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The Case for Oversampling EE247 Lecture 24 Oversampled ADCs – Why oversampling? – Pulse-count modulation – Sigma-delta modulation • 1-Bit quantization • Quantization error (noise) spectrum • SQNR analysis • Limit cycle oscillations –2nd order ΣΔ modulator • Dynamic range • Practical implementation – Effect of various nonidealities on the ΣΔ performance EECS 247 Lecture 24: Oversampling Data Converters © 2005 H. K. Page 1 The Case for Oversampling Nyquist sampling: Signal fs “narrow” “Nyquist” DSP transition fs >2B +δ ADC B Freq AA-Filter Sampler Oversampling: fs >> fN?? Signal “wide” Oversampled ADC DSP transition fs= MfN B Freq AA-Filter Sampler • Nyquist rate fN = 2B • Oversampling rate M = fs/fN >> 1 EECS 247 Lecture 24: Oversampling Data Converters © 2005 H. K. Page 2 Nyquist v.s. Oversampled Converters Antialiasing |X(f)| Input Signal frequency fB Nyquist Sampling f B fs 2fs frequency fS ~2fB Anti-aliasing Filter Oversampling fB fs frequency fS >> 2fB EECS 247 Lecture 24: Oversampling Data Converters © 2005 H. K. Page 3 Oversampling Benefits • No stringent requirements imposed on analog building blocks • Takes advantage of the availability of low cost, low power digital filtering • Relaxed transition band requirements for analog anti-aliasing filters • Reduced baseband quantization noise power • Allows trading speed for resolution EECS 247 Lecture 24: Oversampling Data Converters © 2005 H. K. Page 4 ADC Converters Baseband Noise Δ • For a quantizer with step size and sampling rate fs : – Quantization noise power distributed uniformly across Nyquist bandwidth ( fs/2) Ne(f) NB -fs /2 -fB fB f s /2 – Power spectral density: 22⎛⎞Δ ==e1 N(f)e ⎜⎟ fs ⎝⎠12 fs – Noise is aliased into the Nyquist band –fs /2 to fs /2 EECS 247 Lecture 24: Oversampling Data Converters © 2005 H. K. Page 5 Oversampled Converters Baseband Noise ff BB⎛⎞Δ2 ==1 SN(f)dfBe∫∫⎜⎟ df −− ffBB⎝⎠12 fs Ne(f) Δ2 ⎛⎞ = 2fB ⎜⎟ NB 12⎝⎠ fs = where for fBs f /2 Δ2 = SB0 -fs /2 -fB fB f s /2 12 ⎛⎞2f S SS==BB0 BB0⎜⎟ ⎝⎠fMs f where M==s oversampling ratio 2fB EECS 247 Lecture 24: Oversampling Data Converters © 2005 H. K. Page 6 Oversampled Converters Baseband Noise ⎛⎞2f S SS==BB0 BB0⎜⎟ ⎝⎠fMs f where M==s oversampling ratio 2fB 2X increase in M Æ 3dB reduction in SB Æ ½ bit increase in resolution/octave oversampling To increase the improvement in resolution: Embed quantizer in a feedback loop ÆPredictive (delta modulation) ÆNoise shaping (sigma delta modulation) EECS 247 Lecture 24: Oversampling Data Converters © 2005 H. K. Page 7 Pulse-Count Modulation Nyquist V (kT) in ADC t/T 0 1 2 Oversampled V (kT) in ADC, M = 8 t/T 0 1 2 Mean of pulse-count signal approximates analog input! EECS 247 Lecture 24: Oversampling Data Converters © 2005 H. K. Page 8 Pulse-Count Spectrum Magnitude f • Signal: low frequencies, f < B << fs • Quantization error: high frequency, B … fs / 2 • Separate with low-pass filter! EECS 247 Lecture 24: Oversampling Data Converters © 2005 H. K. Page 9 Oversampled ADC Predictive Coding + vIN ADC _ DOUT Predictor • Quantize the difference signal rather than the signal itself • Smaller input to ADC Æ Buy dynamic range • Only