This book provides practical and comprehensive coverage of the theory and techniques behind alias–free digital signal processing.
Digital Alias–free Signal Processing is ideal for practising engineers and researchers working on the development of digital signal processing applications at extended frequencies. It is also a valuable reference for electrical and computer engineering graduates taking courses in signal processing or digital signal processing.
- Analyses issues of sampling, randomised and pseudo–randomised quantisation and direct and indirectly randomised sampling.
- Examines periodic and hybrid sampling, including information on processing algorithms and potential limitations imposed by signal dynamics.
- Sets out leading methods and techniques for complexity reduced designs, in particular designs of large aperture sensor arrays, massive data acquisition and compression from a number of signal sources, as well as complexity–reduced processing of non–uniform data.
- Presents examples of engineering applications using these techniques including spectrum analysis, waveform reconstruction and the estimation of various parameters, emphasising the importance of the technique for developing new technologies.
- Links DASP and traditional technologies by mapping them into embedded systems with standard inputs and outputs.
Frequently Used Symbols and Abbreviations.
1 Introduction: Signal Digitizing and Digital Processing.
1.1 Subject Matter.
1.2 Digitizing Dictates Processing Preconditions.
1.2.1 Connecting Computers to the Real–life World.
1.2.2 Widening of the Digital Domain.
1.2.3 Digital Signal Representation.
1.2.4 Complexity Reduction of Systems.
1.3 Approach to the Development of Signal Processing Systems.
1.4 Alias–free Sampling Option.
1.4.1 Anti–aliasing Irregularity of Sampling.
1.4.2 Sparse Nonuniform Sampling.
1.4.3 Nonuniform Sampling Events.
1.5 Remarks in Conclusion.
Part 1 Digitizing.
