Applications of Hadamard Transform to Gas
Chromatography/Mass Spectrometry and Liquid
Cheng-Huang Lin,*,†Takashi Kaneta,‡Hung-Ming Chen,†Wen-Xiong Chen,†Hung-Wei Chang,†
and Ju-Tsung Liu§
Department of Chemistry, National Taiwan Normal University, 88 Section 4, Tingchow Road, Taipei, Taiwan,
Department of Applied Chemistry, Graduate School of Engineering, and Division of Translational Research, Center for
Future Chemistry, Kyushu University, Motooka, Fukuoka 819-0395, Japan, and Forensic Science Center, Military
Police Command, Department of Defense, Taipei, Taiwan
Successful application of the Hadamard transform (HT)
technique to gas chromatography/mass spectrometry
(GC/MS) and liquid chromatography/mass spectrometry
(LC/MS) is described. Novel sample injection devices
were developed to achieve multiple sample injections in
both GC and LC instruments. Air pressure was controlled
by an electromagnetic valve in GC, while a syringe pump
and Tee connector were employed for the injection device
in LC. Two well-known, abused drugs, 3,4-methylene-
dioxy-N-methylamphetamine (MDMA) and N,N-dimeth-
yltryptamine (DMT), were employed as model samples.
Both of the injection devices permitted precise successive
injections, resulting in clearly modulated chromatograms
encoded by Hadamard matrices. After inverse Hadamard
transformation of the encoded chromatogram, the signal-
to-noise (S/N) ratios of the signals were substantially
improved compared with those expected from theoretical
values. The S/N ratios were enhanced ∼10-fold in HT-
GC/MS and 6.8 in HT-LC/MS, using the matrices of 1023
and 511, respectively. The HT-GC/MS was successfully
applied to the determination of MDMA in the urine sample
of a suspect.
In separation science, the limit of detection (LOD) for analytes
is an important issue. Thus, much research has focused on the
development of sensitive detectors and improvements in the
sensitivity of detectors for separation techniques such as gas
chromatography (GC), liquid chromatography (LC), and capillary
electrophoresis (CE). These studies have resulted in the develop-
ment of sophisticated detectors. Therefore, in some cases, further
improvement in LOD is physically limited by the detection
principle. The use of mathematical methods is one alternative for
the improvement of the LOD and resolution of analytical instru-
mentation. In chromatographic separations, correlation methods
have been employed for improving the LOD. In 1970, Smit
demonstrated correlation gas chromatography in which a sample
was injected into a column according to a pseudorandom binary
sequence (PRBS).1Correlation chromatography is a powerful
method for reduction of the LOD. Therefore, correlation chro-
matography has been applied to not only GC2–5but also to LC6–11
and to CE.12,13The Hadamard transform (HT) technique is a
technique that is analogous to the correlation method. The HT
technique has been applied in many fields, including time-of-flight
mass spectrometry,14–16Raman,17,18fluorescence imaging,19–21ion
mobility spectrometry,22,23and NMR.24,25In our previous
studies,26–29Hadamard transformation has been successfully
applied to capillary electrophoretic separations where the signal-
* Corresponding author. E-mail: email@example.com.
†National Taiwan Normal University.
§Forensic Science Center.
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10.1021/ac800201r CCC: $40.75 2008 American Chemical Society
Published on Web 06/21/2008
Analytical Chemistry, Vol. 80, No. 15, August 1, 2008
to-noise (S/N) ratios were substantially improved by sample
injections according to the PRBS. Nonconventional sampling
techniques including correlation and HT methods have been
reviewed by Kaljurand and Smit.30
With respect to chromatographic separations, some studies
have reported on the use of HT for GC31and LC.32The advantages
of the HT technique using pseudorandom injections in GC have
also been suggested but only by means of computer simulation.31
Recently, Trapp reported high-throughput multiplexing GC using
the HT method.33
Conversely, in LC, the only report of HT relates to the
application of HT to UV absorption detection, in which the
Hadamard mask was placed in front of a photomultiplier tube to
simplify the measurement of the spectra for the analytes.32To
date, there has been no report on the use of HT based on multiple
injections according to PRBS in LC, although substantial improve-
ment would be expected.
