Quantization:
PCM:
The major steps involved in PCM are sampling, quantizing, and encoding.
Some Advantages associated with PCM are:
The disadvantages of PCM includes:
Advantages of PCM:
Disadvantage of PCM:
Concept:
The number of levels for an n-bit PCM system is given by:
L = 2n
We can also state that the number of bits for a given quantization level will be:
n = log2 L
Calculation:
The number of levels given = 4096, i.e. L = 4096
The number of bits used will be:
n = log2 (4096)
= log2 (2^{12})
= 12 log2 (2)
n = 1
The bandwidth of PCM is given by:
\(BW=n{f_s}\)
n = number of bits to encode
fs = sampling frequency
Concept:
The number of levels for an n-bit PCM system is given by:
L = 2n
We can also state that the number of bits for a given quantization level will be:
n = log_{2} L
Also, the bandwidth of PCM is given by:
\(BW=n{f_s}\)
n = number of bits to encode
fs = sampling frequency
Calculation:
For L = 4 quantization levels, the number of bits n = log2 4 = 2 bits. The bandwidth is, therefore:
B.W. = 2 fs
Similarly, For L = 64 quantization levels, the number of bits n = log2 64 = 6 bits. The bandwidth is, therefore:
B.W. = 6 fs
Clearly, the Bandwidth is increased by 3 times.
The bandwidth requirement of a telephone channel is approximately 3 kHz.
Consider the following:
1. Pulse-position modulation.
2. Pulse-code modulation.
3. Pulse-width modulation.
Which of the above communications are not digital?
PAM (Pulse Amplitude Modulation): In analog modulation, if the amplitude of a pulse or duration of a pulse is varied according to the instantaneous values of the baseband modulating signal, then such a technique is called as Pulse Amplitude Modulation (PAM). This is similar to AM modulation in which the carrier amplitude varies according to the modulating signal.
PWM (Pulse Width Modulation): It is a process in which output signals of low frequency are generated from the input pulse of high frequency. This is similar to the FM modulation technique.
PPM (Pulse Position Modulation): Pulse Position Modulation (PPM) is an analog modulating scheme in which the amplitude and width of the pulses are kept constant, while the position of each pulse, with reference to the position of a reference pulse, varies according to the instantaneous sampled value of the message signal. This makes it similar to a Phase modulation.
PCM (Pulse Code Modulation): It is a technique by which an analog signal gets converted into digital form to have signal transmission through a digital network. ∴ This belongs to a digital modulation scheme.
Concept:
The bandwidth of a PCM system for an encoded signal sampled at a frequency of f_{s} is given by:
B.W. = n f_{s}
f_{s} = Sampling frequency
n = number of bits used for encoding.
n is related to the number of quantization levels (L) as:
L = 2^{n}
or n = log_{2}L
Calculation:
Since the sampling frequency is not mentioned, we'll assume it to be sampled at the Nyquist rate, i.e.
f_{s} = 2f_{m}
f_{m} = Maximum frequency present at the modulating signal.
∴ For the given bandlimited signal with a frequency 4 kHz, the sampling frequency will be:
fs = 2 × 4 = 8 kHz
With L = 256, the number of bits will be:
n = log_{2} 256 = log_{2 }2^{8}
n = 8 bits
Now for 48 such channels, the required bandwidth will be:
B.W. = 48 × n × f_{s} = 48 × 8 × 8000
B.W. = 3.072 MHz
In order to complete analog to digital conversion, each sample value is mapped to a discrete level (represented by a sequence of bits) in a process called quantization.
In a B-bit quantizer, each quantization level is represented with B bits, so that the number of levels equals 2^{B}
Number of levels = 2^{5} = 32
Each level represents 1 V, so for 32 levels = 32 V
On account of modulation, voltage involved is 27.39 V
The quantization error = 0.39 V
Concept:
The transmission rate of a PCM signal is given by:
R_{b} = nf_{s}
n = number of bits to encode a sample
f_{s} = Sampling frequency.
