EE5355 Discrete Transforms and their Applications

INSTRUCTOR: Dr. K.R. RaoFall 2015, Test 1

Tuesday, 22 September 20155:30 – 6:50 PM (1 hour and 20 minutes)

(CLOSED BOOK, CLOSED NOTES)

INSTRUCTIONS:1. Closed books and closed notes.2. Calculator is allowed.3. Please show all the steps in your works. 4. You can work problems in any order.

At the end please rearrange as 6, 7, 8, and 9.5. Please print your name and student ID.6. No cheating, no talking.

Name: _______________________________

Student ID: ___________________________

1.a. In developing the KLT for 2D-signals it is suggested that the row and the column statistics of

a 2D-signal be assumed to be independent of each other. Also row and column statics are considered to be identically distributed. Why? Explain clearly the advantages. [10]

b. Define KLT.List the advantages and disadvantages of KLT.Explain why is it necessary to rearrange the Eigenvectors such that the corresponding Eigenvalues are in the order, λ1 ≥ λ2 ≥ λ3 ≥ λN [10]

2.a. Given sequences of length M, x0, x1, x2, ... , xM-1 and length N, y0, y1, y2, … , yN-1. Explain with a

block diagram how the aperiodic convolution of these two sequences can be obtained via FFT/IFFT approach. [20]

3.a. Show how radix-2, DIT-FFT for N=2n , (n=integer) can be developed, use N=8. Start with {x0 x1

x2 x3 x4 x5 x6 x7} (No proof. No derivation). Only graphical format. What are the advantages of this? Why is it called DIT (Decimation in Time)? [15]

b. If XF(k) is DFT of x(n), find DFT of (-1)n x(n) in terms of XF(k). (Definitions of x(n) and XF(k) given at the end) [05]

4.

a. Show that, X F( N2 +k )=XF∗¿(N2 −k)¿, k=0 ,1 ,2 ,… .., N /2

Name this property. For x(n) real N-point DFT, what is the implication of this? [15]b. The figure below requires 2 adds and 2 multiplies. Rearrange this flowgraph such that it

requires only 1 multiply and 2 adds. [05]

5.a. Show that:

∑n=0

N−1

¿ x (n )∨¿2¿ = 1N ∑k=0

N−1

¿ X F (k )∨¿2¿

Energy is invariant under orthogonal mapping. [10]

b. Given x(n), where n=0,1,2,…….N-1, with T as the sampling interval and XF(k), is the transform sequence in the DFT domain, where, k=0,1,2,…….N-1. Explain clearly what XF(k) represents. [10]

END OF TEST

For your information:1D-DFT and 1D-IDFT are defined as below

XF(k)= ∑n=0

N−1

x (n).WNnk where, k=0,1,2,…….N-1x(n)= ∑

k=0

N−1

X F(k).WN-nk where, n=0,1,2,…….N-1WN = e− j2 Π /N , Nth root of unity.