A Hadamard matrix on the order of n, Hn, is an n × n of +1’s
and -1’s with the property of the scalar product of any two distinct
rows being 0. Thus, Hnmust satisfy the following equation,
order of n. A fundamental equation of the Hadamard transforma-
tion is given by
Tis the transpose of Hnand Inis the unit matrix on the
where η is a series of data, i.e., the observed chromatogram,
encoded by a cyclic S-matrix, S, which is the (n - 1) × (n - 1)
matrix consisting of “zero” and “one” elements, and C is a series
of data representing a chromatogram. A cyclic S-matrix on the
order of (n - 1) is obtained by omitting the first row and column
of Hnand then changing +1’s to 0’s and -1’s to +1’s. To encode
the chromatogram, C, a sample and eluent are introduced into a
column according to the PRBS derived from the cyclic S-matrix.
When the elements of the PRBS are “one” and “zero”, sample
and eluent plugs are introduced into the column, respectively. As
a result, the encoded chromatogram, η, is obtained. The encoded
chromatogram is decoded to the chromatogram, C, by multiplying
an inverse matrix of S, S-1, as follows.
Consequently, the decoded chromatogram shows improvement
in the S/N ratio (Fellgett advantage).
In both correlation and HT methods, the key technology, based
on multiple input techniques according to PRBS, is the injection
device, which permits the continuous introduction of a sample.
In CE, two kinds of injection methods, i.e., electrokinetic injec-
tion29and optically gated injection,26,27have been employed to
achieve successive injections. Multiple injection devices for GC
have also been developed for correlation GC, in which the solenoid
valve,1,2cylindrical slide valve,3and fluidic logic gate4were used.
Conversely, in correlation LC, the input signals modulated by
PRBS were generated by valve systems6–9,11and by an electro-
chemical concentration modulator.10In this study, we developed
novel injection devices for HT-GC/MS and HT-LC/MS, respec-
tively. The injection devices, which are quite simple, permit precise
sample introduction resulting in clearly modulated chromatograms
at the command of PRBS. Two well-known, abused drugs, 3,4-
methylenedioxy-N-methylamphetamine (MDMA) and N,N-dim-
ethyltryptamine (DMT), were selected as model samples. The
design of injection devices, details of experimental conditions for
HT-GC/MS and HT-LC/MS, and the determination of a drug in
an actual sample are reported herein.
Reagents. 3,4-Methylenedioxy-N-methylamphetamine (MDMA),
N,N-dimethyltryptamine (DMT), and suspects’ urine samples were
generously donated by the Military Police Command, Forensic
Science Center, Taiwan. All the other chemicals were of analytical
grade and were obtained from commercial sources.
Extraction Procedure and Safety. A volume of 1 mL of the
urine sample was made alkaline by the addition of excess K2CO3.
The free bases were then extracted into 4 mL of a hexane/CH2Cl2
(3:1, v/v) solution by mixing for 1 min. After shaking for 5 min,
0.1 mL of acetic anhydride was added to derivatize the MDMA.
After centrifugation, the upper layer (3 mL) was collected, and
this organic phase was then evaporated to dryness. The residue
was dissolved in 0.5 mL of methanol for subsequent GC/MS
experiments. The proper and safe handling of urine samples
followed the regulations of the Department of Health, Executive
Apparatus. A gas chromatograph (GC 5890 Hewlett-Packard,
Avondale, PA) equipped with a mass spectrometer (Hewlett-
Packard 5972 mass selective detector) was used to detect the
analytes. A capillary column (30 m × 0.25 µm i.d.) with an HP-
5MS (cross-linked 5% PH ME siloxane) bonded stationary phase
film, 0.25 µm in thickness (Agilent Technologies), was used. The
inlet temperature was maintained at 250 °C and the column oven
was also held at 250 °C (carrier gas: helium, flow rate 1.2 mL/
min operating in either the splitless or split mode). The mass
spectrometry conditions were as follows: ionization energy, 70 eV;
and ion source temperature, 110 °C. The selected ion monitor
(SIM) mode was used for MDMA and DMT by selecting ion peaks
at m/z ) 162 and 58, respectively. The dwell value was set at 80;
10 dots/s could be recorded. Data were collected using Hewlett-
Packard Chem-Station software with transfer to an ASCII text file.
