Module 2

Transforms; Fourier Series and Fourier Transforms; Convergence of Fourier Transforms; Properties of CT transforms.

Related Video and Web Links

Sl no

Link Topic covered

1. http://freevideolectures.com/Course/2252/ Fourier Transform and its applications

2. http://www.youtube.com/watch?v=1JnayXHhjlg Introduction to Fourier Transform

3. http://see.stanford.edu/see/courseinfo.aspx?coll=84d174c2-d74f-493d-92ae-c3f45c0ee091

The Fourier transform applications

4. http://www.infocobuild.com/education/audio-video-courses/electronics/ee261-fourier-transform-stanford.html

Fourier Transform introduction

5. http://www.thefouriertransform.com/ A thorough tutorial on FT and FS

6. http://www.cosmolearning.com/courses/the-fourier-transforms-and-its-applications/video-lectures/

30 video lectures by Prof Brad Osgood, Stanford University

7. http://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011/video-lectures/

Lecture series by Alan.P.Openheim, MIT

8. http://www.learnerstv.com/video/Free-video-Lecture-4374-engineering.htm

Fourier Series video

9. http://www.fourier-series.com/ Audio-Video lectures on FS

10. web.cecs.pdx.edu/~mperkows/CAPSTONES/.../L11-FourierProperties.

Fourier Series Properties with examples

11. www.geophysik.uni-muenchen.de/~igel/.../L06_spectra_applications

Application of FS to seismology

12. www.ele.uri.edu/courses/ele436/CFTSpecial Fourier Transform of special functions

13. web.mit.edu/22.058/www/documents/Fall2002/lectures/lecture3 Applications of FT

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Assignment I – Objective type

1) The set of rational numbers is countably infinite. (True/False).

2) Consider the signal 𝑥[𝑛] = [0,−1,1,3,6,−5,4].If the signal is thought of as a vector, then its dimension is _______.

3) Let 𝑥(𝑡) = 𝑒𝑗𝜔𝑘𝑡 be an input given to a Linear Shift Invariant system. Then the output will be of the form _______.

4) The inner product of 𝑠𝑖𝑛𝑥 and 𝑐𝑜𝑠2𝑥 over the interval [−𝜋,𝜋] is:

a) 2π b) π c) 0 d) 1

5) Define Eigen signal of a given system.

6) Which of the following signals are periodic? a) 𝑥(𝑛) = (−1)𝑛 b) 𝑥(𝑡)=2π+1 c) 𝑥(𝑛) = 𝑒𝑗𝜔𝑛

d) 𝑥(𝑡) = 𝑒−𝑗𝜋𝑘𝑡

7) ∫ 𝑒𝑗2𝜋𝑡2𝜋0 𝑒𝑗4𝜋𝑡 𝑑𝑡=___________.

8) If the set of harmonically related complex exponentials is to form an orthonormal set. Then the normalization factor is _____.

9) The Fourier series coefficients for sin(𝜔𝑡) are ______.

10) Determine which of the periodic waveforms shown have Fourier series coefficients which are purely real:-

a)

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b)

c)

11) Let x[n] be a real and odd periodic signal with period N=7. If the Fourier coefficients X(k) are such that: X(15)=j , X(16)=4j and X(17)=3j. Find i ) X(0) ii)X(-1) 12) Let x(t) be a signal with a fundamental frequency ω. Consider 𝑦(𝑡) = 𝑥(2 − 𝑡) + 𝑥(𝑡 − 2).Then the fundamental frequency of y(t) in terms of ω is ________.

13) If X(k) is the Fourier coefficients of x(t) then the coefficients Y(k) for the signal Re{x(t)} is _______.

14) The Fourier series of sin2 𝑥 is: _____________.

15) Let x1(t) and x2(t) have a fundamental frequency ω1 and ω2 respectively. Given that x2(t) = x1(t − 1) + x1(t + 1), what is the relation between ω1 and ω2?

