# Complementary Sequences In DSP Based Testing Of Electronics Biology Essay

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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).

## Keywords:

Digital Filter, Frequency Response, Matched Filter, Autocorrelation, Complimentary sequences

## 1 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).

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 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.

## 2 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 (we say that a is bi-polar). Define the a-periodic auto-correlation Function (AACF) of a by

(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:

(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 where ï¸ is the primitive qth root of unity, which yields so-called polyphase 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)

(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)

Figure1 (a) Practical implementation of Equations (2), (6) and (7). (b) Pulse like output c of length 2N-1.

Figure 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 the output c in Figure 1(a) will consists of a single pulse at the time index N-1 (delay k= N-1) and zero else where, for a sequence of length N. It is shown in Figure 1(b).

## 3 BASIC PRINCIPLES

The principle of the proposed technique is best illustrated with the help of Figure 2 which is a further development of Figure 1.

Figure 2 Principle of the measurement scheme

Referring to Figure 2 the pulse like output c now excites the system under test. The system is assumed to have weighting sequence h and z-transform 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 Figure 2, will be the discrete impulse response of the system and its Fourier transform will be the required frequency response H(f) of the system under test.

Figure 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 Figure 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 Figure 3 is introduced to avoid any interference between the responses of the two GC sequences and correctly obtain their sum.

## 4 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 Figure 3 the noise component of the output of filter matched to the sequence a will have variance given by

(9)

Similarly the noise component of the output of filter matched to the sequence b will have a variance given by

(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

(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.

## 5 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. Gole 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. Moreoer, Barker codes of length greater than 13 do not exist [5].

Figure 4 Weighting sequence measurement for system

with transfer function H(s) = K/(s+5)

Figure 4 shows the impulse response of the system with transfer function H(s) = K/(s+5), measured using Barker sequence and Gole sequence. The system here has a large time constant that exceeds the duration of the input test sequence. It can be seen from the figures that the side lobes in the response of the matched filter to the test sequence cause considerable error in the measurement. The weighting sequence measured using Gole Complementary Sequence has no errors and is identical to the actual weighting sequence of the system.

Figure 5 Weighting sequence measurement for system

with transfer function H(s) = K/(s+30)

Figures 5 show 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. Here the system time constant is smaller than the length of the input test sequence. It is seen that sidelobes in the autocorrelation function of the Barker sequence still corrupt the measurements and introduce considerable sidelobe error.

## 6 CONCLUSIONS

The method of measuring the frequency response of electronic products discussed in this paper makes use of Gole Complementry 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 least 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.