Phase de-trending of modulated signals

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For a generated or measured modulated signal, the phase relationship between the tones remains constant but the phase of the carrier can be arbitrary. Moreover, when comparing phases of the multi-tone between different measurements, the phase values will change for each measurement. This paper explains how to align several measurements of a modulated signal.

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Phase de-trending of modulated signals

  1. 1. Global optimization technique for phase de-trending Global optimization technique for phase de-trending G. Pailloncy, NMDG nvIntroductionA modulated signal with a carrier frequency fc and presenting H harmonics can be represented by: H x t = ∑ a h t. e j 2 h f c t (1) h=−Hwith ah(t), the modulating signal and a−h t =a ∗ t  , the complex conjugate of ah(t). hThe modulating signal ah(t) is a complex signal that may be expressed in a general way as: a h t =I h t− jQ h t (2)In case of a modulating signal composed of 2N+1-tones and with a modulation frequency fm, ah(t) can be ex-pressed as: N a h t = ∑ Ah , k e j2  k f m t Ah , k ∈ℂ (3) k =−NWhen measuring with a sampler-based Large-Signal Network Analyzer (LSNA), such a modulated signal isdown-converted using a sampler. Due to the effect of the internal local oscillator (LO), a phase offset may oc-curred Ф. Moreover, as no trigger is used when capturing the down-converted signal with the ADCs, a delay τmay appear between the different measured experiments.An i-th measured experiment can then be expressed as: H N x i t =x t− i , i =  ∑ ∑ Ah ,k e− j2 h f c k f m i e j h  e j2  h f i c k f m t (4) h =−H k=− NIn the frequency domain, the modulated signal can be expressed as: H N  X i  f = Fourier { x i t }∣ f =  ∑ ∑ X i ,h , k   f −h f c k f m  (5) h=−H k=−Nwith − j 2  h f c k f m i j h i X i , h , k = Ah , k e e (6)Our purpose, in this report, is to align the different experiments taking the first experiment x 0 t  as the refer- ence (τ0 = 0, Ф0 = 0) by correcting for the delay τi and phase offset Фi between them.© 2009 NMDG NV 1
  2. 2. Global optimization technique for phase de-trendingIn the following, a global optimization technique to extract the delay τi and phase offset Фi is described and ap-plied to a set of modulated signal measurements.Correction of the delay and phase offset between experimentsThe above equation (6) may be rewritten as: X i , h , k = Ah , k e− j 2  k f e j h−2  f = Ah , k e− j 2  k f e j h i  i i  i m c m 0i (7)Applying a Least Square Estimator, the τi and Ф0i values that minimize the following function around the funda-mental (h=1), need to be found for each experiment: N min S =  i , 0i ∑  X 0,1, k − X i , 1,k e j2 k f m  i − j 0i e  . X 0,1,k − X i ,1, k e j2  k f m i − j 0i ∗ e  (8) k=−N 1 2 1The function S is first computed for a set of 10 values both for τi ( [ , .. ] ) and for Ф0i ( 10f m 10f m f m 1 2 1 [ , .. ] ) , and the pair of { τi, Ф0i} values that gives the minimum result is selected as initial 20  20  2 guess values for the Least Square Estimator.One may then correct each experiment for the delay and phase offset using the extracted τi and Ф0i: X i , h , k = X i , h , k e j 2 k f i − j h0i m e (9)ResultsA set of 10 experiments of the measured output current of a commercially available FETis used. The FET is ex-cited by a 3-Tones modulated signal with 1GHz fundamental frequency and 50048.8 Hz modulation frequency.The power spectrum of the measured current is plotted on Figure 1.The set of 10 experiments without any phase alignment is shown in time domain on Figure 2.After applying the above global optimization technique with first experiment as reference, the results shown onFigure 3.To verify further the algorithm, the difference in time domain between the reference (first experiment) and thealigned second experiment is plotted on Figure 4.ConclusionIn this article,a global optimization technique to align a set of modulated signal experiments has been describedmathematically and tested.2 © 2009 NMDG NV
  3. 3. Global optimization technique for phase de-trending - 60 - 80 È i2È HdBL - 100 - 120 - 140 Freq HGHzL 0.998 0.999 1 1.001 1.002 Figure 1: Power Spectrum of measured output current i2 of FET (3 Tones excitation, 50 Tones measured each side, 1GHz fundamental frequency, 50048.8Hz Modulation frequency, Vg=-0.7V, Vd=2V, Pin=2 dBm) 10 5 i2 HmAL 0 -5 - 10 Time HusL 0 10 20 30 40 Figure 2: 10 measured experiments of output current waveform (2 periods) of FET at fundamental frequency (3 Tones excitation, 50 Tones measured each side, 1GHz fundamental frequency, 50048.8Hz Modulation frequency, Vg=-0.7V, Vd=2V, Pin=2 dBm)© 2009 NMDG NV 3
  4. 4. Global optimization technique for phase de-trending 10 5 i2 HmAL 0 -5 - 10 Time HusL 0 10 20 30 40 Figure 3: Result after global optimization alignment (first experiment as reference) 15 10 5 D i2 H u AL 0 -5 - 10 - 15 Time HusL 0 5 10 15 20 Figure 4: Error between second experiment and reference (1 period)4 © 2009 NMDG NV

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