Preview

fourier series

Better Essays
Open Document
Open Document
1690 Words
Grammar
Grammar
Plagiarism
Plagiarism
Writing
Writing
Score
Score
fourier series
Fourier series
From Wikipedia, the free encyclopedia
Fourier transforms
Continuous Fourier transform
Fourier series
Discrete-time Fourier transform
Discrete Fourier transform
Fourier analysis
Related transforms

The first four partial sums of the Fourier series for a square wave
In mathematics, a Fourier series (English pronunciation: /ˈfɔərieɪ/) decomposes periodic functions or periodic signals into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or complex exponentials). The study of Fourier series is a branch of Fourier analysis.
Contents
[hide]

Definition[edit]
In this section, s(x) denotes a function of the real variable x, and s is integrable on an interval [x0, x0 + P], for real numbers x0 and P. We will attempt to represent s in that interval as an infinite sum, or series, of harmonically related sinusoidal functions. Outside the interval, the series is periodic with period P. It follows that if s also has that property, the approximation is valid on the entire real line. The case P = 2π is prominently featured in the literature, presumably because it affords a minor simplification, but at the expense of generality.
For integers N > 0, the following summation is a periodic function with period P:

Using the identities:

Function s(x) (in red) is a sum of six sine functions of different amplitudes and harmonically-related frequencies. Their summation is called a Fourier series. The Fourier transform, S(f) (in blue), which depicts amplitude vs frequency, reveals the 6 frequencies and their amplitudes. we can also write the function in these equivalent forms:
:
where:

When the coefficients (known as Fourier coefficients) are computed as follows:[7]

approximates on and the approximation improves as N → ∞. The infinite sum, is called the Fourier series representation of The Fourier series does not always converge, and even when it does converge

You May Also Find These Documents Helpful

  • Better Essays

    a mathematical method that is used to create all the notes of the musical scale from the harmonic series; this scale is called the Pythagorean Scale. The naturally happening harmonics are whole number multiples. To get the other notes of the scale, we must use fractions.…

    • 1141 Words
    • 5 Pages
    Better Essays
  • Better Essays

    References: 1. J. Callahan, D. Cox, K. Hoffman, D. O’Shea, H. Pollatsek, and L. Senechal, Calculus in Context: The Five College Calculus Project, W.H. Freeman, 1995. 2. B. Cha, Transcendental Functions and Initial Value Problems: A Different Approach to Calculus II, College Math. J. 38 (2007) 288–296. 3. W. Boyce and R. DiPrima, Elementary Differential Equations, 9th ed., Wiley, 2008. 4. E. F. Krause, Taxicab Geometry: An Adventure in Non-Euclidean Geometry, Dover, 1987. 5. C. C. Maican, Integral Evaluations Using the Gamma and Beta Functions and Elliptic Integrals in Engineering: A Self-Study Approach, International Press, 2005. 6. W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, 1986. 7. R. M. Young, Execursions in Calculus, Mathematical Association of America, 1992. Summary Differential equations offers one approach to defining the classical trigonometric functions sine and cosine that parameterize the unit circle. In this article, we adapt this approach to develop analogous functions that parameterize the unit squircle defined by x 4 + y 4 = 1. As we develop our new theory of “squigonometry” using only elementary calculus, we will catch glimpses of some very interesting and deep ideas in elliptic integrals, non-euclidean geometry, number theory, and complex analysis. WILLIAM E. WOOD has recently joined the Mathematics Department at the University of Northern Iowa. He enjoys thinking about various problems across mathematics and somehow turning them all into geometry problems. He lives in Cedar Falls with his wife, cats, and board game collection.…

    • 4026 Words
    • 17 Pages
    Better Essays
  • Satisfactory Essays

    Discrete Mathmatics IP 2

    • 259 Words
    • 3 Pages

    Mathematical sequences can be used to model real life applications. Suppose you want to construct a movie theater in your town. The number of seats in each row can be modeled by the formula C_n = 16 + 4n, when n refers to the nth row, and you need 50 rows of seats.…

    • 259 Words
    • 3 Pages
    Satisfactory Essays
  • Satisfactory Essays

    The properties of the 6 trigonometric functions: sin (x), cos (x), tan(x), cot (x), sec (x) and csc (x) include the domain, range, period, asymptotes and amplitudes.…