The average was calculated for each 30 dots that were treated as
one bin to fit the HT calculation. The LC/MS system (Finnigan
LCQ Classic LC/MS/MS) consisted of a Constametric 4100
solvent delivery system (LDC Analytical, Gelnhausen,Germany),
a manual injection valve from Shimadzu, a reversed phase column
(Jasco C18 T-5, 5 µm, 15 cm × 4.6 mm i.d.; Nacalai Tesque, Kyoto,
Japan), and an electrospray ionization (ESI) probe operated in
the positive ion mode. The mass signal was recorded under the
SIM mode, where the data recording speed was ∼0.63 dot/s. The
Xcalibur data system was used for collecting data, which were
transferred to an ASCII text file. The scan mode used was SIM;
(28) Hata, K.; Kichise, Y.; Kaneta, T.; Imasaka, T. Anal. Chem. 2003, 75, 1765–
(29) Hata, K.; Kaneta, T.; Imasaka, T. Anal. Chem. 2004, 76, 4421–4425.
(30) Kaljurand, M.; Smit, H. C. Chemom. Intell. Lab. Syst. 2005, 79, 65–72.
(31) Kaljurand, T.; Ku ¨llik, E. Chromatographia 1978, 11, 328–330.
(32) Brayan, J. G.; Malcolme-Lawes, D. J.; Mew, C. D.; Xie, S. J. Autom. Chem.
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(33) Trapp, O. Angew. Chem., Int. Ed. 2007, 46, 5609–5613.
Analytical Chemistry, Vol. 80, No. 15, August 1, 2008
the capillary temperature and spray voltage were set to 300 °C
and 4.5 kV, respectively. The tube lens offset and capillary voltage
were set at -10 and 6 V, respectively; sheath gas and auxiliary
gas flow rates were 70 and 20 (arb), respectively. The HT-GC-
and HT-LC-chromatograms were calculated using the LabView
program, as described previously.34
RESULTS AND DISCUSSION
Injection Device. Figure 1 shows a schematic diagram of the
injection device for HT-GC/MS. A sample solution was replaced
in the reservoir of a stainless tube (o.d.,1/4in.; length, 5 in.).
The sample solution was introduced into a glass liner through a
capillary (i.d., 50 µm) by air injection pressure from a compressed
air cylinder. A personal computer turned the electromagnetic valve
on and off, according to a series of Hadamard codes. When the
electromagnetic valve was opened, high-pressure air was im-
mediately squeezed into the reservoir, leading to the introduction
of the sample solution through the capillary into the glass liner
where the capillary was immersed in the sample solution. The
injection volume of the sample solution could be adjusted by
changes in the air pressure, capillary i.d., capillary length, and
injection time. The air pressure (ranged from 1.6 to 1.9 atm) was
inversely proportional to the injection time (ranged from 6 to 3 s).
The optimized condition was obtained when the air pressure was
1.67 atm and the injection time was set at 3 s, using a short
capillary (i.d., 50 µm; 5 cm in length). The injection volume was
estimated at ∼100 nL for a single injection (stability: intraday, 92.7
± 7.4 nL; interday, 95.7 ± 7.4 nL). The injected volume was
recognized by comparing the relationship between a single peak
obtained from a regular GC injector (0.1 µL of sample solution)
and average of the peaks obtained from the Hadamard injection.
The time gap between successive injections was less than 1 s,
which depended on air pressure, inner pressure of the glass liner,
and the size of leaking tunnel, respectively. The glass liner usually
remained at high temperature (∼250 °C), so that the ejective
solution, containing the analytes and solvent, were evaporated
immediately. When the electromagnetic valve was closed, the
inner pressure of the glass liner (i.e., carrier gas, helium) pushed
back the sample, resulting in interruption of the sample introduc-
tion by the carrier gas. The flow rate of helium could be set at
0.6-1.2 mL/min, leading to a 0-8 psi head pressure. Once the
carrier gas pushed the sample solution back and entered into the
reservoir, excess air escaped from a small leaking tunnel (i.d.,
0.4 mm), and was then drained by a fan to keep the reservoir
pressure equal to the atmospheric pressure (waiting for the next
injection). Thus, the injection device permitted sample injections
according to PRBS in GC.
