Numerical

 We want to digitize the human voice.

 What is the bit rate, assuming 8 bits per sample?

Solution:

 The human voice normally contains frequencies from 0 to 4000 Hz.

 So the sampling rate and bit rate are calculated as follows:

𝑺𝒂𝒎𝒑𝒍𝒊𝒏𝒈 𝑹𝒂𝒕𝒆 = 𝟒𝟎𝟎𝟎 × 𝟐 = 𝟖𝟎𝟎𝟎 𝒔𝒂𝒎𝒑𝒍𝒆𝒔
𝑩𝒊𝒕 𝑹𝒂𝒕𝒆 = 𝟖𝟎𝟎𝟎 × 𝟖 = 𝟔𝟒, 𝟎𝟎𝟎 = 𝟔𝟒 𝒌𝒃𝒑𝒔

1 Dr. R.K.Mugelan, Asst. Prof. (Sr), SENSE, VIT

Pulse code modulation (PCM)

PCM Decoder

 To recover an analog signal from a digitized signal we follow the following steps:

1. We use a hold circuit that holds the amplitude value of a pulse till the next pulse arrives.

2. We pass this signal through a low pass filter with a cutoff frequency that is equal to the
highest frequency in the pre-sampled signal.

3. The higher the value of L, the less distorted a signal is recovered.

2 Dr. R.K.Mugelan, Asst. Prof. (Sr), SENSE, VIT

Pulse code modulation (PCM)

3 Dr. R.K.Mugelan, Asst. Prof. (Sr), SENSE, VIT

Pulse code modulation (PCM)
 Recovery of a sampled sine wave for different sampling rates

4 Dr. R.K.Mugelan, Asst. Prof. (Sr), SENSE, VIT

Pulse code modulation (PCM)
 Quantizing error:The difference between the input and output of a quantizer
e(t)x
ˆ(t)x(t)

Process of quantizing noise

Qauntizer
Model of quantizing noise
y  q(x)
AGC x(t ) xˆ (t )
x(t ) xˆ (t )
x
e(t )

+
e(t)  The Noise Model is
xˆ(t)  x(t) an approximation!

5 Dr. R.K.Mugelan, Asst. Prof. (Sr), SENSE, VIT

Pulse code modulation (PCM)
 Quantizing error:

 Granular or linear errors happen for inputs within the dynamic range of quantizer

 Saturation errors happen for inputs outside the dynamic range of quantizer
 Saturation errors are larger than linear errors (AKA as “Overflow” or “Clipping”)

 Saturation errors can be avoided by proper tuning of AGC

 Saturation errors need to be handled by Overflow Detection!

 Quantization noise variance: 

E
2
{[
xq
(
x
q 

)]
} e(
x
)p
(x
)
dx
 2



2 2 2
Lin
Sat

L
/2
12 2

2
2
Lin
q
p(
xl)
q l
l Uniform q. Lin
2

q
l
0 12 12

6 Dr. R.K.Mugelan, Asst. Prof. (Sr), SENSE, VIT

Pulse code modulation (PCM)
Derivation of Quantization Error / Noise Power:
 The difference between the input and output signal is called Quantization error or
Quantization Noise.

 Consider an input signal ‘m(t)’ of continuous amplitude in the range

𝑉𝑝 = 𝑚𝑚𝑎𝑥 , −𝑉𝑝 = − 𝑚𝑚𝑎𝑥
𝑉𝑝 − (−𝑉𝑝 ) 2𝑉𝑝
𝑆𝑡𝑒𝑝 𝑆𝑖𝑧𝑒 ∆ = = , 𝑤ℎ𝑒𝑟𝑒 𝐿 𝑖𝑠 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑒𝑣𝑒𝑙𝑠
𝐿 𝐿
 If m(t) is normalised to 1 i.e., 𝑉𝑝 = 1 , −𝑉𝑝 = −1
2
 Then the 𝑆𝑡𝑒𝑝 𝑆𝑖𝑧𝑒 ∆ =
𝐿

7 Dr. R.K.Mugelan, Asst. Prof. (Sr), SENSE, VIT

Pulse code modulation (PCM)
 Then the quantization error ‘q’ is assumed to be uniformly distributed random variable

