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 Elétrica e Depto. de Eletrônica COPPE/EE/Federal University of Rio de Janeiro, Caixa Postal 68504, Rio de Janeiro, 21945970 Brazil

1

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 20th 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 signal processing 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 as physiological 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 reducing noise 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

2

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 of this 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 the level 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 arithmetic operations 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 is equivalent 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.

5

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 sinω , that is a periodic signal since sin(ω = sinω . There are periodic signals that are harmonically related, meaning that they consist of a set of periodic signals whose fundamental frequencies are all multiples of a single positive frequencyω . For example, the periodic signals sin kω , for any integer k, is called the k-th harmonic of sinω . Also in signal 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 informationω and their harmonics are required, whereas in the time domain they are represented by a continuous function of time.

t0 t)20 π+ t0

0

t0 t0

0

6

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 be recovered from the sampled signal by the interpolation formula

)(tx

T/cf )(nx

nT=

c

f s 1=

cf f2)(nx

[ ]∑∞

−∞= −−

=n s

s

nTtnTtsin

nxtx2)(

2)()()(

ωω

(1)

with T

fssπ

πω22 == . 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 an efficient 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 the frequency 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:

)(txx∗

)()( iTttxi −= δ

)(tx)()()( txtxt i×=

)(txi

)(tx∗

)(tx

)(121)(* ljjXTT

ljT

jXT

eX sl

al

j ωωπωω ∑∑

∞

−∞=

∞

−∞=

+=

+= , (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

)(* ωjeX )()( txeX j ⇔ω

Tπ2

s =ω .

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 ωω > and

2sc ωω < , respectively. It can be concluded that in the latter case the original spectrum is preserved in the frequency range andω . Figure 6-b however shows that when the sampled signal has frequency components above

cω− c

2sω (half the sampling frequency), the repeated spectra

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overlap and result in distortion at frequencies around 2sω . This is aliasing, and undesirable in most cases, as a frequency component (sayω ) originally above 1 2sω will be transferred to a frequency an equal distance below 2sω 1.

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 to sampling. This filter eliminates frequency-components above its cut-off frequency, which may then be taken as ω . In practice the filter only attenuates high frequency components to a level at which they 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 3–5 times ω (which is 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 2sω− and 2sω of the spectra shown in Fig. 6-c, and that sets the spectrum outside the range 2sω− to 2sω to zero. The frequency and impulse responses of the analog filter with these characteristics are shown in Fig. 7. The impulse response of this filter is

ttsinT lp

π

ω )(= (3)

1 More rigorously, ω will be transferred to a frequency ω − (which is a negative frequency). However, for real

signals (i.e. those without an imaginary component) spectra are symmetric (as illustrated in the figure), and ω will be

aliased to ω − .

1 sω1

1

1ωs

8

(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.

9

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

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

the sum of the overlapping spectra.

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

Figure 6. The effect of sampling on the spectrum.

10

Figure 7. Ideal lowpass filter.

→aω means analog frequency.

→lpω−

means cutoff frequency of a lowpass filter.

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

→)(th impulse response of an ideal lowpass filter.

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The parameter ω is the cutoff frequency of the designed lowpass filter. The frequency ω is chosen to ensure that all the desired frequencies, and no others, are included - i.e.,

lp lp

2lpc ω << sωω . It may be noted that the impulse response of equation (21) is noncausal (its impulse response precedes the input impulse; when applied, this filter would requires future as well as past samples of the input signal), and as such cannot be implemented. In practice, it is possible to design continuous-time filters that give good approximations to the frequency response in Fig. 7.

EXAMPLE 2. The acquisition of a blood pressure signal.

A blood pressure (BP) signal contains clinically relevant components up to about 20 Hz. In the current example it is also known that the signal is contaminated by noise at the mains frequency (50Hz) and other noise (mainly below 50 Hz) may also be present. What is the minimum sampling rate required? The sampling rate has to be above 100Hz, as the highest frequency present is 50 Hz. It is the maximum frequency present in the signal, not the maximum frequency of interest, which determines the minimum sampling rate required. If a sampling rate below 100 Hz is employed, the mains (and other) noise may be aliased into the clinically relevant frequency band of BP signal and could contaminate the BP signal (see figure below). If the noise (mains interference) is removed prior to sampling, by a low-pass (anti-alias) filter with a cut-off frequency at say 20 Hz, the sampling rate could be reduced to a value above 40 Hz, without significant aliasing occurring. This was carried out below, and a sampling rate of 67 Hz was chosen.

27 28 29 30

80

90

100

110

120

130

140

150

160

time (s)

mm

Hg

Figure 8. A recording of arterial blood pressure, contaminated by mains noise (50Hz).

12

0 20 40 60 80 10010-8

10-6

10-4

10-2

100

frequency (Hz)

norm

aliz

ed u

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)

norm

aliz

ed u

nits

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

nxm

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]

mm

Hg

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

14

N

nxP

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 µV2), and therefore rather difficult to interpret; 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 of human 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:

15

nPPSNR = (6)

where P is the power of the signal, and Pn the 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):

⋅=

ndB P

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.

16

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)

17

Figure 14. Discrete-time signal representation.

4. 1 Linear Time-Invariant Systems

The main class of DSPS is the linear, time-invariant (LTI) and causal filters. A linear digital filter is one that obeys the following expression

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

]

(9)

for any sequences and , and any arbitrary constants γ and γ . 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 causal digital 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 δ . The filter response when excited by such sequence is denoted by and it is referred to as impulse response of 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<n

∑∞

∞−=−=

kknkxnx )()()( δ (12)

and the response of an LTI digital filter to can then be expressed by )(nx

[ ] ∑∑∑∞

∞−=

∞

∞−=

∞

∞−=

=−=−=

−=

kkknhnxknhkxknFkxknkxFny )(*)()()()()()()()( δδ (13)

as can readily be seen by extending the linearity condition, given earlier. The summation

is called the convolution sum (denoted by ), and relates the

output sequence of a digital filter to its impulse response and to the input sequence . The expression above can also be rewritten as

∑∞

∞−=

−==k

knhkxnhnxny )()()(*)()( ∗

)(nh )(nx

18

∑∞

∞−=

−=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

∫∞

∞−= ω

πωω deeXnx njj )(

21)( (15)

where

∑∞

−∞=

−=n

jj enxeX ωω )()( (16)

