UNIT 02 FOURIER TRANSFORM ANALYSIS | Basic Signal and Systems -AKTU

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Important Questions For Basic Signals and Systems : 
*Unit-01     *Unit-02    
*Unit-03    *Unit-04 
*Unit-05    *Short-Q/Ans
*Question-Paper with solution 21-22 

Q1. Define Fourier series. Also state the conditions for the existence of Fourier series. 

Ans.
A. Fourier series: The phrase “Fourier series” refers to the linear combination of orthogonal functions used to describe signals over a given period of time. 

B. Existence of Fourier series (Dirichlet’s condition):

1. Dirichlet’s conditions are the circumstances in which a periodic signal can be represented by a Fourier series. 

2. They are as follows: 

• i. The function x(t) must only have one value. 
• ii. There are only a finite number of maxima and minima for the function x(t).. 
• iii. There are a finite number of discontinuities in the function x(t). 
• iv. The function x(t) is absolutely integrable over one period, that is 

Q2. Briefly explain the three forms of Fourier series. 

Ans. Three forms of Fourier series are :

1. Trigonometric Fourier series: The trigonometric Fourier series representation of a periodic signal x(t) with fundamental period T, is given by: 

2. Harmonic Fourier series: Another form of Fourier series representation of a real periodic signal x(t) with fundamental period T₀ is.

3. Exponential Fourier series: 

i. The Fourier series that is utilized the most is the exponential Fourier series. 

ii. This uses a weighted sum of the complex exponential functions to represent the function x(t). 

Q3. Obtain the trigonometric Fourier series for the half wave rectified sine wave. 

Ans. 

Q4. Explain fourier transform of single sided exponential pulse. 

Ans. 

Taking fourier transform 

Q5. Derive the Fourier transforms of the following functions: 

i. rect (t)

ii. e^-|t|

iii. sin 2t

Ans. 

Q6.  Use fourier transform to find the output voltage V₀ in the Fig. 

Ans. 

Q7. Define DTFS. Also write the different properties of Discrete time Fourier series? 

Ans. A. DTFS (Discrete time fourier series): For a discrete-time signal x[n], DTFS of x[n] is defined as, 

Eq. (2.18.1) and (2.18.2) are called discrete time fourier series pairs. 

B. Properties of discrete time fourier series: 

1. Linearity property: 

Let x₁[n] and x₂[n] be two According periodic signals with fundamental period N₀ to linearity property the linear combinations of these two signals is also periodic with the same fundamental frequency N₀.

2. Time shifting property: 

If x[n] is time shifted by n₀, then the periodicity of x[n – n₀] is same as x[n].

3. Time reversal property: 

If x[n] with fundamental period N_o, is the time reversal, the fundamental period is not changed but the Fourier coefficient changes its sign.   

4. Multiplication property: 

5. Conjugation property:

According to this property, the discrete time Fourier coefficient of x*[n] conjugate and time reversal of that of x[n], 

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