Description

EE2100: Matrix Theory
Quiz – 1

Please Note :
1. For numerical questions, answers within the accuracy of ±2% are considered good enough to receive full points.
2. If there are any inconsistencies in grading, please indicate it as a comment in the corresponding page on Canvas.

Group 1
(One among the following set of questions)
  x
 
1. (2 points) Consider a =  y  ∈ R3. Compute the angle that a makes with the standard basis vector e3.
 
  z
  x
 
2. (2 points) Consider a =  y  ∈ R3. Compute the angle that a makes with the standard basis vector e2.
 
  z
Group 2
(Understanding the role of vectors in operations related to signals)
1. Let x ∈ RN and y ∈ RN denote a finite discrete time signal of length N. (Note that the discrete-time signals are usually indexed from 0). The auto-correlation function of a signal x[n] (typically denoted as Rxx) is defined as
Rxx[k] = Xx[n]x[n − k] (3)
n
On the other hand, the cross-correlation between two signals x[n] and y[n] is defined as
] (4)
Finally, the correlation coefficient Cxy (which can be considered as a measure of similarity between two signals) is defined as
(5)
(a) (2 points) Compute the auto-correlation coefficient Rxx[0] for a discrete-time signal of length 8, which, when represented as a vector x ∈ R8, is given by
8
x = Xαiei (6)
i=1
(Note: To calculate the exact numerical answer, substitute the value of αi given in the question).

(b) (2 points) Compute the cross-correlation coefficient Rxy[0] for discrete-time signals (of length 8), which, when represented as vectors, are given by
x and y (9)
(Note: To calculate the exact numerical answer, substitute the value of αi, βi given in the question).
(c) (2 points) Compute the correlation coefficient Cxy for discrete-time signals (of length 8), which, when represented as vectors, are given by
x and y (12)
(Note: To calculate the exact numerical answer, substitute the value of αi, βi given in the question).
(d) (2 points) What is the maximum and minimum limit of the correlation coefficient Cxy.
Group 3
(The basic idea behind Grahm-Schmidt Algorithm)
 
1
1. (2 points) Let x =   and y. Compute x · (y − Projxy). α