A random variable X with uniform density in the interval 0 ≤ X ≤ 1 ...

Question

A random variable X with uniform density in the interval 0 ≤ X ≤ 1 is quantized as follows:

If 0 ≤ X ≤ 0.3,     Xq = 0

If 0.3 < X ≤ 1,     Xq = 0.7

      

where X_q is the quantized value of X. The root mean square value of the quantization noise is:

• (a) 0.573
• (b) 0.198
• (c) 2.205
• (d) 0.266

Solution

Given

• Random variable X ∼ Uniform(0,1)
• Non-uniform quantizer:

Xq =

{

  0,   0 ≤ X ≤ 0.3

  0.7, 0.3 < X ≤ 1

}

      

Quantization error: e = X − X_q

Required: e_rms = √E[e²]

Step 1: PDF of X

Since X is uniformly distributed over [0,1]:

f_X(x) = 1 ,   0 ≤ x ≤ 1

Step 2: Mean Square Error

Region 1: 0 ≤ X ≤ 0.3 (X_q = 0)

e = X

E[e²]₁ = ∫₀⁰·³ x² dx

       = [x³ / 3]₀⁰·³

       = 0.009

      

Region 2: 0.3 < X ≤ 1 (X_q = 0.7)

e = X − 0.7

E[e²]₂ = ∫₀·³¹ (x − 0.7)² dx



Let u = x − 0.7

Limits:

x = 0.3 → u = −0.4

x = 1   → u = 0.3



E[e²]₂ = ∫₋⁰·⁴⁰·³ u² du

       = [u³ / 3]₋⁰·⁴⁰·³

       = (0.027 + 0.064) / 3

       = 0.03033

      

Step 3: Total Mean Square Error

E[e²] = 0.009 + 0.03033 = 0.03933

Step 4: RMS Quantization Error

e_rms = √0.03933 ≈ 0.198

Final Answer

e_rms ≈ 0.198

Correct option: (b) 0.198