# Trigonometric Fourier Series

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**.

## Fourier series representation in Trigonometric form

**Fourier series in trigonometric form**can be easily derived from its exponential form. The complex exponential Fourier series representation of a periodic signal x(t) with fundamental period T

_{o}is given by Since sine and cosine can be expressed in exponential form. Thus by manipulating the exponential Fourier series, we can obtain its Trigonometric form.

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
a_{0} 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

**a**3. If x(t) is half symmetric function i.e. x (t) = -x(t ± T

_{0}= 0, a_{k}= 0 and_{0}/2), then

**a**4. Linearity 5. Time shifting 6. Time reversal 7. Multiplication 8. Conjugation 9. Differentiation 10. Integration 11. Periodic convolution

_{0}= 0, a_{k}= b_{k}= 0 for k even,#### 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.