 A continuous random variable is said to be uniformly distributed over an interval (a,b) as

shown below,

 The PDF of ‘x’ is given by

𝑓𝑥 𝑥 = 0 𝑓𝑜𝑟 𝑥 ≤ 𝑎
1
= 𝑓𝑜𝑟 𝑎 < 𝑥 ≤ 𝑏
𝑏−𝑎
= 0 𝑓𝑜𝑟 𝑥 > 𝑏

8 Dr. R.K.Mugelan, Asst. Prof. (Sr), SENSE, VIT

Pulse code modulation (PCM)
 Similarly, the PDF of the quantization error ‘q’ can be written as
−∆
𝑓𝑄 𝑞 = 0 𝑓𝑜𝑟 𝑞 ≤
2
1 −∆ ∆
= 𝑓𝑜𝑟 <𝑥≤
∆ 2 2
∆
= 0 𝑓𝑜𝑟 𝑥 >
2

 For this to be true, the incoming signal should not overload the quantizer.

9 Dr. R.K.Mugelan, Asst. Prof. (Sr), SENSE, VIT

Pulse code modulation (PCM)

 Mean of the quantization error and its variance 𝜎𝑞2 is same as the mean square value,
𝜎𝑞2 = 𝐸 𝑞 2

 Variance of 𝑓𝑥 𝑥 is given by
𝑏

𝜎𝑥2 = 𝑥−𝜇 2𝑓
𝑥 𝑥 𝑑𝑥
𝑎

 Where 𝜇 = 𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒 𝐸(𝑥) or the mean value

10 Dr. R.K.Mugelan, Asst. Prof. (Sr), SENSE, VIT

Pulse code modulation (PCM)
 Similarly, the variance of 𝑓𝑄 𝑞 is given by
∆ ∆
2 2
𝜎𝑞2 = 𝑞−𝜇 2𝑓
𝑄 𝑞 𝑑𝑞 = 𝑞 2𝑓
𝑄 𝑞 𝑑𝑞
−∆ −∆
2 2

 Since mean value or 𝐸 𝑞 = 𝜇 = 0

∆
1 2
 Substituting the value of 𝑓𝑄 𝑞 , 𝜎𝑞2 = 𝑞 2 𝑑𝑞
∆ −∆
2

∆𝟐
 Quantization Noise power 𝝈𝟐𝒒 (𝑵𝒒 ) =
𝟏𝟐

11 Dr. R.K.Mugelan, Asst. Prof. (Sr), SENSE, VIT

Pulse code modulation (PCM)
Derivation of maximum signal to quantization noise ratio
 The number of bits per sample ‘R’ and the quantization levels ‘L’ are related as
2𝑉𝑝 2𝑉𝑝
𝐿= 2𝑅 ; 𝑅 = 𝑙𝑜𝑔2 𝐿; ∆= ; ∆= 𝑅
𝐿 2
4𝑉𝑝2
𝟐
∆2 2 𝑽
= 𝐿 = 𝟐
𝒑
𝑵𝒒 =
12 12 𝟑𝑳
∆𝟐 𝑳𝟐
𝑽𝟐𝒑 =
𝟒
∆𝟐 𝑳𝟐 ∆𝟐 𝑳𝟐
𝑺𝒊𝒈𝒏𝒂𝒍 𝑷𝒐𝒘𝒆𝒓
= 𝑺𝑵𝑹 𝒒 = 𝟒 𝑵𝒒 = 𝟒 = 𝟑𝑳𝟐
𝑵𝒐𝒊𝒔𝒆 𝑷𝒐𝒘𝒆𝒓 ∆2
12
12 Dr. R.K.Mugelan, Asst. Prof. (Sr), SENSE, VIT
Pulse code modulation (PCM)
 If ‘P’ is the average power of the message signal, then

𝑆 𝑃 𝑃 3𝑃 2𝑅
=
𝑁 𝑁𝑞 = = × 2
𝑉𝑝2 𝑉𝑝2
3𝐿2
 If input 𝑉𝑝 and power ‘P’ is normalised, then 𝑆 𝑁 = 3 × 22𝑅
𝑆 𝑆
 In decibels, = 10𝑙𝑜𝑔10
𝑁 𝑑𝐵 𝑁 𝑑𝐵