16) In Q 10, The Fourier series coefficients of x2(t) in terms of the Fourier series coefficients of x1(t) are: a) (X1[k] − jX1[−k])𝑒−𝑗𝜔1𝑘 b) (X1[−k] − jX1[k])𝑒−𝑗𝜔1𝑘 c) (X1[k] + jX1[−k])𝑒−𝑗𝜔1𝑘 d)None of the above

17) Consider the signal 𝑥(𝑡) = 4 cos(100𝜋𝑡) sin (1000𝜋𝑡) with a fundamental period T=1/50.The non zero Fourier coefficients are:

a) X[-4],X[4],X[-7],X[7]

b) X[3],X[-3],X[4],X[-4]

c) Neither a nor b

18) Convergence is square norm implies and is implied by point wise convergence. (True/False)

19) Does the signal 𝑥(𝑡) = 𝑡𝑠𝑖𝑛 �1𝑡� satisfy Diriclet’s conditions?(Yes/No).

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20) Consider the signal 𝑥(𝑡) = �0, 0 < 𝑥 < 25, 2 < 𝑥 < 40, 4 < 𝑥 < 6

and 𝑥(𝑡 + 6) = 𝑥(𝑡).Then the Power of 𝑥(𝑡)

is

a) 10 b) 25 c) 50/6 d) 75/6

21) In Q.21 the Fourier series of 𝑥(𝑡) at the point t=4 converges to the value _____. 22) Gibbs phenomenon occurs when the Fourier series of a signal 𝑥(𝑡) a) Converges uniformly at point b) Does not converge at a point c) Does not converge uniformly d) Does not exist 23) The Fourier series of signals with finite energy (in one period):- a) Converges b) converges uniformly c) diverges d)cannot be concluded 24) Give an example of a function for which the Fourier series fails to converge. 25) The Diriclet conditions for convergence are: a) Necessary b) Sufficient c) Both necessary and sufficient 26) Which of the following functions are absolutely integrable?

a)𝑡2 b) 𝑒−𝑡𝑢(𝑡) c)𝑥−1 d)log 𝑡

27) What is the Inverse Fourier transform of a rect function?

28) The Fourier transform of 𝛿(𝑡 − 5) is_______. 29) 𝑥(𝑡) is a positive rectangular pulse from 𝑡 = −1 to 𝑡 = +1.The value of ∫ |𝑋(𝑗𝜔)|2𝑑𝜔∞

−∞ is:- a) 2 b) 4 c) 2π d) 4π 30) Let 𝑥(𝑡) = 𝑟𝑒𝑐𝑡(𝑡 − 1

2).If 𝑠𝑖𝑛𝑐 = sin(𝜋𝑡) /𝜋𝑡.Then the Fourier transform of 𝑥(𝑡) is:-

a)𝑠𝑖𝑛𝑐(𝜔2𝜋

) b)𝑠𝑖𝑛𝑐 �𝜔2𝜋� sin (𝜔

2) c) neither a nor b

31) Does every periodic function that can be represented as a Fourier series have a Fourier transform?

32) Does the signal 𝑒−𝑎𝑡𝑢(−𝑡) have a Fourier Transform?

33) State the duality property of the Fourier Transform.

34) Let 𝑥(𝑡) = 𝑟𝑒𝑐𝑡(𝑡 − 12) , then the Fourier Transform of 𝑥(𝑡) + 𝑥(−𝑡) is:

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a)𝑠𝑖𝑛𝑐(𝜔2𝜋

) b)2𝑠𝑖𝑛𝑐(𝜔2𝜋

) c)𝑠𝑖𝑛𝑐 �𝜔2𝜋� sin (𝜔

2𝜋) d)2𝑠𝑖𝑛𝑐 �𝜔

2𝜋� cos (𝜔

2𝜋)

35) State the conditions for convergence of the Fourier Transform. 36) The Fourier Transform of 𝑒− 𝑡 2𝑢(𝑡) is ___________. 37) The Fourier Transform of 𝑡𝑒−2𝑡𝑠𝑖𝑛4𝑡 is________. 38) If 𝑥(𝑡) is real valued then 𝑋(𝑗𝜔) = 𝑋∗(𝑗𝜔)[True/False] 39) If 𝑥(𝑡) is real and odd, then 𝑋(𝑗𝜔) is _______ and ________. 40) Find the Fourier Transform of 𝑡𝑒−4𝑡 ∗ sin (6𝜋𝑡) 41) Consider the system F as shown. What is w (t)?

