1. The 4N quantization errors are correlated with the sequence {x(n)}.
a) True
b) False
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)}
2. How is the variance of the quantization error related to the size of the DFT?
a) Equal
b) Inversely proportional
c) Square proportional
d) Proportional
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.
3. Every fourfold increase in the size N of the DFT requires an additional bit in computational precision to offset the additional quantization errors.
a) True
b) False
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
4. What is the variance of the output DFT coefficients |X(k)|?
a) \(\frac{1}{N}\)
b) \(\frac{1}{2N}\)
c) \(\frac{1}{3N}\)
d) \(\frac{1}{4N}\)
Explanation: We know that the variance of the signal sequence is (2/N)2/12=\(\frac{1}{3N^2}\)
Now the variance of the output DFT coefficients |X(k)|=N.\(\frac{1}{3N^2} = \frac{1}{3N}\)
5. What is the signal-to-noise ratio?
a) σX2.σq2
b) σX2/σq2
c) σX2+σq2
d) σX2-σq2
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
6. How many number of bits are required to compute the DFT of a 1024 point sequence with a SNR of 30db?
a) 15
b) 10
c) 5
d) 20
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
7. How many number of butterflies are required per output point in FFT algorithm?
a) N
b) N+1
c) 2N
d) N-1
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.
8. . What is the value of the variance of quantization error in FFT algorithm, compared to that of direct computation?
a) Greater
b) Less
c) Equal
d) Cannot be compared
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.
9. How many number of bits are required to compute the FFT of a 1024 point sequence with a SNR of 30db?
a) 11
b) 10
c) 5
d) 20
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.
10. The general linear constant coefficient difference equation characterizing an LTI discrete time system is?
a) y(n)=-\(\sum_{k=1}^N a_k y(n-k)+\sum_{k=0}^N b_k x(n-k)\)
b) y(n)=-\(\sum_{k=0}^N a_k y(n-k)+\sum_{k=0}^N b_k x(n-k) \)
c) y(n)=-\(\sum_{k=1}^N a_k y(n)+\sum_{k=0}^N b_k x(n) \)
d) None of the mentioned
Explanation: We know that, the general linear constant coefficient difference equation characterizing an LTI discrete time system is given by the expression
y(n)=-\(\sum_{k=1}^N a_k y(n-k)+\sum_{k=0}^N b_k x(n-k)\)