)( ωjeX 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

)( ωje

)( ωjeX

X )(nx

∑−=

∞→∞<

N

NnNnx )(lim (17)

The z transform provides an extension of the concept underlying the Fourier transform, as it can be employed in a wider class of sample sequences than the Fourier transform. The transform of a sequence is defined as

z)(nx

∑∞

∞−=

−=n

nznxzX )()( (18)

where z is a complex number. The transfer function of a digital filter is then defined as the ratio of the transform of the output sequence to the transform of the input signal, i.e., z z

)()()(

zXzYzH = (19)

Taking the transform of both sides of the convolution expression of equation (14), i.e., z

n

n n k

n zknxkhzny −∞

−∞=

∞

−∞=

∞

−∞=

−∑ ∑ ∑ −= )()()( (20)

and substituting variables , )( knl −=

∑∑ ∑∞

−∞=

−∞

−∞=

∞

−∞=

−− =ln k

kn zlxzkhzny 1)()()( (21)

The following relation among the transforms of the output Y(z), of the input X(z) and of the impulse response H(z) of a digital filter is therefore obtained:

z

19

)()()( zXzHzY = (22)

Hence, the transfer function of an LTI digital filter is the transform of its impulse response, and may therefore also be used to describe or define the filter: .

zH )()( nhz ⇔

In equation (22), leads to the Fourier transform, and describes the changes in amplitude introduced by the DSPS to a complex exponential input signal. Such a function of ω is called the frequency response of the digital filter, and it corresponds to the transfer function evaluated on the unit circle of the plane (

ωjez = )( ωjeH

)(zHz 1=z ). In general, is a complex-valued

function, which can be expressed in polar form as )ωj(eH

)()()(ωωω jeHjjj eeHeH ∠= (23)

where )( ωjeH and are called the magnitude response and phase response, respectively, of the digital filter.

)(ωj

eH∠

4.2 Difference Equation

In defining a DSPS above, the summation could involve an infinite number of terms, in which case it can of course not be practically implemented. However, in some cases this problem can be overcome by using past input and output samples in a recursive formulation of the LTI system. A general DSPS can thus be described according to the following difference equation.

∑ ∑= =

=−−−N

i

M

lli lnxbinya

0 0

0)()( (24)

A difference equation represents a system that is linear, time-invariant and causal, if the auxiliary conditions required to solve the underlying difference equation correspond to its initial conditions, and the system is initially relaxed. In other words, the internal past signals of the difference equation are considered zero up to the instant an input x(n) starts. In non-recursive digital filters the output is a function of past input samples. In general, they can be described by equation (20) if that is and , for , ..., N. 10 =a 0=ia 1=i

∑=

−=M

ll lnxbny

0

)()( (25)

These filters have Finite Impulse Response and are known as FIR filters. The digital filter implemented by the above equation is called a nonrecursive filter since there is no feedback of the output in the computation of its present value. For , equation (20) can be rewritten as 10 =a

∑∑==

−+−−=M

ll

N

ii lnxbinyany

01)()()( (26)

(note the first sum now begins at i=1, rather than 0). These filters have, in most cases, infinite impulse responses (i.e., when ) and are therefore known as IIR filters. The filter implementation given by the above difference equation is called recursive.

0)( ≠nh ∞→n

EXAMPLE 6. Time and frequency-domain representations of a signal, and an IIR digital filter.

20

The ECG signal may be contaminated by movement artefact and by the electrical activity of other muscles near the heart, as already mentioned above. High-pass filters, which suppress low-frequency activity, and low-pass filters, which suppress high-frequency activity may be used to selectively enhance the frequencies in which the ECG predominates, and attenuate those frequency bands, in which the noise is strongest. The resulting signals are then more suitable of posterior analysis, for example to determine the heart rate, which can be very difficult in very noisy raw data, as illustrated below.

21

0 1 2 3 4 5 6 7 8 9 10-1

-0.5

0

0.5

1

Am

plitu

de [m

V]

0 1 2 3 4 5 6 7 8 9 10-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Am

plitu

de [m

icro

V]

Time [s]

22

Figure 15. The original signal is filtered using a 'Butterworth' high-pass filter (0.7 Hz) with coefficients a=[1 -3.9954 5.9862 -3.9862 0.9954] and b = [0.9977 -3.9908 5.9862 -

3.9908 0.9977] (the filter coefficients of equation (20) are here expressed as vectors). Clearly M=N=4. The high-pass filter removes low frequencies (below 0.7 Hz, in this case), and thus the relatively slow base-line drift introduced by the movement of the

electrodes and the patient.

0 1 2 3 4 5 6 7 8 9 10-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Am

plitu

de [m

V]

Time [s]

Figure 16. The ECG signal has most of its power in the frequency band up to about 15 Hz, whereas the EMG has much of its power beyond that range. By applying a low-pass filter, with a cut-off frequency at 15 Hz, the EMG can be attenuated, and the ECG is thus enhanced. This was applied here, using an IIR filter with the following coefficients: a = [1 -3.9015 5.7093 -3.7140 0.9062] and b = [0.1202 0.4809 0.7214 0.4809 0.1202] 10-6 . If this signal is compared to the raw data, it may be noted that the heart-beats are now

vastly clearer, and almost all (except that at around 8 s) can be identified.

23

5. Transfer Functions and Structures

In this section different transfer function representations and realizations of LTI filters are introduced. The different structures provide relative advantages in terms of analysis and implementation for specific applications.

5.1 FIR Filters

The general transfer function of an FIR filter in the transform representation is given by z

∏∑==

−− −==M

ll

M

l

Mll zzzHzbzH

000 )()( (27)

A filter with the above transfer function will always be stable, i.e. an input of finite amplitude will lead to an output that is also finite. This can also be inferred from the location of the poles in the z-plane (the two-dimensional plane formed by plotting the real part of z along the abscissa, and the imaginary part along the ordinate). The poles correspond to the points in the z-plane where the denominator of is zero; their counterparts are the zeros, where the numerator becomes zero. For FIR filters, all poles are at the origin ( ), and the zeros are the z

)(zH0=z l in the equation above.

