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Spectral Transformation of Discrete Time Signal Insertion of sampled signal in Fourier formula: X(!) = +1 Z X ¡1 = X Normalization of frequency to sampling frequency: Stephan Henzler Mixed-Signal-Electronics 2012/13 n n x(nT ) x(nT )±(t ¡ nT )e ¡j!t dt +1 Z ¡1 = X x(nT )e ¡j!nT n e ¡j!t ±(t ¡ nT )dt X() = X x[n]e ¡jn X(z) = X x[n]z ¡n n FT of discrete sequence generalization z = e j = e j!T FT of sampled signal = !T = 2¼f n z-Transformation fs 18
Inverse Z-Transformation Methods for computation of inverse z-transformation – Tabular method – Decomposition in partial fractions and tabular method – Power series method (coefficients correspond to time sequence) Formal inversion: x[n] = 1 2¼ I c X(z)zn¡1 dz = X residues of X(z)z n¡1 at poles within C Stephan Henzler Mixed-Signal-Electronics 2012/13 19
Page 1 and 2: Mixed-Signal-Electronics PD Dr.-Ing
Page 3 and 4: Stephan’s ambition for this cours
Page 5 and 6: Administratives Lecture: Stephan He
Page 7 and 8: The Macroscopic World is Purely Ana
Page 9 and 10: Topics of MSE Course Structure of
Page 11 and 12: Recommended Literature Analog Integ
Page 13 and 14: Constraints of Mixed Signal Circuit
Page 15 and 16: Generic Structure of Mixed-Signal S
Page 17: Representation of Discrete Time Sig
Page 21 and 22: Representation of Discrete Time Sig
Page 23 and 24: low-pass low-pass low-pass Aliasing
Page 25 and 26: amplitude fsampling = 1 f signal,2
Page 27 and 28: Anti Aliasing Filter No brick wall
Page 29 and 30: Practical Sampling: Sample & Hold
Page 31 and 32: Xsh(!) = ¡ 1 j!¿ Sampling with Fi
Page 33 and 34: generalization X(s) = Relation Betw
Page 35 and 36: Downsampling II Mathematically dow
Page 37 and 38: Upsampling Stephan Henzler Mixed-Si
Page 39 and 40: Fractional Sample Rate Conversion
Page 41 and 42: Example for Simple Anti Aliasing Fi
Page 43 and 44: Transistor Description (for hand ca
Page 45 and 46: Chapter 2 Sample and Hold Circuits
Page 47 and 48: Sample & Hold Circuit Stephan Henzl
Page 49 and 50: Sample & Hold: Clock Feed-Through 3
Page 51 and 52: 5. 6. Clock jitter Sample & Hold Ci
Page 53 and 54: Impact of Jitter on S&H Performance
Page 55 and 56: Noise due to jitter: Jitter Effect
Page 57 and 58: S&H with Correlated Double Sampling
Page 59 and 60: Noise of Sampled Signals: kT/C-Nois
Page 61 and 62: Closed Loop Track & Hold Circuit 1
Page 63 and 64: Closed Loop Track & Hold Circuit 3
Page 65 and 66: Summary Sample & Hold Circuits Step
Page 67 and 68: Resistor Realizations in MOS Techno
Page 69 and 70:
Resistor Realizations in MOS Techno
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Switched Capacitor Resistor Emulati
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Resistor Equivalents Stephan Henzle
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Parameter Variations in SC Circuits
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Implementation of Integrated Capaci
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Vertical Parallel Plate Capacitor S
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Limitations of SC-Equivalence Steph
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Limitations of SC-Equivalence Switc
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Switched Capacitor Integrators Step
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z-Transformation Approximation on
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Analysis of Switched-Capacitor Circ
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Impact of Parasitic Capacitances C
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Switched Capacitor Amplifier Stepha
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SC Amplifier: Output Signal Stephan
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Superposition Theorem in SC Circuit
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Ideal Digital-to-Analog Conversion
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DAC Gain Error Stephan Henzler Mixe
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Ideal Analog-to-Digital Conversion
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Quantization in A/D-Converters Step
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Quantization in A/D-Converters Sig
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ADC Gain Error Stephan Henzler Mixe
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Non-Linearity in Data Converters St
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Nonlinearity in Data-Converters Ste
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DAC Nonlinearity: Differential Nonl
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ADC Nonlinearity: Integral Nonlinea
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Dynamic ADC Measurement ADC conver
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Harmonic Distortion Caused by Nonli
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Effective Number of Bits An ideal
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Other Useful Figures in Mixed-Signa
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Non-Linearity Gain Compression Co
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Superposition of Two Frequencies St
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Non-Linearity Cross Modulation Co
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Non-Linearity Intermodulation Two
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Non-Linearity: Further Reading Rec
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Chapter 5 Nyquist Rate Digital-to-A
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max value bus voltage follower stri
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Elmore Delay (cont.): Path Resistan
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Elmore Delay (cont): Delay Approxim
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Stephan Henzler Mixed-Signal-Electr
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Normalized Delay 300 250 200 150 10
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Stephan Henzler Mixed-Signal-Electr
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Monotonicity in Binary Weighted DAC
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Implementation of Binary Weighted D
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High number of bits - large area Hi
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Page 159 and 160:
R-2R-Ladder Network Stephan Henzler
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Page 163 and 164:
SC-Amplifier with Controllable Capa
