Trigonometric Fourier Series

We have already covered the Fourier series in its exponential form. Now, let’s explore another version: the Trigonometric Fourier series.

Fourier series representation in Trigonometric form

Fourier Series in Trigonometric Form: The trigonometric Fourier series can be derived from its exponential form. The complex exponential Fourier series for a periodic signal x(t) with a fundamental period T_o is given by

Since sine and cosine can be expressed exponentially, we can derive the trigonometric Fourier series by manipulating the exponential Fourier series.

The trigonometric Fourier series representation of a periodic signal x (t) with fundamental period T, is given by

Where a_k and b_k are Fourier coefficients given by

a₀ is the dc component of the signal and is given by

Properties of Fourier series

1. If x(t) is an even function i.e. x(- t) = x(t), then b_k = 0 and

2. If x(t) is an even function i.e. x(- t) = – x(t), then a₀ = 0, a_k = 0 and

3. If x(t) is half symmetric function i.e. x (t) = -x(t ± T₀/2), then a₀ = 0, a_k = b_k = 0 for k even,

4. Linearity

5. Time shifting

6. Time reversal

7. Multiplication

8. Conjugation

9. Differentiation

10. Integration

11. Periodic Convolution

Relationship between coefficients of exponential form and coefficients of trigonometric form

When x (t) is real, then a, and b, are real, we have

Effect of Shifting Axis of the Signal

• On shifting the waveform to the left right with respect to the reference time axis t = 0 only the phase values of the spectrum changes but the magnitude spectrum remains same.
• On shifting the waveform upward or downward w.r.t time axis changes only the DC value of the function.