When the poles of a transfer function are within the unit circle (i.e. 1<z ), the filter is stable; this clearly is the case for the FIR filters. A further advantage of FIR filters is that they can be designed to have linear phase. FIR filters have a linear phase response if and only if its impulse response is symmetric or anti-symmetric, that is

. Such filters may introduce delay in the output signal (with respect to the input), but all frequencies are delayed by the same amount. These filters thus distort the features of the signal less than filters that are not linear in their phase-response.

)()( nMhnh −±=

FIR filters can be designed by using optimization packages, or by using approximations to ideal infinite-impulse responses, cut to finite length by the application of different types of tapered windows. The main drawback of FIR filters is that in order to satisfy demanding magnitude specifications, FIR filters require a relatively high number of multiplications, additions and storage elements (i.e. large M). These facts make FIR filters potentially more expensive than IIR filters in applications where the arithmetic operations or storage elements are costly, or need to be limited in number. It is mainly the benefit of achieving linear phase that motivates the widespread use of FIR filters.

24

EXAMPLE 7. FIR filters in a hearing aid.

The first example in this chapter referred to the selective amplification of the sound in the frequencies in which the patient has the greatest hearing loss. The desired frequency response is obtained from measurements of the patient's hearing loss, and the filter gain chosen to compensate for this. FIR filters are well suited to this task.

0 2000 4000 6000 8000 10000-5

0

5

10

15

20

25

frequency (Hz)

gain

[dB

]

desired frequency responseFIR filter response

Figure 17. The desired frequency-response of the filter is determined at a number of frequencies (*). In this example, the strongest amplification is required around 1500 Hz. With a practical FIR filter, the desired response is approximated (solid line). Here the

sampling rate was fs=22050 Hz, and the filter-length N=201.

0 2 4 6 8x 10-3

0

1

2

3

4

5

time [s]

a.u.

Figure 18. The impulse response corresponding to the frequency response shown in the

previous figure. The symmetry of the impulse response ensures that the system has a linear phase response.

25

5.2 IIR Filters

The general transfer function for IIR filters is shown below

∑∑

=−

=−

+== N

ii

i

M

ll

l

za

zbzXzYzH

1

0

1)()()( (28)

The above equation can be rewritten in the following alternative forms

∏∏

∏∏

=

=−

=−

=−

−

−=

−

−= N

i i

M

l lMNN

i i

M

l l

pz

zzzH

pz

zzHzH

0

00

01

01

0)(

)(

)1(

)1()( (29)

where and are the zeros and poles of the transfer function, respectively. The above equation shows that while the FIR filter only has poles at the origin, IIR filters can have poles located at the arbitrary locations .

lz ip

ip 1<ip is however required to guarantee stability for the IIR transfer function.

EXAMPLE 8. IIR filters to extract alpha activity from EEG signal

The EEG signal is the electrical signal recorded from the surface of the scalp, and reflects the electrical activity of the brain. It is most commonly used in the diagnosis and investigation of epilepsy. In normal adult subjects, the most noticeable feature is the alpha-rhythm: this is an oscillation of the signal in the range from about 8 – 13 Hz, which is observed most strongly at the back of the head, when the subject is at rest, with eyes closed (but not asleep). In the example below, an IIR filter is used to enhance the alpha activity in an EEG signal that contains mains interference (50 Hz), artifact and noise (including some EEG components from outside the range of interest).

26

17 18 19 20 21

-60

-40

-20

0

20

40

60

80

Time [s]

Am

plitu

de [m

icro

V]

Figure 19. Using and IIR filter with coefficients a=[1 -7.43 24.44 -46.33 55.38 -42.75 20.82 -5.84 0.72] and b =10-4 [0.12 0 -0.48 0 0.720 -0.48 0 0.12] (for the design of

filters, see example 9, below), very rapid change and slow variations are eliminated, and only the signal components in the alpha frequency band are maintained.

0 10 20 30 40 50 60

10-2

10-1

100

101

Time [s]

Am

plitu

de [m

icro

V]

Figure 20. The spectrum of the original (thin line) and filtered signal (bold line). In the

latter, the activity outside the alpha band (including the noise at 50 Hz) are strongly attenuated.

27

5.3 Digital Filter Realizations

From equation (24), it can be seen that digital filters use three basic types of operations: • Delay - used to store previous samples of the input and output signals, in addition to internal states of the filter, which are used in subsequent calculations. • Multiplier - used to implement multiplication operations of a particular digital filter structure. • Adder - used to implement additions and subtractions in the digital filter.

5.3.1 FIR Filter Structures

A general FIR filter structure is shown in Fig. 21. This is called the direct-form nonrecursive structure for FIR filters.

Figure 21. Direct form nonrecursive structure.

The impulse response of linear-phase FIR filters are either symmetric or antisymmetric, that is . It is possible to exploit this, and reduce the number of multiplications required

in implementing the filter. )()( nMhnh −±=

5.3.2 IIR Filter Realizations

The general IIR transfer function can be written as in equation (28). The numerator in this transfer function can be implemented by using an FIR filter. The denominator entails the use of a recursive structure. The cascade of these realizations for the numerator and denominator is shown in Fig. 22. If the recursive part is implemented first, the realization in Fig. 22 can be transformed into the structure in Fig. 23, for the case . In the latter clearly requires only half the delay elements used in the former.

MN =

28

Figure 22. Recursive structure in direct form.

MN =Figure 23. Direct form structure for .

There are many alternative structures for the implementation of FIR and IIR digital filters [2]-[6]. Each has distinct features, advantages and disadvantages related to their practical implementation, but a detailed discussion lies beyond the scope of this article.

EXAMPLE 9. Filter design.

There are a number of methods for designing IIR digital filters. The most popular are the Butterworth, Chebyshev and Elliptic filters. Details of these are beyond the scope of this article, but some illustrative examples are given below. In general, flat frequency response in the pass and stop-bands are desired, as well as a sharp transition between these bands (sharp cut-off). The phase response would ideally be linear, but this can only be approximated with IIR filters.

29

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

frequency

gain

p=2p=4p=6

Figure 24. Frequency responses for Butterworth filters of order p=2, 4 and 6 (for

Butterworth filters, M=N=p). The sampling rate was fs=1 Hz, and the cut-off frequency 0.1 Hz. The filter coefficients for p=2 were found to be a=[1, -1.143, 0.4128],

b=[0.067455, 0.13491, 0.067455].

0 10 20 30

-0.05

0

0.05

0.1

0.15

0.2

0.25

samples

p=2p=4p=6

Figure 25. Impulse responses of the Butterworth filters shown above (since the impulse

response is of infinite length, only the beginning is shown).

