HEART-RATE ADAPTIVE MATCH FILTER

BASED PROCEDURE FOR AUTOMATIC DETECTION OF

T-WAVE ALTERNANS FROM 24-HOUR ECG RECORDINGS

Issues Related to Filter Implementation

Laura Burattini, Silvia Bini and Roberto Burattini

Department of Biomedical, Electronics and Telecommunication Engineering

Polytechnic University of Marche, 60131 Ancona, Italy

Keywords: T-wave alternans, Heart-rate adaptive-match filter, Repolarization analysis.

Abstract: Twenty-four hour T-wave alternans (TWA) analysis is a promising approach for risk stratification, which

still remains unpractical because TWA identification algorithms are complex and require long computation

time (CT). The aim of the present study was to test the applicability to 24-hour ECG recordings of our

heart-rate adaptive match filter (AMF) which allows TWA detection by submitting ECG data to a band-pass

filter centered at the TWA fundamental frequency f

TWA

, equal to a half heart rate. Two implementations are

possible: 1) the passing-band is adapted to a varying f

TWA

value (FA_AMF), and 2) the filter band is fixed

while conditioning the ECG data (SA_AMF). Simulated ECG tracings, characterized by no TWA or by

different kinds of TWA, and 24-hour ECG recordings from healthy subjects and coronary artery disease

patients were used to identify the fastest of these two implementations. Our results yielded the conclusions

that the CT of our AMF-based procedure is independent of the amount of TWA present in the tracing, but

depends on ECG sample length and filter implementation. If filter-design tools are available while

performing ECG analysis, the FA_AMF implementation is to be preferred because its CT is about one third

of SA_AMF CT.

1 INTRODUCTION

T-wave alternans (TWA) is an ECG phenomenon

consisting of every-other-beat changes in the T-

wave morphology. After Adam et al. (1984) reported

the existence of microvolt TWA, too small in

amplitude to be visually detected at standard display

scales, increasing evidence has been found of a link

between TWA and vulnerability to life-threatening

ventricular arrhythmias (Adam et al., 1984;

Rosenbaum et al., 1994; Verrier et al., 1994;

Bloomfield et al., 2004; Narayan, 2007; Chow et al.,

2008; Gold et al., 2008). Original observations of

this link arose from applications of fast Fourier

transform spectral method (Smith et al., 1988) for

TWA detection from ECG tracings of populations at

high risk for ventricular arrhythmias. To reach a

target heart-rate and meet data stationarity

requirements, Fourier analysis was applied to short-

term ECG tracings (typically 128 beats) recorded

under strictly controlled conditions, such as pacing

(Smith et al., 1988; Rosenbaum, 1994) or exercise

(Estes et al., 1997; Hohnloser et al., 1998; Gold et

al., 2000; Klingenheben et al., 2000; Hennersdorf et

al., 2001). Later on, time-domain based techniques,

with less restrictive requirements than Fourier

analysis, were proposed for more flexible TWA

analysis extended to long-term (20-minute to 24-

hour) ambulatory recordings (Nearing et al., 1991;

Burattini et al., 1999; Nearing et al., 2002; Martínez

et al., 2005; Burattini et al., 2006). These techniques

highlighted a strongly non-stationary nature of TWA

(Nearing et al., 1991; Martínez et al., 2006; Burattini

et al., 2008a and b). Especially, 24-hour Holter ECG

analysis was suggested as a promising approach for

risk stratification relative to cardiac arrest and

arrhythmic death in relatively low-risk subjects, such

as postmyocardial infarction patients (Verrier et al.,

2003, Sakaki et al., 2009). Analysis of long-term

ECG recordings for TWA identification is costly in

computation time (CT). This may limit real-time

analysis of ambulatory ECG recordings, unless an

optimized fast-running algorithm is set up.

401

Burattini L., Bini S. and Burattini R. (2010).

HEART-RATE ADAPTIVE MATCH FILTER BASED PROCEDURE FOR AUTOMATIC DETECTION OF T-WAVE ALTERNANS FROM 24-HOUR ECG

RECORDINGS - Issues Related to Filter Implementation.