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Thermometer Code Converters (method
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Hybrid Converter Architectures Step
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Charge Scaling DAC Compatibel with
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Analog-to-Digital Converter Familie
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Dual-Slope Analog-to-Digital Conver
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Iterative Analog-to-Digital Convert
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Converter with Successive Approxima
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Converter with Successive Approxima
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Modified SAR Algorithm Stephan Henz
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Charge Redistribution SAR Converter
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Charge Redistribution SAR Converter
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Hybrid SAR Converters Search can b
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Detailed SAR Architecture Let’s
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Binary Weighted SAR Stephan Henzler
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Algorithmic Analog-to-Digital Conve
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Page 197 and 198:
Illustration in Robertson Diagram 2
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Page 201 and 202:
Voltage Doubling in Algorithmic Con
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Voltage Doubling in Algorithmic Con
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Pipelined ADC 1 Going for pipeline
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High-Speed ADC: Pipeline Processing
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Impact of Comparator Offset on ADC
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Impairments in SAR Algorithms Find
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Redundant SAR Algorithms Stage qua
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Subranging Principle quantization e
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Linear Model of Subranging ADC Step
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Linear Model of Subranging ADC Eff
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General Pipeline SAR ADC Stephan He
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Background Calibration for Stage Ga
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Summary Pipeline ADC Stage errors
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implemented as parallel connection
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thermometer code one-hot code trans
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Bubble Correction in Flash Converte
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Interpolating Flash Converter Steph
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Kickback Effect Especially in flas
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Folding ADC: Operation Principle St
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High-Speed ADC: Parallel Processing
Page 241 and 242:
Chapter 7 Comparators Stephan Henzl
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Static Characteristics of Comparato
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Operational Amplifier as Comparator
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Comparator Propagation Delay Linea
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Track & Latch Circuit I Stephan Hen
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Principle of Track-and-Latch Stage
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Latched Comparators I Standard arc
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Track & latch circuit: Latched Comp
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Current Mode (CML) Latch Combines
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Elimination of Memory Effect Prech
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Chapter 8 Oversampled Converters Si
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Linear Model of Quantization Noise
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Quantization Noise and Oversampling
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Feasibility of Oversampling Goal:
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First Order Noise Shaping Stephan H
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First Order Noise Shaping Stephan H
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Calculation of Signal-to-Noise Rati
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SC Implementation of 1. Order ΣΔ
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Comparison Nyquist Rate vs. SD AD N
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Calculation of Transfer Functions S
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Simulation with Matlab/Simulink Ste
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Output Spectrum of ΣΔ-Modulator f
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Input Level Dependent SNR 2. Order
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Tones Stephan Henzler Mixed-Signal-
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Dithering Reduce the risk of tones
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maximum noise peak Tones input DC l
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STF of 3. Order ΣΔ-Modulator Step
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Cascaded ΣΔ-Modulator (Multi stAg
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3. Order MASH Structure Stephan Hen
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hist(S1) 12000 10000 8000 6000 4000
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hist(S1) 2.5 2 1.5 1 0.5 0 -1 -0.8
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ΣΔ-Analog-to-Digital Conversion S
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Sinc-Filter Basic low-pass filter a
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Sinc Filter Transfer Functions Step
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Implementation of Sinc-Filter Steph
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ΣΔ-Digital-to-Analog Conversion S
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Continuous Time Sigma-Delta Convert
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Comparison of Discrete- and Continu
Page 317:
Web References Murmann. VLSI Data
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