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The Power Spectral Density of Phase Noise and Jitter: Theory, Data Analysis, and Experimental Results by Gil Engel

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AN-1067 APPLICATION NOTE One Technology Way • P. O. Box 9106 • Norwood, MA 02062-9106, U.S.A. • Tel: 781.329.4700 • Fax: 781.461.3113 • www.analog.com The Power Spectral Density of Phase Noise and Jitter: Theory, Data Analysis, and Experimental Results by Gil Engel INTRODUCTION GENERAL DESCRIPTION Jitter on analog-to-digital and digital-to-analog converter sam- There are numerous techniques for generating clocks used in pling clocks presents a limit to the maximum signal-to-noise electronic equipment. Circuits include R-C feedback circuits, ratio that can be achieved (see Integrated Analog-to-Digital and timers, oscillators, and crystals and crystal oscillators. Depend- Digital-to-Analog Converters by van de Plassche in the References ing on circuit requirements, less expensive sources with higher section). In this application note, phase noise and jitter are defined. phase noise (jitter) may be acceptable. However, recent devices The power spectral density of phase noise and jitter is developed, demand better clock performance and, consequently, more time domain and frequency domain measurement techniques costly clock sources. Similar demands are placed on the spectral are described, limitations of laboratory equipment are explained, purity of signals sampled by converters, especially frequency and correction factors to these techniques are provided. The synthesizers used as sources in the testing of current higher theory presented is supported with experimental results applied performance converters. In the following section, definitions to a real world problem. of phase noise and jitter are presented. Then a mathematical derivation is developed relating phase noise and jitter to their frequency representation. The frequency domain representations, or power spectral densities, are shown to directly provide a measure of phase noise/jitter. The theory developed is associated with analog-to-digital and digital-to-analog converters. A spectrum analyzer and an oscilloscope are used to measure a variety of signals. Finally, theory is coupled with experimental results applied to an AD9235 analog-to-digital converter (ADC). Rev. 0 | Page 1 of 20 AN-1067 Application Note TABLE OF CONTENTS Introduction ...................................................................................... 1 Test Equipment ............................................................................... 10 General Description ......................................................................... 1 Oscilloscopes ............................................................................... 10 Revision History ............................................................................... 2 Spectrum Analyzers ................................................................... 10 Definitions ......................................................................................... 3 Proof ............................................................................................. 11 Phase Noise ................................................................................... 3 Lab Results ....................................................................................... 12 Jitter ................................................................................................ 3 Signal 1 ......................................................................................... 12 Power Spectral Density .................................................................... 5 Signal 2 ......................................................................................... 12 Example 1 ...................................................................................... 6 Signal 3 ......................................................................................... 13 Example 2—Phase Noise ............................................................. 6 High Speed Converter ................................................................... 15 Example 3—Jitter .......................................................................... 7 Conclusion....................................................................................... 17 Application to Converters ............................................................... 8 References .................................................................................... 17 Example 1 ...................................................................................... 9 REVISION HISTORY 4/10—Revision 0: Initial Version Rev. 0 | Page 2 of 20 Application Note AN-1067 DEFINITIONS Phase noise and jitter have various interpretations. In the context The function Φ(t) can be composed of frequency components of this application note, phase noise and jitter are defined as not related to ωt, for example, thermal noise, shot noise, and 1/f follows: noise (flicker noise). However, in most cases, it is modeled as Consider the sinusoidal signal, Gaussian noise (see Frequency Synthesizers Theory and Design Third Edition by Manassewitsch in the References section). sin(ωt + A) (1) Similarly, a sample clock can be considered a periodic square where: wave with rising and falling edges repeating at a fixed time ω = 2πf. interval, τ, such that f is the desired frequency. A is a constant phase offset. τ = 1/f (3) 1.5 1.0 0.8 0.6 1.0 0.4 0.2 t) 0.5 ω 0 sin( –0.2 –0.4 0 –0.6 –0.8 –0.5 –1.0 0 0.5 1.0 1.5 2.0 0 /2 3 /2 2 PERIOD ( ) 0 ≤ ωt ≤ 2 08932-003 08932-001 Figure 3. Sampling Clock Figure 1. Normalized Sinusoidal Signal PHASE NOISE JITTER Phase noise is defined as an arbitrary function Φ(t) such that Jitter can be defined as an additive time variation Δ(t) to the Equation 1 becomes fixed interval τ, giving sin(ωt + A + Φ(t)) (2) τ + Δ(t) = 1/f + Δ(t) (4) 1.5 1.0 0.8 0.6 1.0 0.4 0.2 (t)) θ 0.5 t + 0 ω –0.2 sin( –0.4 0 –0.6 –0.8 –0.5 –1.0 0 0.5 1.0 1.5 2.0 0 /2 3 /2 2 PERIOD ( ) 0 ≤ ωt ≤ 2 08932-004 08932-002 Figure 4. Sampling Clock with Jitter Figure 2. Sinusoidal Signal with Phase Noise Likewise, Δ(t), is typically characterized as Gaussian noise. Rev. 0 | Page 3 of 20 AN-1067 Application Note 0 Noise analysis is straightforward above 5 kHz until the active –20 devices are limited at high frequency. Noise below 5 kHz exceeds –40 the shot noise and thermal noise. This noise varies inversely with frequency and is identified as 1/f noise. Figure 5 shows a –60 typical noise spectrum of an oscillator (see Manassewitsch in –80 the References section). –100 –120 NORMALIZED POWER(dB) –140 –160 –180 100 105 1010 1015 1020 FREQUENCY (MHz) 08932-005 Figure 5. Single-Sideband Noise Spectrum of an Oscillator Rev. 0 | Page 4 of 20 Application Note AN-1067 POWER SPECTRAL DENSITY Time domain signals have a direct relation to the frequency 1.0 domain through the Fourier transform (see Discrete-Time Signal 0.9 Processing by Oppenheim in the References section). The 0.8 Fourier transform can be viewed as the magnitude and phase 0.7 spectrum of a signal. A signal’s power can also be viewed in the 0.6 frequency domain. The power spectrum or power spectral 0.5 density is given by 0.4 Syy(ω) = Y(ω) × Y’(ω) (5) 0.3 where Y(ω) is the Fourier transform of y(t). POWER NORMALIZED TO 1 0.2 As stated previously in the Definitions section, Φ(t) can be any 0.1 arbitrary undesired signal. To simplify this analysis, Φ(t) is set 0 to a single frequency. Consider the following: 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 (Hz) × 104 08932-006 Φ(t) = θdsinωmt (6) Figure 6. Power Spectral Density, Syy(ω) Such that Equation 2 becomes Figure 6 is a plot of y(t) = sin(ωct + θdsinωmt) (7) Syy(ω) = Y(ω) × Y’(ω) The result is a phase-modulated signal, y(t), with maximum where Y(ω) is the Fourier Transform of y(t). phase deviation in radians, θd, at a frequency, fm, with ωm = 2πfm, and no offset, A = 0. Syy(ω) displays the magnitude of the power at the frequency, f. The power spectral density of the signal, y(t), modulated by a The Jacobi-Anger expansion (see Concise Encyclopedia of single frequency, fm, only has components at fc and fm with Mathematics by Weisstein in the References section) states that Bessel-squared magnitudes. iz cos( ) n in The higher order Bessel coefficients attenuate very quickly. A e i Jn () z e (8) n log power scale provides better dynamic range, showing the or higher order components in the same view as the large carrier component. The log of Syy(ω) is given by the following equation: e iz sin( ) J() z e in (9) n Lpy(ω) = 10log10(Syy(ω)) (13) n can be manipulated with the help of Euler’s identity to give and is shown in Figure 7. 0 cos(z sin ) J0 ( z ) 2 J2n ( z )cos(2 n ) (10) n1 –20 and –40 –60 sin(z sin ) 2 J2n 1 ( z )cos[(2 n 1) ] (11) n1 –80 (dB) where the Jn(z) factors are Bessel functions of the first kind. –100 Using trigonometric identities, Equation 7, Equation 10, and –120 Equation 11 can be manipulated to give –140 0 d c y(t) = J (θ )sin(ω t) –160 + J1(θd)[sin(ωc+ ωm)t – sin(ωc − ωm)t] 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 2 d c m c m + J (θ )[sin(ω + 2ω )t – sin(ω − 2ω )t] (Hz) × 104 08932-007 + J3(θd)[sin(ωc+ 3ωm)t – sin(ωc − 3ωm)t] Figure 7. Log of Power Spectral Density, Syy(ω) + … (12) Additional terms are now clearly visible. As the phase deviation From Equation 12, it can be seen that y(t) has a first-order Bessel increases, the magnitude at the carrier frequency decreases and component at the carrier frequency, fc, and Bessel-weighted the magnitude of the modulation terms increases. 500 mrad of signals at multiples of the modulation frequency, fm, offset from phase deviation reduces the carrier
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  • Image Denoising by Autoencoder: Learning Core Representations

