Department of ECE, CMR Institute of Technology Jan-May, 2014: Digital Communication Sem 6- A & B, Prof. R. Muralishankar Assignment # 1 – Review of Digital Signal Processing Submission Due: 27 Jan-2013 Note:  Explain your notations and steps explicitly and clearly. Write neatly and legibly.  You will be expected to answer all the assignment questions INDEPENDENTLY. You may be asked to answer any question of the assignment in the class.  Use A4 Sheets only.  An incomplete assignment is not acceptable for submission. ~~~~~ xxx ~~~~~

1. State and prove Parseval’s theorem. 2. If a real signal Φ(𝑡) has a Fourier Transform Φ(𝑓), find the Fourier Transform of Φ(𝑇 − 𝑡). 3. For a real signal Φ(𝑡), prove that Φ∗ (𝑓) = Φ(−𝑓). 4. Show that Fourier Transform of an impulse train is also an impulse train. 5. Show that the multiplication of two signals in the time domain results in the convolution of their individual Fourier Transforms. 6. Find the Fourier Transform of the following pulse. Draw the magnitude spectrum and the phase spectrum. 𝑥(𝑡)

A

T

-T/2

T/2

t

7. Find the Fourier Transform of the following pulse. Draw the magnitude spectrum and the phase spectrum. 𝑥(𝑡) A

-T/2

T/2 -A

t

8. Find the Fourier Transform of the following pulse. Draw the magnitude spectrum and the phase spectrum. 𝑥(𝑡)

A

T/4

T/2

t

-A . . . . . . . .

9. Prove that if a signal 𝑔(𝑡) has Fourier Transform G(𝑓) , then 𝑔(𝑡 − 𝑇) has Fourier Transform G(𝑓)exp(−𝑗2𝜋𝑓𝑇). 10. Prove that if a signal 𝑔(𝑡) has a Fourier transform G(𝑓) then exp(𝑗2𝜋𝑓𝑐 𝑡)𝑔(𝑡) has Fourier transform 𝐺(𝑓 − 𝑓𝑐 ). 11. Find the Fourier Transform of sinc(2𝑤𝑡). 12. Find the Fourier transform of the raised cosine pulse signal defined by 1 + cos(𝜋𝑡) , −1 < 𝑡 < 1 𝑓(𝑡) = � . 0, 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒 13. Use convolution in the frequency domain to prove the following trigonometric identities: a. 2 cos 2 (𝜔0 𝑡) = 1 + cos(2𝜔0 𝑡) b. 2 sin2 (𝜔0 𝑡) = 1 − cos(2𝜔0 𝑡) c. 2 cos(𝜔1 𝑡) cos(𝜔2 𝑡) = cos�(𝜔1 + 𝜔2 )𝑡� + cos�(𝜔1 − 𝜔2 )𝑡� 14. Sketch the following convolution operations: a. 𝐴𝛿(𝑡) ∗ 𝐵𝛿(𝑡 − 𝑡0 ) b. 𝐴𝛿(𝑡 + 𝑡0 ) ∗ 𝐵𝛿(𝑡 − 𝑡0 ) c. 𝐴{𝛿(𝑡 + 𝑡0 ) + 𝛿(𝑡 − 𝑡0 )} ∗ 𝐵{𝛿(𝑡 + 𝑡0 ) + 𝛿(𝑡 − 𝑡0 )} 𝑡

15. Find the Fourier transform P1 (f) of a pulse given by 𝑝1 (𝑡) = sin(8𝜋𝑡) Π �2� , where Π(𝑡) ≅ �

1,

|𝑡| ≤

1

Also find the Fourier series expansion of the 0, otherwise periodic waveform given by 𝑝(𝑡) = ∑∞ 𝑛=−∞ 𝑝1 (𝑡 − 4𝑛) 16. Find the convolution of 𝑥(𝑡) with 𝑥(𝑇 − 𝑡) where 𝑥(𝑡) is shown in Fig.16. 2.

𝑥(𝑡) A

T/2 -A . . . .

Fig. 16.

T

t