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Transcript
Sigma Delta A/D Converter e[n] x(t) Bandlimited to fo Sampler fs x[n] Modulator fs y[n] Decimation Filter 2 fo 16 bits Digital Analog fs Over Sampling Ratio = 2f 2fo is Nyquist frequency o Transfer function for an Lth order modulator given by L 1 Y ( z ) X ( z ) 1 z E ( z ) Modulator Characteristics Y ( z) • Highpass character for noise transfer function: E ( z) • In-band noise power is given by 2 f o no erms 2 L 1 f s L • • • • L 0.5 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 L 2 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 H ( z) 1 1 1 z K 1 1 z 2 N 1 z 1 N N K 1 z N K • Bit-width of the stage is given by K log 2 N b ;‘b’ is the output of the modulator • Decimation in stages to ease hardware implementation • Typically, K L 1 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!!)