HARMONIC AND INTERMODULATION PERFORMANCE OF ENVELOPE DETECTORS

Analytical expressions are obtained for predicting the amplitudes of the harmonics and 
intermodulation products at the output of an envelope detector excited by a signal 
formed of a carrier plus a number of sidebands. These expressions are in terms of the 
ordinary Bessel functions with arguments dependent on the modulations indecies. 
Comparison between results obtained using the proposed technique and previously 
published results is performed to establish the accuracy of the proposed technique.


INTRODUCTION
Envelope detectors are widely used in electronic systems [1,2].While envelope detector circuits are generally simple, their analysis is not.This is attributed to the nonlinear characteristics of the diode(s) and the reactive components involved [3][4][5][6].Of particular interest here is the prediction of the harmonic and intermodulation performance of envelope detectors [7][8][9][10].This problem is of special interest in color- television reception where a vestigial sideband signal is demodulated by an envelope detector [8][9][10].
Using numerical methods, the nonlinear differential equation of a simple envelope detector can be solved in the time-domain.Then, applying FFT the output spectrum resulting from an input formed of a SSB plus carrier can be obtained [7].The method can be easily extended to predict the intermodulation performance of an envelope detector excited by a SSB plus carrier plus interfering signals.
On the other hand, through direct integration of the envelope detector output, closed-form analytical expressions, in terms of the complete elliptic integrals of the first and second kinds, can be obtained for the dc and harmonic contents [8].Alternatively, through the repeated use of power series expansions, approximate power series expressions can be obtained for the dc and harmonic contents of the envelope detector output [10].In both cases the input signal to the envelope detector is formed of a SSB plus carrier.
While the use of closed-form expressions, for evaluating the output spectrum of the envelope detector, is more attractive than numerically- based methods, it appears that the expressions obtained in [8] and [10] cannot be used for evaluating the intermodulation performance of the envelope detector.
Thus, there is a need for a new technique for predicting the output spectrum of an envelope detector excited by a carrier plus a number of sidebands.The technique would be attractive if it can yield closedform analytical expressions for the output dc, harmonics and intermodulation components.It is the major intention of this paper to present such a technique.

ANALYSIS
Figure 1 shows the phasor diagram of an input signal formed of a carrier plus a number of sidebands.At the input of the envelope detector, this signal can be represented by vi(t) V sin o + mk sin (Wo k)   k=l (1) where Wo is the carrier frequency, Wo-Wk is the frequency of the k th sideband and V is the amplitude of the carrier.The output of the tort.,,x

FIGURE
Carrier and sidebands phasors at the input.
envelope detector can be expressed as where Ok COk t.
Eq. ( 2) can be rewritten in the form 2mimj cos (Oi-Oj)) ( where and Eq. ( 3) is in the form y (1 + x)1/2 (4) Eq. ( 4) can be approximated by a Fourier-series of the form N y= Co+ (Cn cos (--x)+fln sin (2-x)) where the parameters Co, cn and fin can be obtained using FFT algorithms or curve-fitting techniques.Alternatively, following the procedure described in [11] and [12], first the offset at x 0 is removed and the resulting function is mirror imaged to obtain a complete period of the periodic function f(x)=y- 1.Secondly, a number of data points is chosen and connected using straight line segments joind end to end as shown in Figure 2. Denoting the slope of each segment by "Yt, the parameters OZ n and n can be expressed as [11, 12]   FIGURE 2 The function of Eq. ( 4) after removing the offset at x 0 and mirror imaging.
On '71 --'7L-1 + ('7l+1 --'7l) COS Xl+I (6) 2 (nn') 2 1=1 /3n 2 (n 7r) 2 ('71+1 'Tt) sin 2 Xl+I (7) k,l=l where T is the period of the periodic function.The parameter '7o can be easily obtained by calculating the area under the curve in Figure 2 using any numerical integration method.Thus, '70 can be expressed as 1(1 From ( 6) and ( 7) one can see that calculation of the parameters Cn and /3, requires only simple mathematical operations, without recourse to sophisticated algorithms for FFT or curve-fitting techniques.Also, inspection of ( 6) and ( 7) suggests that as n becomes infinite, the Fourier-series parameters c, and/3 always approach zero.In fact, the number of terms in ( 5) can be increased until the inclusion of the next term is seen to make a negligible contribution towards a best fit criterion; for example the minimum relative mean-square (RRMS) error.Table I shows the parameters of the first 21 terms of (5) used for approximating (4).
Using the parameters of Table I and Eq. ( 5) calculations were made and are summarized in Table II which shows the change of the RRMS error with the number of terms of the Fourier-series.It is obvious that increasing the number of terms improves the accuracy of the Fourier- series approximation.

HARMONIC AND INTERMODULATION PRODUCTS
One of the potential application of the proposed model of ( 5) is in the prediction of the amplitudes of the harmonics and intermodulations  products at the output of the envelope detector.Thus, combining (4) and (5) and using the trigonometric identities sin (z cos b) 2 (J1 (2) cos t J3 (z) cos 3 q ---... and cos (z cos ) Jo (z) 2 J2 cos 2 + 2 J4 cos 4 where Jk(z) is the Bessel function of order k, and after simple mathematical manipulations it is easy to show that the amplitude of an output product of frequency E :=1 6k a;k + I 6ij 0" and order i=l,j=l,j>i I kl / r _l,j_,j>ilSej l, where 6k and 6/ are positive, negative integers or zeros, will be given by n=l k=l i=l,j=l,j>i Note that a unity is introduced into (11) to restore the removed offest at x 0. Eqs. ( 9) and (10) can be used for calculating the amplitudes of the harmonics and intermodulation products at the output of an envelope detector excited by a carrier plus a number of sidebands.

SPECIAL CASE
In this section the special case of an input signal formed of a carrier plus a single sideband will be considered in detail and the results will be compared with previously published results to establish the accuracy of the proposed technique.Under these conditions, the dc output component will be g(1 + m2) 1/2 + ao + n=l OnJo al (12)   Ref. [8]  Ref. [10]   N= 31 N 21 N= 11 Vdc/V Using ( 12)-( 14) the output dc, fundamental and second-harmonic components can be calculated for any value of m.The results obtained for m 1, together with the results obtained using the techniques pro- posed in [8] and [10] are shown in Table III.It appears from Table III that the results obtained using the present technique are in excellent agreement with the results obtained using previously published techniques.This establishes the accuracy of the present technique.
CONCLUSION By approximating the envelope detector characteristic of Eq. ( 2), using a Fourier-series, analytical expressions can be obtained for the harmonic and intermodulation performance of an envelope detector excited by a signal formed of a carrier plus a number of sidebands.The Fourier-series coefficients and the amplitudes of the output components can be evaluated using simple calculations without recourse to numerical integration or FFT.Comparison with previously published results confirms the validity of the proposed technique.

TABLE II
Variation of the RRMS with the number of terms N

TABLE III
Comparison between previously published and present results