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QUESTION: 1

Which of the following parameters are required to calculate the correlation between the signals x(n) and y(n)?

Solution:

Explanation: Let us consider x(n) be the input reference signal and y(n) be the reflected signal.

Now, the relation between the two signals is given as y(n)=αx(n-D)+w(n)

Where α-attenuation factor representing the signal loss in the round-trip transmission of the signal x(n)

D-time delay between the time of projection of signal and the reflected back signal

w(n)-noise signal generated in the electronic parts in the front end of the receiver.

QUESTION: 2

The cross correlation of two real finite energy sequences x(n) and y(n) is given as:

Solution:

Explanation: If any two signals x(n) and y(n) are real and finite energy signals, then the correlation between the two signals is known as cross correlation and its equation is given as

QUESTION: 3

Which of the following relation is true?

Solution:

Explanation: we know that, the correlation of two signals x(n) and y(n) is

QUESTION: 4

What is the cross correlation sequence of the following sequences?

x(n)={….0,0,2,-1,3,7,**1**,2,-3,0,0….}

y(n)={….0,0,1,-1,2,-2,**4**,1,-2,5,0,0….}

Solution:

Explanation:

QUESTION: 5

Which of the following is the auto correlation of x(n)?

Solution:

Explanation: We know that, the correlation of two signals x(n) and y(n) is

QUESTION: 6

What is the relation between cross correlation and auto correlation?

Solution:

Explanation:

We know that, a^{2}r_{xx}(0)+b^{2}r_{yy}(0)+2abr_{xy}(l) ≥0

=> (a/b)^{2}r_{xx}(0)+r_{yy}(0)+2(a/b)r_{xy}(l) ≥0

Since the quadratic is nonnegative, it follows that the discriminate of this quadratic must be non positive, that is 4[r^{2}_{xy}(l)- r_{xx}(0) r_{yy}(0)] ≤0 =>|r_{xy}(l)|≤√(r_{xx(0)}.r_{yy}(0)).

QUESTION: 7

The normalized auto correlation ρxx(l) is defined as:

Solution:

Explanation: If the signal involved in auto correlation is scaled, the shape of auto correlation does not change, only the amplitudes of auto correlation sequence are scaled accordingly. Since scaling is unimportant, it is often desirable, in practice, to normalize the auto correlation sequence to the range from -1 to 1. In the case of auto correlation sequence, we can simply divide by r_{xx} (0). Thus the normalized auto correlation sequence is defined as ρ_{xx}(l)= (r_{xx} (l))/(r_{xx} (0)).

QUESTION: 8

Auto correlation sequence is an even function.

Solution:

Explanation: Let us consider a signal x(n) whose auto correlation is defined as r_{xx} (l).

We know that, for auto correlation sequence r_{xx} (l)=r_{xx} (-l).

So, auto correlation sequence is an even sequence.

QUESTION: 9

What is the auto correlation of the sequence x(n)=a^{n}u(n), 0<a<l?

Solution:

Explanation:

QUESTION: 10

If x(n) is the input signal of a system with impulse response h(n) and y(n) is the output signal, then the auto correlation of the signal y(n) is:

Solution:

Explanation: r_{yy}(l)=y(l)*y(-l)

=[h(l)*x(l)]*[h(-l)*x(-l)] =[h(l)*h(-l)]*[x(l)*x(-l)] =r_{hh}(l)*r_{xx}(l).

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