International Journal of Engineering & Technology IJET-IJENS Vol:10 No:02 17

103502-4848 IJET-IJENS © April 2010 IJENS I J E N S

Abstract— This paper presents a scheme for rapid

measurement of frequency characteristics of electronic products

using DSP techniques. The use of Complementary sequences

instead of the Barker sequences completely removes the sidelobe

error and provides exact measurement of the product characteristics in the absence of noise. DSP based testing allows

us to send all the test frequencies through the device under test.

Each test tone in the output waveform can then be separated

from other tones using an appropriate digital filter. The gain and

phase measurements at each frequency can then be calculated without running many separate tests and thus considerably

reducing the measurement time and consequently the time to

market (TTM).

Index Term— Digital Filter, Frequency Response, Matched

Filter, Autocorrelation, Complimentary sequences.

I. INTRODUCTION

Time to market (TTM) is the length of time it takes from a

product being conceived until its being available for sale.

TTM is important in industries where products are outmoded

quickly like the electronic industry [1-3]. TTM is one of the

important factors that affect the profit margin and the product

cost in the electronic industry and is required to be as short as

possible. For this reason test and measurement of electronic

products has grown into a highly specialized field of electrical

and electronic engineering. Effective short time tests for

testing fabricated circuits and devices are to be designed in

order to reduce TTM.

Most of the performance measures of electronic products are

represented in terms of their frequency response. Hence

measurement of frequency response of electronic devices

/components continues to be an important step in their

production. It is important that such measurements are carried

out effectively and quickly in order to reduce TTM and also

maintain desirable measurement accuracy. Frequency

response of a linear system is the Fourier transform of its

weighting sequence (discrete impulse response). Thus the

measurement of frequency response of linear devices

/components is essentially the measurement of their discrete

impulse response (weighting sequence).

Farid Ghani is with the Universiti Malaysia Perlis, 01000, Kangar, Perlis,

Malaysia; (phone: + 604-9853944; fax: +604-9851695; e-mail: faridghani@unimap.edu.my ; faridghani@rediffmail.com )

Noor Shafiza Mohd Tamim is with the Universiti Malaysia Perlis, 01000,

Kangar, Perlis, Malaysia; (e-mail: noorshafiza@unimap.edu.my ).

A method for rapid measurement of discrete weighting

sequence involving finite duration sequences has recently been

proposed. The system under test is here perturbed with a

carefully chosen sequence of finite duration. The response of

the system to the sequence is then processed with an

appropriate digital filter to give the required weighting

sequence [4,5]. Two different filters namely the digital

matched filter and the optimum inverse filter have been

proposed and sequences like Barker and Huffman sequences

have been considered [5]. The method has the advantage that

the measurements are carried out rapidly. A delay of L clock

periods only is incurred and L can be as small as the sequence

length. This contrasts with the use of most commonly used

periodic pseudo random test signals where it is necessary, in

principle at least, to wait for the system under test to reach its

periodic steady state before starting measurements, which

must then be taken over at least one whole sequence

period[6,7]. There is greater freedom of choice of sequence

length in the case of finite length sequences; the sequence can

be of any length provided that it has sufficient energy and its

autocorrelation sidelobes are low enough. In contrast, with a-

periodic test sequences, the period must exceed the effective

duration of the weighting sequence that is to be measured.

The use of finite duration sequences instead of the PRBS

considerably reduces the measurement time and hence the

time to market. However, the side-lobes in the autocorrelation

of the test sequence introduce measurement errors in addition

to those due to noise, and considerably limit the use of the

technique. In this paper an alternative technique is proposed

for the measurement of discrete weighting sequence of linear

systems. The system under test is here perturbed with two

Golay complementary sequences of finite duration[8,9,10].

The response of the system to these sequences is then

processed with appropriate digital matched filters and added

together to give the required weighting sequence. The use of

Golay complementary sequences and digital matched filters

totally removes the sidelobe errors and provide hundred

percent measurement accuracy in the absence of noise.