Figure 2 shows a schematic diagram for the HT-LC/MS
injection device. In this case, the sample solution was placed in a
homemade reservoir (a brass injector; i.d., 6.0 mm), and1/16in.
(i.d., 0.13 mm) pipes were used in the system. A sample solution
was pushed out by a commercial syringe pump (model 22 syringe
pump; Harvard apparatus), which was controlled by a personal
computer through an RS232-port. When the flow rate was set at
5.4 µL/min, the volume of sample solution was estimated to be
∼0.3 µL/injection (stability: intraday, 0.34 ± 0.03 µL; interday, 0.30
± 0.04 µL). Meanwhile, a commercial HPLC pump was used to
provide eluent (mobile phase: acetonitrile/water) for separation.
The sample solution and eluent interflowed in a Tee connector
and alternately entered the separation column. Once the separa-
tion was finished, the effluent entered the mass spectrometer by
means of the ESI process. The pressure of the syringe pump was
adjusted to be greater than that of the HPLC pump. Thus, when
the syringe pump was operated, only the sample solution could
pass through the Tee connector. Conversely, when the syringe
pump was stopped (the sample injection was interrupted), only
the eluent could pass through the Tee connector. Thus, multiple
sample injections were accomplished by controlling the operation
of the syringe pump. The initial pressure of the HPLC pump was
adjusted to less than 200 psi. When an ESI mode was used, the
pressure increased to 500 psi. Hence, the optimal pressure of the
syringe pump was set at 500-550 psi. The eluent used in this
experiment was water and acetonitrile (50:50, v/v), pumped at a
rate of 1.0 mL/min. The time gap between the sample solution
and the eluent depended on the viscosity of solvent used and the
pump pressure, respectively. At this point, the optimized time gap
ranged from 6 to 11 s. The separation efficiency of DMT was
influenced by the acidity of the mobile phase. When a basic mobile
phase was used, the peak of DMT was broadened, whereas when
0.1% formic acid was added to the mobile phase, good separation
efficiency was obtained. On the other hand, if the ratio of water
(34) Hata, K.; Kaneta, T.; Imasaka, T. Electrophoresis 2007, 28, 328–334.
Figure 1. Schematic diagram of HT-GC/MS injection device.
Figure 2. Schematic diagram of HT-LC/MS injection device.
Analytical Chemistry, Vol. 80, No. 15, August 1, 2008
was higher than 50% in volume, or methanol was used instead of
acetonitrile, the inner pressure was increased (>1000 psi), which
made the sample injection and the ESI process difficult.
Hadamard Transformation. MDMA is a semisynthetic en-
tactogen of the phenethylamine family and is most commonly
known today by the street name “ecstasy.” DMT is a naturally
occurring tryptamine and human neurotransmitter. It is well-
known that phenethylamines (such as MDMA) and tryptamines
(such as DMT) provide a strong imine fragment (m/z ) 58) under
electron impact (EI; 70 eV) mode. Thus, optimum derivatization
for MDMA is needed. The derivatization was performed via a
reaction with acetic anhydride, leading to characteristic mass
fragments of MDMA (m/z ) 58, 100, and 162). Figure 3A shows
a typical GC/MS chromatogram of the MDMA derivative obtained
by single injection, based on the SIM mode (ion peak at m/z )
162 was selected for monitoring). The concentration of MDMA
was 10 µg/mL before derivatization; yield was 60-70%. The period
of sample injection was 3.010 s; sample injection volume was ∼0.1
µL in a 10:1 split mode. The characteristic MDMA peak in
methanol, as well as a tiny impurity (marked as “/”), were found.
The S/N ratio appeared to be poor in the chromatogram obtained
by single injection. Under the same experimental conditions, when
an HT injection was performed (n ) 1023), the S/N ratio was
dramatically improved to 9.13-fold, as shown in Figure 3B. The
inset shows the raw data of the total ion current (TIC) (selecting
the ion at m/z ) 162) chromatogram before inverse HT was
applied. The total operation time was more than 120 min,
suggesting that the HT-GC/MS injection device works very well.
If a faster electromagnetic valve and higher air pressure is used,
the operation time can be shortened and the use of a splitless
mode is also possible. Another test sample, DMT, was also
examined using this approach. Since DMT is a tertiary amine, its
derivatization is difficult, so an imine fragment (m/z ) 58) was
selected for monitoring. Table 1 shows the relationship between
the order of matrix, the enhancement of the S/N ratio, and the
mass conditions for analysis of MDMA and DMA, respectively.
