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

A. Time delay
B. Attenuation factor
C. Noise signal
D. All of the mentioned

Answer: D

Let us consider x(n) to be the input reference signal and y(n) to 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.

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

A. \(r_{xy}(l)=\sum_{n=-\infty}^{\infty}x(n)y(n-l)\) where l=0,±1,±2,…

B. \(r_{xy}(l)=\sum_{n=0}^{\infty}x(n)y(n-l)\) where l=0,±1,±2,…

C. \(r_{xy}(l)=\sum_{n=-\infty}^{\infty}x(n)y(n-l)\) where -∞<l<∞

D. none of the mentioned

Answer: A

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

rxy(l)=\(\sum_{n=-\infty}^{\infty} x(n)y(n-l)\) where l=0,±1,±2,…

3. Which of the following relation is true?

A. rxy(l)= rxy(-l)
B. rxy(l)= ryx(l)
C. rxy(l)= ryx(-l)
D. none of the mentioned

Answer: C

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

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….}

A. {10,9,19,36,-14,33,0,7,13,-18,16,7,5,-3}
B. {10,-9,19,36,-14,33,0,7,13,-18,16,-7,5,-3}
C. {10,9,19,36,14,33,0,-7,13,-18,16,-7,5,-3}
D. {10,-9,19,36,-14,33,0,-7,13,18,16,7,5,-3}

Answer: B

We know that rxy(l)=\(\sum_{n=-\infty}^{\infty} x(n)y(n-l)\) where l=0,±1,±2,…

At l=0, rxy(0)=\(\sum_{n=-\infty}^{\infty} x(n)y(n)\)=7

For l>0, we simply shift y(n) to the right relative to x(n) by ‘l’ units, compute the product sequence and finally, sum over all the values of product sequence.

We get rxy(1)=13, rxy(2)=-18, rxy(3)=16, rxy(4)=-7, rxy(5)=5, rxy(6)=-3, rxy(l)=0 for l≥7

similarly for l<0, shift y(n) to left relative to x(n)

We get rxy(-1)=0, rxy(-2)=33, rxy(-3)=-14, rxy(-4)=36, rxy(-5)=19, rxy(-6)=-9, rxy(-7)=10, rxy(l)=0 for l≤-8
So, the sequence rxy(l)= {10,-9,19,36,-14,33,0,7,13,-18,16,-7,5,-3}

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

A. rxy(l)=x(l)*x(-l)
B. rxy(l)=x(l)*x(l)
C. rxy(l)=x(l)+x(-l)
D. None of the mentioned

Answer: A

We know that, the correlation of two signals x(n) and y(n) is rxy(l)=\(\sum_{n=-\infty}^{\infty} x(n)y(n-1)\)

Let x(n)=y(n) => rxx(l)=\(\sum_{n=-\infty}^{\infty}x(n)x(n-l)\) = x(l)*x(-l)

6. What is the energy sequence of the signal ax(n)+by(n-l) in terms of cross correlation and auto correlation sequences?

A. a2rxx(0)+b2ryy(0)+2abrxy(0)
B. a2rxx(0)+b2ryy(0)-2abrxy(0)
C. a2rxx(0)+b2ryy(0)+2abrxy(1)
D. a2rxx(0)+b2ryy(0)-2abrxy(1)

7. What is the relation between cross correlation and auto correlation?

A. |rxy(l)|=\(\sqrt{r_{xx}(0).r_{yy}(0)}\)
B. |rxy(l)|≥\(\sqrt{r_{xx}(0).r_{yy}(0)}\)
C. |rxy(l)|≠\(\sqrt{r_{xx}(0).r_{yy}(0)}\)
D. |rxy(l)|≤\(\sqrt{r_{xx}(0).r_{yy}(0)}\)

Answer: D

We know that, a2rxx(0)+b2ryy(0)+2abrxy(l) ≥0
=> (a/B.2rxx(0)+ryy(0)+2(a/B.rxy(l) ≥0
Since the quadratic is nonnegative, it follows that the discriminate of this quadratic must be non positive, that is 4[r2xy(l)- rxx(0) ryy(0)] ≤0 => |rxy(l)|≤\(\sqrt{r_{xx}(0).r_{yy}(0)}\).

8. The normalized auto correlation ρxx(l) is defined as _____________

A. \(\frac{r_{xx}(l)}{r_{xx}(0)}\)

B. –\(\frac{r_{xx}(l)}{r_{xx}(0)}\)

C. \(\frac{r_{xx}(l)}{r_{xy}(0)}\)

D. None of the mentioned

Answer: A

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 rxx (0). Thus the normalized auto correlation sequence is defined as

ρxx(l)=\(\frac{r_{xx}(l)}{r_{xx}(0)}\).

9. Auto correlation sequence is an even function.

A. True
B. False

Answer: A

Let us consider a signal x(n) whose auto correlation is defined as rxx (l).
We know that for auto correlation sequence rxx (l)=rxx (-l).
So, auto correlation sequence is an even sequence.

10. What is the auto correlation of the sequence x(n)=anu(n), 0<a<l?

A. \(\frac{1}{1-a^2}\) al (l≥0)

B. \(\frac{1}{1-a^2}\) a-l (l<0)

C. \(\frac{1}{1-a^2}\) a|l|(-∞<l<∞)

D. All of the mentioned

Answer: D

rxx(l)=\(\sum_{n=-\infty}^{\infty} x(n)x(n-l)\)

For l≥0, rxx(l)=\(\sum_{n=l}^{\infty} x(n)x(n-l)\)

=\(\sum_{n=l}^{\infty} a^n a^{n-l}\)

=\(a^{-l}\sum_{n=l}^{\infty} a^{2n}\)

=\(\frac{1}{1-a^2}a^l\)(l≥0)

For l<0, rxx(l)=\(\sum_{n=0}^{\infty} x(n)x(n-l)\)