Computation of Noise Spectral Density in Switched Capacitor Circuits using the MixedFrequencyTime Te

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Computation of Noise Spectral Density in Switched
Capacitor Circuits using the Mixed-Frequency-Time
Technique

• Vinita Vasudevan and Ramakrishna Mokkapati
• Indian Institute of Technology Madras

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Outline

• Existing methods
• Algorithm for computation of noise spectral
  density
• Methods of solution
• Results
• Advantages and limitations

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Previous Work
Use the Adjoint Network Technique Okumura et
al,1993,Telichevsky et al, 1996,Yuan and Opal,
2000

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Previous work Contd
Find the impulse response h(t,t)
Rice, 1970 Toth, Yusim,Suyama - 1999
Assuming white noise input
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Noise Algorithm - Definitions
Expected power spectral density, PSD,
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DefinitionsContd.
x(t) is a stationary process
x(t) is a non-stationary process
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Circuit Equations
x(t) state vector, v(t) large signal
excitation
If noise xn(t), is regarded as a perturbation
A(t) Jacobian at steady state
B(t) Spectral intensity of noise sources
W(t) Wiener process
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Noise Equations
Define
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Application toSwitched Capacitor Circuits

• Linear periodically switched ( for noise
  analysis)
• Output noise is cyclo-stationary with a
  periodequal to clock period

wc clock frequency
wm measurement frequency
(Steady state)
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Computation of PSD

• K(t) contains a tone at ? and spectral
  componentsat ? ? n?c
• To compute the PSD, we need the component d0
  of KN(t)
• Shooting Newton technique or the
  mixed-frequency-time technique

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Shooting Newton Method

• Assumes K(t) is periodic This occurs for
  example,if ? All components of K(t) are
  periodic with period
• Start with an initial guess K(0)
• Integrate equations for the cross-spectral
  density for one output period
• Correct K(0) Use Newton method
• Integrate for one more cycle usingcorrected K(0)

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Cross-spectral Density from Shooting Newton Method
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Mixed Frequency-Time Method

• Sample K(t) at t0 and tTc
• Solve using Newton method
• Only two integrations over a clock cycle are
  required to compute the power spectral densityat
  a particular frequency
• Very efficient especially when ?ltlt ?cor ? ? ?c

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Equivalent circuits
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Results Low Pass Filter
Average CPU Time per frequency MFT 17.86
seconds SN 80.38 seconds
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Results Second order bandpass filter
Average CPU Time per frequency MFT 19.07
seconds SN 846.87 seconds
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Advantages and Limitations

• Algorithm can be used for all circuits in which
  noise is regarded as a perturbation
• Noise can be stationary/cyclo-stationary
• Can be easily integrated into a circuit simulator
• All noise sources considered simultaneously
• Aliasing sidebands not considered seperately
• For an N node network, N(N1)/2 equations for the
  covariance matrix
• Additional filtering networks for 1/f noise

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C2
V
AV
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