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

Modulator Characteristics Highpass character for noise transfer function: In-band noise power is given by n o falls by 3(2L+1) for doubling of Over Sampling Ratio L+0.5 bits of resolution for doubling of Over Sampling Ratio n o essentially is uncorrelated for Dithering is used to decorrelate quantization noise

Implementation 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 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

Example Implementation

Decimation 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 N 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 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 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!!)