Tones 5 th order Σ modulator DC inputs Tones Dither kt/c noise EECS 47 Lecture : Oversampled ADC Implementation B. Boser 5 th Order Modulator /5 /6-/64 b b b b X / /4 /4 /8 kz - -z - I kz - -z - I k3z - -z - I3 k4z - -z - I4 k5z - -z - I5 3 4 5 6 I_ I_ I_3 I_4 I_5 a a a3 a4 a5 a a / a3 / a4 /4 a5 /4 7 Q + / - Stable input range ~ -.3 +.3 Comparator Y see L_L5_sim.mdl and L_L5.m EECS 47 Lecture : Oversampled ADC Implementation B. Boser
Output Spectrum [dbwn] / Int. Noise [dbv] 5-5 - 5 th Order Noise Shaping Tones at f s /-Nf in exceed input Output Spectrum Integrated Noise (3 averages) -5 5 5 Frequency [Hz] x 5 Input:.V, sinusoid 5 point DFT 3 averages EECS 47 Lecture : Oversampled ADC Implementation B. Boser 3 In-Band Noise Output Spectrum [dbwn] / Int. Noise [dbv] 5-5 - Output Spectrum Integrated Noise (3 averages) In-Band quantization noise: db! -5 3 4 5 Frequency [Hz] x 4 EECS 47 Lecture : Oversampled ADC Implementation B. Boser 4
Output Spectrum [dbwn] / Int. Noise [dbv] 5-5 - 5 th Order Noise Shaping 5dB stopband attenuation needed to attenuate unwanted f s /-Nf in components down to the in-band quantization noise level Output Spectrum Integrated Noise (3 averages) -5 5 5 Frequency [Hz] x 5 Input:.V, sinusoid 5 point DFT 3 averages EECS 47 Lecture : Oversampled ADC Implementation B. Boser 5 Out-of-Band vs In-Band Signals A digital (low-pass) filter with suitable coefficient precision can eliminate out-of-band quantization noise No filter can attenuate unwanted in-band components without attenuating the signal We ll spend some time making sure the components at f s /-Nf in will not mix down to the signal band But first, let s look at the modulator response to small DC inputs (or offset) EECS 47 Lecture : Oversampled ADC Implementation B. Boser 6
Output Spectrum [dbwn] / Int. Noise [dbv] 5-5 - Σ Tones Output Spectrum Integrated Noise (3 averages) 6kHz khz -5 3 4 5 Frequency [Hz] x 4 mv DC input (V full-scale) Simulation technique: A random st input randomizes the noise and enables averaging. Without the small tones are not visible. EECS 47 Lecture : Oversampled ADC Implementation B. Boser 7 Limit Cycles Representing a DC term with a /+ pattern e.g. 3 + 3 + 3 + 3 + 3 + + 3 4 5 44444444 44444444 3 444444444 444444444 3 Spectrum f s f s f s 3 K EECS 47 Lecture : Oversampled ADC Implementation B. Boser 8
Limit Cycles Fundamental f δ = f V DC s VDAC mv = 3MHz V = 6kHz Tone velocity df δ dv DC f = V s DAC = 3kHz/V EECS 47 Lecture : Oversampled ADC Implementation B. Boser 9 Σ Tones Output Spectrum [dbwn] / Int. Noise [dbv] 5-5 - 6kHz Output Spectrum Integrated Noise (3 averages) -5.47.475.48.485.49.495.5 Frequency [Hz] x 6 EECS 47 Lecture : Oversampled ADC Implementation B. Boser
Σ Tones Tones follow the noise shape The fundamental of a tone that falls into a quantization noise null disappears V DC = V FB f f.5khz = V 3MHz = 3.5mV δ s EECS 47 Lecture : Oversampled ADC Implementation B. Boser Σ Tones Output Spectrum [dbwn] / Int. Noise [dbv] 5-5 - Output Spectrum Integrated Noise (3 averages) -5 3 4 5 Frequency [Hz] x 4 3.5mV DC input EECS 47 Lecture : Oversampled ADC Implementation B. Boser