30

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

frequency

gain

p=2p=4p=6

Figure 26. Frequency responses for Chebyshev filters of order p=2, 4 and 6 (as with

Butterworth filters, M=N=p). Chebyshev filters show ripple in the pass-band, and this was specified as 0.5 dB (i.e. the pass-band ripple produces pass-band gains in the

range from 0.944 to 1). The sampling rate was fs=1 Hz, and the cut-off frequency 0.1 Hz. The filter coefficients for p=2 were found to be a=[1, -1.0349, 0.42929] and

b=[0.093093, 0.18619, 0.093093].

0 10 20 30 40 50 60

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

samples

p=2p=4p=6

Figure 27. The impulse response for the Chebyshev filters shown above.

Clearly, the Butterworth filters have a smoother frequency response in the pass-band, but the Chebyshev filters give a sharper transition between the pass and stop-bands, for the same filter-orders. Higher order filters give sharper transitions, with longer impulse responses. Sharper transitions may be desirable in order to separate wanted and unwanted frequency components, but they can lead to oscillations (ringing) in the output signals of the filter, near sharp transitions in the signal. This is evident in the impulse responses, particularly for the higher order Chebyshev filters, which have the sharper transitions.

31

6. Discrete Fourier Transform

In this section the discrete Fourier Transform (DFT) is described, following closely the approach in [6]. The DFT is a key tool in digital signal processing, as it transforms a finite-length discrete, time-domain signal into its frequency domain representation, and can readily be implemented on digital machines. Very efficient algorithms in terms of computational complexity have been developed, which are collectively known as the fast Fourier transforms (FFTs). A discussion of the FFT falls however beyond the scope of this article, but can be found in most signal processing textbooks [2]-[6]. A discrete-time signal can be represented in the frequency domain through the Fourier transform defined earlier as

( ) ∑∞

−∞=

−=n

njj enxeX ωω )( (30)

This representation is a function of a continuous variable ω , involving the sum of an infinite number of terms, and is therefore not suitable for manipulation by a digital computer. An alternative frequency representation for finite length sequences may be defined, by sampling ( )ωjeX at the

discrete points kNkπ2

=ω , leading to

( ) ( )∑∞

−∞=

−=

k

jj kN

eXeX πωδωω 2ˆ (31)

This corresponds to multiplication in the frequency domain by a train of pulses, which is equivalent to convolution in the time-domain:

)(2)(ˆ 1 nxkN

Fnxk

∗

−= ∑

∞

−∞=

− πωδ (32)

However

( )∑∑∞

−∞=

∞

−∞=

− −=

−

lk

NlnNkN

F δπ

πωδ

221 (33)

and thus

( ) ∑∑∞

−∞=

∞

−∞=

−=∗

−=ll

NlnxNnxNlnNnx )(2

)(2

)(ˆπ

δπ

(34)

This corresponds to a periodic repetition of , every N samples. Thus for )(nx ( )ωjeX sampled at intervals of , a finite-length signal of M samples in duration can be recovered perfectly from , provided .

N/2π )(nx)(ˆ nx NM ≤

)(ˆ2)( nxN

nx π= , for (35) 10 −≤≤ Nn

32

If M is larger than N, then successive repetitions of overlap in , and can no longer be restored. The correspondence between this and time-domain sampling / aliasing is clearly evident.

)(nx )(ˆ nx )(nx

The time-domain sequence can be expressed as a function of the samples of )(nx ( )ωjeX , as follows

( ) ∑∞

−∞=

−

=

k

kN

jj kN

eXeX πωδ

πω 2ˆ

2

(36)

Applying the inverse Fourier transform to the above equation, we obtain

( )

knN

jN

k

kN

j

nj

k

kN

jnjj

eeX

dekN

eXdeeXnx

ππ

πω

π πωω

π

ωπ

ωδπ

ωπ

21

0

2

2

0

2

0

2

21

221ˆ

21)(ˆ

∑

∫ ∫ ∑−

=

∞

−∞=

=

−

==

(37)

Then, provided , can be found as NM ≤ )(nx

∑−

=

=

1

0

221)(N

k

knN

jkN

jeeX

Nnx

ππ

(38)

Assuming has non-zero samples within the interval , it follows that )(nx 10 −≤≤ Nn

∑−

=

=

1

0

22

)(N

k

knN

jkN

jenxeX

ππ

, for (39) 10 −≤≤ Nk

The signal is related to the as )(nx )(ˆ nx

)]()()[(ˆ2)( NnununxN

nx −−=π (40)

where u is a unit step ( ), and u therefore a rectangular window of unit amplitude in the range 0 . Since multiplication in the time domain corresponds to a convolution in the frequency domain

)(n 0,1;0,0 ≥=<= nnNn <≤

)()( Nnun −−

( ) ( )

( )

∗=

−−∗=

−−

2)1(

2

2ˆ1

)}()({ˆ221

Njj

jj

esin

NsineX

N

NnunuFeXN

eX

ωω

ωω

ω

ω

ππ

(41)

Then by substituting equation (36) in equation (41), we obtain

33

( ) ∑∞

−∞=

−

−−

−

−

∗

=

k

NkN

jkN

jj e

Nksin

kNsineX

NeX

)1(2

2

2

21 πωπω

πω

πω

(42)

i.e. can be obtained from the frequency domain samples )( ωjeX

=

kN

jj eXeXπ

ω2

)(ˆ by

interpolation. Now defining

Nj

N eWπ2

−= (43)

The definitions of the DFT and its inverse (IDFT) become

1

0

( ) ( ).N

knN

n

X k x n W−

=

= ∑ , for (44) 0 1k N≤ ≤ −

1N −

0

1( ) ( ). knN

n

x n X k WN

−

=

= ∑ , for 0 (45) 1n N≤ ≤ −

EXAMPLE 10. An EEG signal and its DFT . )(nx )(kX

Figure 28, 29, 30 and 31 show an example of the DFT of a segment of an EEG signal (taken form the electrodes O2/A2, and showing strong alpha activity). The definition of the DFT leads to a spectrum that shows symmetry around the mid-point (N/2), when the signal x(n) only has real components (i.e. the imaginary part of is zero). The frequency index k may also be converted into true frequency (e.g. in Hz), by considering the sampling rate and the length of the signal analyzed.

