When the source symbol probabilities are equal the ML and MAP generates

When the source symbol probabilities are equal the ml

This preview shows page 4 - 7 out of 62 pages.

source symbol probabilities or assumes all are equal. When the source symbol probabilities are equal the ML and MAP generates identical decisions. In the figure for an example of ML, the received signal is r , the output alphabet consists of S i ’s and the distances according to a defined metric (usually Euclidean distance) are d i ’s. ML would decide that S 2 is sent because it has the smallest distance from the output signal. On the other hand MAP could have another decision since the symbol probabilities are used in the decision. But it can be said, with a reasonable confidence, that MAP decision would be the same if 2 , ) ( ) ( 2 i S P S P i .
Image of page 4
151227621 DIGITAL COMMUNICATIONS 5 6. 25.12.2004 2 nd midterm Describe distance metrics and derive correlation metrics from it. Solution Distance metrics is a measure of the distance/closeness of two points in signal space which is usually represented as Cartesian coordinate system. The square of Euclidean distance between two points, r and s m , in Cartesian coordinate system is given as N k mk k m s r D ! 2 ) ( ) , ( s r where, when comparing, the square and the distance itself do the same jobs. Expanding it, we’d have N n mn N n mn n N n n m s s r r D 1 2 1 1 2 2 ) , ( s r or in norms form 2 2 2 ) , ( m m m D s s r r s r . Since 2 r term is common for all s m , and does not have any effect when comparing distances, we can remove it. The negative of the remaining, 2 2 ) , ( m m m C s s r s r is called the correlation metrics since it involves the correlation (inner product) of r and s m . ) , ( m C s r should be maximized to minimize ) , ( m D s r . 2 m s term represents the signal energy and can also be removed in cases of equal energy signals. 7. 25.12.2004 2 nd midterm A binary antipodal signaling source has P (-1)=0.7 and P (1)=0.3. Find the output ensemble of the BSC through which the output of the source is fed if P e ( s i | s j ; i j )=0.01 by forming the channel transition matrix first. Solution The channel transition matrix for the BSC is formed as 99 . 0 01 . 0 01 . 0 99 . 0 Q . The output probabilities of output symbols are 3 . 0 7 . 0 99 . 0 01 . 0 01 . 0 99 . 0 Qz v and 304 . 0 696 . 0 v . Although the signals would probably be distorted, we can expect that the output symbols are reasonably close to input symbols which are -1 and 1, forming the alphabet as A={-1, 1} and concluding our search for the ensemble (A, v ).
Image of page 5
151227621 DIGITAL COMMUNICATIONS 6 8.04.02.2005 Final Exam Show that the following set is orthonormal. Solution For two signals to be orthogonal it is required that ) ( , 0 ) ( ) ( j i dt t t j i 0 ) ( ) ( 2 1 2 1 1 0 2 1 2 1 2 1 2 1 1 0 2 1 2 1 2 1 t t dt dt dt t t (orthogonal) 0 ) ( ) ( 3 1 dt t t (by inspection) (orthogonal) 0 ) ( ) ( 3 2 dt t t (by inspection) (orthogonal) The energies of the signals are 1 ) ( 2 0 2 2 1 2 1 dt dt t 1 ) ( 2 2 dt t (this one is the same as the previous one because of the square) 1 ) ( 3 2 2 3 dt dt t Conclusion : The signals are orthogonal and their individual energies equal to unity. Therefore this set is an orthonormal set. 9. 04.02.2005 Final Exam Determine if the following code is linear. 11011 11 10100 10 01111 01 00000 00 Solution In order for a code to be linear j i c c must also be a symbol in the set.
Image of page 6
Image of page 7

You've reached the end of your free preview.

Want to read all 62 pages?

  • Fall '18
  • Mr. Bhullar
  • Hamming Code, Error detection and correction, Parity bit

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Stuck? We have tutors online 24/7 who can help you get unstuck.
A+ icon
Ask Expert Tutors You can ask You can ask ( soon) You can ask (will expire )
Answers in as fast as 15 minutes
A+ icon
Ask Expert Tutors