Fourier Series Content Periodic Functions l Fourier Series
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Fourier Series
Content Periodic Functions l Fourier Series l Complex Form of the Fourier Series l Impulse Train l Analysis of Periodic Waveforms l Half-Range Expansion l Least Mean-Square Error Approximation l
Fourier Series Periodic Functions
The Mathematic Formulation l Any function that satisfies where T is a constant and is called the period of the function.
Example: Find its period. Fact: smallest T
Example: Find its period. must be a rational number
Example: Is this function a periodic one? not a rational number
Fourier Series
Introduction l Decompose a periodic input signal into primitive periodic components. A periodic sequence f(t) t T 2 T 3 T
Synthesis DC Part Even Part Odd Part T is a period of all the above signals Let 0=2 /T.
Orthogonal Functions l Call a set of functions { k} orthogonal on an interval a < t < b if it satisfies
Orthogonal set of Sinusoidal Functions Define 0=2 /T. We now prove this one
Proof m n 0 0
Proof m=n 0
Orthogonal set of Sinusoidal Functions Define 0=2 /T. an orthogonal set.
Decomposition
Proof Use the following facts:
Example (Square Wave) f(t) 1 -6 -5 -4 -3 -2 - 2 3 4 5
Example (Square Wave) f(t) 1 -6 -5 -4 -3 -2 - 2 3 4 5
Example (Square Wave) f(t) 1 -6 -5 -4 -3 -2 - 2 3 4 5
Harmonics DC Part Even Part Odd Part T is a period of all the above signals
Harmonics Define , called the fundamental angular frequency. Define , called the n-th harmonic of the periodic function.
Harmonics
Amplitudes and Phase Angles harmonic amplitude phase angle
Fourier Series Complex Form of the Fourier Series
Complex Exponentials
Complex Form of the Fourier Series
Complex Form of the Fourier Series
Complex Form of the Fourier Series
Complex Form of the Fourier Series If f(t) is real,
Complex Frequency Spectra |cn| amplitude spectrum n phase spectrum
Example f(t) A t
Example A/5 -120 -15 0 -80 -10 0 -40 -5 0 0 40 5 0 80 10 0 120 15 0
Example A/10 -120 -80 -40 -30 0 -20 0 -10 0 0 40 80 120 10 0 20 0 30 0
Example f(t) A t 0
Fourier Series Impulse Train
Dirac Delta Function and Also called unit impulse function. 0 t
Property (t): Test Function
Impulse Train 3 T 2 T T 0 T 2 T 3 T t
Fourier Series of the Impulse Train
Complex Form Fourier Series of the Impulse Train
Fourier Series Analysis of Periodic Waveforms
Waveform Symmetry l Even l Odd Functions
Decomposition l Any function f(t) can be expressed as the sum of an even function fe(t) and an odd function fo(t). Even Part Odd Part
Example Even Part Odd Part
Half-Wave Symmetry and T/2 T
Quarter-Wave Symmetry Even Quarter-Wave Symmetry T/2 T Odd Quarter-Wave Symmetry T/2 T
Hidden Symmetry l The following is a asymmetry periodic function: A T l T Adding a constant to get symmetry property. A/2 T T A/2
Fourier Coefficients of Symmetrical Waveforms l The use of symmetry properties simplifies the calculation of Fourier coefficients. – – – Even Functions Odd Functions Half-Wave Even Quarter-Wave Odd Quarter-Wave Hidden
Fourier Coefficients of Even Functions
Fourier Coefficients of Even Functions
Fourier Coefficients for Half-Wave Symmetry and T/2 T The Fourier series contains only odd harmonics.