    • 371 Words
    • 2 Pages
    Satisfactory Essays
  • Satisfactory Essays

    Sine, Cosine Function

    • 530 Words
    • 3 Pages

    Notice that the sine goes through the origin and travels to a maximum at (π/2, 1). Then, it travels down through (π, 0) to a minimum at (3π/2, -1). Finally the sine travels back up to (2π, 0). Then the sine wave will continue this same process again. Thus, the period of the sine function is 2π. Its amplitude is 1. Recall that sin (-x) = -sin x. This means that the sine function is odd, or it is symmetric to the origin.…

    • 530 Words
    • 3 Pages
    Satisfactory Essays
  • Good Essays

    Graphs 8

    • 493 Words
    • 2 Pages

    The Fourier Series is the sum of all the waves within a waveform, in other words, it separates all the waves in the waveform into its basic components. For instance, take the waveforms of the flute and oboe. I will use these…

    • 493 Words
    • 2 Pages
    Good Essays
  • Good Essays

    Data Communications

    • 1027 Words
    • 5 Pages

    Figure 1 shows the frequency domain function for a single square pulse. The signal pulse could represent a digital 1 in a communications system. Note that an infinite number of higher frequencies of decreasing magnitudes is needed to represent the single pulse. What implication does that have for a real digital transmission system?…

    • 1027 Words
    • 5 Pages
    Good Essays
  • Good Essays

    Question 2.2 – Accounting and Cash Flows: Why is it that the revenue and cost figures shown on a standard income statement may not be representative of the actual cash inflows and outflows that occurred during a period?…

    • 804 Words
    • 4 Pages
    Good Essays
  • Good Essays

    let the polynomial fn(x)= a0 + a1x + a2x2 + ... + anxn approximate f(x) near x=0,…

    • 573 Words
    • 3 Pages
    Good Essays
  • Satisfactory Essays

    physics

    • 298 Words
    • 2 Pages

    If one wants to know something about the wavelengths that make up a particular type of radiation (i.e., the spectrum…

    • 298 Words
    • 2 Pages
    Satisfactory Essays
  • Powerful Essays

    Operational Amplifiers

    • 10121 Words
    • 41 Pages

    circuit design. It is most commonly used in amplifier and analog signal processing circuits in the frequency…

    • 10121 Words
    • 41 Pages
    Powerful Essays
  • Satisfactory Essays

    Samurai Research Paper

    • 426 Words
    • 2 Pages

    The medieval period was a period of gradual mathematical development. In other ways it was a period of great philosophical shifts, not so much on the surface as the Roman Church dominated much of philosophy and all of religion but underneath, the old Aristotelian views began to erode. Though it would dominate education for many more centuries, certain notions began to be be admitted. Most particularly, we see a lively discussion of the infinite, actual and potential. Many geometric shapes are…

    • 426 Words
    • 2 Pages
    Satisfactory Essays
  • Good Essays

    Seasonal Component- Regular pattern of up and down fluctuations. Due to weather, customs, etc. Occurs within a single unit (length).…

    • 1093 Words
    • 4 Pages
    Good Essays
  • Good Essays

    The Beth Kennedy series are a series of cozy mystery novels by the American author Laura Alden best known for her versatile and unique cozies. The debut novel in the Beth Kennedy series was the 2010 published Murder at the PTA that both critics and fans agreed was one of her most spectacular of works. The novel would go on to spawn four more titles in the series as the author published a novel each year until 2014. Just like in her Bookmobile Cat Mystery Series, Alden writes some insanely mysterious and suspenseful storylines that include powerful, likable, and highly memorable characters. The lead character in the series is Beth Kennedy the titular character, who is a single mother who runs a children’s bookstore in the small town of Rynwood,…

    • 963 Words
    • 4 Pages
    Good Essays
  • Good Essays

    low pass filter

    • 1351 Words
    • 6 Pages

    Basically, an electrical filter is a circuit that can be designed to modify, reshape or reject all unwanted frequencies of an electrical signal and accept or pass only those signals wanted by the circuits designer. In other words they "filter-out" unwanted signals and an ideal filter will separate and pass sinusoidal input signals based upon their frequency.…

    • 1351 Words
    • 6 Pages
    Good Essays