In Proceedings of the Third International Conference on Bio-inspired Systems and Signal Processing, pages 401-408

DOI: 10.5220/0002694604010408

 SciTePress

To automatically detect stationary, as well as

time-varying TWA, we have recently proposed a

heart-rate adaptive match filter (AMF) based

procedure, which was tested by applications to a)

simulated ECG tracings, characterized by no TWA

or by different kinds of TWA, and b) ECG

recordings (up to 20-min long) from healthy subjects

and cardiac disease patients (Burattini et al., 2006,

Burattini et al., 2008a and b, Burattini et al., 2009a

and b). This procedure allows TWA detection by

submitting ECG data to bandpass filtering with

narrow passing band centered around the TWA

fundamental frequency f

TWA

(which, by definition,

equals a half heart rate), so that the TWA signal is

provided as output. Adaptation of the AMF to heart-

rate, and thus to f

TWA

, can be accomplished by two

different implementations. The first adapts the AMF

passing-band to the current f

TWA

value (adaptation at

the filter level; FA_AMF); the second assumes a

fixed filter band for conditioning the ECG data

(adaptation at the signal level; SA_AMF).

The aim of the present study was to set-up a low-

CT procedure suitable for real-time TWA analysis

by testing the applicability of our AMF-based

procedure to 24-hour ECG recordings, and to

analyze pros and cons of its FA_AMF and SA_AMF

implementations.

2 METHODS

2.1 Heart-Rate Adaptive Match Filter

Implementations

Our AMF is conceptually a bandpass filter (Burattini

et al., 2006 and 2008a and b), having a passing band

of 0.12 Hz (2·df=0.12 Hz implies df=0.06 Hz) wide

centered at the TWA fundamental frequency (f

TWA

When applied to a 128-beat ECG tracing, f

TWA

defined by the following equation:

MRR2

TWA

⋅

(1)

where MRR denotes mean RR interval, over 128

beats.

2.1.1 FA_AMF Implementation

After defining ω

=2π(f

TWA

-df) and ω

=2π(f

TWA

+df),

the AMF is implemented as a 6

order Butterworth

bandpass filter, whose transfer function H

(ω) is

expressed as follows (Burattini et al., 2008):

)(H

⎟

⎠

⎞

⎜

⎝

⎛

⎟

⎠

⎞

⎜

⎝

⎛

⋅

⎟

⎠

⎞

⎜

⎝

⎛

=ω

(2)

The input of the AMF is a 128-beat ECG signal

(ecg(t), t denoting time) potentially affected by

TWA. Its output is a constant phase and, possibly,

amplitude-modulated sinusoid, which represents

TWA and is, then, denominated ‘TWA signal’

(twa(t)). Consequently, if h

(t) is the impulse

response function associated to H

(ω), twa(t) is

given by the following equation:

(

)

(

)()

thtecgttwa

∗

(3)

This implementation procedure is graphically

displayed in Fig. 1. When TWA is analyzed from

long-term ECG recordings by recursively (for

example every 10 s) extracting a 128-beat ECG

string from the entire recording (Burattini et al.,

2008), the passing band has to be relocated at f

TWA

which is a function of heart rate (eq. 1).

Consistently, this AMF implementation was called

filter-adapting AMF (FA_AMF). Specific steps for

TWA analysis are as follows:

1. Extraction of a 128-ECG string, ecg(t), from the

long-term recording.

2. f

TWA

computation from mean heart rate (eq. 1).

3. Setting of the 6

order Butterworth passing-band

at f

TWA

(eq. 2).

4. Filtering of ecg(t) with FA_AMF, according to

eq. 3, to obtain twa(t).

5. Exit, if the end of long term tracing is reached;

otherwise restart from step 1 after 10 seconds

time-increase.

Figure 1: Filter-adapting AMF implementation

(FA_AMF).

2.1.2 SA_AMF Implementation

The impulse response function h

(t), associated to

the transfer function of eq. 2, can be synthesized by

using an appropriate designed low-pass filter.

Specifically, it is possible to show (see Appendix)

that h

(t) can be expressed as:

BIOSIGNALS 2010 - International Conference on Bio-inspired Systems and Signal Processing

402

()

tcos)t(h2)t(h

TWALPBP

ω=

(4)

where ω

TWA

=2πf

TWA

, and h

(t) is the impulse

response of a lowpass filter having ω

=2π·df cut-off

frequency. Indeed, the basic idea is to translate the

passing band around 0 Hz, so that it becomes

independent of f

TWA

, and move the heart-rate

dependency into the cosine (translation in the

frequency domain corresponds to modulation in the

time domain).