In-band tones look like signals Σ Tones Can be a big problems in some applications E.g. audio even tones with power below the quantization noise floor can be audible Tones near f s / can be aliased down into the signal band Since they are often strong, even a small alias can be a big problem We will look at mechanisms that alias tones in the next lecture First let s look at dither as a means to reduce or eliminate inband tones EECS 47 Lecture : Oversampled ADC Implementation B. Boser 3 Dither DC inputs can of course be represented by many possible bit patterns Including some that are random but still average to the DC input The spectrum of such a sequence has no tones How can we get a SD modulator to produce such randomized sequences? EECS 47 Lecture : Oversampled ADC Implementation B. Boser 4
Dither The target DR for our audio SD is 6 Bits, or 98dB Let s choose the sampling capacitor such that it limits the dynamic range: = ( V ) 9.8 FS DR = kbt C kbt C = DR ( V ) FS k T B ( V) = 5.5pF v n = kbt C = 9µV EECS 47 Lecture : Oversampled ADC Implementation B. Boser 5 Dither Output Spectrum [dbwn] 5-5 - No dither With dither mv DC input -5 3 4 5 Frequency [Hz] x 4 EECS 47 Lecture : Oversampled ADC Implementation B. Boser 6
Dither Output Spectrum [dbwn] 5-5 - No dither With dither -5.47.475.48.485.49.495.5 Frequency [Hz] x 6 Dither at an amplitude which buries the inband tones has virtually no effect on tones near f s / EECS 47 Lecture : Oversampled ADC Implementation B. Boser 7 kt/c Noise So far we ve looked at noise added to the input of the SD modulator, which is also the input of the first integrator Now let s add noise also to the input of the second integrator Let s assume a 4pF sampling capacitor This gives.4 x 3µV rms noise (two uncorrelated 3µV samples per clock) EECS 47 Lecture : Oversampled ADC Implementation B. Boser 8
kt/c Noise Output Spectrum [dbwn] / Int. Noise [dbv] 5-5 - No noise st Integrator nd Integrator -5 3 4 5 Frequency [Hz] x 4 mv DC input Noise from nd integrator smaller than st integrator noise shaped Why? EECS 47 Lecture : Oversampled ADC Implementation B. Boser 9 kt/c Noise vn b b b b X kz - -z - I kz - -z - I k3z - -z - I3 k4z - -z - I4 k5z - -z - I5 3 4 5 6 I_ I_ I_3 I_4 I_5 a a a a a3 a3 a4 a4 a5 a5 7 Q Comparator Y Noise from st integrator is added directly to the input Noise from nd integrator is first-order noise shaped Noise from subsequent integrators is attenuated even further Especially for high oversampling ratios, only the first or integrators add significant thermal noise. This is true also for other imperfections. EECS 47 Lecture : Oversampled ADC Implementation B. Boser
Dither Output Spectrum [dbwn] / Int. Noise [dbv] 5-5 - No noise st Integrator nd Integrator -5.47.475.48.485.49.495.5 Frequency [Hz] x 6 No practical amount of dither eliminates the tones near f s / EECS 47 Lecture : Oversampled ADC Implementation B. Boser Full-Scale Inputs With practical levels of thermal noise added, let s try a 5kHz sinusoidal input near full-scale (.3V) No distortion is visible in the spectrum -Bit modulators are intrinsically linear But tones exist at high frequencies to the oversampled modulator, a sinusoidal input looks like two slowly alternating DCs hence giving rise to limit cycles EECS 47 Lecture : Oversampled ADC Implementation B. Boser
Full-Scale Inputs 5 Output Spectrum [dbwn] -5 - Output Spectrum Integrated Noise (3 averages) -5 3 4 5 Frequency [Hz] x 4 EECS 47 Lecture : Oversampled ADC Implementation B. Boser 3 Full-Scale Inputs 5 Output Spectrum [dbwn] -5 - Output Spectrum Integrated Noise (3 averages) -5 5 5 Frequency [Hz] x 5 EECS 47 Lecture : Oversampled ADC Implementation B. Boser 4
