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Signal to Quantization noise ratio derivation

Let $m(t)$= $A_m$ cos 2$\pi$ $f_mt$ be the signal which is undergoing quantization.

Then for this signal, the  power will be  $A_m^{2}$/2

i.e $P_s$ =$A_m^{2}/2$

Suppose if we are quantizing a signal with n-bits , then the step size(i.e difference between quantization steps) will be

$q$=  $\frac {V_{max}-V_{min}}{2^{n}}$$\frac {2A_m}{2^{n}}$

The maximum quantization error can be q/2. So it is usually assumed that quantization noise is uniformly distributed from -q/2 to q/2. Hence the probability density function is a rectangle with height 1/q and extends it’s base from -q/2 to +q/2. Note that the area under PDF curve should be equal to 1.

Quantization noise power is calculated as follows

$P_q$ = $\frac{1}{q}\int_{-q/2}^{q/2}x^2dx$ = $\frac{q^2}{12}$

Signal to quantization noise ratio is a ratio of signal power $P_s$ to the quantization noise power $P_q$

SQNR= $\frac{P_s}{P_q}$ =1.5 * $2^{2n}$

$SQNR_{dB}$= 10 log {1.5 * $2^{2n}$}

$SQNR_{dB}$ =1.76 +6.02n

For any doubts regarding derivation, please use comment section.

GATE 2016 EC questions and answers – Forenoon session

1)Despite the new medicine’s ______________ in treating diabetes, it is not ______ widely

a) effectiveness — prescribed                                   b) availability —– used

c) prescription — available                                      d)acceptance ——- proscribed

Correct Ans : a

2) In a huge piles of apples and oranges, both ripe and unripe mixed together, 15% are unripe fruits. Of the unripe fruits, 45% are apples. Of the ripe one’s , 66 % are oranges. If the pile contains a total of 5692000 fruits, how many of them are apples?

a) 2029198          b) 2467482          c)  2789080         d) 3577422

correct Ans : a

3) The policeman asked the  victim of a theft, ” What did you __ ?”

A) loose b) lose  c) loss d) louse

correct Ans : b

4) Which of the following is correct with respect to grammar and usage?

Mount Everest is _________

a) the highest peak in the world

b) highest peak in the world

c) one of highest peak in the world

d) one of the highest peak in the world
Correct Ans : a

5)  Michael lives 10km away from where I live. Ahmed lives  5 km away and Susan lives 7 km away from where I live. Arun is farther away from Ahmed but closer than Susan from where I live. From the information provided here, what is one possible distance (in km) at which I live from Arun’s place?

a) 3.00        b) 4.99         c)  6.02        d) 7.01

Correct Ans : c

6) In a world filled with uncertainty, he was glad to have many good friends. He had always assisted them in times of need and was confident that they would reciprocate. However, the events of last week proved him wrong

Which of the following inferences is/are logically valid  and can be inferred from the above passage

i) His friends were always asking him to help them.

ii) He felt that when in need of help, his friends would let him down.

iii) He was sure that his friends would help him  when in need.

iv) His friends didn’t help him last week.

A) i and ii          B)  iii and iv   C) iii only       d) iv only

correct Ans : C

7)

correct Ans : c

8) Leela is older than her cousin Pavithra. Pavithra’s brother Shiva is older than Leela. When Pavithra and shiva are visiting Leela, all three are likely to play Chess. Pavithra wins more often than Leela does.

Which of the following statements must be TRUE ?

a) When Shiva plays chess with  Leela and Pavithra, he often loses.

b) Leela is the oldest of three

c) Shiva is a better chess player than Pavithra.

d) Pavithra is the youngest of the three.

Ans : d

9) A person moving through a tuberculosis prone zone has a 50% probability of becoming infected. However 30% of infected people develop the disease. What percentage of people moving through tuberculosis prone zone remains infected but does not show symptoms of disease?

a)  15    b) 33   c) 35     d)  37

——————–Electronics and Communication Questions and Answers————————-

2) A continuos time sinusoid of frequency 33Hz is multiplied with periodic dirac impulse train of frequency 46 Hz. The resulting signal is passed through an ideal analog low pass filter with cut-off frequency of 23 Hz. The fundamental frequency(inHz)  of the output is ______________

Ans : 13 Hz.

When 33 Hz and 46 Hz are multiplied with each other, we get signals of frequency 13 Hz and 79 Hz so on. Since we are passing the output through low pass filter, it allows only low frequency signals till it’s cut off frequency(23 Hz). So only 13 Hz is passed.

