+++
title = "Understanding Digital Signal Processing"
author = ["Dehaeze Thomas"]
draft = true
+++
- Tags
- [IRR and FIR Filters]({{< relref "irr_and_fir_filters.md" >}}), [Digital Filters]({{< relref "digital_filters.md" >}})
- Reference
- (Lyons 2011)
- Author(s)
- Lyons, R.
- Year
- 2011
Discrete Sequences And Systems
Discrete Sequences And Their Notation
Signal Amplitude, Magnitude, Power
Signal Processing Operational Symbols
Introduction To Discrete Linear Time-Invariant Systems
Discrete Linear Systems
Time-Invariant Systems
The Commutative Property Of Linear Time-Invariant Systems
Analyzing Linear Time-Invariant Systems
{{< figure src="/ox-hugo/lyons11_lti_impulse_response.png" caption="<span class="figure-number">Figure 1: LTI system unit impulse response sequences. (a) system block diagram. (b) impulse input sequence \(x(n)\) and impulse reponse output sequence \(y(n)\)." >}}
{{< figure src="/ox-hugo/lyons11_moving_average.png" caption="<span class="figure-number">Figure 2: Analyzing a moving average filter. (a) averager block diagram; (b) impulse input and impulse response; (c) averager frequency magnitude reponse." >}}
Periodic Sampling
Aliasing: Signal Ambiguity In The Frequency Domain
{{< figure src="/ox-hugo/lyons11_frequency_ambiguity.png" caption="<span class="figure-number">Figure 3: Frequency ambiguity; (a) discrete time sequence of values; (b) two different sinewaves that pass through the points of discete sequence" >}}
Sampling Lowpass Signals
{{< figure src="/ox-hugo/lyons11_noise_spectral_replication.png" caption="<span class="figure-number">Figure 4: Spectral replications; (a) original continuous signal plus noise spectrum; (b) discrete spectrum with noise contaminating the signal of interest" >}}
{{< figure src="/ox-hugo/lyons11_lowpass_sampling.png" caption="<span class="figure-number">Figure 5: Low pass analog filtering prior to sampling at a rate of \(f_s\) Hz." >}}
The Discrete Fourier Transform
\begin{equation}
X(f) = \int_{-\infty}^{\infty} x(t) e^{-j2\pi f t} dt
\end{equation}
\begin{equation}
X(m) = \sum_{n = 0}^{N-1} x(n) e^{-j2 \pi n m /N}
\end{equation}
Understanding The Dft Equation
Dft Symmetry
Dft Linearity
Dft Magnitudes
Dft Frequency Axis
Dft Shifting Theorem
Inverse Dft
Dft Leakage
Windows
Dft Scalloping Loss
Dft Resolution, Zero Padding, And Frequency-Domain Sampling
Dft Processing Gain
The Dft Of Rectangular Functions
Interpreting The Dft Using The Discrete-Time Fourier Transform
The Fast Fourier Transform
Relationship Of The Fft To The Dft
Hints On Using Ffts In Practice
Derivation Of The Radix-2 Fft Algorithm
Fft Input/Output Data Index Bit Reversal
Radix-2 Fft Butterfly Structures
Alternate Single-Butterfly Structures
Finite Impulse Response Filters
An Introduction To Finite Impulse Response (Fir) Filters
Convolution In Fir Filters
Lowpass Fir Filter Design
Bandpass Fir Filter Design
Highpass Fir Filter Design
Parks-Mcclellan Exchange Fir Filter Design Method
Half-Band Fir Filters
Phase Response Of Fir Filters
A Generic Description Of Discrete Convolution
Analyzing Fir Filters
Infinite Impulse Response Filters
An Introduction To Infinite Impulse Response Filters
The Laplace Transform
The Z-Transform
Using The Z-Transform To Analyze Iir Filters
Using Poles And Zeros To Analyze Iir Filters
Alternate Iir Filter Structures
Pitfalls In Building Iir Filters
Improving Iir Filters With Cascaded Structures
Scaling The Gain Of Iir Filters
Impulse Invariance Iir Filter Design Method
Bilinear Transform Iir Filter Design Method
Optimized Iir Filter Design Method
A Brief Comparison Of Iir And Fir Filters
Specialized Digital Networks And Filters
Differentiators
Integrators
Matched Filters
Interpolated Lowpass Fir Filters
Frequency Sampling Filters: The Lost Art
Quadrature Signals
Why Care About Quadrature Signals?