By substituting eq. 4 into eq. 3 (and recalling that

cos(x-y)=cos x cos y + sin x sin y ):

()

() () ( )()

() ( ) ( )()

() ( ) ( ) ( )

()()()()

dτthsinecgtsin2

dτthcosecgtcos2

dsintsinτthecg2

dcostcosτthecg2

dττ-tωcosτthτecg2

tωcost2htecg

twat

LPTWA

TWA

LPTWA

TWA

TWATWALP

TWALP

TWA

τ−τωτω+

+τ−τωτω=

=ττωω−τ+

+ττωω−τ=

=−=

=∗=

∫

∞+

∞

∞+

∞

∞+

∞

∞+

∞

∞+

∞−

Consequently,

()

()()()()()

[]

()()()()()

[]

thtsintecgtsin2

thtcostecgtcos2

twat

LPTWATWA

∗ωω+

+∗ωω=

(5)

Based on this implementation, graphically shown in

Fig. 2, when TWA is recursively analyzed from

long-term ECG recordings, the passing band of the

two lowpass filters (which are identical) keeps

constant, while their input and output signals are

modulated to adapt to heart rate variations. This

AMF implementation is referred to as signal-

adapting AMF (SA_AMF), and each one of the

lowpass filters incorporated is set-up as a 3

order

Butterworth filter. Specific steps for TWA analysis

are as follows:

1. ‘A priory’ design of a 3

order Butterworth

lowpass filter with a cutoff frequency set at a

fixed df value.

2. Extraction of a 128-ECG string, ecg(t), from the

long-term recording.

3. Computation of ω

TWA

=2πf

TWA

, being f

TWA

defined by eq. 1.

4. Filtering of ecg(t) with SA_AMF, according to

eq. 5, to obtain twa(t).

5. Exit, if the end of long term tracing is reached;

otherwise restart from step 2 after 10 seconds

time-increase.

Figure 2: Signal-adapting AMF implementation

(SA_AMF).

2.2 FA_AMF vs. SA_AMF

Simulated and clinical ECG tracings (described in

2.3) were analysed by extracting 128 beats every 10

seconds (Burattini et al., 2008). Comparison

between FA_AMF and SA_AMF based TWA

detection procedures was performed in terms of

computation time (CT).

2.3 ECG Data

2.3.1 Simulated 128-Beat ECG Tracings

Basic simulated ECG tracing consisted of an 128-

fold repeated real and clean ECG complex sampled

at 200 samples/s. The RR interval was 0.750 s. Thus,

TWA fundamental frequency (f

TWA

) was 0.67 Hz ,

that is 1/(0.750×2 s) or 0.5 cycles per beat. T wave

was identified in a 160 ms window centred around

the T-wave apex. The endpoints samples of the T-

wave window (T

onset

and T

offset

) were localized 120

ms and 280 ms after the R peak.

All simulated 128-ECG tracings are displayed in

Fig. 3. The first simulated ECG (NO_TWA)

represents the ideal case of a tracing not affected by

any kind of variability (basic cardiac complex

repeated with no changes). The second simulated

ECG (STATIONARY_TWA) was affected by a

stationary 10 μV TWA. Eventually, the third and the

forth simulated ECG tracings were affected by time-

varying (non-stationary) TWA, respectively

characterized by a smoothed step profile

(STEP_TWA; with 24 beats or 18 s transition time-

duration), and by a sinusoidal profile

HEART-RATE ADAPTIVE MATCH FILTER BASED PROCEDURE FOR AUTOMATIC DETECTION OF T-WAVE

ALTERNANS FROM 24-HOUR ECG RECORDINGS - Issues Related to Filter Implementation

403

(SINUSOID_TWA; with the sinusoid period of 60

beats or 45 s, and amplitude of 5 μV, yielding to a

maximum TWA amplitude of 10 μV).

Simulated long-term (20-minute and 24-hour)

tracings were obtained by repetition of the basic

128-beat ECG simulated strings.