42) The impulse response of the filter shown below has which of the following properties?

(a) Real valued (b) Causal (c) Even (d) Complex valued (e) Odd

43) The above filter can be classified as: a) Low-pass b) High-pass c) Band-Pass d) Band reject

44) The ideal filters are realizable (True/False)

45) State Parseval’s Theorem.

46) The Fourier transform of 𝑥(𝑡 − 𝑎) is _______.

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47) Given that the Fourier transform of 𝑥(𝑡) is 𝑋(𝑗𝜔) , find the Fourier transform of 𝑑(𝑒𝑗𝜔𝑡)𝑑𝑡

. 48) The Fourier transform of 𝑒−𝑡𝑢(𝑡) is 1

1+𝑗𝜔.Then the Fourier transform of 𝑡2𝑒−𝑡𝑢(𝑡) is

________.

49) The signals 𝑥(𝑡) and 𝑥(𝑎𝑡) have the same energy. (True/False).

50)𝑥(0) = 12𝜋 ∫ 𝑋(ω)dω∞

−∞ . (True/False). 51) If 𝑥(𝑡) is real valued then:- a)𝑋(𝜔) is real (b)𝑋(𝜔) = 𝑋(−𝜔) (c)𝑋(𝜔) = 𝑋∗(−𝜔) (d) 𝑋(𝜔) is purely imaginary 52) Let X(ω) be the Fourier Transform of x(t).Then the inverse Fourier Transform of 𝑗 𝑑𝑋(𝜔)

𝑑𝜔 is

_________.

53) If 𝑥(𝑡) is even then X(ω) is also even (True/False).

54) Determine which out of (a) and (b) correspond to real valued functions.

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Assignment II

1) Let the impulse response of a LSI system be:-

ℎ(𝑡) = �1, 0 < 𝑡 < 60, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

If the input to the system is 𝑥(𝑡) = 𝑒−𝑗𝜔𝑡 then the output will be of the form 𝐻(𝑗𝜔)𝑒−𝑗𝜔𝑡. Find 𝐻(𝑗𝜔)

2) Prove that the trigonometric functions are orthogonal over every interval of 2π.

3) Find the Fourier series coefficients for the following signals:

a)𝑥(𝑡) = sin (10𝜋𝑡 + 𝜋6

)

b) 𝑦(𝑡) = 1 + 𝑐𝑜𝑠2𝜋𝑡

c)𝑧(𝑡) = 𝑥(𝑡)𝑦(𝑡)

4) Determine the Fourier series of the signal shown:

5) Suppose x(t) is periodic with period T and is specified in the interval 0 < t < T/4 as shown

Sketch x(t) for its entire period if it’s Fourier series has only odd harmonics and x(t) is an even function

6) Find the Fourier series of the following signal

a) 𝑥(𝑡) = �1, 0 < 𝑡 < 40, 4 < 𝑡 < 8 and 𝑥(𝑡 + 8) = 𝑥(𝑡)

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7) Prove that the set of all functions which are absolutely integrable can be represented by a Fourier series.

8) State and explain the Diriclet conditions for convergence of the Fourier series. If a signal does not satisfy Diriclet’s conditions, can the signal still be represented in terms it’s Fourier series?

9) Find the Inverse Fourier transform of the signal 𝑋(𝑗𝜔).The Magnitude and Phase plots of the signal is as shown

10) Consider the impulse train as shown below. Find the Fourier series and the Fourier Transform of the impulse train.

