rms value provides a reasonable measure of the 'average signal

amplitude', whereas the power is more difficult to interpret. Note that the calculations

were performed excluding the sharp spike at the beginning of each muscle-activation.

This is probably an artifact, caused by the small movement of the electrodes on the skin.

Many signals, including most from biomedical origin, are contaminated by noise. The signal-to-

noise ratio gives the ratio of signal power, to that of the noise:

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nP

PSNR= (6)

wherePis the power of the signal, andPnthe power of the noise-component. Because the order of

magnitude of signal and noise powers can be very dissimilar, a logarithmic scale is often employed,

with the result expressed in dB (Deci-Bell):

=

n

dBP

PSNR 10log10 (7)

Of course, the definition of signal and noise is application dependent: what is signal for one

problem, may be noise for another. For example, an ECG (heart-signal) may be contaminated by

EMG (muscle) noise; for other studies, it is the EMG which is of interest, and the ECG is

considered as noise (or artefact).

EXAMPLE 5. An electrocardiogram (ECG) signal, with increasing levels of noise.

The ECG signal is the electrical signal generated by the heart. It shows clear peaks during

each cardiac cycle, and is used in the diagnosis of a wide range of heart-diseases. It is

also the most commonly used signal for monitoring the heart electrical activity.

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Figure 13. An ECG signal contaminated by noise of progressively increasing power, with

signal to noise ratios decreasing form SNR=(no noise top row) to SNR=10, 2 and 1(bottom row). In the last line, the signal is almost completely obscured by the noise.

Digital Signal Processing Systems relate input signals to outputs. The relationship between the

output sequence of this system and may be represented by the operator as)(ny )(nx F

)]([)( nxFny (8)

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Figure 14. Discrete-time signal representation.

4. 1 Linear Time-Invariant Systems

The main class of DSPS is the linear, time-invariant(LTI) and causalfilters. A linear digital filter

is one that obeys the following expression

[ ] [ ] )()()()( nxFnxFnxnxF bbaabbaa +=+ [ ]

]

(9)

for any sequences and , and any arbitrary constants and . Thus if the input is the

weighted sum of two (or more) signals, the output is a similarly weighted sum of the responses to

the individual inputs. A digital filter is said to be time-invariant when its response to an input

sequence remains the same, irrespective of the time instant that the input is applied to the filter.

That is, if then

)(nxa

)(ny=

)(nxb a b

)]([ nxF

)()]([ 00 nnynnxF = (10)

for all integers and (where is the delay, in samples, of the signals). A causaldigital filter is

one whose response does not anticipate the behavior of the excitation signal. Therefore, for any two

input sequences and such that for , the corresponding responses of

the digital filter are identical for , that is,

n

x

on

)

0n

n

(na )(nxb )()( nxnx ba = 0nn

0n

[ ] [ )()( nxFnxF ba = , for n , (11)0n

irrespective of each signal's characteristics for n . An initially relaxed, linear, time invariant

digital filter is fully characterized by its response to the impulse (or unit sample) sequence .The filter response when excited by such sequence is denoted by and it is referred to as

impulse responseof the digital filter. Observe that if the digital filter is causal, then for

. An arbitrary input sequence can be expressed as a sum of delayed and weighted impulse

sequences, i.e.,

0n>

)(n

0

)(nh

)( =nh0


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=

=k

knxkhny )()()( (14)

In order to analyze, describe and design digital filters, the Fourier and the z transforms are very

convenient and powerful tools. A large class of sequences (e.g. signals and impulse responses) can

be expressed in the form

=

deeXnx njj )(2

1)( (15)

where

=

=n

jj enxeX )()( (16)

)( jeX is called the Fourier transform of the sequence . represents the signal in the

time-domain, and gives the same signal in the frequency domain, and the two can, in

general, be calculated from each other according to the above pair of equations. A sufficient but not

necessary condition for the existence of the Fourier transform is that the sequence is

absolutely summable, i.e.,

)(nx )(nx

)( je

)( jeX

X )(nx

=

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