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Index: Introduction to Digital Filters with Audio Applications

Introduction to Digital Filters with Audio Applications

Sinusoids as Eigenfunctions of LTI Systems

Sinusoids as Eigenfunctions of LTI Systems

Proof Using Complex Variables

A sinusoid is any function of the form A sin(ω t+φ), where t is the independent variable, and A, ω, φ are fixed parameters of the sinusoid called the amplitude, (radian) frequency, and phase, respectively. Sinusoidal motion is produced by any 'pure' vibration, such as that of an ideal tuning fork or mass-spring system.  —  Click for https://ccrma.stanford.edu/~jos/mdft/Sinusoids.html

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Index: Introduction to Digital Filters with Audio Applications

Introduction to Digital Filters with Audio Applications

Sinusoids as Eigenfunctions of LTI Systems

Sinusoids as Eigenfunctions of LTI Systems

Proof Using Complex Variables

Proof Using Trigonometry

We want to show it is always possible to solve

$$\begin{array}{}\text{(1)}& A\cos (\omega t+\varphi )=A_1\cos (\omega t+{\varphi }_1)+A_2\cos (\omega t+{\varphi }_2)+\cdots +A_N\cos (\omega t+{\varphi }_N)\end{array}$$

for $A$ and $\varphi$, given $A_i,{\varphi }_i$ for $i=1,\dots ,N$. For each component sinusoid, we can write

$$\begin{array}{rcl}{A}_i\cos (\omega t+{\varphi }_i)& =& A_i\cos (\omega t)\cos ({\varphi }_i)-A_i\sin (\omega t)\sin ({\varphi }_i)\\ \text{(2)}& & =& [A_i\cos ({\varphi }_i)]\cos (\omega t)-[A_i\sin ({\varphi }_i)]\sin (\omega t)\end{array}$$

Applying this expansion to Eq.(A.2) yields

$$\begin{array}{rcl}[A\cos (\varphi )]\cos (\omega t)& -& [A\sin (\varphi )]\sin (\omega t)\\ & =& [\sum _{i=1}^N{A}_i\cos ({\varphi }_i)]\cos (\omega t)-[\sum _{i=1}^N{A}_i\sin ({\varphi }_i)]\sin (\omega t).\end{array}$$

Equating coefficients gives

$$\begin{array}{rcl}A\cos (\varphi )& =& \sum _{i=1}^N{A}_i\cos ({\varphi }_i)\stackrel{\mathrm{\Delta }}{=}x\\ \text{(3)}& A\sin (\varphi )& =& \sum _{i=1}^N{A}_i\sin ({\varphi }_i)\stackrel{\mathrm{\Delta }}{=}y,\end{array}$$

where $x$ and $y$ are known. We now have two equations in two unknowns which are readily solved by (1) squaring and adding both sides to eliminate $\varphi$, and (2) forming a ratio of both sides of Eq.(A.4) to eliminate $A$. This gives

$$\begin{array}{rcl}A& =& \pm \sqrt{x^2+y^2}\\ \varphi & =& {\tan }^{-1}\left(\frac{y}{x}\right)\end{array}$$

for any values of $A_i$ and ${\varphi }_i$. Since ${\tan }^{-1}(y/x)={\tan }^{-1}(-y/-x)$, we have $\varphi \in [-\pi /2,\pi /2)$. To impose $A\ge 0$ and $\varphi \in [-\pi ,\pi )$, a four-quadrant arctangent$(y,x)$ must be used, normally written atan2(y,x) in computer languages.

Check Your Understanding

Exercise 1 Multiple Choice

In the trigonometric proof, one arrives at the two equations A cos(φ) = x and A sin(φ) = y with x and y known. How is the amplitude A obtained?

A = x + y

A = ±√(x² + y²), by squaring and adding both equations to eliminate φ

A = x/y, by dividing the equations

A = √(x² − y²)

Check Answer Show Hint

Use the identity cos²φ + sin²φ = 1.

One combination of the equations removes φ; the other removes A.

Exercise 2 Multiple Choice

Why must a four-quadrant arctangent (atan2(y,x) in computer languages) be used to recover the phase φ, instead of the ordinary arctangent of y/x?

Because the ordinary arctangent is numerically unstable

Because tan⁻¹(y/x) cannot distinguish (y, x) from (−y, −x), returning φ only in [−π/2, π/2), while atan2 gives φ in [−π, π) with A ≥ 0

Because atan2 also returns the amplitude A

Because the ordinary arctangent works only for positive y

Check Answer Show Hint

The ratio y/x loses the individual signs of y and x.

A phase in the second or third quadrant cannot come out of a plain arctangent.