3) The propagation constant of lossy transmission line is (2+j5) $m^-1$ and it’s characteristic impedance is 50+j0 $\mho$ at $\omega$ = $10 ^6$ rad/s. The value of line constants L,C,R and G  respectively are

a) L=200 $\mu$H/m, C=0.1 $\mu$ F/m , R=50 $\Omega$ , G=0.02 S/m

b) L=250 $\mu$H/m, C=0.1 $\mu$ F/m , R=100 $\Omega$ , G=0.04 S/m

c) L=200 $\mu$H/m, C=0.2 $\mu$ F/m , R=100 $\Omega$ , G=0.02 S/m

d) L=250 $\mu$H/m, C=0.2 $\mu$ F/m , R=50 $\Omega$ , G=0.04 S/m

Options

a

b

c

d

Correct Ans : b

4) Consider the plot of $f(x)$ versus x as shown below.

Options

A

B

C

D

Correct Ans : C

5)  Consider a two port network with  2*2 transmission matrix  T= {A ,   B , C,   D}         . If the network is reciprocal, then

a) $T^-1=T$                              b)  $T^2=T$

C) Determinant(T)=0              d) Determinant(T)= 1

Correct Ans : d

6) Which one of the following is an eigen function of the class of all continuous time, linear, time-invariant systems?

a) $e^{j\omega_0t}$ u(t)                          b) cos $\omega_ot$

c) $e^{j\omega_0t}$                                d) sin $\omega_ot$

Options

a

b

c

d

Correct Ans : c

7) Consider the following statements for metal oxide semiconductor field effect transistor (MOSFET)

P: As channel length reduces, off current increases.

Q: As channel length reduces, output resistance increases

R: As channel length reduces, threshold voltage remains constant

S: As channel length reduces, on current increases.

Which of the above statements are INCORRECT?

a) P and Q           b) P and S           c)  Q and  R              d) R and S

Correct Ans : C

9)

Correct Ans : A

Explanation : For intrinsic semiconductor, fermi level is at the centre. For n-type semiconductor, the fermi level is near to conduction band because of the presence of more electrons than holes. In the above diagram, the fermi level is close to conduction band than valence band, hence among given options, A is the correct answer.

10) Match the inferences X,Y and Z about a system, to the corresponding properties of a elements of a first  column in Routh’s table of the system characteristic equation

X: The system is stable ..                                                    P: ………… when all the elements are positive

Y: The system is unstable                                                   Q:…………. when any one element is zero

Z: The test breaks down                                                    R:  …………… When there is change in sign of coefficients

Ans : X-P, Y-R, Z-Q

11)

Given the following statements about a function $f$ : R ->R, select the right option

P: If $f(x)$ is continuous at $x$= $x_0$, then it is also differentiable at x= $x_0$
Q: If $f(x)$ is continuous at  $x$= $x_0$, then it may not be differentiable at x= $x_0$
R: If $f(x)$  is differentiable at $x$= $x_0$, then it is also continuous at x= $x_0$

A) P is true, Q is false, R is false           B) P is false, Q is true, R is true
C) P is false, Q is true, R is false         D) A) P is true, Q is false, R is true

Correct Ans : B

13) The amplitude of sinusoidal carrier is modulated by a single sinusoid to obtain the amplitude modulated signal

$S(t)$= 5 cos 1600$\pi$t +20 cos 1800$\pi$t+5 cos 2000$\pi$t. The value of modulation index is

Ans : 0.5

This is a standard AM equation. Here A_c=20, uA_c/2=5  . Therefore u(modulation index) =10/20=0.5

14) A continuous time function $x$(t) is periodic with period T. The function is sampled uniformly  with a sampling period $T_s$ . In which of the following cases is the sampled signal periodic.?

a) T= $\sqrt2$ $T_s$                                           b) T=1.2 $T_s$

c) Always                                                                                     d) Never

Ans: b

15)  Consider the constant current source shown in the figure below. Let 𝛽 represent the current gain of the transistor.

The load current $I_o$ through $R_L$ is  ___________

a) $I_o$ = $\frac{\beta+1}{\beta} \frac{V_{ref}}{R}$

b) $I_o$ = $\frac{\beta}{\beta+1}\frac{V_{ref}}{R}$

c) $I_o$ = $\frac{\beta+1}{\beta}\frac{V_{ref}}{2R}$

d) $I_o$ = $\frac{\beta}{\beta+1}\frac{V_{ref}}{2R}$

Solution :

Voltage at Non Inverting terminal is –> $V_{cc}$$V_{ref}$

For a Op-Amp under Negative feedback, voltage at inverting terminal is equal to voltage at Non-inverting terminal.

Therefore Voltage at Inverting terminal is –> $V_{cc}$  –  $V_{ref}$

In the circuit diagram, we can see that, $I_E$= $\frac{V_{cc} - (V_{cc} - V_{ref})}{R}$

In this circuit,  $I_o$ is nothing but collector current  $I_c$ of the PNP transistor.

We know that

$I_c$= $\alpha$ $I_E$

$I_o$ = $\frac{\beta}{\beta+1}\frac{V_{ref}}{R}$

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