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4 Dez

In [1, p. 60 exercise 3.3] we are asked to prove that the complex multivariate Gaussian PDF reduces to the complex univariate Gaussian PDF if N=1.

**Solution:**
The complex multivariate Gaussian PDF is given by [1, p. 44, (3.13)] :

with denoting an dimensional complex vector, which is composed of the real dimensional vectors and . The univariate complex gaussian probability density function is given by [1, p. 44, (3.10)] :

with where is a one dimensional complex variable with , and being real variables. For in (1) we obtain for the covariance matrix the following relation:

and thus . The determinant of an single element matrix is per definition the element value itself ([2, p. 326]), thus for . Replacing and in (1) while setting we obtain:

which is equal to (2). QED.

[1] Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”, Prentice Hall, ISBN: 0-13-598582-X.

[2] Lawrence J. Corwin and Robert H. Szczarba: “Calculus in Vector Spaces”, Marcel Dekker, Inc, 2nd edition, ISBN: 0824792793.

(1) |

with denoting an dimensional complex vector, which is composed of the real dimensional vectors and . The univariate complex gaussian probability density function is given by [1, p. 44, (3.10)] :

(2) |

with where is a one dimensional complex variable with , and being real variables. For in (1) we obtain for the covariance matrix the following relation:

and thus . The determinant of an single element matrix is per definition the element value itself ([2, p. 326]), thus for . Replacing and in (1) while setting we obtain:

which is equal to (2). QED.

[1] Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”, Prentice Hall, ISBN: 0-13-598582-X.

[2] Lawrence J. Corwin and Robert H. Szczarba: “Calculus in Vector Spaces”, Marcel Dekker, Inc, 2nd edition, ISBN: 0824792793.

## 3 Responses for "Steven M. Kay: “Modern Spectral Estimation – Theory and Applications”,p. 60 exercise 3.3"

hi dear Panagiotis Chatzichrisafis,

your solutions were really usefull. but i need some other solutions of this book( Modern Spectral Estimation ).

could you please help me?

the problems are:

4.1, 4.8, 5.1, 5.3, 5.9, 5.13, 6.1, 6.4, 6.5, 6.10, 6.11

Hi Forooz,

i am happy to hear that some of the solutions i post are helpfull.

I have currently reviewed solutions up to exercize 5.2 , but i haven’t transferred them into digital form. Please be patient, I’ll try to post asap some solutions of the problems

you asked for, but i can’t promise any dates. May i ask what your profession is ?

Kind regards

P. Chatzichrisafis

Hi- just wanted to thank you for these solutions, they were really helpful! Keep up the good work!

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