Fourier synthesis of a periodic signal with integer harmonics is defined as: wit
ID: 2969720 • Letter: F
Question
Fourier synthesis of a periodic signal with integer harmonics is defined as:
with the Fourier coefficients ak and bk compound for k = 0, 1, 2,... through
Compute and plot the Fourier coefficients for the following 2pi-periodic functions
If each ak and bk for the square wave gets multiplied with 1/k (when k>= 1), how does the resulting function look like?
What does a0 represent, and why does a0 never equal to 0 for the square wave and the constant?
Why are generally all ak = 0 with the exception of some a0?
What symmetries can you identify?
What is the decay behavior of the coefficients as the functions more closely resemble a sine function?
Fourier synthesis of a periodic signal with integer harmonics is defined as: f(t) = a0/2 + infinity k = 1 ak cos(kt) + bksin(kt) with the Fourier coefficients ak and bk compound for k = 0, 1, 2,... through f(t) = 1 for -infinityExplanation / Answer
1) a0 = 2pi/pi = 2 ak =0, bk=0 2) a0 = ak = 0 b1 = 1 rest bk=0 3) a0 = 0 bk=0 a4 = 1 rest all ak=0 4) a0 = 1, ak = 0 bk = {1-cos(k*pi)}/k*pi If each ak and bk is multiplied by 1/k then function also get multiplied by 1/k so its amplitude decreases. a0 represent dc or mean value. Since square or constant mean can't be zero its a0 is not zero.
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