Moreover, Golay complementary sequences being binary, they

provide the maximum sequence energy for a given length and

hence maximize the signal to noise ratio when the

measurements are carried out in noisy conditions. These

advantages are achieved with only a slight increase in the

measurement time. MATLAB and SIMULINK is used to

implement the proposed measurement scheme and to study its

performance for two different systems. The performance of

the proposed technique is also compared with the one that uses

Complementary Sequences in DSP Based Testing of Electronic Systems

Farid Ghani, Fellow IET, Fellow IETE, Fellow NTF, C.Eng and Noor Shafiza Mohd Tamim

International Journal of Engineering & Technology IJET-IJENS Vol:10 No:02 18

103502-4848 IJET-IJENS © April 2010 IJENS I J E N S

finite length sequences and digital filters. Since the frequency

response of a system is the Fourier transform of its impulse

response, this paper concentrates on the measurement of the

weighting sequence or the discrete impulse response of

electronic devices /components rather than the frequency

response itself. Once the discrete impulse response is

measured the frequency response is obtained by taking its

Fourier transform.

II. COMPLEMENTARY SEQUENCES

A complementary pair of sequences (CS pair) satisfies the

useful property that their out-of-phase a-periodic

autocorrelation coefficients sum to zero [8-10]. Let {a} = (a0

a1 . . . aN-1) be a sequence of length N such that ia {+1,-1}

(we say that a is bi-polar). Define the a-periodic auto-

correlation Function (AACF) of a by

N-k-1

a i i+k

i=0

a a 0 k N-1ρ (k)= (1)

In defining ρa(k) we have considered only the positive values

of delay k. It may be noted that ρa(k) is an even function of

delay k and ρa(k) = ρa(-k). Let b be defined similarly to a. The

pair (a;b) is called a Golay Complementary Pair (GCP) if:

-k -k

ak bk

-k -k

ak bk

z z k=N-1

z z N-1

ρ + ρ = 1,

ρ + ρ = 0, k (2)

Each member of a GCP is called a Golay complementary

sequence (GCS, or simply Golay sequence). Note that the

definition (2) can be generalized to non-binary sequences. For

instance, ai and bi can be selected from the set h0 0 2 -1, , . . . , }{ξ ξ ξ where is the primitive q

th root of unity,

which yields so-called poly-phase Golay sequences. Since the

response of a digital matched filter to the sequence it matches

with, is the AACF of the sequence, it is clear from Equation

(2) that if the responses of the GCP to their respective match

filters are added together element by element then the sum

would be a single pulse of magnitude unity. Let

fa = { fa0, fa1, . . . faN-1} (3)

and fb = { fb0, fb1, . . . fbN-1} (4)

be the discrete weighting sequence of length N of the matched

filters matched to the Golay sequences a and b, respectively.

Let A(z) ,B(z), ρa(z), and ρb(z) be the respective z-transform

of sequences a, b, AACF ρa and ρb. If Fa(z) and Fb(z), is the

z-transform of fa and fb, then from Equation (1)

ρa(z) = A(z)*Fa(z); ρb(z) = B(z)*Fb(z) (5)

From Equations (2) and (5)

-k -k -k

ak bk

-k -k

ak bk

z z k = N-1

z z N-1

ρ + ρ = z ,

ρ + ρ = 0, k (6)

Let c be the output sequence given by

c = ρa + ρb (7)

The z-transform C(z) of the output is then

C(z) = z-(N-1)

(8)

fa

Filter Matched

to GCS a

fb

Filter

Matched to

GCS b

+

Input GCS a

Input GCS b

Pulse like

Output c

Delay k

ck

N-10 2N-1

1

(a)

(b)

ρa

ρb

Fig. 1. (a) Practical implementation of Equations (2), (6) and (7). (b) Pulse

like output c of length 2N-1.

Fig. 1(a) shows diagrammatically the implications of

Equations (2) , (6) and (7). Golay Complementary Sequences

(GCS) a and b excite their respective matched filters fa and fb.

The output from these filters is added element by element to

give the resultant output c as in Equation (7). It can be seen

that for a sequence of length N, the output c in Fig. 1(a) will

consists of a single pulse at the time index N-1 (delay k= N-1)

and zero elsewhere. It is shown in Fig. 1(b).

III. BASIC PRINCIPLES

The principle of the proposed technique is best illustrated

with the help of Fig. 2 which is a further development of Fig.

1.

fa

Filter

Matched to

GCS a

fb

Filter

Matched to

GCS b

+

Input

GCS a

Input

GCS b

Pulse like

Output cρa

ρb

h

System

Under Test

System

Weighting

Sequence h

Fourier

Transform

Frequency

response

H(f)

Fig. 2. Principle of the measurement scheme.