2 Randomization as a Tool.
2.1 Randomized Versus Statistical Signal Processing.
2.2 Accumulation of Empirical Experience.
2.2.1 Using Monte Carlo Methods for Signal Processing.
2.2.2 Polarity Coincidence Methods.
2.2.3 Stochastic Ergodic Method.
2.2.4 Stochastic Computing.
2.2.6 Generalized Scheme of Randomized Digitizing.
2.3 Discovery of Alias–free Signal Processing.
2.3.1 Early Academic Research in Randomized Temporal Sampling.
2.3.2 Early Research in Randomized Spatial Signal Processing.
2.3.3 Engineering Experience.
2.4 Randomization Leading to DASP.
2.4.1 DASP Mission.
2.4.2 Demonstrator of DASP Advantages and Limitations.
2.5 Some of the Typically Targeted Benefits.
3 Periodic Versus Randomized Sampling.
3.1 Periodic Sampling as a Particular Sampling Case.
3.1.1 Generalized Sampling Model.
3.2 Spectra of Sampled Signals.
3.2.1 Spectra of Periodically Sampled Signals.
3.2.2 Spectra of Randomly Sampled Signals.
3.3 Aliasing Induced Errors at Seemingly Correct Sampling.
3.4 Overlapping of Sampled Signal Components.
3.5 Various Approaches to Randomization of Sampling.
4 Randomized Quantization.
4.1 Randomized Versus Deterministic Quantization.
4.1.2 Input Output Characteristics.
4.1.3 Rationale of Randomizing.
4.2 Deliberate Introduction of Randomness.
4.2.1 Various Models.
4.3 Quantization Errors.
4.3.1 Probability Density Function of Errors.
4.3.2 Variance of Randomly Quantized Signals.
4.4 Quantization Noise.
4.4.1 Covariance between the Signal and Quantization Noise.
5 Pseudo–randomized Quantizing.
5.1 Pseudo–randomization Approach.
5.2 Optimal Quantizing.
5.2.1 Single–threshold Quantizing.
5.2.2 Multithreshold Quantizing.
5.2.3 Implementation Approaches.
5.3 Input Output Relationships.
5.4 Quantization Errors.
5.5 Quantization Noise.
5.5.1 Covariance between Signal and Quantization Noise.
5.5.2 Spectrum of the Pseudo–randomized Quantization Noise.
5.5.3 Noise Reduction by Oversampling.
5.6 Some Properties of Quantized Signals.
6 Direct Randomization of Sampling.
6.1 Periodic Sampling with Jitter.
6.2 Additive Random Sampling.
6.3 Sampling Function.
6.4 Elimination of Bias Errors.
7 Threshold–crossing Sampling.
7.1 Sampling at Input and Reference Signal Crossings.
7.1.1 Level–crossing Sampling.
7.1.2 Time–variant Threshold Crossings.
7.2 Representing Signals Using Timing Information.
7.3 Sine–Wave Crossings.
7.3.1 Recovery of Signal Sample Values.
7.3.2 Various Realizations.
7.4 Remote Sampling Based on Sine–Wave Crossings.
7.5 Advantages and Disadvantages.
8 Derivatives of Periodic Sampling.
8.1 Phase–shifted Periodic Sampling.
8.1.1 Dependence of Aliasing on the Sampling Phase.
8.1.2 Reconstruction of Sampled Signals.
8.2 Periodic Sampling with Random Skips.
8.2.1 General Model.
8.2.2 Typical Use.
8.3 Compensation Effect.
8.3.1 Display of Fourier Transforms.
8.3.2 Observing the Aliasing Processes.
8.4 Generation of Randomized Sampling Pulse Trains.
8.4.1 Basic Approach.
8.4.2 Practical Experience.
9 Fuzzy Aliasing.
9.1 Meaning of the DFT of a Nonuniformly Sampled Signal.
9.2 Concept of Fuzzy Aliasing.
9.2.1 Generic Periodic Sampling with Random Skips.
9.2.2 Primary and Secondary Aliasing.
9.2.3 Decomposition of Sampling Point Processes.
9.3 Anatomy of Fuzzy Aliasing.
9.3.1 Tracking of Particular Contributions.
9.3.2 Incomplete Compensation of Aliases.
9.3.3 Aliasing at Multiple Frequencies.
9.4 Object Lesson.
10 Hybrid Sampling.
10.1 Hybrids of Periodic and Random Sampling.
10.1.1 Basic Approach.
10.1.2 Arrangements for Sample Value Processing.
10.2 Hybrid Double Sampling.
10.2.1 Providing for Short Sampling Intervals.
10.2.2 Double Periodic Sampling with Jitter.
10.2.3 Double Additive Pseudo–random Sampling.
10.2.4 Periodic Additive Pseudo–random Sampling.
10.3 Mixing Hybrid Sampling with Periodic Sampling.
10.4 Comments in Conclusion.
Part 2 Processing.