2.3.2 Clinical ECG Tracings

Clinical tests were performed on 24-hour Holter

ECG recordings from 3 healthy (H) subjects and 3

coronary artery disease (CAD) patients. All subjects

pertain to the Intercity Digital Electrocardiology

Alliance (IDEAL) Study conducted following the

required rules for human subjects’ research

principles, according to the Declaration of Helsinki,

as well as to Title 45, U.S. Code of Federal

Regulations, Part 46, Protection of Human Subjects,

Revised November 13, 2001, effective December

13, 2001. The IDEAL protocol was approved by

Research Subject Review Board of the University of

Rochester. ECG tracings were acquired using the

SpaceLab-Burdick digital Holter recorder

(SpaceLab-Burdick, Inc., Deerfield, WI; sampling

frequency 200 samples/s).

2.4 Statistics

Each ECG tracing (either 128-beat, 20-minute or 24-

hour long) was analyzed five times, and mean and

standard deviation values of the CT were reported

(0.1 s resolution) and compared using Student’s t-

test. Associations between quantities was evaluated

using the correlation coefficient (r). Statistical

significance was assumed at P<0.05. Analysis were

performed on PC (Intel® Core™ Quad CPU Q9300

@ 2.50GHz, 3GB of RAM) using the MATLAB 7.0

development environment.

3 RESULTS

Fig. 4 shows the TWA signals obtained from both

FA_AMF-based (dotted line) and SA_AMF-based

(solid line) procedures when the input is represented

by the simulated data. For all simulated conditions

(namely NO_TWA, STATIONARY_TWA,

STEP_TWA, SINUSOID_TWA), the TWA-signal

outputs of FA_AMF implementation are not

distinguishable from the corresponding SA_AMF

ones. Thus, solid lines superimpose to dotted line in

the four panels of Fig.4.

Table 1 shows CT values of the two competing

filter implementations, in relation to different

simulated tracings lengths and to the four simulated

conditions specified above. FA_AMF column of this

table clearly shows that, for any given ECG tracing-

length (either 128 beats, or 20 minutes or 24 hours),

the CT is irrespective of TWA presence and kind.

This is true also for the SA_AMF implementation,

as judged from CT values in the related column of

Table 1. Comparison between corresponding CT

values of columns FA_AMF and SA_AMF, at 0.1 s

time resolution, shows no significant differences for

128-beat ECG analysis, while the FA_AMF

procedure is significantly faster than the SA_AMF

when applied to long-term (20-minute and 24-hour)

tracings. In absolute terms, for 20-minute ECGs, the

mean CT of FA_AMF differs by less than 2 seconds

compared to the FA_AMF implementation (1.2±0.0

s vs. 3.0±0.0; P<0.001). For 24-hour ECGs, the

SA_AMF mean CT (203.3±2.9 s) and the FA_AMF

mean CT (73.5±1.5 s) differ in such a way that the

ratio between the two is 2.77±0.03.

Mean (±SD) values for the FA_AMF and

SA_AMF implementations in individual H and CAD

cases are compared in Table 2. In the three H-

subjects, the MRR (0.74±0.08 s) is similar to that

(0.75 s) used in simulated cases. Also the mean CT

values of 24-hour simulations for FA_AMF

(73.4±0.8) and SA_AMF (203±2) are comparable

with the corresponding mean CTs from the three H

subjects (75.5±6.1 and 201±15, respectively).

On average, the ratio between SA_AMF CT and

FA_AMF CT is 2.67±0.13, a value comparable

(P<0.05) with that found in 24-hour simulated cases.

MRR variation between 0.660 and 1.130 s, over all

H and CAD cases, compared with corresponding CT

variations of SA_AMF and FA_AMF, suggests a

MRR-CT correlation, which is characterized by

r=0.99 and P<0.05 for both implementations.

4 DISCUSSION

Our AMF–based TWA identification algorithm is

substantially a bandpass filter which identifies TWA

by filtering out every ECG frequency component but

the TWA typical one (f

TWA

). This method has been

applied to 128-beat ECG recordings (Burattini et al.,

2006, 2008a and 2008b, 2009a) and to 20-minute

recordings (Burattini et al., 2008a and 2008b,

2009b). The latter study involved a recursive

application of the AMF procedure to 128-beat ECG

strings extracted every 10 seconds. On this basis,

from a theoretical point of view, our method appears

suitable for application to longer term tracings such

as 24-hour ECGs. Nevertheless, practical, routine

BIOSIGNALS 2010 - International Conference on Bio-inspired Systems and Signal Processing

404

applications of this technique to long recordings is

potentially limited by the CT required to identify

TWA. This possibility was tested in the present

study by analyzing the CT of two different

implementations of the AMF.

