Continuous Time Periodic Signals Using Fourier Series

In the present article, you will find the study notes onFourier Series & Representation of Continuous Periodic Signal which will cover the topics such asFourier Theorem, Fourier Coefficients, Fourier Sine Series, Fourier Cosine Series, Orthogonality Relations of Fourier Series Exponential Fourier Series Representation, Even & Odd Symmetry

Periodic Functions

The Periodic functions are the functions which can be defined by the relation f(t + P) = f(t) for all t. For example, if f(t) is the amount of time between sunrise and sunset at a certain latitude, as a function of time t, and P is the length of the year, then f(t + P) = f(t) for all t, since the Earth and Sun are in the same position after one full revolution of the Earth around the Sun.

Fourier Series

Any arbitrary continuous time signal x(t) which is periodic with a fundamental period T_o, can be expressed as a series of harmonically related sinusoids whose frequencies are multiples of fundamental frequency or first harmonic. In other words, any periodic function of (t) can be represented by an infinite series of sinusoids called as Fourier Series.

Periodic waveform is expressed in the form of Fourier series, while non-periodic waveform may be expressed by the Fourier transform.

The different forms of Fourier series are given as follows.

(i) Trigonometric Fourier series

(ii) Complex exponential Fourier series

(iii) Polar or harmonic form Fourier series.

Trigonometric Fourier Series

Any arbitrary periodic function x(t) with fundamental period T₀ can be expressed as follows

This is referred to as trigonometric Fourier series representation of signal x(t). Here, ω₀ = 2π/T₀ is the fundamental frequency of x(t) and coefficients a₀, a_n, and b_n are referred to as the trigonometric continuous time Fourier series (CTFS) coefficients. The coefficients are calculated as follows.

Fourier Series Coefficient

From equation (ii), it is clear that coefficient a₀ represents the average or mean value (also referred to as the dc component) of signal x(t).

In these formulas, the limits of integration are either (–T₀/2 to +T₀/2) or (0 to T₀). In general, the limit of integration is any period of the signal and so the limits can be from (t₁ to t₁ + T₀), where t₁ is any time instant.

Trigonometric Fourier Series Coefficients for Symmetrical Signals

If the periodic signal x(t) possesses some symmetry, then the continuous time Fourier series (CTFS) coefficients become easy to obtain. The various types of symmetry and simplification of Fourier series coefficients are discussed as below.

Consider the Fourier series representation of a periodic signal x(t) defined in equation below as,

Even Symmetry: x(t) = x(–t)

If x(t) is an even function, then product x(t) sinω_ot is odd and integration in equation (iv) above becomes zero. i.e b_n = 0 for all n and the Fourier series representation is expressed as

For example, the signal x(t) shown in below figure has even symmetry so b_n = 0 and the Fourier series expansion of x(t) is given as

The trigonometric Fourier series representation of even signals contains cosine terms only. The constant a₀ may or may not be zero.

Odd Symmetry: x(t) = –x(–t)

If x(t) is an odd function, then product x(t) cosω_ot is also odd and integration in equation (iii) above becomes zero i.e. a_n = 0 for all n. Also, a₀ = 0 because an odd symmetric function has a zero-average value. The Fourier series representation is expressed as

For example, the signal x(t) shown in below figure is odd symmetric so a_n = a₀ = 0 and the Fourier series expansion of x(t) is given as

Fourier Sine Series

The Fourier Sine series can be written as

  ------(2)

• Sum S(x) will inherit all three properties:
• (i): Periodic S(x +2π)=S(x);    (ii): Odd S(−x)=−S(x);    (iii):  S(0) = S(π)=0
• Our first step is to compute from S(x), the term b_k that multiplies sinkx.