11 Data Acquisition.
11.1 Data Acquisition from Wideband Signal Sources.
11.1.1 Practical Results Confirming the Theory.
11.1.2 Sampling with Reduced Uncontrolled Jitter.
11.2 Application of Hybrid Double Sampling.
11.3 Pseudo–randomized Multiplexing.
11.4 Massive Data Acquisition.
11.4.1 Specifics of Multichannel Data Acquisition.
11.4.2 Reconfigurable Distributed Structure ADC.
12 Quantizing–specific Signal Parameter Estimation.
12.1 Theoretical Limits.
12.1.1 Minimal Observation Time.
12.1.2 Sufficient Number of Signal Samples.
12.1.3 Influence of Quantization Errors.
12.1.4 Estimation of Periodic Signal Parameters.
12.2 Optimal Estimation.
12.2.1 Minimizing the Number of Signal Samples.
12.2.2 Simplifying Hardware.
12.2.3 Minimizing Bit Flow.
12.2.4 Deviations from Optimal Conditions.
12.3 Specifics Related to Pseudo–randomized Quantizing.
12.3.1 Avoiding Processing of the Dither Process.
12.3.2 Simplified Processing of the Dither Process.
12.4 Estimation of the Absolute Mean Value.
12.4.1 Electronic Device.
12.4.2 Estimation Errors.
12.5 Estimation of the Mean Power.
12.5.1 Estimation Efficiency.
12.6 Errors Due to Randomized Sampling.
12.6.1 Absolute Mean Value Estimate.
12.6.2 Mean Power Estimate.
12.6.3 Overall Estimation Errors.
13 Estimation of Correlation Functions.
13.1 Multiplication of Quantized Signals.
13.1.1 Expected Value of Multiplied Quantized Signals.
13.1.2 Variance of Multiplication Results.
13.1.3 Optional Approaches.
13.2 Correlation Analysis of Pseudo–randomly Quantized Signals.
13.2.1 Estimation Procedure.
13.2.2 Essential Relationships.
13.2.3 Implementation Issues.
13.3 Correlation Analysis of Pseudo–randomly Sampled Signals.
14 Signal Transforms.
14.1 Problem of Matching Signal Processing to Sampling.
14.2 Bases of Signal Transforms.
14.2.1 Required Properties of the Transform Bases.
14.2.2 Transforms by Means of a Finite Number of Basis Functions.
14.3 Orthogonal Transforms.
14.3.1 Analog Processing.
14.3.2 Digital Processing.
14.4 Discrete Unorthogonal Transforms.
14.5 Conversion of Unorthogonal Transforms.
15 DFT of Nonuniformly Sampled Signals.
15.1 Problems Related to Sampling Irregularities.
15.1.1 Alternative Approaches to DFT.
15.1.2 Best–fitting Procedure Versus Direct DFT.
15.1.3 Sample Values Partly Fitting to Any Frequency.
15.2 Cross–interference Corrupting DFT.
15.3 Exploitation of FFT.
15.3.1 Application of FFT for Processing Nonuniformly Sampled Signals.
15.3.2 Fast Transforms of Signals Sampled at Sine–Wave Crossing Instants.
15.4 Revealing the Essence of the Fourier Coefficient Estimation.
16 Complexity–reduced DFT.
16.1 Potential Gains from Application of Rectangular Function Sets.
16.1.1 Use of Orthogonal Rectangular Functions.
16.1.2 Reduction in the Computational Burden for DFT.
16.2 Complexity–reduced DFT Exploiting Rectangular Functions.
16.2.1 Essentials of the Method.
16.2.2 Mathematical Description.
16.2.3 Digital Implementation.
16.3 Computer Simulations of the Rectangular Function–based DFT.
16.4 Fast DFT of Sine–Wave Crossings.
17 Spatial Data Acquisition and Processing.
17.1 Sensor Array Model.
17.2 Temporal and Spatial Spectra of Array Signals.
17.2.1 When Signal Source Frequencies Do Not Overlap.
17.2.2 When Signal Source Frequencies Overlap.
17.2.3 Aliasing in the Spatial Domain.
17.4 Signal Direction of Arrival Estimation.
17.5 Pseudo–randomization of Sensor Arrays.
17.5.1 Complexity Reduction of Arrays.
17.5.2 Pseudo–randomization of Array Signal Processing.
18 Adapting Signal Processing to Sampling Nonuniformities.
18.1 Cross–interference Coefficients.
18.2 Taking the Cross–interference into Account.
18.3 Achievable Improvement and Typical Problems.
18.4 Parallel Computing Approach.
18.4.1 Decomposition of the Signal Sample Value Sequence.
18.4.2 Adapting the Estimation for Each Signal Sample Value Subset.
18.4.3 Data Aggregation.
18.5 Mapping of the Cross–interference Coefficients.
18.5.1 Required Frequency Resolution.
18.5.2 Coefficient Mapping Versus On–line Calculations.
19 Estimation of Object Parameters.
19.1 Measuring the Frequency Response of Objects.
19.2 Test Signal Synthesis from a Sparsely Periodically Sampled Basis Function.
19.2.1 Synthesis in the Case of Monoharmonic Basis.
19.2.2 Synthesis in the Case of Multifrequency Basis.
19.3 Test Signal Synthesis from a Nonuniformly Sampled Basis Function.
19.3.1 Spectrum of the Synthesized Signal.
19.3.2 Multifrequency Signal Synthesis.
19.3.3 Amplitude Equalization.
19.4 Synthesis of Narrowband and Wideband Signals.
19.5 Measuring Small Delays and Switching Times.
19.6 Bioimpedance Signal Demodulation in Real–time.
19.6.1 Typical Conditions for Bioimpedance Signal Forming.
19.6.2 Complexity Reduction of Bioimpedance Signal Demodulation.
20 Encapsulating DASP Technology.
20.1 Linking Digital Alias–free Signal Processing with Traditional Methods.
20.1.1 Generic Model of the Embedded DASP Systems.
20.1.2 Various DASP System Embedding Conditions.
20.2 Algorithm Options in the Development of Firmware.
20.2.1 Sequential Exclusion of Signal Components.
20.2.2 Iterative Variable Threshold Calculations of DFT and IDFT.
20.2.3 Algorithms Adapted to the Sampling Irregularities.
20.2.4 Comparison of Algorithm Performance.
20.3 Dedicated Services of the Embedded DASP Systems.
20.4 Dedicated Services Related to Processing of Digital Inputs.
20.4.1 Approach to Data Compression.
20.4.2 Data Compression for One–Dimensional Signals.
20.4.3 Data Compression for Two–Dimensional Signals.
20.4.4 Providing for Fault Tolerance.
20.5 Reducing the Quantity of Sensors in Large–aperture Arrays.
20.5.1 Adapting Signal Processing to Pseudo–random Positions of Sensors.