Continuous Time Fourier Series

CONTINUOUS TIME FOURIER SERIES

Trigonometric Form of Fourier Series

The prove that the signal x(t), which is a summation of sine and cosine functions of frequencies 0, ω₀, 2ω₀,...kω₀, is a periodic signal with period T. By changing a_ns and b_ns, we can construct any periodic signal with period T. If k → ∞ in the expression for x(t), we obtain the fourier series representation of any periodic signal x(t). That is, any periodic signal can be represented as an infinite sum of sine and cosine functions which themselves are periodic signals of angular frequencies 0, ω₀, 2αω₀,..., kω₀, this set of harmonically related sine and cosine function, i.e., sin nω_{0 t} , and cos nω_{0 t} , n = 0, 1, ... forms a complete orthogonal set over the interval t₀ to t₀+T where T = 2π/ω₀.

The infinite series of sine and cosine terms of frequencies 0, ω₀, 2αω₀,..., kω₀, is known as trigonometric form of Fourier series and can be written as:

Evaluation of Fourier Coefficients of the Trigonometric Fourier Series

The constants  are called Fourier coefficients. To evaluate a₀, we shall integrate both sides of the equation for x(t) over one period (t₀+T) at an arbitrary time to thus,

we know that  since the net areas of sinusoids over complete periods are zero for any non zero integer n and any time t0. Hence each of the integrals in the above summation is zero.

Thus, we obtain

To evaluate a_n and b_n, we can use the following results.

To find Fourier coefficient a_n, multiply the equation for x(t) by cos mω₀t and integrate over one period. that is,

The first and third integrals in the above equation are equal to zero and the second is equal to T/2 where m = n. Therefore,

To find b_n multiply both sides of equation for x(t) by sin mω₀t and integrate over one period. Then

The first and second integrals in the above equation are zero, and the third integral is equal to T/2 when m = n. Thus, we have

a₀, a_n and b_n are called trigonometric fourier series coefficients.

A periodical signal has the same Fourier series for the entire interval -∞ to ∞ as for the interval t₀ to t₀+T, since the same function repeats after every T seconds. The Fourier series expansion of a periodic function is unique irrespective of the location of t₀ of the signal.

Problems

1. Find the Fourier series expression of the half wave rectified sine wave shown in figure.

Solution:

The periodic waveform shown in figure with period T = 2π

Therefore, the trigonometric Fourier series is :

2. Obtain the trigonometric Foureir series for the waveform shown in figure.

Solution :

The waveform shown in figure with a period T = 2 π

Thetrigonometic Fourier series is :

Cosine Fourier Series

The trigonometric Fourier series of x(t) contains sine and cosine terms of the same frequency.

Substituting the values,

The cosine form is also called the Harmonic form Fourier series or polar form Fourier series.

Problem 1:

Find the cosine Fourier series for the waveform shown in figure.

Solution:

The waveform is given by

Trigonometric Fourier series

are the trigonometric Fourier series co efficients.

Cosine form of Representation

Note:

When even or minor symmetry exists, the trigometric Fourier series coefficient are,

Problem 1:

Find the Fourier series expression for the waveform shown in figure.

Solution:

The given waveform is periodic with period T = 2π. For the computational convenience choose one cycle of waveform from -π to π.

Fundamental frequency 

The waveform is described by,

The given waveform has even symmetry because x(t) = x(− t)

The trigometric Fourier series is,

Problem 2:

Obtain the Fourier components of the periodic rectangular waveform shown in figure

Solution:

The waveform shown in figure is a periodic waveform with period T.

Fundamental frequency ω₀ = 2π/T

The given function has even symmetry because

The trigonometric Fourier series is :

Odd or Rotation Symmetry:

When odd or rotation symmetry exists, the trigonometric Fourier series coefficients are:

Problem 1:

Obtain the trigonometric fourier series for the waveform shown in figure.

Solution:

The waveform shown in figure is periodic with period = T. For computational sim plicity. Consider the period from -T/2 to T/2

Fundamental frequency ω₀ = 2π/T

The waveform has odd symmetry because x(t) = -x(-t)

The trigonometric Fourier series is :

Exponential Fourier Series

The function x(t) is expressed as a weighted sum of the complex exponential functions.

Note:

1. The Magnitude line spectrum is always an even function of n.

2. The phase line spectrum is always an odd function of n.

Problem 1:

Obtain the exponential Fourier series for the waveform shown in figure. Also draw the frequency spectrum.

Solution:

The periodic waveform shown in figure with a period T=2π can be expressed as:

Exponential Fourier series :

The Frequency spectrum for the given waveform is :

Problem 2:

Find the exponential Fourier series for the rectified sine wave shown in figure.

Solution:

The waveform shown in figure with a period T = 2

The period of rectified sine wave is T = 1

The fundamental frequency of the rectified

The exponential Fourier series:

The exponential Fourier series is :