As seen in Table 1, the enhancement of the S/N ratio increases
with increasing order of the Hadamard matrix, as expected from
theoretical observation, although the experimental values are
slightly smaller than the theoretical ones. In order to evaluate
the applicability of the present method to an actual sample, a
profile of MDMA extracted from a urine sample of a suspect was
Figure 3. (A) A typical GC/MS chromatogram of MDMA derivative
obtained by single injection based on the SIM mode (ion peak at m/z
) 162 was selected for monitoring); the concentration of MDMA was
10 µg/mL before derivatization. (B) HT-GC/MS (order of matrix, 1023)
chromatogram of MDMA derivative. Inset, the raw data of TIC (by
selecting the ion at m/z ) 162) chromatogram before inverse
Table 1. Relationship between the Order of Matrix,
Enhancement of the S/N Ratio, and Mass Conditions
matrix order enhancement of S/N ratio
SIM mode: m/z ) 162
SIM mode: m/z ) 58
SIM mode: m/z ) 189
aThe enhancement of the S/N ratio was calculated as the ratio of
the S/N values obtained in the chromatograms, measured by HT-GC/
MS, HT-LC/MS, and a single injection method.
Figure 4. Typical GC/MS chromatograms of a urine derivative from
a suspect based on the SIM mode (ion peak at m/z ) 162 was
selected for monitoring). (A) Single-injection; (B) HT-GC/MS (order
of matrix, 255).
Analytical Chemistry, Vol. 80, No. 15, August 1, 2008
also attempted and was successful, as shown in Figure 4 (frame
A, single injection; B, HT injection; n ) 255). The concentration
level detected in the urine sample from a suspect was 5 µg/mL.
Figure 5A shows a typical LC/ESI-MS chromatogram of DMT
obtained by single injection based on the SIM mode (the parent
ion, [M + H]+ion at m/z ) 189, was selected for monitoring).
The concentration of DMT was 1.0 µg/mL. The S/N ratio was
∼3, i.e., the LOD was estimated to be ∼1.0 µg/mL. Under the
same experimental conditions, when an HT injection was per-
formed (n ) 511), the S/N ratio was improved 6.84-fold, as shown
in Figure 4B. The inset shows the raw data of the TIC (by
selecting the ion at m/z ) 189) chromatogram before inverse
Hadamard transformation. In Table 1, the results obtained by HT-
LC/MS are also summarized. As with HT-GC/MS, the enhance-
ment of the S/N ratio increases with the order of the Hadamard
matrix. The results indicate that the present injection device, with
a simple design, permits precise multiple injections for the HT-
In this study, we developed novel injection devices for HT-
GC/MS and HT-LC/MS. The utility of the HT-GC/MS and HT-
LC/MS injection devices was demonstrated using two well-known
abused drugs (MDMA and DMT) as model compounds. In both
cases, the devices permitted precise sample injections continu-
ously, resulting in substantial improvement in the S/N ratio
through the application of the Hadamard transformation. The
enhancement factors of the S/N ratios were in good agreement
with the theoretical values. On the other hand, the present HT-
LC/MS method is based on a regular C-18 HPLC column. In the
near future, if nanocolumns can be used, the sample injection
volume can be decreased and a higher order of matrix can be
applied. Thus, the present methods have a variety of applications
and could potentially be used in practical trace analysis.
This work was supported by grants from the National Science
Council of Taiwan under Contract No. NSC 95-2113-M-003-016-
MY3. T.K. acknowledges support by the Japan-Taiwan Joint
Research Program from the Interchange Association, Japan.
Received for review January 28, 2008. Accepted May 15,
Figure 5. (A) A typical LC/MS chromatogram of DMT obtained by
single injection based on the SIM mode (ion peak at m/z ) 189 was
selected for monitoring); the concentration of DMT was 1.0 µg/mL.
(B) HT-LC/MS (order of matrix, 511) chromatogram of DMT. Inset,
the raw data of TIC (by selecting the ion at m/z ) 189) chromatogram
before inverse Hadamard transformation.
Analytical Chemistry, Vol. 80, No. 15, August 1, 2008