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PrefaceOutlineBook Series OverviewAcknowledgmentsErrata§1The Simplest Lowpass Filter§1Introduction§1What is a Filter?§1Why learn about filters?§1.2The Simplest Lowpass Filter§1.2Definition of the Simplest Low-Pass§1.3Finding the Frequency Response§1.3.1Sine-Wave Analysis§1.3.2Mathematical Sine-Wave Analysis§1.3Amplitude Response§1.3Phase Response§1.4An Easier Way§1.4Complex Sinusoids§1.4Complex Amplitude§1.4Phasor Notation§1.4Complex Sinusoids as Circular Motion§1.4.5Rederiving the Frequency Response§1Summary§2Matlab Filter Analysis§2.1Matlab Filter Implementation§2.2Matlab Sine-Wave Analysis§2.3Complex Sine-Wave Analysis§2.4Practical Frequency-Response Analysis§3Analysis of a Digital Comb Filter§3Difference Equation§3Signal Flow Graph§3Software Implementation in Matlab§3Sample-Level Implementation in Matlab§3Software Implementation in C++§3Software Implementation in Faust§3Impulse Response§3.7Transfer Function§3Frequency Response§3Amplitude Response§3Phase Response§3.11Pole-Zero Analysis§3.12Alternative Realizations§3.12.1First-Order Parallel Sections§3.12Parallel, Real, Second-Order Sections§3.12Parallel Second-Order Signal Flow Graph§3.12Series, Real, Second-Order Sections§3Summary§4Linear Time-Invariant Filters§4.1Definition of a Signal§4.2Definition of a Filter§4.3Examples of Digital Filters§4.4Linear Filters§4.4Scaling:§4.4Superposition:§4.4Real Linear Filtering of Complex Signals§4.5Time-Invariant Filters§4Showing Linearity and Time Invariance§4.7Dynamic Range Compression§4.7Why Dynamic Range Compression is Nonlinear§4.8A Musical Time-Varying Filter Example§4.9Analysis of Nonlinear Filters§4Conclusions§5Time Domain Representations§5.1Difference Equation§5.2Signal Flow Graph§5.3Causal Recursive Filters§5.4Filter Order§5.5Direct-Form-I Implementation§5.6Impulse-Response Representation§5.7Filter Stability§5.8Impulse Response Example§5.9Implications of Linear-Time-Invariance§5.10Convolution Representation§5.10Convolution Representation Summary§5.11FIR Digital Filters§5.11FIR impulse response§5.11Convolution Representation of FIR Filters§5.11The ``Finite'' in FIR§5.11Causal FIR Filters§5.11FIR Transfer Function§5.11FIR Order§5.11FIR Software Implementations§5.12Transient and Steady State Response§5.12FIR Example§5.12IIR Example§5.12.3Transient and Steady-State Signals§5.12.4Decay Response, Initial Conditions Response§5.12.5Complete Response§5Summary and Conclusions§6Transfer Function Analysis§6.1The Z Transform§6.2Existence of the Z Transform§6.3Shift and Convolution Theorems§6.3Shift Theorem§6.3Convolution Theorem§6Z Transform of Convolution§6Z Transform of Difference Equations§6.6Factored Form§6.7Series and Parallel Transfer Functions§6.7Series Case§6.7Parallel Case§6.7Series Combination is Commutative§6.8Partial Fraction Expansion§6.8Example§6.8Complex Example§6.8.1PFE to Real, Second-Order Sections§6.8.2Inverting the Z Transform§6.8.3FIR Part of a PFE§6.8.3Example: The General Biquad PFE§6.8.4Alternate PFE Methods§6.8.5Repeated Poles§6.8.5Dealing with Repeated Poles Analytically§6.8.5Example§6.8.5Impulse Response of Repeated Poles§6.8.5So What's Up with Repeated Poles?§6.8.6Alternate Stability Criterion§6.8.7Summary of the Partial Fraction Expansion§6.8.8Software for Partial Fraction Expansion§6.8.8Example 2§6.8.8Polynomial Multiplication in Matlab§6.8.8Polynomial Division in Matlab§7Frequency Response Analysis§7.1Frequency Response§7.2Amplitude Response§7.3Phase Response§7.4Polar Form of the Frequency Response§7.4Separating the Transfer Function Numerator and Denominator§7.5Frequency Response as a Ratio of DTFTs§7.5.1Frequency Response in Matlab§7.5.2Example LPF Frequency Response Using freqz§7.6Phase and Group Delay§7.6.1Phase Delay§7.6.2Phase Unwrapping§7.6.3Group Delay§7.6.3Derivation of Group Delay