Trigonometric Fourier series are used to represent periodic signals in the form of a sum of sine and cosine functions. These series are used in many engineering and mathematical applications, such as signal processing, control theory, and harmonic analysis. Trigonometric Fourier series are named after French mathematician and physicist Joseph Fourier, who developed the theory in the early 19th century. The series is a representation of a signal or function as an infinite sum of sinusoidal components. Each component of the Fourier series has a different frequency and amplitude and can be used to analyze signals with different frequencies.
We have already discussed the Fourier series in exponential form. In this article we will discuss another form of Fourier series i.e. Trigonometric Fourier series. Trigonometric Fourier series is a summation of a periodic function using sine and cosine waves.
Let us consider a periodic function f(x) with period T. The trigonometric Fourier series for this function can be written as f(x) = a_0 + ∑_{n=1}^∞ (a_n cos(nωx) + b_n sin(nωx))
where a_0 is the average value of f(x) over the period T, a_n and b_n are the coefficients of the cosine and sine terms respectively, and ω = 2π/T is the angular frequency.
The coefficients of the trigonometric Fourier series can be determined using the following equations:
a_0 = 1/T ∫_0^T f(x) dx
a_n = (2/T) ∫_0^T f(x) cos(nωx) dx
b_n = (2/T) ∫_0^T f(x) sin(nωx) dx
Once the coefficients are determined, the trigonometric Fourier series can be used to represent the periodic signal or function. This representation can be used to analyze the signal or function for different frequencies and amplitudes.
The Trigonometric Fourier Series is a representation of a periodic signal or function as an infinite sum of sinusoidal components. It is determined by calculating the coefficients of the cosine and sine terms, which are given by the integrals of the signal or function multiplied by cosine or sine, respectively. Once the coefficients are determined, the series can be used to analyze the signal or function for different frequencies and amplitudes.
The trigonometric Fourier series of a periodic signal x (t) with period T can be obtained by manipulating the exponential Fourier series.
The Fourier coefficients ak and bk for the trigonometric Fourier series of the periodic signal x (t) with period T can be calculated.
The dc component (a0) of the signal for the trigonometric Fourier series can be calculated.
Properties of Fourier series
If x(t) is an even function, then bk = 0 and the Fourier coefficients ak can be determined.
If x(t) is an odd function, then a0 = 0, ak = 0 and the Fourier coefficients bk can be determined.
If x(t) is a half-symmetric function, then a0 = 0, ak = bk = 0 for k even and the Fourier coefficients can be calculated for k odd.
4. Linearity property of Fourier series i.e. a0 = a1 + a2 + a3 + ….
The linearity property of Fourier series states that a0 = a1 + a2 + a3 + … and the Fourier coefficients can be determined.
5. Time shifting property of Fourier series i.e. x (t – t0) = a0 + a1 cos (ωt0) + b1 sin (ωt0) + a2 cos (2ωt0) + b2 sin (2ωt0) + ….
The time shifting property of Fourier series states that x (t – t0) = a0 + a1 cos (ωt0) + b1 sin (ωt0) + a2 cos (2ωt0) + b2 sin (2ωt0) + …, and the Fourier coefficients can be calculated.
6. Time reversal property of Fourier series i.e. x(-t) = a0 + a1 cos(ωt) + b1 sin(ωt) + a2 cos(2ωt) + b2 sin(2ωt) +… .
The time reversal property of Fourier series states that x(-t) = a0 + a1 cos(ωt) + b1 sin(ωt) + a2 cos(2ωt) + b2 sin(2ωt) +… and the Fourier coefficients can be determined.
7. Multiplication property of Fourier series i.e. x(t)y(t) = a0 + a1 cos(ωt) + b1 sin(ωt) + a2 cos(2ωt) + b2 sin(2ωt) +…
The multiplication property of Fourier series states that x(t)y(t) = a0 + a1 cos(ωt) + b1 sin(ωt) + a2 cos(2ωt) + b2 sin(2ωt) +… and the Fourier coefficients can be calculated.
8. Conjugation property of Fourier series i.e. x*(t) = a0* + a1 cos(ωt) + b1 sin(ωt) + a2 cos(2ωt) + b2 sin(2ωt) +… .
The conjugation property of Fourier series states that x*(t) = a0* + a1 cos(ωt) + b1 sin(ωt) + a2 cos(2ωt) + b2 sin(2ωt) +… and the Fourier coefficients can be determined.
9. Differentiation property of Fourier series i.e. x'(t) = a1ω sin(ωt) – b1ω cos(ωt) + a2 2ω sin(2ωt) – b2 2ω cos(2ωt) +…
The differentiation property of Fourier series states that x'(t) = a1ω sin(ωt) – b1ω cos(ωt) + a2 2ω sin(2ωt) – b2 2ω cos(2ωt) +… and the Fourier coefficients can be calculated.
10. Integration property of Fourier series i.e. x(t) = a0t + a1 sin(ωt)/ω + b1 cos(ωt)/ω + a2 sin(2ωt)/2ω + b2 cos(2ωt)/2ω +…
The integration property of Fourier series states that x(t) = a0t + a1 sin(ωt)/ω + b1 cos(ωt)/ω + a2 sin(2ωt)/2ω + b2 cos(2ωt)/2ω +… and the Fourier coefficients can be determined.
11. Periodic Convolution property of Fourier series i.e. x(t) = a0 + a1 cos(ωt) + b1 sin(ωt) + a2 cos(2ωt) + b2 sin(2ωt) +…
The periodic convolution property of Fourier series states that x(t) = a0 + a1 cos(ωt) + b1 sin(ωt) + a2 cos(2ωt) + b2 sin(2ωt) +… and the Fourier coefficients can be calculated.
The relationship between the coefficients of the exponential form and the coefficients of the trigonometric form is given by: a0 = c0/2, a1 = (c1+c-1)/2, b1 = (c1 – c-1)/2j, a2 = (c2+c-2)/2, b2 = (c2 – c-2)/2j, and so on.
When x (t) is real, then a, and b, are real, we have
Shifting the waveform to the left or right with respect to the reference time axis t = 0 only affects the phase values of the spectrum, while the magnitude spectrum remains unchanged. Moving the waveform up or down with respect to the time axis only affects the DC value of the function.