≤ 10𝑙𝑜𝑔10 3 × 22𝑅
≤ 10𝑙𝑜𝑔10 3 + 10𝑙𝑜𝑔10 22𝑅 𝑺
≤ 𝟒. 𝟖 + 𝟔𝑹 𝒅𝑩
≤ 4.8 + 2𝑅 × 10 × 0.3 𝑵 𝒅𝑩

13 Dr. R.K.Mugelan, Asst. Prof. (Sr), SENSE, VIT

Pulse code modulation (PCM)
 For full-load Sinusoidal Signal with peak amplitude Am
𝑉2
 Power 𝑃 = , where V = RMS value
𝑅

𝐴𝑚 2
𝑃=
2

 The normalised power P, when resistance R=1

𝐴2𝑚
𝑃=
2
𝐴2𝑚
3𝑃 3× 3
𝑺 = × 22𝑅 = 2 × 22𝑅 = × 22𝑅 = 𝟏. 𝟓 × 𝟐𝟐𝑹
𝑵 𝑉2 𝐴2𝑚 2
𝑝

14 Dr. R.K.Mugelan, Asst. Prof. (Sr), SENSE, VIT

Pulse code modulation (PCM)
 Expressing in dB,
𝑆 𝑆
= 10𝑙𝑜𝑔10 = 10𝑙𝑜𝑔10 (1.5 × 22𝑅 )
𝑁 𝑑𝐵 𝑁 𝑑𝐵

= 10𝑙𝑜𝑔10 1.5 + 10𝑙𝑜𝑔10 22𝑅

= 1.76 + 2𝑅 × 10 × 0.3
= 1.8 + 6𝑅
𝑺
 Therefore, for Sinusoidal Signal = 𝟏. 𝟖 + 𝟔𝑹 𝒅𝑩
𝑵 𝒅𝑩

15 Dr. R.K.Mugelan, Asst. Prof. (Sr), SENSE, VIT

Pulse code modulation (PCM)
Uniform quantization

 When the quantization levels are uniformly distributed over the full amplitude range of the
input signal, the quantizer is called an uniform or linear quantizer.

 In uniform quantization, the step size between quantization levels remains the same
throughout the input range.

Non-uniform quantization

 If the quantizer characteristic is nonlinear, then the quantization is known as non-uniform

quantization.

 In non-uniform quantization, the step size is not constant.

 The step size is variable, depending on the amplitude of input signal.

16 Dr. R.K.Mugelan, Asst. Prof. (Sr), SENSE, VIT

Pulse code modulation (PCM)
 Uniform quantization V
output w2(t)

-V V
input w1(t)

-V
Region of operation
For L=2n levels, step size :
 = 2V /2n = V(2-n+1)
17 Dr. R.K.Mugelan, Asst. Prof. (Sr), SENSE, VIT
Pulse code modulation (PCM)
output w2(t)
 Uniform quantization: Quantization Error
V

-V V
input w1(t)

-V

Error, e
/2
-/2 input w1(t)
18 Dr. R.K.Mugelan, Asst. Prof. (Sr), SENSE, VIT
Pulse code modulation (PCM)
Companding

 The non-uniform quantization is practically achieved through a process called companding.

 The word Companding is a combination of Compressing and Expanding, which means that
it does both.

 This is a non-linear technique used in PCM which compresses the data at the transmitter and
expands the same data at the receiver.

 The effects of noise and crosstalk are reduced by using this technique.

19 Dr. R.K.Mugelan, Asst. Prof. (Sr), SENSE, VIT

Pulse code modulation (PCM)
Nonuniform quantizer

Discrete Uniform digital

samples Compressor Quantizer signals

••••
Channel
••••

Decoder Expander output

received
digital
20
signals Dr. R.K.Mugelan, Asst. Prof. (Sr), SENSE, VIT
Pulse code modulation (PCM)

21 Dr. R.K.Mugelan, Asst. Prof. (Sr), SENSE, VIT

Pulse code modulation (PCM)

compression+expansion companding

y  C(x) x̂
x(t ) y (t ) yˆ (t ) xˆ (t )

x ŷ
Compress Qauntize Expand
Channel

22 Dr. R.K.Mugelan, Asst. Prof. (Sr), SENSE, VIT