as Modulation Delay§7.6.4Group Delay Examples in Matlab§7.6.5Vocoder Analysis§7.6.6Numerical Computation of Group Delay§8Pole-Zero Analysis§8.1Filter Order = Transfer Function Order§8.2Graphical Amplitude Response§8.3Graphical Phase Response§8.4Stability Revisited§8.4.1Computing Reflection Coefficients§8.4Step-Down Procedure§8.4Testing Filter Stability in Matlab§8.5Bandwidth of One Pole§8.6Time Constant of One Pole§8.7Unstable Poles--Unit Circle Viewpoint§8.7Geometric Series§8.7One-Pole Transfer Functions§8.8Poles and Zeros of the Cepstrum§8.9Conversion to Minimum Phase§8Hilbert Transform Relations§9Implementation Structures§9.1The Four Direct Forms§9.1.1Direct-Form I§9.1.1Two's Complement Wrap-Around§9.1.2Direct Form II§9.1.2More about Potential Internal Overflow of DF-II§9.1.3Transposed Direct-Forms§9.1.4Numerical Robustness of TDF-II§9.2Series and Parallel Filter Sections§9.2.1Series Second-Order Sections§9.2.1Matlab Example§9.2.2Parallel First and/or Second-Order Sections§9.2.2First-Order Complex Resonators§9.2.2Real Second-Order Sections§9.2.2Implementation of Repeated Poles§9.2.3Formant Filtering Example§9.2.4Butterworth Lowpass Filter Example§9.2.5Summary of Series/Parallel Filter Sections§10Filters Preserving Phase§10.1Linear-Phase Filters§10.2Zero-Phase Filters§10.2\(\pi\)-Phase Filters§10.2Phase \(\pi\) in the Stopband§10.2.1Example Zero-Phase Filter Design§10.2.2Elementary Zero-Phase Filter Examples§10.3Odd Impulse Responses§10.4Symmetric Linear-Phase Filters§10.4.1Simple Linear-Phase Filter Examples§10.4.2Software for Linear-Phase Filter Design§10.5Antisymmetric Linear-Phase Filters§10.6Forward-Backward Filtering§10Phase Distortion at Passband Edges§11Minimum-Phase Filters§11Definition of Minimum Phase Filters§11.2Minimum-Phase Polynomials§11.3Maximum Phase Filters§11.3Example§11.4Minimum Phase Means Fastest Decay§11.5Minimum-Phase/Allpass Decomposition§11.6Linear Phase Audio Filters§11.7Creating Minimum PhaseConclusion§ABackground Fundamentals§A.1Signal Representation and Notation§A.1Units§A.1Sinusoids§A.1Spectrum§A.2Complex and Trigonometric Identities§A.2Complex Numbers§A.2The Exponential Function§A.2Trigonometric Identities§A.2Trigonometric Identities, Continued§A.2Half-Angle Tangent Identities§A.3Sinusoids as Eigenfunctions of LTI Systems§A.3Proof Using Trigonometry§A.3Proof Using Complex Variables§A.3.3Phasor Analysis§BElementary Audio Digital Filters§B.1Elementary Filter Sections§B.1.1One-Zero§B.1.2One-Pole§B.1.3Two-Pole§B.1.3Resonator Bandwidth in Terms of Pole Radius§B.1.4Two-Zero§B.1.5Complex Resonator§B.1.5Two-Pole Partial Fraction Expansion§B.1.6The BiQuad Section§B.1Biquad Software Implementations§B.2Allpass Filter Sections§B.2.1The Biquad Allpass Section§B.2.2Allpass Filter Design§B.3DC Blocker§B.3.1DC Blocker Frequency Response§B.3DC Blocker Software Implementations§B.4Low and High Shelf Filters§B.4Exercise§B.5Peaking Equalizers§B.6Time-Varying Two-Pole Filters§B.6.1Normalizing Two-Pole Filter Gain at Resonance§B.6.2Constant Resonance Gain§B.6.3Peak Gain Versus Resonance Gain§B.6.4Constant Peak-Gain Resonator§B.6Four-Pole Tunable Lowpass/Bandpass Filters§CAllpass Filters§C.1Allpass Examples§C.2Paraunitary Filters§C.3MIMO Allpass Filters§C.3.1Paraunitary MIMO Filters§C.3.1MIMO Paraconjugate§C.3.1MIMO Paraunitary Condition§C.3.1Properties of Paraunitary Systems§C.3.1Properties of Paraunitary Filter Banks§C.3Paraunitary Filter Examples§DLaplace Transform Analysis§D.1Existence of the Laplace Transform§D.2Analytic Continuation§D.3Relation to the z Transform§D.4Laplace Transform Theorems§D.4.1Linearity§D.4.2Differentiation§D.5Laplace Analysis of Linear Systems§D.5.1Moving Mass§D.5.2Mass-Spring Oscillator Analysis§EAnalog Filters§E.1Example Analog Filter§E.2Capacitors§E.2Mechanical Equivalent of a Capacitor is a Spring§E.3Inductors§E.3Mechanical