V ref Interference Dither successfully removes in-band tones that would corrupt the signal The high-frequency tones in the quantization noise spectrum will be removed by the digital filter following the modulator What if some of these strong tones are demodulated to the base-band before digital filtering? Why would this happen? EECS 47 Lecture : Oversampled ADC Implementation B. Boser 5 AM Modulation x (t) x (t) y(t) x x x ( t) = X cos( ωt) ( t) = X cos( ω t) X X ( t) x ( t) = [ cos( ω t + ω t) + cos( ω t ω t) ] EECS 47 Lecture : Oversampled ADC Implementation B. Boser 6
AM Modulation in DAC V ref y(t) DAC v(t) y V v ( t) ref = D out = V + mv f / square wave ( t) = y( t) = ± V ref s = fundamental +.5% of spectrum at f / s EECS 47 Lecture : Oversampled ADC Implementation B. Boser 7 AM Modulation in DAC D OUT spectrum V ref spectrum interferer convolution yields sum of red and green, mirrored tones and noise appear in band f s / f s EECS 47 Lecture : Oversampled ADC Implementation B. Boser 8
V ref Interference Output Spectrum [dbwn] 5-5 - 6dB ( db/db) V e-6v.v -5 3 4 5 Frequency [Hz] x 4 EECS 47 Lecture : Oversampled ADC Implementation B. Boser 9 V ref Interference Simulation are for specified amounts of f s / interference in the DAC reference As predicted interference demodulates the high-frequency tones Since the high frequency tones are strong, a small amount (µv) of interference suffices to create huge base-band tones Stronger interference (mv) rises the noise floor also Amplitude of demodulated tones is proportional to interference EECS 47 Lecture : Oversampled ADC Implementation B. Boser 3
V ref Interference Output Spectrum [dbwn] 5-5 - V e-6v.v -5 3 4 5 Frequency [Hz] x 4 Output Spectrum [dbwn] 5-5 - V e-6v.v -5.47.48.49.5 Frequency [Hz] x 6 Symmetry of the spectra at f s / and DC confirm that this is AM modulation EECS 47 Lecture : Oversampled ADC Implementation B. Boser 3 V ref Tone Velocity Output Spectrum [dbwn] 5-5 - V in V ref.5khz/mv = 6mV / mv.6v DC = V DC.V + mv f s / square wave -5 3 4 5 Frequency [Hz] x 4 EECS 47 Lecture : Oversampled ADC Implementation B. Boser 3
V ref Tone Velocity The velocity of AM demodulated tones is half that of the native tone Such differences help debugging of real silicon How clean does the reference have to be? EECS 47 Lecture : Oversampled ADC Implementation B. Boser 33 V ref Interference Output Spectrum [dbwn] / Int. Noise [dbv] 5-5 - Output Spectrum Integrated Noise (3 averages) Tone dominates noise floor w/o thermal noise -5 3 4 5 Frequency [Hz] x 4 EECS 47 Lecture : Oversampled ADC Implementation B. Boser 34
V ref Interference db of clock-to-v ref isolation is not sufficient for digital audio applications Achieving this level of performance requires careful engineering Getting an accurate requirement is the first (and an essential) step See E. Swanson, N. Sooch, and D. Knapp, Method for Reducing Effects of Electrical Noise in Analog-to-Digital Converter, U.S. Patent 4746899, 988 for more ideas EECS 47 Lecture : Oversampled ADC Implementation B. Boser 35 Summary Our stage model can drive almost all capacitor sizing decisions Gain scaling kt/c noise Dither Dither removes effectively in-band tones Actual tonality determined by demodulation of limit cycles near f s / Next we will add relevant component imperfections, e.g. Real capacitors aren t perfect Real opamps aren t ideal We ll model nonlinearities in the Σ system next time EECS 47 Lecture : Oversampled ADC Implementation B. Boser 36