# Digital Signal Processing Questions and Answers – Analysis of Quantization Errors

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This set of Digital Signal Processing Multiple Choice Questions & Answers (MCQs) focuses on “Analysis of Quantization Errors”.

1. If the input analog signal is within the range of the quantizer, the quantization error eq (n) is bounded in magnitude i.e., |eq (n)| < Δ/2 and the resulting error is called?
a) Granular noise
c) Particulate noise
d) Heavy noise

Explanation: In the statistical approach, we assume that the quantization error is random in nature. We model this error as noise that is added to the original (unquantized) signal. If the input analog signal is within the range of the quantizer, the quantization error eq (n) is bounded in magnitude
i.e., |eq (n)| < Δ/2 and the resulting error is called Granular noise.

2. If the input analog signal falls outside the range of the quantizer (clipping), eq (n) becomes unbounded and results in _____________
a) Granular noise
c) Particulate noise
d) Heavy noise

Explanation: In the statistical approach, we assume that the quantization error is random in nature. We model this error as noise that is added to the original (unquantized) signal. If the input analog signal falls outside the range of the quantizer (clipping), eq (n) becomes unbounded and results in overload noise.

3. In the mathematical model for the quantization error eq (n), to carry out the analysis, what are the assumptions made about the statistical properties of eq (n)?
i. The error eq (n) is uniformly distributed over the range — Δ/2 < eq (n) < Δ/2.
ii. The error sequence is a stationary white noise sequence. In other words, the error eq (m) and the error eq (n) for m≠n are uncorrelated.
iii. The error sequence {eq (n)} is uncorrelated with the signal sequence x(n).
iv. The signal sequence x(n) is zero mean and stationary.
a) i, ii & iii
b) i, ii, iii, iv
c) i, iii
d) ii, iii, iv

Explanation: In the mathematical model for the quantization error eq (n). To carry out the analysis, the following are the assumptions made about the statistical properties of eq (n).
i. The error eq (n) is uniformly distributed over the range — Δ/2 < eq (n) < Δ/2.
ii. The error sequence is a stationary white noise sequence. In other words, the error eq (m)and the error eq (n) for m≠n are uncorrelated.
iii. The error sequence {eq (n)} is uncorrelated with the signal sequence x(n).
iv. The signal sequence x(n) is zero mean and stationary.
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4. What is the abbreviation of SQNR?
a) Signal-to-Quantization Net Ratio
b) Signal-to-Quantization Noise Ratio
c) Signal-to-Quantization Noise Region
d) Signal-to-Quantization Net Region

Explanation: The effect of the additive noise eq (n) on the desired signal can be quantified by evaluating the signal-to-quantization noise (power) ratio (SQNR).

5. What is the scale used for the measurement of SQNR?
a) DB
b) db
c) dB
d) All of the mentioned

Explanation: The effect of the additive noise eq (n) on the desired signal can be quantified by evaluating the signal-to-quantization noise (power) ratio (SQNR), which can be expressed on a logarithmic scale (in decibels or dB).
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6. What is the expression for SQNR which can be expressed in a logarithmic scale?
a) 10 $$log_{10}⁡\frac{P_x}{P_n}$$
b) 10 $$log_{10}⁡\frac{P_n}{P_x}$$
c) 10 $$log_2⁡\frac{P_x}{P_n}$$
d) 2 $$log_2⁡\frac{P_x}{P_n}$$

Explanation: The signal-to-quantization noise (power) ratio (SQNR), which can be expressed on a logarithmic scale (in decibels or dB) :
SQNR = 10 $$log_{10}⁡\frac{P_x}{P_n}$$.

7. In the equation SQNR = 10 $$log_{10}⁡\frac{P_x}{P_n}$$. what are the terms Px and Pn are called ___ respectively.
a) Power of the Quantization noise and Signal power
b) Signal power and power of the quantization noise
c) None of the mentioned
d) All of the mentioned

Explanation: In the equation SQNR = $$10 log_{10}⁡\frac{P_x}{P_n}$$ then the terms Px is the signal power and Pn is the power of the quantization noise

8. In the equation SQNR = 10 ⁡$$log_{10}\frac{P_x}{P_n}$$, what are the expressions of Px and Pn?
a) $$P_x=\sigma^2=E[x^2 (n)] \,and\, P_n=\sigma_e^2=E[e_q^2 (n)]$$
b) $$P_x=\sigma^2=E[x^2 (n)] \,and\, P_n=\sigma_e^2=E[e_q^3 (n)]$$
c) $$P_x=\sigma^2=E[x^3 (n)] \,and\, P_n=\sigma_e^2=E[e_q^2 (n)]$$
d) None of the mentioned

Explanation: In the equation SQNR = $$10 log_{10}⁡ \frac{P_x}{P_n}$$, then the terms $$P_x=\sigma^2=E[x^2 (n)]$$ and $$P_n=\sigma_e^2=E[e_q^2 (n)]$$.

9. If the quantization error is uniformly distributed in the range (-Δ/2, Δ/2), the mean value of the error is zero then the variance Pn is?
a) $$P_n=\sigma_e^2=\Delta^2/12$$
b) $$P_n=\sigma_e^2=\Delta^2/6$$
c) $$P_n=\sigma_e^2=\Delta^2/4$$
d) $$P_n=\sigma_e^2=\Delta^2/2$$

Explanation: $$P_n=\sigma_e^2=\int_{-\Delta/2}^{\Delta/2} e^2 p(e)de=1/\Delta \int_{\frac{-\Delta}{2}}^{\frac{\Delta}{2}} e^2 de = \frac{\Delta^2}{12}$$.

10. By combining $$\Delta=\frac{R}{2^{b+1}}$$ with $$P_n=\sigma_e^2=\Delta^2/12$$ and substituting the result into SQNR = 10 $$log_{10}⁡ \frac{P_x}{P_n}$$, what is the final expression for SQNR = ?
a) 6.02b + 16.81 + $$20log_{10}\frac{R}{σ_x}$$
b) 6.02b + 16.81 – $$20log_{10}⁡ \frac{R}{σ_x}$$
c) 6.02b – 16.81 – $$20log_{10}⁡ \frac{R}{σ_x}$$
d) 6.02b – 16.81 – $$20log_{10}⁡ \frac{R}{σ_x}$$

Explanation: SQNR = $$10 log_{10}⁡\frac{P_x}{P_n}=20 log_{10} \frac{⁡σ_x}{σ_e}$$
= 6.02b + 16.81 – ⁡$$20 log_{10}\frac{R}{σ_x}$$dB.

11. In the equation SQNR = 6.02b + 16.81 – $$20log_{10} ⁡\frac{R}{σ_x}$$, for R = 6σx the equation becomes?
a) SQNR = 6.02b-1.25 dB
b) SQNR = 6.87b-1.55 dB
c) SQNR = 6.02b+1.25 dB
d) SQNR = 6.87b+1.25 dB

Explanation: For example, if we assume that x(n) is Gaussian distributed and the range o f the quantizer extends from -3σx to 3σx (i.e., R = 6σx), then less than 3 out o f every 1000 input signal amplitudes would result in an overload on the average. For R = 6σx, then the equation becomes
SQNR = 6.02b+1.25 dB.

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