Learning Objectives:

1. Calculate the Infinite Fourier transform, Fourier Sine and Cosine transform of elementary functions from the definition.
2. Demonstrate their understanding of the shifting theorems, Fourier integral theorems, Inverse Fourier sine and cosine transforms by applying them to appropriate examples.
3. Calculate the Finite Fourier cosine and sine transform and apply it in solving boundary value problems.

• Fourier Transform is a mathematical transformation employed to transform signal between time (or spatial) domain and frequency domain.
• It is a tool that breaks a waveform (a function or signal) into an alternate representation, characterized by sine and cosines.
• It shows that any waveform can be re-written as the weighted sum of sinusoidal functions.
• Given an image 'a' and it Fourier transform 'A':
  • The Forward transform goes from the spatial somain (either continuous or discrete) to the frequency domain which is always continuous.
  • The Inverse goes from the frequency domain to the spatial domain.

Solved Example: 9995-01

A signal x(t) has a Fourier transform X($\omega$). If x(t) is a real and odd function of t, them X($\omega$) is: (GATE ECE 1999)

A. A real and even function of $\omega$

B. An imaginary and odd function of $\omega$

C. An imaginary and even function of $\omega$

D. A real and odd function of $\omega$

If 𝑓(𝑡) is real and even then 𝐹($\omega$) is real Even → 𝑓(𝑡) = 𝑓(−𝑡) 𝐹($\omega$) = 𝐹(−$\omega$) Real → 𝑓(−$\omega$) = 𝑓∗($\omega$) Or 𝐹($\omega$) = 𝐹∗($\omega$)

If 𝑓(𝑡) is real and odd 𝐹($\omega$) is pure imaginary odd → 𝑓(𝑡) = −𝑓(−𝑡) 𝐹($\omega$) = −𝐹(−$\omega$)

Correct Answer: B

Solved Example: 9995-02

Consider a signal defined by: \(x\left( t \right) = \left\{ {\begin{array}{*{20}{c}} {{e^{~j10t}}}&{for\left| t \right| \le 1}\\ 0&{for\left| t \right| > 1} \end{array}} \right.\) Its Fourier Transform is: (UPPCL AE EE Nov 2019 Shift I)

A. \(\dfrac{{2\sin \left( {\omega - 10} \right)}}{{\omega - 10}}\)

B. \(\dfrac{{2{e^{j10}}\sin \left( {\omega - 10} \right)}}{{\omega - 10}}\)

C. \(\dfrac{{2sin\omega }}{{\omega - 10}}\)

D. \(\dfrac{{{e^{j10\omega }}2sin\omega }}{\omega }\)

Correct Answer: A

Solved Example: 9995-03

Let f(t) be a continuous-time signal and let F(ω) be its Fourier Transform defined by

\(F\left( \omega \right) = \mathop \smallint \limits_{ - \infty }^\infty f\left( t \right){e^{ - j\omega t}}dt\)

\(g\left( t \right) = \mathop \smallint \limits_{ - \infty }^\infty F\left( u \right){e^{ - ju t}}du\)

What is the relationship between f(t) and g(t)? (GATE EE 2014 Shift I)

A. g(t) would always be proportional to f(t)

B. g(t) would be proportional to f(t) if f(t) is an even function

C. g(t) would be proportional to f(t) only if f(t) is a sinusoidal function

D. g(t) would never be proportional to f(t)

Correct Answer: B

Solved Example: 9997-01

The trigonometric Fourier series of an even function of time does not have: (GATE ECE 1996)

A. The DC term

B. Sine terms

C. Cosine terms

D. Odd harmonic terms

For periodic even function, the trigonometric Fourier series does not contain the sine terms (odd functions)

It has DC term and cosine terms of all harmonics.

Correct Answer: C

Solved Example: 9997-02

The Fourier series of a real periodic function has only

P. Cosine terms if it is even

Q. Sine terms if it is even

R. Cosine terms if it is odd

S. Sine terms if it odd

Which of the above statements are correct? (GATE ECE 2009)

A. P and S

B. P and R

C. Q and S

D. Q and R

The Fourier series for a real periodic function has only cosine terms if it is even and sine terms if it is odd

Correct Answer: A