    Image Denoising by Autoencoder: Learning Core Representations

    Image Denoising by AutoEncoder: Learning Core Representations Zhenyu Zhao College of Engineering and Computer Science, The Australian National University, Australia, [email protected] Abstract. In this paper, we implement an image denoising method which can be generally used in all kinds of noisy images. We achieve denoising process by adding Gaussian noise to raw images and then feed them into AutoEncoder to learn its core representations(raw images itself or high-level representations).We use pre- trained classifier to test the quality of the representations with the classification accuracy. Our result shows that in task-specific classification neuron networks, the performance of the network with noisy input images is far below the preprocessing images that using denoising AutoEncoder. In the meanwhile, our experiments also show that the preprocessed images can achieve compatible result with the noiseless input images. Keywords: Image Denoising, Image Representations, Neuron Networks, Deep Learning, AutoEncoder. 1 Introduction 1.1 Image Denoising Image is the object that stores and reflects visual perception. Images are also important information carriers today. Acquisition channel and artificial editing are the two main ways that corrupt observed images. The goal of image restoration techniques [1] is to restore the original image from a noisy observation of it. Image denoising is common image restoration problems that are useful by to many industrial and scientific applications. Image denoising prob- lems arise when an image is corrupted by additive white Gaussian noise which is common result of many acquisition channels. The white Gaussian noise can be harmful to many image applications. Hence, it is of great importance to remove Gaussian noise from images.