This document discusses quantization in digital signal processing. It defines quantization as the process of converting continuous amplitude signals to discrete amplitude levels. Quantization introduces quantization noise, which can be modeled as having a uniform probability distribution between +/- Δ/2, where Δ is the quantization step size. The signal-to-noise ratio increases exponentially with more quantization bits. Quantization can be uniform, where the step size is constant, or non-uniform, where lower amplitudes have smaller steps to reduce error. Compression techniques like μ-law are used prior to non-uniform quantization.

1. Fundamentals of Communication Systems
(ECE 252)
Lecture 7 :

2. 2
Outlines
• Quantization :
o Quantization process
o Quantization noise
o Types of quantization

3. 3
Recall( How to digitize)
• Our message (e.g. voice,video.. etc) is continuous both in time and amplitude
• To convert it into digital, we should do the following:
o Choose certain time samples “Sampling”
o Each time sample is approximated to discrete levels “Quantization”.
o Representthe discrete value by digital symbols (e.g. 0 1 ) “Encoding”
0 𝑇
𝑇𝑠
𝑡
0
𝑡
𝑚(𝑡)
𝑚𝛿(𝑡)
0 𝑇
𝑇𝑠
𝑡
0100 ,1000, 0011, …..

4. 4
What is quantization
• Continuous signals (e.g. voice)have a continuous values.
• No necessary to transmit all these ranges as human ear for example can detect infinite
differences.
• Thus, message of continuous amplitude → is converted to → signal of discrete amplitude.
• In other words, amplitudes are rounded to discrete values (i.e. as you round decimal
numbers to integer numbers).
• The process is called quantization and defined as transforming time sampled message
𝒎(𝒏𝑻𝒔) to discrete amplitudes 𝒗(𝒏𝑻𝒔) taken froma finite set of amplitudes.

5. 5
Quantization process
• 𝒎(𝒏𝑻𝒔) : input sampled message
• 𝒈𝒌: decision levels (thresholds)
• 𝒗(𝒏𝑻𝒔) : representation/ reconstructionlevels
Quantizer
(𝒈𝒌)
𝑚(𝑛𝑇𝑠) 𝑣(𝑛𝑇𝑠)
𝑔6
𝑔5
𝑔1
𝑔2
𝑔3
𝑔4
𝑣1
𝑣2
𝑣3
𝑣4
𝑣5
𝑣6
𝑣7

6. 6
Outlines
• Quantization :
o Quantization process
o Quantization noise
o Types of quantization

7. 7
Quantization noise
• Since, the signal (𝑚) is transformed(approximated) to discrete levels (𝑣), the error/
difference(q)between output and input (i.e. 𝒒 = 𝒗 − 𝒎 ) can be consideredas a noise.
• The noise value (q) ranges from −𝚫/𝟐 to +𝚫/𝟐
𝑚
𝑣
𝑞

8. 8
Quantization noise (cont.)
• If the number of levels are big, one can safely assume that all values of 𝒒 are likely to occur
with same probability.
• Since, the summation of the probability density function should be = 1, the probability
distribution function is given by:
• The mean value of 𝒒 = 0.
• The rootmean square of the quantization noise is givenby:
𝝈𝟐
= න
−Δ/2
Δ/2
𝒇𝒒 𝒒 . 𝒒𝟐
𝒅𝒒
𝝈𝟐 = 𝚫𝟐/𝟏𝟐
−Δ/2 Δ/2
1/Δ
𝑓
𝑞(𝑞)
𝑞

9. 9
SNR of quantized signal
• Recall :
o Quantization noise has range from −𝚫/𝟐 to +𝚫/𝟐.
o where 𝚫 is the quantization step.
o Maximum noise amplitude is 𝚫/𝟐.
o Corresponding noise variance is 𝝈𝟐 = 𝚫𝟐/𝟏𝟐.
• If 𝑹 bits are used to representthe quantization levels,
number of levels “possible words” = 𝟐𝑹
.
• If the message has amplitude range : −𝒎𝒑 𝒕𝒐 𝒎𝒑, the quantization step can be givenby:
𝚫 =
𝟐𝐦𝐩
𝟐𝑹
• Thus, the corresponding SNR (Ratio of signal power “𝑺𝒐” to noise power):
𝑺𝑵𝑹 = 𝟑
𝑺𝒐
𝒎𝒑
𝟐.
𝟐𝟐𝑹
• Increasing number of bits leads to exponential increase in the SNR.

10. 10
SNR of quantized sinusoidal signal
• For sinusoidal message of peak amplitude 𝒎𝒑, the corresponding 𝑺𝒐 is givenby:
𝑺𝒐 = 𝟎. 𝟓𝒎𝒑
𝟐
• Thus, the corresponding SNR (Ratio of signal power “𝑺𝒐” to noise power):
𝑺𝑵𝑹 =
𝟑
𝟐
𝟐𝟐𝑹
• In decibel Scale (𝑺𝑵𝑹𝒅𝑩 = 𝟏𝟎𝒍𝒐𝒈𝟏𝟎𝑺𝑵𝑹𝒓𝒂𝒕𝒊𝒐):
𝑺𝑵𝑹𝒅𝑩 = 𝟏. 𝟖 + 𝟔𝑹

11. 11
Outlines
• Quantization :
o Quantization process
o Quantization noise
o Types of quantization

12. 12
Types of quantization
• Quantization can be classifiedof:
o Uniform:step 𝚫 is constant for all amplitude ranges
o Non-uniform:step 𝚫 is not constant
• Quantizer also can be also classifiedof :
Midtread
Decision levels
Midrise
Decision levels

13. 13
Non-uniform quantization
• Range of human voices is big : Ratio between loud voice to weak voice is 1000:1.
• Loud sounds are less likely to happen compared to lower voices.
• Thus, it is more efficient(in terms of reducing the quantization error/noise)to:
o Quantize lower voices with smaller quantization levels
o Quantize higher voices with larger quantization levels
• A non-uniform quantization == Signal compressing followedby uniformquantization.
• Example for such compressionis 𝝁- law:
• Compression:
o Expand low values of input signal to wider range in the O/P.
o Limit higher input signals to smaller range
• Increasing the value 𝝁 increases such compression, which decreases
error for lower values and increase the error for high input values
• A compromise shouldbe applied. (Typical value is 𝝁 = 𝟐𝟓𝟓)
Non-uniform quantization
𝑣 =
log 1 + 𝜇 |𝑚|
log(1 + 𝜇)

14. 14
Non-uniform quantization (Cont.)
• Another type of compressionis 𝑨 type
• To restore the original signal @ the receiver,a complementary process is used
• Such block called the expander.

15. The end