11) Find the Fourier transform of the following signals

a)𝛿(𝑡 − 5)

b)𝑒−𝑎𝑡𝑢(𝑡),𝑎 > 0

c) sin(ωt)

12) Consider the impulse train:

Let x(t) be the signal shown below

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Show that the periodic signal y(t) obtained by periodically repeating x(t) is given by y(t)=x(t)*p(t). Find the Fourier transform of y(t).

13) Given the relationships : 𝑦(𝑡) = 𝑥(𝑡) ∗ ℎ(𝑡)

𝑔(𝑡) = 𝑥(3𝑡) ∗ ℎ(3𝑡)

And given that 𝑥(𝑡) has the Fourier transform 𝑋(𝑗𝜔) and ℎ(𝑡) has the transform 𝑋(𝑗𝜔), show that:- 𝑔(𝑡) = 𝐴𝑦(𝐵𝑡)

Determine the values of A and B.

14) Determine the Fourier transform of 𝑒−|𝑡|

a) Using the appropriate property, find the Fourier transform of 𝑡𝑒−|𝑡|

b) Using the result of (a) and the duality property, find the Fourier transform of 4𝑡(1+𝑡2)2

15) By first expressing the triangular signal in the Figure as the convolution of a rectangular pulse with itself, determine the Fourier transform of 𝑥(𝑡)

16) The output of a causal LTI system is related to the input 𝑥(𝑡) by the differential equation: 𝑑𝑦(𝑡)𝑑𝑡

+ 𝑦(𝑡) = 𝑥(𝑡) Determine 𝐻(𝑗𝜔) and sketch its magnitude and phase spectrum Find 𝑌(𝑗𝜔) 𝑎𝑛𝑑 𝑦(𝑡) 𝑖𝑓 𝑥(𝑡) = 𝑒−𝑡𝑢(𝑡)

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17) Find the Fourier transform of x(t) in the following ways:- a) Directly using integration b) Using the time differentiation property c) Using the Fourier transform of the rect( ) function and convolution property

18) Determine the energy in the signal 𝑥(𝑡) for which the Fourier transform 𝑋(𝑗𝜔) is given by the figure below

19) State and prove the convolution property of the Fourier transform

20) Find the response of the system whose impulse response is given by ℎ(𝑡) = 𝑒−𝑎𝑡𝑢(𝑡) to the signal 𝑥(𝑡) = 𝑒−𝑏𝑡𝑢(𝑡) using the convolution property.

21) x(t) is as shown. If y(t) = x �t2� . Then Find Y(jω) and plot its magnitude and phase spectra.

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22) Determine the energy in the signal 𝑥(𝑡) if 𝑋(𝑗𝜔) is a shown.

23) Using mathematical induction show that for

24) Check the validity of the following:-

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True or False

1. Transformations are performed normally on the dependent variable. 2. There are some signals which cannot be represented using Fourier Series. 3. A signal is a vector with infinite dimensions. 4. The set of integers is countably finite. 5. The set of real numbers is countably infinite. 6. The complex exponential signal is an Eigen signal. 7. The spacing between adjacent frequency components becomes larger as the time period

increases. 8. The frequency spectrum of a single pulse, is also a pulse. 9. Fourier Transforms do not exist for signals which have discontinuities. 10. The FT of a linear combination of two signals is same as the linear combination of their

respective transforms. 11. A small local change in the time domain signal will cause a small local change in the

frequency spectrum. 12. The Fourier Coefficient magnitudes of the signal x(t-to) is scaled when compared to the

coefficients of x(t). 13. When the input is a constant signal , the FT does not converge. 14. The frequency spectrum of the impulse function has one component at f=0. 15. The FT of an even signal is odd. 16. Energy calculated form time domain is same as energy calculated from frequency

domain. 17. Convolution between two periodic signals diverge. 18. Rxx(0) is energy of the signal. 19. If x(t) is periodic, tx(t) is also periodic. 20. Two different time domain signals cannot have the same frequency components.