Referring to Fig. 2 the pulse like output c now excites the

system under test. The system is assumed to have weighting

sequence h and z-transfer function H(z). Since the input to

the system is a single pulse except for a possible time shift of

some known clock periods, then the system output in Fig. 2,

will be the discrete impulse response of the system and its

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Fourier transform will be the required frequency response H(f)

of the system under test.

fa

Filter

Matched to

GCS a

fb

Filter

Matched to

GCS b

+

Input

GCS a

Input

GCS b

h

System

Under Test

Fourier

Transform

z-(N+M)

Frequency

Response H(f)

System Weighting Sequence h

z-(N+M)

S1

S2

N=length of Sequences a and b; M= Memory of system under test

Switch S1 closes at t=0 and opens at t= (M+N-1)T

Switch S2 closes at t=(M+N)T and opens at t= (2M+N-1)T

T = bit duration

Fig. 3. Actual configuration of system and filters

In order to keep the system in its linear mode of operation, it is

not desirable to excite it with a single pulse of large magnitude

which is essential if the accuracy of measurement is to be

improved in the presence of noise. The arrangement of Fig. 2

is, therefore, not desirable for practical reasons. However,

since the entire arrangement is linear, the overall transfer

function remains unaltered if the arrangement of Fig. 3 is

used. In this case, the input to the system is a sequence of

pulses rather than a single pulse of large magnitude. The delay

of (N+M) clock instants in Fig. 3 is introduced to avoid any

interference between the responses of the two Golay

sequences and correctly obtain their sum.

IV. ACCURACY OF MEASUREMENT

Inaccuracy in the weighting sequence measurement is

introduced due to the system noise that perturbs the output of

the digital matched filter and produces error. If the system

noise is assumed to be a sequence of purely random numbers

(sampled data white noise) of variance σi2, then referring to

Fig. 3 the noise component of the output of filter matched to

the sequence a will have variance given by

N-1

2 2 2

oa i ai

i=0

σ = σ f (9)

Similarly the noise component of the output of filter matched

to the sequence b will have a variance given by

N-1

2 2 2

ob i bi

i=0

σ = σ f (10)

If the two outputs are uncorrelated then the upper limit to the

total error due to system noise will have a variance given by

2 2 2

0 oa obσ =σ +σ (11)

It may be noted that in the absence of system noise there will

be no measurement error and the measured weighting

sequence will be identical to the actual weighting sequence of

the system

V. RESULTS

The proposed measuring scheme employing Gole

sequences is implemented using MATLAB and SIMULINK to

measure the weighting sequence of a first order system with

two different values of the time constant. Golay sequences of

length 26 are used for the measurement [8,9]. The

measurement is carried out in the absence of noise. For the

sake of comparison the weighting sequences of the same

systems are also measured using finite length sequence and

matched filter [5]. Barker sequence of length 13 is used for

these measurements as this sequence is a binary sequence and

has the best possible AACF. Moreover, Barker codes of length

greater than 13 do not exist [5].

Fig. 4. Weighting sequence measurement for system with transfer function

H(s) = K/(s+5)

Fig. 5. Weighting sequence measurement for system with transfer function

H(s) = K/(s+30)

Fig.s 4 and 5 show the impulse response of systems with

transfer function H(s) = K/(s+5) and H(s) = K/(s+30)

measured using Barker sequence and Gole sequence,

respectively. The time axis is normalized in the two fig.s. In

Fig. 4, the system has a large time constant that exceeds the

duration of the input test sequence. It can be seen from the

fig.s that the side lobes in the response of the matched filter to

the test sequence cause considerable error in the measurement.

On the other hand the impulse response measured using

complementary codes has no sidelobe errors.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Time

Am

plitu

de

Impulse Response

Barker Sequence

Gole Sequence

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Time

Am

plitu

de

Impulse Response

Barker Sequence

Gole Sequence

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103502-4848 IJET-IJENS © April 2010 IJENS I J E N S

Fig. 5 shows the impulse response of the system with transfer

function H(s) = K / (s+30). The adverse effect of the

sidelobes in the autocorrelation function of the Barker

sequence, on the measurement can be clearly seen. On the

other hand, the results obtained using Gole sequence are

virtually error free. It is seen that sidelobes in the

autocorrelation function of the Barker sequence still corrupt

the measurements and introduce considerable sidelobe error.

