FS Lecture Notes
Page 1: Periodic Functions and Fourier Series
Page 2: Mathematics: Relationship Between Taylor and Fourier Series
Periodic time-series described by a function with period 2π
Two types of series:
Taylor series: Expands as a linear combination of polynomials around t=0
Fourier series: Expands as a linear combination of sinusoids for t in [-π, π]
Page 3: Trigonometry Review - Sinusoids
Basic properties of sinusoids:
Amplitude: Maximum value or height from the midline.
Frequency (f): Number of cycles per unit time, given by f = 1/T, where T is the period.
Phase: Horizontal shift of the sinusoid from the origin.
Page 4: The Fourier Series
Proposed by Fourier in 1807.
A periodic waveform f(t) can be expressed as an infinite series of simple sinusoids:
Construction of the exact waveform from sinusoids by summing them up.
Period (T): The smallest value satisfying f(t + nT) = f(t).
Page 5: Periodic Functions
A function f(q) is periodic if defined for all real q and
There exists a positive number T such that f(q + T) = f(q).
Page 6: Simple Fourier Series Representation
No specific information, just a placeholder on this page.
Page 7: Simple Fourier Series Graph Sketch
Graph representation concerning f(0) = 0 and points at intervals of T.
Page 8: Symmetry in Periodic Functions
No specific information, just a placeholder on this page.
Page 9: Fourier Series Expansion
f(q) can be expressed as:
Trigonometric series: f(q) = a0 + sum( an * cos(nq) + bn * sin(nq)) for n from 1 to infinity.
Period is generally represented as 2π.
Page 10: Fourier Series Representation Continuation
Similar representation as in Page 9 for better understanding of harmonic components using cosine and sine functions.
Page 11: Fourier Coefficients for Cosine Waves
Example function evaluation using cosine waves at multiple intervals.
Page 12: Fourier Coefficients for Sine Waves
Similar function evaluation for sine components across the periods.
Page 13: Coefficient Determination
How to establish coefficients a_n and b_n by integrating functions.
Introduction to useful integration identities for Fourier coefficients.
Pages 14-16: Useful Integrations
Present useful integrations establishing 0 integrals for orthogonality conditions between sine and cosine terms for periodic functions.
Emphasizes the zero result when integrating over the full period ( -π to π).
Page 17: Determining Coefficient a_0
Integrate f(q) over one period to find a_0, which represents the average value of the function over that period.
Page 18: Integration to Determine Coefficients
Describes the integration process across one cycle to establish coefficients for both sine and cosine terms in Fourier series expansion.
Page 19: Average Value of function f(q)
The coefficient a_0 is determined using average value of f(q) over its period.
Page 20: Integration Clarifications
Reinforcement of integration limits when establishing coefficients a_n and b_n.
Page 21: Fourier Series Coefficients
Description of how to revert coefficients based on the integrals to include different forms of representation for coefficients a_n and b_n.
Page 22: Another Average Value Explanation
Explains averaging through integrals on both sides, reiterating earlier concepts regarding Fourier series lengths.
Page 23: Coefficient a_n Determination
Multiply original function by cos(nq) and integrate to solve for a_n across its periodic limits.
Page 24: Sequential Integration
Simplification of finding coefficients by evaluating integral parts for establishing Fourier coefficients pattern.
Page 25: Continued Integration for Coefficients
Reiteration of applying integrals for obtaining terms useful in Fourier coefficients for periodic functions.
Page 26: Simplifying f(m) Terms
How to relate the function evaluated terms when working through Fourier series expansion.
Page 27: Coefficient b_n Determination
Similar method as a_n but using sine component to correlate values with respective function.
Page 28: Integration Parts for Coefficient Determination
Clarifying integrations by parts to discover various sine component coefficient results yielding results for the Fourier series.
Page 29: Continued Integration and Expansions
Establishment of evaluation for sine components and the respective values that contribute to the series.
Page 30: Conclusion of Coefficients Determination
Integration to finalize the coefficients relating back to f(q) for sin terms across the integration.
Page 31: Average Value Relationship
Defining average values in relation to Fourier coefficients across periods.
Page 32: Codifying Coefficients
Adjustment of parameter m in place of n to indicate coefficients on relation lines yielding similarity across segments.
Page 33: Integration Flexibility
Expand on the thinking of integrating Fourier terms over a larger range if beneficial to the equation outcomes from the function.
Page 34: Example Fourier Series Problem
Demonstrative calculation to find Fourier series for provided periodic function across multiple ranges defining its waveform interactions.
Page 35: Integral Setups for Fourier Function
Set basic boundaries for integral calculations processing the values aiding the Fourier series establishment.
Page 36: Integration Strategy for Series
Continued evaluation of the Fourier components to arrive at the correct series establishment.
Page 37: Further Examples of Integration
Steps modeling integrations allowing for development of coefficients linked to certain functions.
Page 38: Odd Symmetry in Terms
Utilizing odd terms to articulate the function through series use with constraints defining evaluations.
Page 39: Even Symmetry in Terms
No terms from the even function contributions back through integration exclusive to cosine components.
Page 40: Final Fourier Series Form
Presentation of Fourier series in practical terms from periodic evaluations.