Suppose S(x)=∑ b_n sinnx. Multiply both sides by sin kx. Integrate from 0 to π in Sine Series in equation (2)

• On the right side, all integrals are zero except for n = k. Here the property of "orthogonality" will dominate. The sines make 90^o angles in function space, when their inner products are integrals from 0 to π.
• Orthogonality for sine Series

 Condition for Orthogonailty:

    ------(3)

• Zero comes, if we integrate the term cos mx  from 0 to π. ⇒ = 0-0=0
• Integrating cosmx with m = n−k and m = n + k proves orthogonality of the sines.
• The exception is when n = k. Then we are integrating (sinkx)² = 1/2 − 1/2 cos2kx

------(4)

• Notice that S(x)sinkx is even (equal integrals from −π to 0 and from 0 to π).
• We will immediately consider the most important example of a Fourier sine series. S(x) is an odd square wave with SW(x) = 1 for 0<x<π. It is an odd function with period 2 π, that vanishes at x=0 and x= π.

Example:

Find the Fourier sine coefficients b_k of the square wave SW(x) as given above .

Solution:

For k =1 ,2,...using the formula of sine coefficient with S(x)=1 between 0 and π:

• Then even-numbered coefficients b_2k are all zero because cos2kπ = cos 0 = 1.
• The odd-numbered coefficients b_k =4/πk decrease at the rate 1/k.
• We will see that same 1/k decay rate for all functions formed from smooth pieces and jumps. Put those coefficients 4/πk and zero into the Fourier sine series for SW(x).

Fourier Cosine Series

The cosine series applies to even functions with C(−x)=C(x) as

 -----(5)

Cosine has period 2π shown as above in the figure two even functions, the repeating ramp RR(x) and the up-down train UD(x) of delta functions.

• That sawtooth ramp RR is the integral of the square wave. The delta functions in UD give the derivative of the square wave. RR and UD will be valuable examples, one smoother than SW, one less smooth.
• First we find formulas for the cosine coefficients a0 and ak. The constant term a₀ is the average value of the function C(x):

-----(6)

• We will integrate the cosine series  from 0 to π. On the right side, the integral of a₀=a₀π (divide both sides by π). All other integrals are zero.

• Again the integral over a full period from −π to π (also 0 to 2π) is just doubled.

Orthogonality Relations of Fourier Series

Since from the Fourier Series Representation we concluded that for a periodic Signal it can be written as

-------(7)

The condition of orthogonality as follow:

Proof of the orthogonality relations:

This is just a straightforward calculation using the periodicity of sine and cosine and either (or both) of these two methods:

Energy in Function = Energy in Coefficients

There is also another  important equation (the energy identity) that comes from integrating (F(x))². When we square the Fourier series of F(x), and integrate from −π to π, all the "cross terms" drop out. The only nonzero integrals come from 1² and cos2 kx and sin2 kx, multiplied by a₀ ²,a_k ² b_k ².

• Energy  in F(x) equals the energy in the coefficients.
• Left hand side is like the length squared of a vector, except the vector is a function.
• Right hand side comes from an infinitely long vector of a's and b's.
• If the lengths are equal, which says that the Fourier transform from function to vector is like an orthogonal matrix.
• Normalized by constants √2π and √π, we have an orthonormal basis in function space.

Complex Fourier Series

• In place of separate formulas for a₀ and a_k and b_k , we may consider one formula for all the complex coefficients c_k.
• So that  the function F(x) will be complex, The Discrete Fourier Transform will be much simpler when we use N complex exponentials for a vector.

The exponential form of Fourier series of a periodic signal x(t) with period T₀ is defined as

where, ω₀ is the fundamental frequency given as ω₀ = 2π /T₀. The exponential Fourier series coefficients c_n are calculated form the following expression

• Since  c₀ = a₀ is still the average of F(x), because e₀ = 1.
• The orthogonality of e^inx and e^ikx is to be checked by integrating.
  

Example:

Compute the Fourier series of f(t), where f(t) is the square wave with period 2π. which is defined over one period by.

The graph over several periods is shown below.

Solution:

Computing a Fourier series means computing its Fourier coef­ficients. We do this using the integral formulas for the coefficients given with Fourier's theorem in the previous note. For convenience we repeat the theorem here.

By applying these formulas to the above waveform we have to split the integrals into two pieces corresponding to where f(t) is +1 and where it is −1.

thus for n ≠ 0 ;

   for n = 0

We have used the simplification cos nπ = (−1)ⁿ to get a nice formula for the coefficients b_n.

This then gives the Fourier series for f(t)

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