VI. CONCLUSIONS

The method of measuring the frequency response of electronic

products discussed in this paper makes use of Golay

complementary sequences and matched filters. It has the

advantage that there are no errors due to the AACF sidelobes

in the measurements. Moreover the measurements are also

carried out rapidly. This contrasts with the use of finite length

test signals like Barker sequences where the sidelobes in the

AACF of the test sequence introduce considerable errors in the

measurements. The proposed method also overcomes the

disadvantages associated with periodic test sequences where it

is necessary, in principle at least, to wait for the system under

test to reach its periodic steady state before starting

measurements, which must then be taken over at leas t one

whole sequence period. There is greater freedom of choice of

sequence length in the case of Complementary sequences; the

sequence can be of any length provided that it has sufficient

energy to result in a high signal to noise ratio in the presence

of system and measurement noise.

REFERENCES [1] B. K. Kenneth, Editor, The PDMA Handbook of New Product

Development, Second Edition, John Wiley & Sons,. 2004, pp. 173-187,

[2] P. G Smith,. and G. M. Merritt , , Proactive Risk Management , Productivity Press, 2002.

[3] P. G Smith and D. G. Reinertsen, Developing Products in Half the

T ime, 2nd Edition, John Wiley and Sons, New York, 1998. [4] Z.-S. Liu, Mar. "QR methods of O(N) complexity in adaptive

parameter estimation", IEEE Trans, on Signal Processing, vol. 43, 1995, pp. 720-729,.

[5] F. Ghani, N.S.M. Tamim " DSP Based Testing of Electronic Products" in Proc. International Conference on Robotics, Vision, Signal processing and Power Applications RoViSP 2009, Langkawi Island, Malaysia9-20 December 2009 (To appear)

[6] A. Milewsky, , , "Periodic sequences with optimal properties for channel estimation and fast start-up equalization", IBM Journal of Research and Development, vol. 27, Sept. 1983, pp. 426-431.

[7] N. Benvenuto, "Distortion analysis on measuring the impulse

response of a system using a cross-correlation method", AT&T Bell Laboratories Technical Journal, vol. 63, Dec. 1984, pp. 2171-2192.

[8] P. Spasojevic,C.N. Georghiades, “Complementary sequences for ISI channel estimation,” IEEE Trans. Inform. Theory, IT -47, pp. 1145–1152, March 2001.

[9] J.A. Davis, J. Jedwab, “Peak-to-mean power control in OFDM,

Golay complementary sequences and Reed-Muller codes,” IEEE Trans. Inform. Theory, IT-45, no. 7, pp.2397–2417, Nov. 1999.

[10] K.G. Paterson, “Generalized Reed-Muller codes and power control in OFDM modulation,” IEEE Trans. Inform. Theory, IT-46, no. 1,

pp. 104–120, Jan. 2000.

Farid Ghani received B.Sc. (Engg) and M.Sc. (Engg) degrees from Aligarh Muslim University, India and M.Sc. and Ph.D. degrees from Loughborough

University Of Technology (U.K). From 1982 to 2007 and for varying durations, he worked as Professor of Communication Engineering at Aligarh Muslim University, India, Professor and Head of the Department of Electronics Engineering, Al-Fateh

University, Libya, and Professor at Universiti Sains Malaysia. He is currently working as Professor in the School of Computer and Communication Engineering ,Universiti Malaysia Perlis, Malaysia. He is actively engaged in research in the general areas of digital and wireless

communication, digital signal processing including image coding and adaptive systems. He has a large number of publications to his credit and is a reviewer for several international research journals. Professor Ghani is Fellow of Institution of Engineering and Technology (IET)

UK, Fellow of Institution of Electronics and Telecommunication Engineers(IETE) India, and Fellow of National Telematic Forum (NTF) India. He is also registered with the Council of Engineers(C.Eng) UK as Chartered

Engineer. Noor Shafiza Mohd Tamim received B. Eng. (Communication and Computer Engineering) in 2004from Universiti Kebangsaan, Malaysia and M. Sc. (Communications Engineering) in 2008 from University of Applied

Science, Germany. She is currently working as lecturer in the School of Computer and Communication Engineering, Universiti Malaysia perils, Malaysia. She teaches courses in digital signal processing , Electronic Communication Engineering, Applied Electrical Engineering,C Programming

to undergraduate engineering students. Her research interests include Signal Detection /Estimation, Analysis and Simulation, Digital Signal Processing.