Equivalent of an Inductor is a Mass§E.4RC Filter Analysis§E.4.1Driving Point Impedance§E.4.2Transfer Function§E.4.3Impulse Response§E.4.4The Continuous-Time Impulse§E.4.5Poles and Zeros§E.5RLC Filter Analysis§E.5.1Driving Point Impedance§E.5.2Transfer Function§E.5.3Poles and Zeros§E.5.4Impulse Response§E.6Relating Pole Radius to Bandwidth§E.7Quality Factor (Q)§E.7.1Q of a Complex Resonator§E.7.2Q of a Real Second-Order Resonator§E.7.2Critical Damping and Related Terms§E.7.3Damping Ratio§E.7.4Decay Time is Q Periods§E.7Q as Energy Stored over Energy Dissipated§E.8Analog Allpass Filters§E.8Lossless Analog Filters§FMatrix Filter Representations§F.1Introduction§FGeneral Causal Linear Filter Matrix§F.3General LTI Filter Matrix§F.4Cyclic Convolution Matrix§F.5Inverse Filters§F.6State Space Realization§F.6State Space Filter Realization Example§F.7Time Domain Filter Estimation§F.7Effect of Measurement Noise§F.7Matlab System Identification Example§GState Space Filters§G.1Markov Parameters§G.2Response from Initial Conditions§G.3Complete Response§G.4Transfer Function of a State Space Filter§G.4Example State Space Filter Transfer Function§G.5Transposition of a State Space Filter§G.6Poles of a State Space Filter§G.7Difference Equations to State Space§G.7Converting to State-Space Form by Hand§G.7Signal Flow Graph to State Space Filter§G.7.3Controllability and Observability§G.7A Short-Cut to Controller Canonical Form§G.7Matlab Direct-Form to State-Space Conversion§G.7.6State Space Simulation in Matlab§G.7Other Relevant Matlab Functions§G.7.8Matlab State-Space Filter Conversion Example§G.8Similarity Transformations§G.9Modal Representation§G.9.1Diagonalizing a State-Space Model§G.9Finding the Eigenvalues of A in Practice§G.9Example of State-Space Diagonalization§G.9Properties of the Modal Representation§G.10Repeated Poles§G.10.1Jordan Canonical Form§G.11Digital Waveguide Oscillator Example§G.11Finding the Eigenstructure of A§G.11Choice of Output Signal and Initial Conditions§GReferences§HLinear Time-Varying Filters§HIntroduction§HDerivation§HSummary§IRecursive Digital Filter Design§I.1Lowpass Filter Design§I.2Butterworth Lowpass Design§I.2Butterworth Lowpass Poles and Zeros§I.2Example: Second-Order Butterworth Lowpass§I.3Bilinear A/D Transformation§I.3.1Bilinear Transformation§I.3.2Frequency Warping§I.3.3Analog Prototype Filter§I.3Examples§I.4Equation-Error Filter Design§I.4.1Equation Error Formulation§I.4Error Weighting and Frequency Warping§I.4Stability of Equation Error Designs§I.4.4An FFT-Based Equation-Error Method§I.4.5Prony's Method§I.4.6The Padé-Prony Method§JMatlab Utilities§J.1Time Plots: myplot.m§J.2Frequency Plots: freqplot.m§J.3Saving Plots to Disk: saveplot.m§J.4Frequency Response Plots: plotfr.m§J.5Partial Fraction Expansion: residuez.m§J.5Method§J.5Example with Repeated Poles§J.6Partial Fraction Expansion: residued.m§J.6Method§J.7Parallel SOS to Transfer Function: psos2tf§J.8Group Delay Computation: grpdelay.m§J.9Matlab listing: fold.m§J.10Matlab listing: clipdb.m§J.11Matlab listing: mps.m and test program§J.11Matlab listing: mps.m§J.11Matlab listing: tmps.m§J.11Matlab diary: tmps.d§J.12Signal Plots: swanalplot.m§J.13Frequency Response Plot: swanalmainplot§KDigital Filtering in Faust and PD§KA Simple Faust Program§KGenerating Faust Block Diagrams§KTesting a Faust Filter Section§KA Look at the Generated C++ code§K.5Generating a Pure Data (PD) Plugin§K.5Generating the PD Plugin§K.5Generating a PD Plugin-Wrapper Abstraction§K.5A PD Test Patch for the Plugin Wrapper§K.6Generating a LADSPA Plugin via Faust§K.7Generating a VST Plugin via Faust§K.7Bypassing Windows§KGenerating a MIDI Synthesizer for PD§KMIDI Synthesizer Test PatchLinks to Online ResourcesBibliographyIndex for this DocumentAbout this document ...

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