)(nx

0 1 2 3 4-100

-50

0

50

100

time (s)

EE

G -

O2/

A2

(mic

roV

)

Figure 28. A segment of an EEG signal (O2-A2 derivation). An oscillation is evident, and from counting the number of cycles per second, a frequency of approximately 11 Hz can be estimated (i.e. this is an oscillation in the EEG’s alpha frequency band). The signal was sampled at 256 Hz and is 5 seconds in duration, hence N=5*256=1280 samples.

34

0 500 10000

1

2 x 104

harmonic(i)

ampl

itude

0 500 1000

-2

0

2

harmonic (i)

phas

e

Figure 29. The output of applying the DFT. The 1280 samples in the time-domain have

been converted into N=1280 samples (horizontal axis) in the frequency domain. Since the output of the DFT gives complex numbers, both the amplitude and phase for each

‘harmonic’ is given. It may be noted that the amplitude spectrum is symmetrical about N/2: |X(i)|=|X(N-i)| (i=1 .. N/2-1). This symmetry arises from the signal having only real-valued samples. The phase also shows symmetry (but with the sign reversed), i.e. φ(i)=-(N-i) (i=1 .. N/2-1). Thus , with φ )()( * iNXiX −= * indicating the complex conjugate. The symmetry is shown more clearly in the detail below. Note that for the signal of length

N=1280, the harmonics are given as i=0 ... 1279 (rather than 1 ... 1280).

40 45 50 55 600

0.5

1

1.5

2 x 10 4

harmonic (i)

ampl

itude

1220 1225 1230 1235 12400

0.5

1

1.5

2 x 104

harmonic (i)

ampl

itude

40 45 50 55 60-3

-2

-1

0

1

2

3

harmonic (i)

phas

e

1220 1225 1230 1235 1240-3

-2

-1

0

1

2

3

harmonic (i)

phas

e

Figure 30. Details of the amplitude and phase spectra, showing the region of the largest peak in the amplitude. The harmonic (i) and its frequency f (in Hz) are related through

Nfi

NTiif s==)( , as may deduced by comparing the definitions of the DFT and the

Fourier transform. The peak in the spectrum at i=53 thus corresponds to the frequency

128025653×

=f =10.6 Hz, which is in agreement with the value expected from visual

inspection of the time-domain signal.

35

0 50 100 0

10

20

30

ampl

itude

(mic

roV)

0 50 100

-2

0

2

frequency (Hz)

phas

e

Figure 31. The amplitude and phase spectrum of the signal, with the abscissa rescaled in Hz. The full frequency range of the signal (i.e. up to ½ of the sampling frequency –

according to the sampling theorem) is shown. The vertical axis has been rescaled in accordance with the Fourier series. Thus the sum of all the 641 sinusoidal components in

the signal (1280/2+1=641, corresponding to frequencies from 0 to 128 Hz) with the amplitudes (in µV) and phases (in rad) given in these spectra, provides perfect

reconstruction of the original digital signal of length N=1280 samples. In particular, the sum of all the sinusoids near the spectral peak at 10.6 Hz will give the alpha rhythm,

whose amplitude is seen to be approximately 50 µV in the original data.

6.1 Some Properties of the DFT

Circular Convolution in Time

If the sequences and are periodic with period , then )(nx )(nh N

1 1

0 0

( ). ( ) ( ). ( ) ( ). ( )N N

n l

x n h n l x n l h l X k H k− −

= =

− = − ↔∑ ∑ (46)

where and are the DFTs of length for and , respectively. It should be noted that in multiplying the DFTs, circular convolution is always the result, i.e. periodicity of the signal is assumed, even if this is not in fact the case for the particular signal under consideration.

)(kX )(kH N )(nx )(nh

Correlation In a similar manner to convolution, the DFT may also be used to calculate the correlation in time between two sequences:

1*

0

( ) ( ) ( ) ( )N

n

h n x l n H k X k−

=

+ →∑ (47)

Again, the resultant correlation sequence assumes that the signal is periodic.

Relationship Between the DFT and the z Transform

The transform of a finite length sequence is z ( )X z ( )x n

( )1

0

( )N

nz

n

X z x n z−

−

=

= ∑ (48)

36

By substituting 2j knNz eπ

= , it follows that

2 21

0

( )Nj kn j kn

N

n

X e x n eπ π− −

=

=

∑ N (49)

The equation above leads exactly to the definition of the DFT as given in (44), which thus is simply given by samples of the transform equally spaced on the unit circle. z In order to obtain the transform from the DFT coefficients , equation (45) is substituted into equation (48), generating

z )(kX

( )1 1 1 1

1

0 0 0 0

1 1

1 10 0

1 1( ) ( ) ( )

1 1 1 ( ) ( )1 1

N N N Nkn n k n

N Nn k k n

kN N NN NN

k kk kN N

X z X k W z X k W zN N

W z z X kX kN W z N W

− − − −− − − −

= = = =

− − −− −

− − − −= =

= =

− −= =

− −

∑ ∑ ∑ ∑

∑ ∑ z

(50)

6.2 Linear Filtering via DFT

When the input signal sequence to a DSPS is available in blocks, the DFT can be used to implement linear convolution, leading to the same result as the serial filtering operation, but with reduced computational complexity.

Linear X Circular Convolutions Supposing that the DFTs are of size , circular convolution between two sequences and is given by

N )(nx )(ˆ nx

1

01

0 1

ˆ( ) ( ) ( )

ˆ ˆ = ( ) ( ) ( ) ( )

N

ln N

l l n

y n x l x n l

x l x n l x l x n l N

−

=

−

= = +

= −

− + − +

∑

∑ ∑ for 0 (51) 1n N≤ ≤ −

The circular convolution can be made equal to the linear convolution between and , if the second sum in the above equation becomes zero. If has duration and has duration , it can be shown that for the linear and circular convolutions to be equivalent

)(nx)(ˆ nx

)(ˆ nx)(nx M K

1N M K≥ + − (52)

This is achieved by appending zeros to the original signal (‘zero-padding’), up to length N. When the input signal sequence is very long, the linear convolution can be performed though the Overlap-and-Add Method. The approach is to describe the signal in non-overlapping blocks

of length as )(nx

( )lx n N

0

( ) ( )ll

x n x n lN∞

=

= −∑ (53)

where

37

( ), for 0( )

0,l

x n lN n Nx n

otherwise+ ≤ ≤

=

1− (54)

The convolution of with another signal can be written as )(nx )(nh

0

0

( ) ( ) ( ) ( )

[ ( ) ( )]

ll

ll

y n y n lN x n h n

x n lN h n lN

∞

=

∞

=

= − = ∗

= − ∗ −

∑

∑ (55)

where represents the block decomposition of the resulting sequence originated from the convolution of h with the l - th block .