The first AMF implementation, referred to as

FA_AMF (Fig. 1), consists of a single 6

order

Butterworth bandpass filter. Since heart-rate is time

variant, the f

TWA,

defined as a half of mean heart rate,

is also time variant. Consequently, the FA_AMF

passing-band has to be recursively adapted to

properly detect TWA. This requires a real-time (i.e.

while analyzing TWA from the ECG) filter setting.

Thus, in the environment where TWA is analyzed

(for example in an ECG analysis machine),

availability of filter-design tools is required, such as

those provided by MATLAB.

The second AMF implementation, referred to as

SA_AMF (Fig. 2), is more complex for it involves

two identical lowpass filters and repeated signal

modulation. The 3

order lowpass Butterworth

implementation of each one of these filters is

independent of heart rate and, thus, of f

TWA

Consequently, it can be designed ‘a priory’ and

imported in the TWA analysis environment, thus

releasing the need of recursive filter setting.

In this study we performed a quantitative

comparison of the two implementations, in terms of

CT, when analyzing 128-beat, 20-minute and 24-

hour simulated ECG recordings, and 24-hour clinical

ECG recordings. Both simulated and clinical

tracings were characterized by the same sampling

Figure 3: Central strings (about 45 s) of simulated 128-

beat ECG tracings. NO_TWA represents the ideal case of

a tracing not affected by any kind of modulation.

STATIONARY_TWA represents an ECG affected by a

stationary TWA of 10 μV. STEP_TWA represents an

ECG affected by TWA that varies from 10 μV to 0 μV

following a smoothed step profile. SINUSOIDAL_TWA

represents an ECG affected by TWA that varies from 10

μV to 0 μV following a sinusoidal profile.

Figure 4: Output TWA signals provided by the FA_AMF

(dotted line) and SA_AMF (solid line) based procedures,

when the inputs are the 128-beat simulated ECG tracings

with: no TWA (NO_TWA), stationary TWA

(STATIONARY_TWA), smoothed step TWA

(STEP_TWA) and sinusoidal TWA (SINUSOID_TWA).

For all the simulated conditions, the two estimated TWA

signals superimpose, so that dotted lines are not visible.

frequency (200 samples/sec), so that the number of

samples in a 24-hour recording is the same.

Results of the simulation study show that, for

any given ECG tracing length, the CT of related

filter implementation is irrespective of TWA

presence and kind. Comparison between

corresponding CT values of FA_AMF and SA_AMF

implementations indicates that, for short-time ECG

recordings, the two are equivalent, while the

FA_AMF procedure is significantly faster when

applied to long-term tracings (Table 1). In the

specific, when analyzing 24-hour recordings, the

ratio between CT of SA_AMF and CT of FA_AMF

was, on average, almost three times larger.

These observations are confirmed by the results

on the 24-hour clinical data. Indeed, the ratio of 2.67

between the CT of SA_AMF and FA_AMF is not

significantly different from that of 2.77 found for the

simulated 24-hour tracings.

In our clinical data, a strong correlation (r=0.99;

P<0.05) was observed between MRR over 24 hours

and corresponding CT of either one of the two

implementations. This result finds an explanation in

that 128-beat ECG are recursively submitted to the

AMF. These ECG segments, although matching in

terms of number of beats, are characterized by a

different length in terms of number of samples. The

longer the RR interval, the higher the number of

samples, and thus, the longer the required CT. This

observation is supported by the facts that simulated

and clinical data are characterized by similar MRR

and similar mean 24-hour CTs for both FA_AMF

and SA_AMF implementations (see Results). The

HEART-RATE ADAPTIVE MATCH FILTER BASED PROCEDURE FOR AUTOMATIC DETECTION OF T-WAVE

ALTERNANS FROM 24-HOUR ECG RECORDINGS - Issues Related to Filter Implementation

405

Table 1: Computation time (CT, seconds) of FA_AMF- and SA_AMF-based TWA detection procedures applied to 128-

beat, 20-minute and 24-hour simulated ECG recordings, in the absence of TWA, or in the presence of different kinds of

TWA. Mean CT±SD values are computed over five algorithm runs.

