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Sigma Delta A/D Converter. e [n]. f s. f s. Decimation Filter. 2 f o. x (t). Sampler. Modulator. x [n]. y [n]. 16 bits. Bandlimited to f o. Digital. Analog. Over Sampling Ratio = 2 f o is Nyquist frequency Transfer function for an L th order modulator given by.

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sigma delta a d converter
Sigma Delta A/D Converter






2 fo






16 bits

Bandlimited to fo



Over Sampling Ratio =

2fo is Nyquist frequency

Transfer function for an Lth order modulator given by

modulator characteristics
Modulator Characteristics
  • Highpass character for noise transfer function:
  • In-band noise power is given by
  • no falls by 3(2L+1) for doubling of Over Sampling Ratio
  • L+0.5 bits of resolution for doubling of Over Sampling Ratio
  • no essentially is uncorrelated for
  • Dithering is used to decorrelate quantization noise
  • Select Over Sampling Ratio and L such that quantization noise is not the limiting factor
  • Switched capacitor circuits
    • easy to build in a digital CMOS process
    • gains and time constants decided by capacitor ratios and clock frequency
  • Fully differential circuits achieve better power supply rejection and common mode noise rejection
  • Analog characteristics are very sensitive to layout
    • layouts are made symmetrical to overcome variations in process
influence of circuit parameters
Influence of Circuit Parameters
  • Infinite DC gain for the integrators is unrealistic
    • Finite DC gain (“integrator leakage”) causes DC offset and increased baseband noise
    • Always build the best possible op-amp for the first integrator
  • Non-linearity in the feedback D/A converter
    • Harmonic distortion in the output signal
    • Possible modulation of the reference voltage (bad!!)
    • A simple 2 level D/A (two switches and a reference voltage) is used
  • Circuit noise is usually the performance limiting factor
    • kT/C noise in the capacitors
    • kTR noise in the resistors and switches
    • Thermal and 1/f noise in the MOSFETS
  • Sample rate conversion from a high rate to Nyquist rate
  • Performed using cascaded digital FIR filters
  • One class of filters used are called CICs (cascaded integrator comb filters) with the transfer function


  • Bit-width of the stage is given by ;‘b’ is the output of the modulator
  • Decimation in stages to ease hardware implementation
  • Typically,
sigma delta d a converters
Sigma Delta D/A Converters
  • Modulator loop is digital
  • Theory and math applicable exactly: quantization error is replaced by truncation error
  • Interpolation filter instead of sampler to raise sample rate
  • Analog part: A 1 bit D/A followed by one or more filters
    • Harder to build than A/D counterparts (!!) (analog part has no feedback loop to take advantage of)
    • Switched capacitor D/As, Current steering D/As are popular
    • Switched capacitor filters followed by a continuous time smoothing filter
    • Tapped delay line FIR filters are also used (tends to be larger in area)
general circuit considerations
General Circuit Considerations
  • Keep analog and digital circuitry on separate power supplies and spaced as far as possible
  • Use the biggest capacitors possible (area and loading on amplifiers are issues)
  • Use the smallest switches possible (lower noise, lower parasitic capacitive coupling)
  • Low thermal and 1/f noise in op-amps
  • Keep signal level as large as possible in the signal path
  • Keep the reference voltage clean (easier said than done!!)