( )ly n lN−( )n ( )lx n

We see that in order to convolve a long with a length- , all that needs to be done is: )(nx )(nhK(a) Divide into length- blocks . )(nx N ( )lx n(b) Zero-pad and each block to length . )(nh ( )lx n 1−+ KN(c) Perform the circular convolution of each block using the length- DFT. )1( −+ KN(d) Add the results according to equation (50).

EXAMPLE 11. DFT and filtering in an EEG signal.

0 50 1000

1

2

3

4

5

frequency (Hz)

ampl

itude

(mic

roV

)

Figure 32. The amplitude spectrum of an EEG signal (temporal region, derivation T6-A2) showing a peak in the α frequency band (8-13 Hz). If the signal is reconstructed

(inverse DFT) using only the frequency components in this band, α activity is enhanced, as shown below.

38

29 29.5 30 30.5

-80

-60

-40

-20

0

20

40

time (s)

ampl

itude

(mic

roV

)

Figure 33. A segment of the original EEG signal (solid line), and the same signal

reconstructed from only its DFT components in the α frequency band (dotted line). The oscillation in the frequency band of 8-13 Hz is enhanced. The slow change during the

artefact in the EEG signal around 29.5 s is eliminated, as are some of the faster variations (higher frequency beta activity, and noise ) seen in the raw data.

A similar effect to the above can be achieved with a linear band-pass filter.

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

frequency [Hz]

gain

Figure 35. Frequency response )( fH of a second-order Butterworth band-pass filter, with cut-off frequencies at 8 and 13 Hz. This filter will preserve activity in the alpha

frequency band, while attenuating other frequency bands.

39

0 0.1 0.2 0.3 0.4 0.5 0.6-0.06

-0.04

-0.02

0

0.02

0.04

0.06

time [s]

h(k)

Figure 36. The impulse response h(k) of a filter (the full response continues to die away after 0.6 s). The sampling rate has to be the same as that of the signal to be filtered, i.e.

256 Hz.

28 29 30 31

-80

-60

-40

-20

0

20

40

time [s]

mic

roV

IPOP (using h)OP (using H)

Figure 37. A segment of the EEG signal (solid line; derivation T6-A2) and the output of the filter, achieved by applying the filter through time-domain convolution (i.e. using h, dotted line), or through frequency-domain multiplication (i.e. using H, dashed line). The

outputs are clearly identical, and the alpha activity in the EEG signal has been enhanced, in a way similar to that achieved with the inverse DFT earlier. Note that the filter was

applied in the forward and reverse direction, in order to correct for the delays and phase-distortion introduced by the IIR filter.

40

0 0.1 0.2 0.3 0.4 0.5 0.6

-30

-20

-10

0

10

20

30

40

time (s)

mic

roV

IPOP (using h)OP (using H)

Figure 38. At the beginning of the same EEG recording shown above, there is a difference between time- and frequency-domain implementation of the filter. The former assumes the signal is zero for time < 0, whereas the latter assumes the signal is periodic. However, the two output signals rapidly

converge, and after some 0.5 seconds are indistinguishable.

41

7. Random Signal Representation

In this section we briefly review some concepts related to random discrete-time signals. Essential tools to analyze random signals are reviewed. Further details on concepts and techniques may be found in references [7]-[8].

7.1 Random Signals

A random variable is a function that assigns a number to the outcome of a given experiment denoted by . Related to this, a stochastic process entails the rules to describe the time evolution of the random variable depending on . As such, it is a function of two variables . The set of all experimental outcomes, called the ensemble, is the domain of . Denote as a sample of the given process with fixed, where if is also fixed, then is a number. For any statistical operator applied to , it means that is fixed and is variable. In this section we will use the notation to represent a random signal.

X

(nx

ρ

)n

ρ ( , )X n ρ)(nxρ

)ρ)

n (nxn ρ

(x Random signals have no precise description of their waveforms. All we can do is to characterize the random signals using measured statistics or apply a probabilistic model. In most cases, the first- and second-order statistics of the random signal (i.e. means, variances and correlations between samples), are sufficient for characterization of the stochastic process. The first- and second-order statistics can also be easily measured, while the effects on these statistics generated by linear filtering are straightforward to analyze. The expected value, or mean value, of the process is defined by

( ) [ ( )]xm n E x n= (57)

The definition of the expected value is expressed as

( )[ ( )] ( )x nE x n p dα α∞

−∞= ∫ α (58)

where is the probability density function (pdf) of at the point (amplitude) α . For appropriate interpretation of the pdf one needs to define the distribution function of a random variable as

( ) ( )x np α )(nx

( ) ( )x nP α = probability of being (59) )(nx α≤

or, alternatively

( ) ( )x nP α = (60) ( ) ( )x np dα

β β−∞∫

The pdf is the derivative of the distribution function

( )( )

( )( ) x n

x n

dPp

dα

αα

= (61)

The autocorrelation function of the process is defined by )(nx

42

( ), ( )( , ) [ ( ) ( )] ( , )x x nr n l E x n x l p d dβα α β α β∞ ∞

−∞ −∞= = ∫ ∫ x l (62)

where is the joint probability density of the random variables and defined as

( ), ( ) ( , )x n x lp α β )(nx )(lx

2( ), ( )

( ), ( )

( , )( , ) x n x l

x n x l

Pp

α βα β

α β

∂=

∂ ∂ (63)

where

( ), ( ) ( , )x n x lP α β = probability of { and } (64) ( )x n α≤ ( )x l β≤

The autocovariance function is defined as

2 ( , ) [( ( ) ( ))( ( ) ( ))] ( , ) ( ) ( )x x x x xn l E x n m n x l m l r n l m n m lσ = − − = − x (65)

where the last equality follows from the definitions of mean value and autocorrelation. For , which is the variance of .

ln =2 2( , ) ( )x xn l nσ σ= )(nx

7.2 Autoregressive Moving Average Process

The process generated by applying an input signal that is a white noise (i.e. uncorrelated random samples) to a system described by a general linear difference equation given by

)(nx

)()()(10

inyjnxnyN

ii

M

jj ab −+−= ∑∑

== (66)

is called an autoregressive moving average (ARMA) process. The coefficients and b are the parameters of the ARMA process. The output signal is also said to be colored noise since the autocorrelation function of is nonzero for a lag different from zero, i.e., for some

.

ia

(lr

i

)(ny)(ny 0) ≠

0≠lFor the case where b for = 1, 2, ..., the resulting process is called autoregressive (AR). The terminology means that the process depends on the present value of the input signal and on a linear combination of past output samples of the process, indicating the presence of a feedback of the output signal. On the other hand, for the case a for i = 1, 2, ..., N the process is described as a moving average (MA) process. This terminology indicates that the process depends on a linear combination of the present and past samples of the input signal.