CT of FA_AMF (s) CT of SA_AMF (s) P (Student’s t test)

128-beat ECGs

NO_TWA 0.1±0.0 0.1±0.0 NS

STATIONARY_TWA 0.1±0.0 0.1±0.0 NS

STEP_TWA 0.1±0.0 0.1±0.0 NS

SINUSOID_TWA 0.1±0.0 0.1±0.0 NS

20-minute ECGs

NO_TWA 1.2±0.0 3.0±0.0 <0.001

STATIONARY_TWA 1.2±0.0 3.0±0.0 <0.001

STEP_TWA 1.2±0.0 3.0±0.0 <0.001

SINUSOID_TWA 1.2±0.0 3.0±0.0 <0.001

24-hour ECGs

NO_TWA 73.1±1.9 201.1±3.4 <0.001

STATIONARY_TWA 74.5±0.6 204.8±2.5 <0.001

STEP_TWA 72.7±1.6 204.6±1.0 <0.001

SINUSOID_TWA 73.5±1.2 202.5±3.2 <0.001

Table 2: Computation time (CT, seconds) of FA_AMF- and SA_AMF-based TWA detection procedures applied to 24-hour

ECG recordings of three H-subjects (H1, H2, and H3) and three CAD patients (CAD1, CAD2, and CAD3). Mean CT±SD

values are computed over five algorithm runs.

MRR (s) CT of FA_AMF (s) CT of SA_AMF (s) P (Student’s t test)

24-hour ECGs

H1 0.660 68.9±1.0 186.1±3.2 <0.001

H2 0.757 76.8±0.8 202.0±3.3 <0.001

H3 0.814 80.9±1.4 215.7±1.1 <0.001

CAD1 0.960 98.5±0.1 241.0±1.4 <0.001

CAD2 1.091 103.9±2.0 281.7±9.7 <0.001

CAD3 1.130 105.9±0.6 303.0±2.6 <0.001

above mentioned high correlation between mean RR

and CT was obtained considering together H-

subjects, who are not supposed to show significant

TWA levels, and CAD-patients, who instead are

known to show increased levels of TWA (Burattini

et al., 2008a and 2009a). This result agrees with the

result, obtained with simulated data, that the CT is

irrespective of TWA presence and kind. Altogether,

our results suggest that the FA_AMF

implementation is preferable as it is faster. This

advantage, however, conflicts with the limitation

relying on the need of filter-design tools that are not

commonly available in clinical environments. On the

other hand, when using the SA_AMF, the CT

assumes, over 24-hour ECGs, values of few minutes

(up to five in our results; Table 1 and Table 2).

These CT values can be considered acceptable for

real time computation in light of the fact that the

operator is made free from the need of filtering

design tools.



5 CONCLUSIONS

In conclusion, our AMF-based procedure for TWA

identification proved to be appropriate for

applications to 24-hour recordings. If filter design

tools are available while performing the analysis, the

FA_AMF implementation is to be preferred because

it allows faster analysis of 24-hour recordings, with

CT ratio between SA_AMF and FA_AMF being

about 2.7. Otherwise, SA_AMF implementation can

be performed, which provides TWA identification

with a CT of a few minutes.



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ALTERNANS FROM 24-HOUR ECG RECORDINGS - Issues Related to Filter Implementation

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APPENDIX

Consider a filter having the following unit impulse

response:

(

)

tcos)t(h2)t(h

TWALP

ω=

(A1)

where h

(t) is the unit impulse response of a

lowpass filter with cutoff frequency df. By taking

the Fourier transform of h(t) we get:

()

2()cos

TWA TWA

LP TWA

jt jt

LP LP

ht te dt

hte hte dt

ωω ωω

+∞

−

−∞

+∞

−− −+

−∞

∫

(A2)

Upon examination of eq. A2, it is seen that the first

summation term is equal to a right-shifted by ω

TWA

version of the LP frequency response, while the

second term corresponds to a left-shifted by ω

TWA

version of the LP frequency response. Thus, the

frequency response associate to h(n) is:

() ( ) ( )

TWALPTWALP

HHH ω+ω

ω−ω=ω

(A3)

The magnitude of the first component in eq. A3 is

seen to equal the magnitude of LP shifted by ω

TWA

radians. That is, it corresponds to the rightmost

passband centered at ω

TWA

. Similarly, the magnitude

of the second term corresponds to the leftmost

passband centered at ω

TWA

. Thus, we can conclude

that the filter, whose frequency response is

represented in eq. A3, can be used to implement our

AMF:

() ()

ω=ω HH

(A4)

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