works if combined with oversampling • 1-Bit digital output • Digital filter computes “average” ÆN-Bit output EECS 247 Lecture 24: Oversampling Data Converters © 2005 H. K. Page 10 Oversampled ADC f = M f f = f +δ s1 N s2 N Signal E.g. Decimator “wide” Pulse-Count f = Mf “narrow” DSP transition s N Modulator transition Analog Sampler Modulator Digital AA-Filter AA-Filter B Freq 1-Bit Digital N-Bit Digital Decimator: • Digital (low-pass) filter • Removes quantization error for f > B • Provides most anti-alias filtering • Narrow transition band, high-order • 1-Bit input, N-Bit output (essentially computes “average”) EECS 247 Lecture 24: Oversampling Data Converters © 2005 H. K. Page 11 Modulator • Objectives: – Convert analog input to 1-Bit pulse density stream – Move quantization error to high frequencies f >>B – Operates at high frequency fs >> fN • M = 8 … 256 (typical)….1024 • Since modulator operated at high frequencies Æ need to keep circuitry “simple” Æ ΣΔ = ΔΣ Modulator EECS 247 Lecture 24: Oversampling Data Converters © 2005 H. K. Page 12 Sigma- Delta Modulators Analog 1-Bit ΣΔ modulators convert a continuous time analog input vIN into a 1-Bit sequence dOUT fs + vIN H(z) _ dOUT DAC Loop filter 1b Quantizer (comparator) EECS 247 Lecture 24: Oversampling Data Converters © 2005 H. K. Page 13 Sigma-Delta Modulators • The loop filter H can be either switched-capacitor or continuous time • Switched-capacitor filters are “easier” to implement + frequency characteristics scale with clock rate • Continuous time filters provide anti-aliasing protection • Loop filter can also be realized with passive LC’s at very high frequencies fs + vIN H(z) _ dOUT DAC EECS 247 Lecture 24: Oversampling Data Converters © 2005 H. K. Page 14 Oversampling A/D Conversion fs fs /M Input Signal Bandwidth Oversampling 1-bit Decimation n-bit Modulator Filter @ f /M B=fs /2M @ fs s fs = sampling rate M= oversampling ratio • Analog front-end Æ oversampled noise-shaping modulator • Converts original signal to a 1-bit digital output at the high rate of (2MXB) • Digital back-end Æ digital filter • Removes out-of-band quantization noise • Provides anti-aliasing to allow re-sampling @ lower sampling rate EECS 247 Lecture 24: Oversampling Data Converters © 2005 H. K. Page 15 1st Order ΣΔ Modulator In a 1st order modulator, simplest loop filter Æ an integrator z-1 H(z) = 1 – z-1 + vIN _ ∫ dOUT DAC EECS 247 Lecture 24: Oversampling Data Converters © 2005 H. K. Page 16 1st Order ΣΔ Modulator Switched-capacitor implementation φ φ 1 2 φ 2 - dOUT 1,0 + Vi +Δ/2 -Δ/2 EECS 247 Lecture 24: Oversampling Data Converters © 2005 H. K. Page 17 1st Order ΔΣ Modulator + vIN _ ∫ Δ ≤ ≤ Δ - /2 vIN + /2 dOUT -Δ/2 or +Δ/2 DAC • Properties of the first-order modulator: – Analog input range is equal to the DAC reference – The average value of dOUT must equal the average value of vIN – +1’s (or –1’s) density in dOUT is an inherently monotonic function of vIN Æ linearity is not dependent on component matching – Alternative multi-bit DAC (and ADCs) solutions reduce the quantization error but loose this inherent monotonicity & relaxed matching requirements EECS 247 Lecture 24: Oversampling Data Converters © 2005 H. K. Page 18 1st Order ΣΔ Modulator Tally of 1-Bit quantization error quantizer Analog input 1 2 3 Δ ≤ ≤ Δ X Q Y - /2 Vin + /2 z-1 -1 1-z 1-Bit digital Sine Wave Integrator Comparator output stream, -1, +1 Instantaneous Implicit 1-Bit DAC quantization error +Δ/2, -Δ/2 (Δ = 2) • M chosen to be 8 (low) to ease observability EECS 247 Lecture 24: Oversampling Data Converters © 2005 H. K. Page 19 1st Order Modulator Signals 1st Order Sigma-Delta X analog input 1.5 X Q Q tally of q-error Y Y digital/DAC output 1 0.5 Mean of Y approximates X 0 Amplitude -0.5 -1 T = 1/fs = 1/ (M fN) -1.5 0 10 20 30 40 50 60 Time [ t/T ] EECS 247 Lecture 24: Oversampling Data Converters © 2005 H. K. Page 20 ΣΔ Modulator Characteristics • Quantization noise and thermal noise (KT/C) distributed over –fs /2 to +fs /2 Æ Total noise within signal bandwidth reduced by 1/M • Very high SQNR achievable (> 20 Bits!) • Inherently linear for 1-Bit DAC • To first order, quantization error independent of component matching • Limited to moderate & low speed EECS 247 Lecture 24: Oversampling Data Converters © 2005 H. K. Page 21 Output Spectrum • Definitely not white! 30 • Skewed towards higher 20 Input frequencies 10 • Notice the distinct tones 0 -10 -20 Amplitude [ dBWN ] • dBWN (dB White Noise) -30 scale sets the 0dB line -40 at the noise per bin of a random -1, +1 -50 0 0.1 0.2 0.3 0.4 0.5 sequence Frequency [ f /fs] EECS 247 Lecture 24: Oversampling Data Converters © 2005 H. K. Page 22 Quantization Noise Analysis Quantization Error e(kT) Integrator −1 = z Σ H(z ) −1 Σ x(kT) 1+ z y(kT) Quantizer Model • Sigma-Delta modulators are nonlinear systems with memory Æ difficult to analyze directly • Representing the quantizer as an additive noise source linearizes the system EECS 247 Lecture 24: Oversampling Data Converters © 2005 H. K. Page 23 Signal Transfer Function −1 z x(kT) H(z ) = Σ Integrator −1 y(kT) 1+ z - H(z) ω Hj()ω = 0 jω Signal transfer function Æ low pass function: ω = 1 HjSig () + s Magnitude 1 ω 0 Yz() Hz () − f Frequency Hz()== = z1 ⇒ Delay 0 Sig Xz() 1+ Hz () EECS 247 Lecture 24: Oversampling Data Converters © 2005 H. K. Page 24 Noise Transfer Function Qualitative Analysis 2 vn v ω i 0 vo 2 Σ veq - jω 22=×⎛⎞f vveq n ⎜⎟ 2 ⎝⎠f0 2 × ⎛⎞f vn ⎜⎟ ⎝⎠f0 ω v vi Σ 0 o - jω f Frequency 2 0 22=×⎛⎞f vveq n ⎜⎟ ⎝⎠f0 ω v • Input referred-noiseÆ zero @ Σ 0 o ω DC (s-plane) vi - j EECS 247 Lecture 24: Oversampling Data Converters © 2005 H. K. Page 25 STF and NTF Quantization Error e(kT) Integrator −1 = z Σ H(z ) − Σ x(kT) 1+ z 1 y(kT) Quantizer Model Signal transfer function: Yz() Hz () − STF== =z 1 ⇒ Delay Xz() 1+ Hz () Noise transfer function: Y (z) 1 − NTF = = = 1− z 1 ⇒ Differentiator E(z) 1+ H (z) EECS 247 Lecture 24: Oversampling Data Converters © 2005 H. K. Page 26 Noise Transfer Function Yz() 1 − NTF== =−1 z 1 Ez() 1+ Hz () jTωω/2− jT /2 −−jTωω jT/2⎛⎞ee− NTF()(1 jω =− e )=2 e ⎜⎟ ⎝⎠2 − ω = 2sin/2ejTjT/2 ()ω =×−−jTωπ/2 j /2 ()ω 2sin/2ee⎣⎦⎡⎤ T = ()ω −−jT()ωπ/2 ⎣⎦⎡⎤2sinTe /2 = where Tf1/ s Thus: ωπ NTF()=2sin f() T /2=2sin() f / fs = 2 Nfy () NTF() f Ne () f EECS 247 Lecture 24: Oversampling Data Converters © 2005 H.
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