0=j j M

0=i

EXAMPLE 12. Evoked potentials.

In response to auditory stimulation by a series of clicks, the EEG signal recorded on the scalp is modified. However, the response is very small compared to the background, spontaneous EEG activity, and hardly evident in the raw data. The response to each click may be considered as one realization of a random process, and an ensemble of responses may thus be built up. The average of these EEG epochs, calculated at each time-instant with respect to the stimulus in accordance with (57) (also known as the coherent average), clearly shows the stimulus response, and is called the auditory evoked potential. This is illustrated below for the auditory brain-stem response. Similar methods are used to measure the response to visual and sensory stimulation.

43

0 0.005 0.01 0.015 0.02 0.025 0.03-1.5

-1

-0.5

0

0.5

1x 104

time [s]

a.u

Figure 39. An ensemble of the EEG signals following each auditory stimulus. The

stimulus-frequency was 33 Hz, and 20 of a total of 800 responses are shown (continuous lines). The sampling rate was 2000 Hz, and the instants of stimulation correspond to time

= 0. The coherent average is also shown (*). Clearly the evoked potential is much smaller than the EEG signal, and only evident once the EEG epochs are averaged.

0 0.005 0.01 0.015 0.02 0.025 0.03

-400

-200

0

200

400

time [s]

a.u

Figure 40. The coherent average shown in detail. The amplitude and latency (delay following the stimulus) are the main features of this signal used in the diagnosis of

auditory disorders. Separate averages calculated over the first 400, and last 400 stimuli confirm that the basic shape of the evoked potential is a consistent feature in the data.

The EEG following each stimulus is a random signal. Each 'realization' of the process is different, but they all share statistical characteristics. The EEG signals following the stimuli are non-stationary, as the mean value changes over time.

44

7.3 Power Spectral Density

Stochastic signals that have mean and autocorrelation functions that are time invariant, are called wide-sense stationary (WSS). These signals are persistent and therefore are not finite-energy signals. On the other hand, signals in this class have finite-power such that the generalized discrete-time Fourier transform can be applied for their analysis. The generalized discrete-time Fourier transform, when applied to a WSS process, leads to a random function of frequency [7]. As an alternative, the autocorrelation functions of most practical stationary random processes have discrete-time Fourier transforms, and these have been found very useful in many applications. This transform defined as

[∑ == − )()()( lrFelreR xlj

xj

xωω ] (67)

is called the power spectral density, where is the autocorrelation of the process represented by . The inverse discrete-time Fourier transform allows us to recover from by

employing the relation

)(lrx

)(nx )(lrx )( ωjx eR

[∫−

−==π

π

ωωω ωπ

)()(21)( 1 j

xljj

xx eRFdeeRlr ] (68)

Note that is a deterministic function of ω . The area under the power-spectral density curve within a selected frequency band can be interpreted as the power of the random process within that band, considering the average outcome of all possible realizations of the process. Thus the power (mean-square value) of the signal is given by the total area under the power spectral density curve

)( ωjx eR

∫−=

π

π

ω ωπ

deRr jxx )(

21)0( (69)

When random signals that represent any single realization of a stationary process are applied as input to a linear and time-invariant filter with impulse response , the following equalities may be derived:

)(nh

∑∞

−∞=

∗=−=n

nhnxinhixny )()()()()( (70)

)()()( hxy lrlrlr ∗= (71)

2

[ ])()()()()( lylxElhlrlr =∗=

ωωω jjj =

)()()( ωωω jjx

jy eHeReR = (72)

xyx (73)

)()()( xyx eHeReR (74)

where , is the power spectral density of the output signal, r is the

cross-correlation of and , and is the cross-power spectral density.

)()( lhlhrh −∗=

(nx

)( ωjy eR

(ny

)(nyx

) ) )( ωjyx eR

45

If the transforms of the autocorrelation and cross-correlation functions exist, we can generalize the definition of power spectral density. In particular, the definition of equation (67) corresponds to the following relation

z

∑ −= nxx znrzR )()( (75)

If the random signal representing any single realization of a stationary process is applied as input to a linear and time-invariant filter with impulse response , the following equalities can be found: )(nh

)()()()( 1−= zHzHzRzR xy (76)

and

)()()( zHzRzR xyx = (77)

where is the transform of h . If we wish to calculate the cross-correlation of and , namely , we can use the inverse transform formula given by

)(zH z)

)(l )(ny)(nx 0(yxr z

[ ] ∫ ∫==z

dzzRzHz

dzzRnxnyE xyx )()(21)(

21)()(

πιπι (78)

where the integration path is a counterclockwise closed contour in the region of convergence of [6]. )(zRyx

EXAMPLE 13. Estimated power spectral density for an EEG signal

0 5 10 15 20 25

-60

-40

-20

0

20

40

60

time [s]

mic

roV

Figure 41. A 28-second segment of EEG signal, recorded in a child in the occipital

region of the head. This signal may be regarded as random.

46

0 1 2 3

1

2

3

4

5

6

7

time [s]

data

-seg

men

t

Figure 42. The same signal as above, split up into 7 consecutive segments of 4 second

duration. Each of these is then regarded as one realization of the random process.

0 5 10 15 20 25

0.51

1.52

x 104

a.u.

0 5 10 15 20 25

2000

4000

6000

frequency [Hz]

a.u.

Figure 43. The DFT (magnitude only) of the whole signal (top line), which shows many small peaks. These reflect the random nature of the signal. The DFTs of the 7 segments

are also shown (thin lines, bottom plot), together with their averages (bold line).