Page 41: Graphical Representation - 4 Terms
Visualization of function plotting approximating through Fourier components using 4 terms to yield results on screen.
Page 42: Graphical Representation - 6 Terms
Update on the function graph depiction after integrating 6 terms for enhanced rendering.
Page 43: Graphical Representation - 8 Terms
Further graphed portrayals adopting 8 terms with elevated accuracy and detail through visual aids.
Page 44: Greater Accuracy with 12 Terms
Extending 12 terms for refined approaches and visual feedback on sinusoidal approximations of periodic functions.
Page 45: Curve Comparison Graph
Display similarities and differences between functions across plotted renderings in various term equations sizing.
Page 46: Further Curve Comparison
Continued comparisons depicting clearer graphs through 12 terms again contrasting with 4.
Page 47: Advanced Comparison with 20 Terms
Graphing between the high detailed 20 term representation against 4 terms expressing the level of adjustment made.
Page 48: Introduction to Even and Odd Functions
Differentiation between types of functions relating to series evaluation methodologies across contexts.
Page 49: Definition of Even Functions
Key characteristics of even functions relating distances across values for evaluation confirmatory of properties discussed.
Page 50: Definition of Odd Functions
Clarity on how odd functions would invert or shift to access sign change during functional application.
Page 51: Representing Even Functions
Clarify that even functions align with cosine waves thus being represented through cosine series explicitly.
Page 52: Representing Odd Functions
Odd functions only conform to sine waves establishing their exclusive representations as previously stated.
Page 53: Expressing Even/Odd Functions as Series
Mathematical representations for odd and even functions through sine and cosine formats as Fourier series constructs.
Page 54: Example of EOF from Series
Example problem highlighting the process for determining the Fourier series for a periodic function with conditions.
Page 55: Fourier Transformation Integrals
Set of integrals to find function outputs f(x) across the prescribed period yielding periodic compliance on outcomes.
Page 56: Integration Application by Parts
Strategy leveraging integration by parts to derive coefficients involved in Fourier evaluations from set functions.
Page 57: Evaluation of Coefficients Properties
Constants determined with conditions placed on odd/even interactions to set functional terms.
Page 58: Series Results Overview
Final Fourier series representation for the derived function being fully disclosed to conclude evaluations.
Page 59: Forms of Arbitrary Period Functions
Generalization to periodic function scenarios with denoted angular rotations indicating transformations occuring.
Page 60: Angular Velocity Definition
Description of angular velocity adjustments in terms of periodic nature denoting transforms alongside f(t).
Page 61: Integral Limit Change Derivation
Transforming integration limits correlating back to time adjustments fitting within Fourier parameters.
Page 62: Mean Value Referencing
More integrated assessments within one period realizing behaviors surrounding Fourier algebra applications.
Page 63: Finalizing Coefficient Detections
Defining a_n through achieved components of Fourier functions across defined periodic adjustments.
Page 64: The Fourier Series Structure
Exposition of Fourier series expressions showcasing how multiple n values interact across a defined time function.
Page 65: Completing Integration Terms Summation
Enables sine wave definitions tackling B coefficient measures over structured frameworks.
Page 66: Additional Example Troubleshooting
Display functions evaluated over different periods showcasing component participations within Fourier measurements across cosine and sine waves.
Page 67: Odd Functions Exploration
Characteristics driven from results yielding Fourier coefficients adjustments across sine terms distinctly.
Page 68: Integration Summation Scope
Usage of summations indicative of their relations enhancing function behavior connectivity for specific terms.
Page 69: Relation and Odd/Even Properties
Result led solutions showcasing even versus odd evaluations across outputs from functions specified.
Page 70: Series Composition Contracts
Summation of main Fourier framework presenting all derived terms correlating back to wave harmonization across their functional limits.
Page 71: The Complex Form of Fourier Series
Breakdown of functions using complex formulations alongside coefficients providing harmonic expression for Fourier interactions distilling easier interpretations.
Page 72: nth Harmonic Expressions
Analysis of nth harmonic components overcoming complex forms to identify variables contributing to transformations.
Page 73: Coefficient Interpretation
Clarification of coefficient behavior with preliminary references to sine and cosine aspects across wave characterizations.
Page 74: Generic Fourier Functions
Presenting general forms translating direct terms across common interactions modeling Fourier outputs defined through proper representations.
Page 75: Overall Fourier Series Representation
Final confirmation establishing base Fourier coefficients struck through harmonic interactions across known results within a single harmonic evaluation.
Page 76: Coefficient Evaluation Techniques
Summarized techniques applied to Fourier coefficient evaluations confirming complex nature outputs against basic periodials.
Page 77: Final Complex Coefficients Derivation
Establish full definitions of complex coefficients illustrating establishment throughout Fourier metrics carrying complex art on sinusoidal measures.
Page 78: Coefficients Nature
Phase relationships presenting evidence back against periodic dimensions establishing iterative developments promoting complex understanding through summaries.
Page 79: Conclusion & Key Findings
Share insights and final conclusions regarding Fourier series expressed coefficient behaviors illustrating functions across periodic rates defined.