A new simplified approach to the error analysis of digitally synthesized signals generated by digital-to-analog converters and its fundamental component measurement without a filter

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2010 Conference on Precision Electromagnetic Measurements (CPEM 2010), pp.

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A new simplified approach to the error analysis of digitally synthesized

signals generated by digital-to-analog converters and its fundamental

component measurement without a filter

Zuliang, Lu; Lu, Huang; Yan, Yang; Lei, Wang; So, Eddy

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265 Monday Tuesday W ednesday Thursday Friday

2010 Conference on Precision Electromagnetic Measurements June 13-18, 2010, Daejeon Convention Center, Daejeon, Korea

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A NEW SIMPLIFIED APPROACH TO THE ERROR ANALYSIS OF DIGITALLY SYNTHESIZED SIGNALS GENERATED BY DIGITAL-TO-ANALOG CONVERTERS AND

ITS FUNDAMENTAL COMPONENT MEASUREMENT WITHOUT A FILTER Lu Zuliang1, Huang Lu1, Yang Yan1, Wang Lei1, and Eddy So2

1

NIM, National Institute of Metrology, Beijing, China

2

NRC, National Research Council, Ottawa, Canada Contact Email: luzl@nim.ac.cn

Abstract

A new simplified approach to the error analysis of digitally synthesized waveforms generated by DACs is described. An experiment to compare only the fundamental component without filter is presented.

Introduction

Digitally synthesized signals generated by digital-to-analog converters (DAC) have been described in the literature [1, 2]. This paper will address only the error analysis of the stepwise-type generation method [1], which provides an analysis of the frequency response of a DAC with important conclusions related to the amplitude of fundamental and harmonic components. This paper presents a different but simplified approach in analyzing the errors of the produced waveforms, which would provide a new perspective on some simple/convenient forms to understand and to calculate the error, especially the phase angle error, as this is a very important information too. From this approach, a new aliasing effect is discussed. Furthermore, a novel experiment for comparison of the fundamental component of the waveform produced by DACs is described in this paper, in which the traditional filter is not used, even if many harmonic components exist in the waveform. The practical output of a DAC basically can be considered as consisting of a perfect wave and a transient signal due to imperfections of the DAC circuitry. The frequency spectrums of the two parts could be separately analyzed and afterwards combined, assuming a linear combination. The transient signal, causing errors, is primarily dependent on the circuit of the DAC. By improving the DAC circuitry, such as the use of a dual DAC circuitry, the transient errors could be reduced. However, the basic characteristic of the ideal perfect wave will not be changed in the recent form of the DAC. This paper will only focus on the perfect

output of the ideal DAC. The original digital signal is limited to the sinusoidal case.

A New Simplified Analysis

Wave Produced by DAC - For simplicity an ideal digital sinusoidal signal is presented as

 

x x

y sin , where the amplitude is 1V. This is defined as “the original signal” here. It is now made of discrete (sampled) signals/points with n equal-intervals in one period, the values of discrete points are represented asy_j sin

 

jh , where the discrete interval is

n

h 2S , j 0,1,2,/,n. The wave produced by DAC becomes:

 

x y

 

jh

z _j sin ,

where jhdx

 

j1h,j 0,1,2,/,

 

n1 . It is a consecutive signal, defined as “the produced wave” here. The relative signals are shown in Fig.1. Obviously, the produced wave signal

z

 

x

is different from the original signal . However, it is important to note that their rms values are the same

 

x

y

2 , 1 sin 2 1 1 0 2 2 0 2

¦

_!

³

 y h n nh xdx n j j S S . (1)

y(x), the original signal

z1(x), the fundamental of z(x)

z(x), the produced wave

266 Monday Tuesday W ednesday Thursday Friday Spectrum Analysis

The Fourier transform of the produced wave signal, , would give the following coefficients:

 

x z

 

_{¦ ³}

 

³

1  0 1 2 0 sin 1 sin 1 n j h j jh j k zx kxdx y kxdx a S S S

 

_{¦ ³}

 

³

1  0 1 2 0 cos 1 cos 1 n j h j jh j k zx kxdx y kxdx b S S S

Since is a consecutive signal, these forms are different from the general DFT. They can further be calculated, when k=1,

 

x z h h a₁ sin( ), and h h b₁ cos( )1 . Thus, the amplitude of the fundamental becomes:

2 / ) 2 / sin( ) cos( 1 2 2 1 2 1 1 h h h h b a c  

This is consistent with the previous results described in [1], which can be further simplified resulting in a more easily understood form as:

2 2 2 1 1 1.6449/ 6 1 n n c |  S |  .

The phase shift of the fundamental component with respect to the original signal can also be calculated:

n a b S M arctan( )  1 1

1 .Thus, the fundamental

component of z(x) can be completely expressed as ) sin( ) 6449 . 1 1 ( ) ( 2 1 n x n x z  S . (2) Further, all harmonics are of the

order ., and the corresponding amplitudes and phase shifts are

/ , 2 , 1 , 1 r t tn k 1 1 1 1 c tn c_tn r r , n tn S M _r1 P .

It should be emphasized that the amplitudes of the harmonics are expressed in terms of the fundamental. This could be interpreted as a phenomena called “pseudo aliasing effect” of a DAC. This is an interesting phenomena that a DAC starting with an original digital sine-wave signal, generates an analog signal, consisting of a new fundamental and all harmonics of the order of with their coefficients of 1 c 1 r tn ) 1 ( 1 tnr

c . Compared with the typical aliasing effect of analog-to-digital converter (ADC), all components in the produced wave are orthogonal to each other with the following relation:

(3) 2 1 2 1 2 1 c c c t tn 

¦

f r

where c is the amplitude of the original signal. This is the other significant feature of the produced wave of . It can be used to evaluate the errors. A DAC can be used to construct an AC voltage standard in which all harmonic components are included, or as a “pure” sinewave voltage standard in which a filter is used to reduce the impact of all harmonics. However, when a filter is used, then the filter will affect the accuracy and/or stability of the fundamental component. A new measurement approach is proposed in this paper, including experimental verification, in which the fundamental is measured in the presence of all the generated harmonics, without the need of using a filter.

) (x

z

Conclusion

The new simplified approach of error analysis of digitally synthesized sinewave generated by a DAC has resulted in a number of important features. The first one is that its effective value is equal to that of the original sinusoidal signal, as shown in (1). The second one is the “pseudo-aliasing effect”. The third one is that the sum of the square of all harmonic amplitudes, including the fundamental of the produced wave, is equal to the square of that of the original signal, as shown in (3). The previous analysis results of the frequency spectrum have been verified and discussed in a more simplified form. The produced wave, referred to the original sinusoidal signal, has an fundamental amplitude error of

2

6449 .

1 n

 , and a phase shift of S n, where n is the number of the equal discrete intervals. The harmonics are only of the orders of tnr1, whose amplitudes are



11.6449 n2





tnr1



with phase shifts of PS/n. Furthermore, a new measurement approach has been proposed to measure only the fundamental component in the presence of all the generated harmonics without the use of a filter will be presented, including experimental verification using commercial instruments of a DAC and a lock-in amplifier, with a result lock-in the order of parts per million.

Reference

[1] N.M. Oldham, “Digital Waveform Synthesis Technique,” NBS Special Publ. 707, pp. 1-13, Oct. 1985

[2] P.S. Wright and J.R. Pickering, “An AC Voltage Standard Based on a PWM DAC,” IEEE Trans.

Instrum. Meas., vol. 48, no. 2, pp. 457-461, April