Analysis:
Given the transmission bandwidth capacity of the binary channel is:
B_{ch} = 36 k bits / sec
The bandwidth of the message signal is f_{m} = 3.2 kHz.
Sampling frequency must be at least:
f_{s} ≥ 2 f_{m}
f_{s} ≥ 2 × 3.2 kHz
f_{s} ≥ 6.4 kHz
Let the quantization level be q, we can write:
q = 2^{n}
n = number of bits required to encode the sample.
Since the channel must support the data rate of the PCM signal, we can write:
R_{b} ≤ R_{ch}
nf_{s} ≤ 36 ---(1)
From the given options, the sampling frequency will be 7.2 kHz, i.e. we can write:
7.2k × n ≤ 36k
n ≤ 5
2^{n} ≤ 2^{5}
q < 32
∴ The appropriate values of the quantizing level and sampling frequencies are q = 32 and f_{s} = 7.2 kHz.
Match List-l with List-II and select the correct answer using the code given below the lists :
List - I | List - II |
A. Pilot carrier | 1. Delta modulation |
B. Tuned circuit | 2. Frequency modulation |
C. Slope oveload | 3. PCM |
D. A to D converter | 4. Single sideband AM |
Pilot carrier:
In communication, a pilot carrier is a single frequency, transmitted over a communication system for synchronization, control, continuity purposes we use it in the SSB (single sideband AM) scheme.
Tuned circuit:
A tuned circuit is nothing but a variable capacitor by varying the voltage we can vary the capacitor, by varying capacitor we can vary the frequency. It is used in FM (frequency modulation scheme)
\(f = \frac{1}{{2\pi \sqrt {LC} }}\)
Slope overload:
When the slope of manage signal \(\frac{d}{{dt}}m\left( t \right) > \frac{{\rm{\Delta }}}{{{T_s}}}\)
Where Δ = steps sign then this problem occurs in the delta modulation (DM) scheme.
NOTE:
When \(\frac{d}{{dt}}m\left( t \right) < \frac{{\rm{\Delta }}}{{{T_s}}}\) then granular noise error (GNE) occur.
A to D converter:
In pulse coded modulation scheme after sampling of signal, we do quantization. The process which is similar to analog to digital conversion.
Pulse modulation is the process of transmitting signals in the form of pulses (discontinuous signal) by using special techniques.
Pulse modulation techniques are divided into two types:
Analog pulse Modulation, which includes:
Digital Pulse Modulation, which includes:
Telemetry may be defined as measurement at a distance.
The pulse telemetry system uses a pulse carrier, which is modulated using one of the pulse modulation techniques like PAM, PWM, PPM, and PCM.
PCM is used in digital telemetry.
An analog voltage in the range 0 to 8 V is divided into 16 equal intervals for conversion to 4-bit digital output. The maximum quantization error (in V) is _________
Concept:
The concept of converting on an analog signal to its digital counterpart is explained with the help of the following diagram:
∴ The maximum quantization is given as:
\({Q_{e\left( {max} \right)}} = \frac{{\rm{\Delta }}}{2}\)
Δ = step size given by:
\({\rm{\Delta }} = \frac{{{V_{max}} - {V_{min}}}}{L}\)
L = Number of levels
Calculation:
With analog input in the range 0 to 8 V and L = 16, the step size will be:
\({\rm{\Delta }} = \frac{{8 - 0}}{{16}} = 0.5\)
Now, the maximum quantization error will be:
\({Q_{e\left( {max} \right)}} = \frac{{0.5}}{2} = 0.25\)
Signal to noise ratio is one of the important parameters to analyze the performance of a communication system. It is, therefore, always desirable to reduce the noise power and increase the signal power for efficient transmission and reception.
The factors that determine the amount of noise present include:
Bandwidth:
Signaling Rate:
Transmitted Power:
In PCM system, the encoder
PCM is a technique by which an analog signal gets converted into digital form to have signal transmission through a digital network.
The major steps involved in PCM is sampling, quantizing, and encoding.
