Signals & Systems - Fourier Series & Fourier Transform Complete Study Notes

Fourier Series Representation

The Periodic functions are the functions which can define by 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 Theorem 

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 another word, any periodic function of (t) can be represented by an infinite series of sinusoids called as Fourier Series.

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

The different forms of the 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 t₂ + 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 disused below.

Consider the Fourier series representation of a periodic signals x(t) defined in equation

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) becomes zero. That is b_n = 0 for all n and the Fourier series representation expressed as

For example, the signal x(t) shown 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 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) 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 number 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 Orthogonality:

    ------(3)

• Zero comes quickly if we integrate the term cosmx  from 0 to π. ⇒ ₀∫^π cosmx dx = 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:

Finding 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 = cos0 = 1.
• The
• odd-numbered coefficients bk =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 a₀²,a_k² b_k².

• Energy in F(x) equals the energy in the coefficients.
• The left-hand side is like the length squared of a vector, except the vector is a function.
• The 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 transforms 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 the 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)

Fourier Transform:

Fourier transform is a transformation technique which transforms non-periodic signals from the continuous-time domain to the corresponding frequency domain. The Fourier transform of a continuous-time non periodic signal x(t) is defined as

where X(jω) is frequency domain representation of the signal x(t) and F denotes the Fourier transformation. The variable ‘ ω’ is the radian frequency in rad/sec. Sometimes X(jω) is also written as X(ω) .

If the frequency is represented in terms of cyclic frequency f (in Hz), then the above equation is written as

Note:

The signal x(t) and its Fourier transform X(jω) are said to form a Fourier transform pair denoted as

Existence of Fourier Transform:

A function x(t) have a unique Fourier transform if the following conditions are satisfied, which are also referred to as Dirichlet Conditions:

Dirichlet Conditions:

(i)  is absolutely integrable. That is,

(ii) x(t) has a finite number of maxima and minima and a finite number of discontinuities within any finite interval.

The above conditions are only sufficient conditions, but not necessary for the signal to be Fourier transformable. For example, the signals u(t),r(t)  and cos (ω₀t) are not absolutely integrable but still possess a Fourier transform.

Magnitude and Phase Spectrum:

The Fourier transform X(jω) of a signal x(t) is in general, complex form can be expressed as

The plot of |X(jω)| versus ω is called magnitude spectrum of x(t) and the plot of versus ω is called phase spectrum. The amplitude (magnitude) and phase spectra are together called Fourier spectrum which is nothing but frequency response of X(jω) for the frequency range  .

Inverse Fourier Transform:

The inverse Fourier transform of X(jω) is given as

This method of calculating the inverse Fourier transform seems difficult as is involves integration. There is another method to obtain inverse Fourier transform using partial fraction. Let a rational Fourier transform is given as

X(jω) can be expressed as a ratio of two factorized polynomial in  jω as shown below.

By partial fraction expansion technique, the above can be expressed as shown below.

where k₁ ,k_{2 ......}k_n calculated depending on whether the roots are real and simple or repeater or complex.

Properties of Fourier Transform:

There are some properties of continuous-time Fourier transform (CTFT) based on the transformation of signals, which are listed below.

a. Linearity:

Linearity property states that the linear combination of signals in the time domain is equivalent to a linear combination of their Fourier transform in the frequency domain.

where a and b are any arbitrary constants.

b. Time Shifting:

The time-shifting property states that delay of t₀ in the time domain is equivalent to multiplication of  with its Fourier transform. It implies that the amplitude spectrum of the original signal does not change but the phase spectrum is modified by a factor of -jωt₀.

c. Conjugation and Conjugate Symmetry:

d. Time Scaling

Time scaling property states that the time compression of a signal in the time domain is equivalent to expansion in the Frequency domain and vice-versa,

e. Differentiation in Time-Domain

The time differentiation property states that differentiation in the time domain is equivalent to multiplication of jω in the frequency domain.

f. Integration in Time-Domain:

g. Differentiation in Frequency Domain:

The differentiation of Fourier transform in the frequency domain is equivalent to multiplication of time-domain signal with -jt .

Differentiation in Frequency Domain

h. Frequency Shifting:

The frequency-shifting property states that a shift of ω₀ in frequency is equivalent to multiplying the time domain signal by 

i. Duality Property:

j. Time Convolution:

Convolution between two signals in the time domain is equivalent to the multiplication of Fourier transforms of the two signals in the frequency domain.

 k. Frequency Convolution:

Convolution in frequency domain (with a normalization factor of 2π) is equivalent to the multiplication of the signals in the time domain.

l. Area Under x(t):

If X(jω) is the Fourier transform of x(t), then,

that is, the area under a time function x(t) is equal to the value of its Fourier transform evaluated at ω= 0

m. Area Under X(jω):

If  X(jω) is the Fourier transform of x(t), then,

n. Parseval's Energy Theorem:

If X(jω) is the Fourier transform of an energy signal x(t). then

where E_x is the total energy of the signal x(t).

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