The Notation Of Complex Numbers
Representing Real Signals Using Complex Phasors
A Few Thoughts On Negative Frequency
Quadrature Signals In The Frequency Domain
Bandpass Quadrature Signals In The Frequency Domain
Complex Down-Conversion
A Complex Down-Conversion Example
An Alternate Down-Conversion Method
The Discrete Hilbert Transform
Hilbert Transform Definition
Why Care About The Hilbert Transform?
Impulse Response Of A Hilbert Transformer
Designing A Discrete Hilbert Transformer
Time-Domain Analytic Signal Generation
Comparing Analytical Signal Generation Methods
10 Sample Rate Conversion
10.1 Decimation
10.2 Two-Stage Decimation
10.3 Properties Of Downsampling
10.4 Interpolation
10.5 Properties Of Interpolation
10.6 Combining Decimation And Interpolation
10.7 Polyphase Filters
10.8 Two-Stage Interpolation
10.9 Z-Transform Analysis Of Multirate Systems
10.10 Polyphase Filter Implementations
10.11 Sample Rate Conversion By Rational Factors
10.12 Sample Rate Conversion With Half-Band Filters
10.13 Sample Rate Conversion With Ifir Filters
10.14 Cascaded Integrator-Comb Filters
11 Signal Averaging
11.1 Coherent Averaging
11.2 Incoherent Averaging
11.3 Averaging Multiple Fast Fourier Transforms
11.4 Averaging Phase Angles
11.5 Filtering Aspects Of Time-Domain Averaging
11.6 Exponential Averaging
12 Digital Data Formats And Their Effects
12.1 Fixed-Point Binary Formats
12.2 Binary Number Precision And Dynamic Range
12.3 Effects Of Finite Fixed-Point Binary Word Length
12.4 Floating-Point Binary Formats
12.5 Block Floating-Point Binary Format
13 Digital Signal Processing Tricks
13.1 Frequency Translation Without Multiplication
13.2 High-Speed Vector Magnitude Approximation
13.3 Frequency-Domain Windowing
13.4 Fast Multiplication Of Complex Numbers
13.5 Efficiently Performing The Fft Of Real Sequences
13.6 Computing The Inverse Fft Using The Forward Fft
13.7 Simplified Fir Filter Structure
13.8 Reducing A/D Converter Quantization Noise
13.9 A/D Converter Testing Techniques
13.10 Fast Fir Filtering Using The Fft
13.11 Generating Normally Distributed Random Data
13.12 Zero-Phase Filtering
13.13 Sharpened Fir Filters
13.14 Interpolating A Bandpass Signal
13.15 Spectral Peak Location Algorithm
13.16 Computing Fft Twiddle Factors
13.17 Single Tone Detection
13.18 The Sliding Dft
13.19 The Zoom Fft
13.20 A Practical Spectrum Analyzer
13.21 An Efficient Arctangent Approximation
13.22 Frequency Demodulation Algorithms
13.23 Dc Removal
13.24 Improving Traditional Cic Filters
13.25 Smoothing Impulsive Noise
13.26 Efficient Polynomial Evaluation
13.27 Designing Very High-Order Fir Filters
13.28 Time-Domain Interpolation Using The Fft
13.29 Frequency Translation Using Decimation
13.30 Automatic Gain Control (Agc)
13.31 Approximate Envelope Detection
13.32 A Quadrature Oscillator
13.33 Specialized Exponential Averaging
13.34 Filtering Narrowband Noise Using Filter Nulls
13.35 Efficient Computation Of Signal Variance
13.36 Real-Time Computation Of Signal Averages And Variances
13.37 Building Hilbert Transformers From Half-Band Filters
13.38 Complex Vector Rotation With Arctangents
13.39 An Efficient Differentiating Network
13.40 Linear-Phase Dc-Removal Filter
13.41 Avoiding Overflow In Magnitude Computations
13.42 Efficient Linear Interpolation
13.43 Alternate Complex Down-Conversion Schemes
13.44 Signal Transition Detection
13.45 Spectral Flipping Around Signal Center Frequency
13.46 Computing Missing Signal Samples
13.47 Computing Large Dfts Using Small Ffts
13.48 Computing Filter Group Delay Without Arctangents
13.49 Computing A Forward And Inverse Fft Using A Single Fft
13.50 Improved Narrowband Lowpass Iir Filters
13.51 A Stable Goertzel Algorithm
Bibliography
Lyons, Richard. 2011.
Understanding Digital Signal Processing. Upper Saddle River, NJ: Prentice Hall.