Fourier Coefficients for Half-Wave Symmetry and
Fourier Coefficients for Even Quarter-Wave Symmetry T/2 T
Fourier Coefficients for Odd Quarter-Wave Symmetry T/2 T
Example Even Quarter-Wave Symmetry T T/2 T/4 1 1 T/2 T/4 T
Example Even Quarter-Wave Symmetry T T/2 T/4 1 1 T/2 T/4 T
Example Odd Quarter-Wave Symmetry T T/2 1 T/4 1 T/2 T/4 T
Example Odd Quarter-Wave Symmetry T T/2 1 T/4 1 T/2 T/4 T
Fourier Series Half-Range Expansions
Non-Periodic Function Representation l A non-periodic function f(t) defined over (0, ) can be expanded into a Fourier series which is defined only in the interval (0, ).
Without Considering Symmetry T l A non-periodic function f(t) defined over (0, ) can be expanded into a Fourier series which is defined only in the interval (0, ).
Expansion Into Even Symmetry l T=2 A non-periodic function f(t) defined over (0, ) can be expanded into a Fourier series which is defined only in the interval (0, ).
Expansion Into Odd Symmetry T=2 l A non-periodic function f(t) defined over (0, ) can be expanded into a Fourier series which is defined only in the interval (0, ).
Expansion Into Half-Wave Symmetry l T=2 A non-periodic function f(t) defined over (0, ) can be expanded into a Fourier series which is defined only in the interval (0, ).
Expansion Into Even Quarter-Wave Symmetry T/2=2 l T=4 A non-periodic function f(t) defined over (0, ) can be expanded into a Fourier series which is defined only in the interval (0, ).
Expansion Into Odd Quarter-Wave Symmetry T/2=2 T=4 l A non-periodic function f(t) defined over (0, ) can be expanded into a Fourier series which is defined only in the interval (0, ).
Fourier Series Least Mean-Square Error Approximation
Approximation a function Use to represent f(t) on interval T/2 < t < T/2. Define Mean-Square Error
Approximation a function Show that using Sk(t) to represent f(t) has least mean-square property. Proven by setting Ek/ ai = 0 and Ek/ bi = 0.
Approximation a function
Mean-Square Error
Mean-Square Error
Mean-Square Error
- Fourier transform of a periodic function
- Half-wave symmetry examples
- Fourier series coefficients
- Fourier series of periodic function
- Orthogonal functions in fourier series
- Fourier series of even and odd functions
- Orthogonality fourier series
- Fourier series and orthogonal functions
- Real content and carrier content in esp
- Dynamic content vs static content
- Chapter 6 the periodic table and periodic law
- Trends in periodic table
- Boron group number
- The periodic table and periodic law chapter 6
- The periodic table and periodic law chapter 6
- Rn content mastery series 2020 proficiency levels
- Trigonometric fourier series
- Formulas series de fourier
- Series de fourier
- Serie de fourier compleja
- Frequency domain to time domain
- Fourier transform equations
- Fourier series
- Use of fourier series
- Fourier series multiplication property
- Full wave rectified sine wave fourier series
- Fourier transform formula
- Fourier transform of multiplication of two signals
- Fourier series equation
- Polar fourier series
- Fourier series multiplication property
- Fourier series half range
- Fourier transform properties
- Fourier transform in image processing
- What is fourier transform
- Fourier's theorem
- Half range fourier series is defined in
- Discrete time fourier series
- Discrete fourier transform
- Discrete time fourier series
- Fourier series circuit analysis
- Fourier transform of dirac delta function proof
- Series complejas de fourier
- Matlab fourier series coefficients
- Series fourier
- Orthogonal series expansion
- Series de fourier
- Time frequency
- Fourier series
- Series de fourier
- Dirichlet conditions
- Series de fourier
- Dft table
- Wolfram fourier series
- What is fourier series
- Pulse train fourier transform
- Fourier transform of impulse train
- Fourier series formula
- Transformata fourier
- Polar fft
- Fourier series
- Half wave rectifier formula
- Fourier transform of product of two functions
- Fourier transform of product of two functions
- Periodic properties of trig functions
- Periodic properties of the trigonometric functions
- Transformations of trigonometric functions
- Maclaurin series vs taylor series
- Heisenberg 1925 paper
- Taylor series of composite function
- Taylor frederick
- Ibm p series
- General
- Series aiding and series opposing
- Arithmetic series vs geometric series