4.1 Fourier series

Calculate the Fourier series representation of the time domain waveform shown in Figure 4.1a. Also, plot the resulting amplitude line spectrum assuming T = 1 10 ^?3 seconds.

Figure 4.1: Fourier series

Solution:

• a ₀ = (1/T) ? ₀ ^Tf(t)dt = 1/T[ ? ₀ ^T/2 10dt ? ? _T/2 ^T 10dt = (10/T)[T/2 ? 0 ? T + T/2] = 0

• a _n = (1/T)[ ? ₀ ^Tf(t)cos(n ? ₀t)dt], let ? ₀T = 2 ?

• a _n = (1/ ?)[ ₀ ^?10cos(n ? ₀t)dt ? ? _? ^{2 ?}10cos(n ? ₀t)dt] = (10/n ?)([sin(n ? ₀t)] ₀ ^? ? [sin(n ? ₀t)] _? ^{2 ?})

• a _n = (10/n ?)[sin(n ?) ? sin0 ? sin(n2 ?) + sin(n ?)]

• a _n = (10/n ?)[2sin(n ?) ? sin(n2 ?)] = 0, for all n

• b _n = (1/T)[ ? ₀ ^Tf(t)sin(n ? ₀t)dt], let ? ₀T = 2 ?

• b _n = (1/ ?)[ ? ₀ ^?10sin(n ? ₀t)dt ? ? _? ^{2 ?}10sin(n ? ₀t)dt] = (10/n ?)([ ?cos(n ? ₀t)] ₀ ^? + [cos(n ? ₀t)] _? ^{2 ?})

• b _n = (10/n ?)[ ?cosn ? + cos0 +...