Encoder:
Important Points
Low Pass Filter :
Sampler:
\({f_s} \ge 2{f_m}\)
Quantizer:
Concept:
For PCM, the noise power (N_{p}) is given by:
\({N_p} = \frac{{{{\rm{\Delta }}^2}}}{{12}}\)
Where \({\rm{\Delta }} = \frac{{{V_{peak - peak}}}}{{{2^n}}}\)
For calculating n, the bit-rate and the bandwidth of message signal is given as:
Bitrate, R_{b} = nf_{s}
f_{s} = 2f_{m} (Nyquist criteria)
Signal power = (RMS value of signal)^{2}
Calculation:
Given
RMS value of signal = 0.1 V
Signal power = (0.1)^{2} = 0.01 ---(1)
Bit rate R_{b} = nf_{s}
R_{b} = n (2f_{m})
50 = n (2 × 5)
n = 5
Step size will be:
\({\rm{\Delta }} = \frac{{{V_{peak - peak}}}}{{{2^n}}} = \frac{{2V}}{{{2^5}}} = \frac{1}{{{2^4}}}\)
Now, the Noise Power will be:
\(\frac{{{{\rm{\Delta }}^2}}}{{12}} = {\left( {\frac{1}{{{2^4}}}} \right)^2}\frac{1}{{12}} = \frac{1}{{{2^8} \times 12}}\)
∴ The required Signal to Noise Ratio (SNR) will be:
\(\frac{{Signal\;power}}{{Noise\;Power}} = \frac{{0.01}}{{\left( {\frac{1}{{{2^8} \times 12}}} \right)}}\)
= 30.72
Concept:
The Signal to Noise (SNR) Ratio for a PCM system is given by:
SNR = (1.8 + 6n) dB
Where ‘n’ is the number of bits per sample.
Calculation:
For n number of bits, the Signal to Noise Ratio will be:
SNR_{1} = (1.8 + 6n) dB
With the increase in one bit ( "n+1" bits), the SNR becomes:
SNR_{2} = (1.8 + 6(n+1)) dB
SNR_{2} - SNR_{1} = 6 dB
Some salient features of a PCM system are:
Explanation:
Companding
It is used in voice signal transmission only.
Based on the characteristics of the compressor, companding is classified into two types.
1) μ law
2) A law
μ law
A law:
The SNR value is more when compared to μ law companding.
Conclusion:
μ-law compander produces more companding at low amplitudes.
Note:
1) The practical value of μ is 255
2) The practical value of A is 87.6
The Signal to Noise (SNR) Ratio for a PCM system is given by:
SNR = 1.8 + 6n dB (where ‘n’ is the number of bits per sample)
For 8 number of bits, the Signal to Noise Ratio will be:
1.8 + 6 × 8 dB
So the SNR would increase by:
1.8 + 6 × 8 - (1.8 + 6 × 7)
= 6 dB
Some salient features of a PCM system are:
The Secondary TDM level provides
(i) 96-channels and 6.312 Mbps data rate in μ – law system.
(ii) 128-channels and 8.448 Mbps data rate in the A-law system.
μ -law is a companding method and it is followed in Japan and the USA.
A-law is a companding method and it is followed in India and European countries.
Companding is a process where low amplitude signals are expanded and high amplitude signals are compressed at the transmitter side and reverse operation is performed at the receiver side.
Companding is two types:
(i) μ-law Companding
(ii) A-law Companding
Concept:
For an 'n-bit' encoder, the number of levels will be:
L = 2^{n}
The range of an encoder in volts can be calculated as:
0 ≤ Range ≤ (2^{n} - 1) × Step size
Step size = Change in amplitude between two successive levels.
Analysis:
For n = 5, the number of levels will be:
L = 2^{5} = 32
Given step size = 1 V, i.e. every successive levels has a difference of 1 V between them.
∴ The range of the encoder will be:
0 V ≤ Range ≤ (32 - 1) × 1 V
0 V ≤ Range ≤ 31 V