Each of the individual DFTs (both for the whole signal, and for the segments) shows large random variations, but the average expresses some statistical characteristics of the EEG signal. One of the most popular methods for estimating the power spectral density (the Welch method) follows the same rationale, thought the square of the magnitude of the DFT (rather than just the magnitude) is averaged. The result is also rescaled in amplitude, which allows the result to be expressed in suitable units (µV2/Hz), and more readily interpreted. The power spectral density estimate is far more amenable to further statistical analysis, than the average of the magnitude of the DFT would be.

47

0 5 10 15 20 25

10

20

30

40

50

60

70

80

90

frequency [Hz]

mic

roV2 /H

z

Figure 44. Power spectral density estimate for the EEG signal. This again shows a strong

peak in the alpha frequency band.

48

EXAMPLE 14. The PSD of a second EEG signal.

75 76 77 78 79 80

-60

-40

-20

0

20

40

time (s)

EEG

(mic

roV

)

0 20 40 600

20

40

60

80

100

120

frequency (Hz)

PSD

(V2 /H

z)

0 20 40 60

10-2

100

102

frequency (Hz)

PSD

(V2 /H

z)

Figure 45. A segment of an EEG signal (total length 93 seconds) taken from a 7-year old girl. The signal was obtained from the occipital region, with the reference electrode at the ear (derivation O2-A2). The sampling rate was 128 Hz. Alpha activity is evident in the time-domain, and appears as a sharp peak in the frequency domain, with a centre

frequency around 10 Hz. The closer the oscillation is to a pure sinusoid, the sharper this peak would become. There is a second peak at very low frequencies, which may be

largely due to artefacts and noise. There is very little power above 20 Hz. A small sharp peak also appears at 52 Hz (more clearly evident on a logarithmic scale). This may be due to aliasing of the third harmonic of mains frequency. In this recording the mains

frequency was 60 Hz. Its third harmonic is 180 Hz. By considering the diagram in Fig. 6, it can be shown that 180 Hz, sampled at 128 Hz, is aliased to 180-128 =52 Hz.

49

EXAMPLE 15. A blood pressure signal, contaminated by mains noise.

135 140 145

100

110

120

130

140

150

160

170

time (s)0 20 40 60 80 10010-8

10-6

10-4

10-2

100

frequency (Hz)

norm

aliz

ed u

nits

dB 0

-20

-40

-60

-80

2 4 6 8 10 1210-8

10-6

10-4

10-2

frequency (Hz)

norm

aliz

ed u

nits

dB 0

-20

-40

-60

-80

Figure 46. A short segment of an arterial blood pressure signal (sampled at 200 Hz), and its spectrum (given on a logarithmic scale). The spectrum has been normalized, such that the maximum value is 1. This shows that most of the signal power is concentrated in the low frequencies (very little power is contained in the band above about 10 Hz – note the logarithmic scale!), but there is also a sharp spike at 50 Hz. This is due to 50 Hz (mains) noise in the signal. The detail of the low frequency spectrum shows peaks at the pulse-

rate (about 1.4 Hz, corresponding to 84 beats per minute), and its harmonics. The higher harmonics (especially) give peaks that are not very sharp. This may be expected from the

natural variability between beats and in the heart-rate.

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8. Implementation of Digital Signal Processing Systems

DSPS can be implemented by software in general-purpose computers, in specialized digital signal processor (DSP) chips, or by hardware in special-purpose logic circuits. The software implementations are suitable for rapid prototyping and provides flexibility in testing and modifying the DSPS characteristics. On the other hand, special-purpose hardware implementations allow for improved performance, at higher-speed and with lower power consumption (the latter is particularly important in battery powered systems). The basic procedure of the software implementation entails the generation of the program code corresponding to the DSPS being implemented, in a high-level language or directly in the assembly language of the processor. A compiler then generates a set of instructions to the processor from the code. Since in general-purpose computers the instructions are executed sequentially, the speed of the DSP is constrained by the execution time of each instruction. As a result, this solution might not be acceptable in applications requiring very high-processing speed and/or fast data input/output interfaces, and for large-scale production. Efficient software implementations of DSPS are usually based on special-purpose microcomputers known as Digital Signal Processors (DSPs). These processors are able to implement sum-of-product operations in a very efficient manner. As has been seen, sum-of-product operations are the main computations required in the implementation of digital signal processing algorithms. The efficient implementation of the multiply-and-accumulate operation, as well as the high-degree of parallelism with which the instructions are executed in a DSP, result in a relatively high input-output throughput rate. Special purpose hardware is also possible for implementing a DSPS. Hardware implementations consist of designing and possibly integrating a digital circuit with logical gates to perform the basic operating blocks inherent to any DSPS such as multiplications, additions and storage elements. On the other hand, special purpose hardware entails high cost of development, which might be offset by a large production.

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9. Summary

This article presents a brief introduction to some of the main concepts of digital signal processing, which is a technology widely used to solve current real life problems. Digital signal processing is one of the main tools in modern technology since it relies on the fast developing digital integrated circuit technology which in turn allows for high performance computing machines at relatively low cost. Each concept discussed is followed by its application in the analysis, synthesis and enhancement of medical signals. The correct analysis of measured signals from a patient relies on the removal of contamination from clinically relevant signals and correct measures of parameters related to these signals. In other cases, diagnosed impairments of a patient can be compensated by properly shaped signals synthesized via external equipments. Currently many electronics equipments in a wide variety of application fields utilize digital signal processing technology.

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McGraw-Hill, 2nd Edition.

3. L. B. Jackson. (1996) Digital Filters and Signal Processing. Boston, MA: Kluwer, 3rd Edition.

4. A. V. Oppenheim, and R. W. Schafer (1989). Discrete-Time Signal Processing. Englewood

Cliffs, NJ: Prentice Hall.

5. S. K. Mitra. (2001) Digital Signal Processing: A Computer-Based Approach. New York, NY: McGraw-Hill, 2nd Edition.

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Analysis and Design. Cambridge, UK: Cambridge University Press.

7. H. Stark and J. W. Woods. (2002) Probability and Random Processes with Applications to Signal Processing. Upper Saddle River, NJ: Prentice Hall, 3rd Edition.

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MA: Kluwer, 2nd Edition.

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