Fourier Transform and its Properties Study notes

1.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

Formulas for GATE Electrical Engineering - Power Systems

Note:

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

2.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) it must be 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.

3.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 ∠jω 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  - infinity to + infinity .

4. 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 repeated or complex.

5. Properties of Fourier Transform:

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

5.a. Linearity:

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

where a and b are any arbitrary constants.

5.b. Time Shifting:

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

5.c. Conjugation and Conjugate Symmetry:

5.d. Time Scaling

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

5.e. Differentiation in Time-Domain

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

5.f. Integration in Time-Domain:

5.g. Differentiation in Frequency Domain:

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

Differentiation in Frequency Domain

5.h. Frequency Shifting:

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

5.i. Duality Property:

5.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.

 5.k. Frequency Convolution:

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

5.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

5.m. Area Under X(jω):

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

5.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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