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Review Questions

1. Why is transformation necessary? 2. Give some examples of transformations in real life. 3. What is the period of an aperiodic signal? 4. What are the conditions satisfied by the inner dot product? 5. Define the inner product for continuous and discrete time signals. 6. Show that H(ω) is the eigenvalue corresponding to exponential signal. 7. Represent H(ω) as an innerproduct. 8. What is Fourier Series representation of a periodic signal? 9. Show that the set of complex exponentials forms an orthogonal set in vector space. 10. Derive expression for the Fourier coefficient. 11. Distinguish between pointwise convergence and convergence in squared form. 12. What are Dirichlet Conditions For Pointwise Convergence ? Give examples of signals

which do not satisfy these conditions. 13. How does the frequency spectrum of a periodic square wave look like, as T tends to

infinity? 14. What happens to the frequency spectrum, when the time period of a signal is very high? 15. Plot the frequency spectrum of a single rectangular pulse, symmetric about the time

origin, t=0. 16. The Fourier Transform is an extended version of the Fourier series. 17. Discuss conditions for the Fourier Transform to exist. 18. What is the meaning of a signal being absolutely integrable? 19. What is duality which exists in Fourier Transform? 20. What is the trigonometric and cosine representation of Fourier Series? 21. How does a local change in the time signal affect the frequency spectrum? 22. What happens to the time domain signal if a small kink is introduced in the frequency

spectrum? 23. The frequency spectrum of the impulse function contains all the frequencies. Comment. 24. How are the coefficients of the Fourier Transform of a periodic signal obtained from the

coefficients of the Fourier Series? 25. Show that the FT of a periodic signal is a train of impulses. 26. How are the FS coefficients of the signals x(t) and x(-t) related? 27. How are the FS coefficients of the signals x(t) and conjugate of x(t) related? 28. Using properties of FT, show that the FT of an even signal is even and odd signal is odd. 29. Modulation makes it possible to transmit different signals on the same channel.

Comment. 30. What is the FT of the convolution of two signals? 31. What is the FT of two signals multiplied? 32. What is energy spectral density? 33. Prove Parseval’s theorem? 34. How are the Fourier Series coefficients of a periodic signal x(t) multiplied by an

aperiodic signal y(t) related to the FS coefficients of x(t)? 35. The circular convolution picks out common frequencies in the two signals convolved.

Prove.

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36. What is the power in the signal y(t) where y(t) is the convolution of two periodic signals x(t) and z(t)?

37. If y(t) =x(t-to) where does the autocorrelation of integral peak and where does the cross correlation integral peak?

38. How can cross correlation be used to find the distance of an object from a location? 39. What is the FT of u(t), u(t-to), e-t u(t) , t e-t u(t) 40. What is energy normalized scaling of the independent variable?

FAQs

1. What is an Eigensignal? 2. What is a complex sinusoid? 3. What is the period of an aperiodic signal? 4. What happens to the spacing between adjacent periodic components as the period of a

signal increases? 5. To what signals do we apply Fourier Transform? 6. When does the frequency spectrum become continuous? 7. How is the FT of x(t-to) related to FT of x(t)? 8. If Ck is the kth Fourier Series coefficient of a signal x(t) what is the kth coefficient of the

FS of x(-t)? 9. What is the FT of an even signal? 10. What is half-wave symmetry in a periodic signal? 11. What is AM? 12. What is FM? 13. What is a band-limited signal? 14. Plot the frequency spectrum of a LP, HP, BP and BS filters. 15. What does multiplication of FTs in frequency domain imply in time domain? 16. What is Parseval’s theorem? 17. What is the FT of impulse function? 18. What is the FT of dx(t)/dt? 19. What is FT of x(at)? 20. Two different signals can have the same Frequency spectrum. Justify.

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