Howtosignalprocessing
- What is the exponential Fourier series?
- How do you find the exponential Fourier series?
- What is the Fourier series of Cos 3x?
- What is the Fourier transform of an exponential?
What is the exponential Fourier series?
A periodic signal can be represented over a certain interval of time in terms of the linear combination of orthogonal functions. If these orthogonal functions are exponential functions, then it is called the exponential Fourier series.
How do you find the exponential Fourier series?
In order to derive the exponential Fourier series, we replace the trigonometric functions with exponential functions and collect like exponential terms. This gives f(x)∼a02+∞∑n=1[an(einx+e−inx2)+bn(einx−e−inx2i)]=a02+∞∑n=1(an−ibn2)einx+∞∑n=1(an+ibn2)e−inx.
What is the Fourier series of Cos 3x?
The reason is that: f(x) = cos(3x) is a perfectly good Fourier series. i.e. f(x) = [latex]\sum a_n \cos nx + b_n \sin nx[/latex] (where a_n = 0 unless n = 3, a_3 = 1, b_n = 0 always). This series obviously equals cos(3x).
What is the Fourier transform of an exponential?
Therefore, the Fourier transform of a single-sided real exponential function is, F[e−atu(t)]=1a+jω Or, it can also be represented as, e−atu(t)FT↔1a+jω
