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Test: Quantization Effects


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15 Questions MCQ Test Signals and Systems | Test: Quantization Effects

Test: Quantization Effects for Electrical Engineering (EE) 2022 is part of Signals and Systems preparation. The Test: Quantization Effects questions and answers have been prepared according to the Electrical Engineering (EE) exam syllabus.The Test: Quantization Effects MCQs are made for Electrical Engineering (EE) 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Quantization Effects below.
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Test: Quantization Effects - Question 1

The effect of round off errors due to the multiplications performed in the DFT with fixed point arithmetic is known as Quantization error. 

Detailed Solution for Test: Quantization Effects - Question 1

Explanation: Since DFT plays a very important role in many applications of DSP, it is very important for us to know the effect of quantization errors in its computation. In particular, we shall consider the effect of round off errors due to the multiplications performed in the DFT with fixed point arithmetic.

Test: Quantization Effects - Question 2

What is the model that has been adopt for characterizing round of errors in multiplication?

Detailed Solution for Test: Quantization Effects - Question 2

Explanation: Additive white noise model is the model that we use in the statistical analysis of round off errors in IIR and FIR filters.

Test: Quantization Effects - Question 3

How many quantization errors are present in one complex valued multiplication?

Detailed Solution for Test: Quantization Effects - Question 3

Explanation: We assume that the real and imaginary components of {x(n)} and {WNkn} are represented by ‘b’ bits. Consequently, the computation of product x(n). WNkn requires four real multiplications. Each real multiplication is rounded from 2b bits to b bits and hence there are four quantization errors for each complex valued multiplication.

Test: Quantization Effects - Question 4

What is the total number of quantization errors in the computation of single point DFT of a sequence of length N?

Detailed Solution for Test: Quantization Effects - Question 4

Explanation: Since the computation of single point DFT of a sequence of length N involves N number of complex multiplications, it contains 4N number of quantization errors.

Test: Quantization Effects - Question 5

What is the range in which the quantization errors due to rounding off are uniformly distributed as random variables if Δ=2-b

Detailed Solution for Test: Quantization Effects - Question 5

Explanation: The Quantization errors due to rounding off are uniformly distributed random variables in the range (-Δ/2,Δ/2) if Δ=2-b. This is one of the assumption that is made about the statistical properties of the quantization error.

Test: Quantization Effects - Question 6

 The 4N quantization errors are mutually uncorrelated.

Detailed Solution for Test: Quantization Effects - Question 6

Explanation: The 4N quantization errors are mutually uncorrelated. This is one of the assumption that is made about the statistical properties of the quantization error.

Test: Quantization Effects - Question 7

. The 4N quantization errors are correlated with the sequence {x(n)}.

Detailed Solution for Test: Quantization Effects - Question 7

Explanation: According to one of the assumption that is made about the statistical properties of the quantization error, the 4N quantization errors are uncorrelated with the sequence {x(n)}.

Test: Quantization Effects - Question 8

How is the variance of the quantization error related to the size of the DFT?

Detailed Solution for Test: Quantization Effects - Question 8

Explanation: We know that each of the quantization has a variance of Δ2/12=2-2b/12.
The variance of the quantization errors from the 4N multiplications is 4N. 2-2b/12=2-2b(N/3).
Thus the variance of the quantization error is directly proportional to the size of the DFT.

Test: Quantization Effects - Question 9

. Every fourfold increase in the size N of the DFT requires an additional bit in computational precision to offset the additional quantization errors.

Detailed Solution for Test: Quantization Effects - Question 9

Explanation: We know that, the variance of the quantization errors is directly proportional to the size N of the DFT. So, every fourfold increase in the size N of the DFT requires an additional bit in computational precision to offset the additional quantization errors.

Test: Quantization Effects - Question 10

 What is the variance of the output DFT coefficients |X(k)|?

Detailed Solution for Test: Quantization Effects - Question 10

Explanation: We know that the variance of the signal sequence is (2/N)2/12=1/(3N2)
Now the variance of the output DFT coefficients |X(k)|=N. 1/(3N^2 2) = 1/3N.

Test: Quantization Effects - Question 11

 What is the signal-to-noise ratio? 

Detailed Solution for Test: Quantization Effects - Question 11

Explanation: The signal-to-noise ratio of a signal, SNR is given by the ratio of the variance of the output DFT coefficients to the variance of the quantization errors.

Test: Quantization Effects - Question 12

 How many number of bits are required to compute the DFT of a 1024 point sequence with a SNR of 30db?

Detailed Solution for Test: Quantization Effects - Question 12

Explanation: The size of the sequence is N=210. Hence the SNR is
10log10(σX22/ σq2)=10 log1022b-20
For an SNR of 30db, we have
3(2b-20)=30=>b=15 bits.
Note that 15 bits is the precision for both addition and multiplication.

Test: Quantization Effects - Question 13

 How many number of butterflies are required per output point in FFT algorithm?

Detailed Solution for Test: Quantization Effects - Question 13

Explanation: We find that, in general, there are N/2 in the first stage of FFT, N/4 in the second stage, N?8 in the third state, and so on, until the last stage where there is only one. Consequently, the number of butterflies per output point is N-1.

Test: Quantization Effects - Question 14

 What is the value of the variance of quantization error in FFT algorithm, compared to that of direct computation?

Detailed Solution for Test: Quantization Effects - Question 14

Explanation: If we assume that the quantization errors in each butterfly are uncorrelated with the errors in the other butterflies, then there are 4(N-1) errors that affect the output of each point of the FFT. Consequently, the variance of the quantization error due to FFT algorithm is given by
4(N-1)( Δ2/12)=N(Δ2/3)(approximately)
Thus, the variance of quantization error due to FFT algorithm is equal to the variance of the quantization error due to direct computation.

Test: Quantization Effects - Question 15

How many number of bits are required to compute the FFT of a 1024 point sequence with a SNR of 30db?

Detailed Solution for Test: Quantization Effects - Question 15

Explanation: The size of the FFT is N=210. Hence the SNR is 10 log1022b-v-1=30
=>3(2b-11)=30
=>b=21/2=11 bits.

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