Bit Rate and Bit Duration Calculation for PCM modulation

The PCM(Pulse Code Modulation) refers to the process of converting analog signal. The analog signal such as speech is usually the input signal. The whole process of converting the analog signal to digital form involves 4 steps-

• Filtering
• Sampling
• Quantization
• Encoding

The Sampling and Quantization is performed by a single physical device called ADC(Analog to Digital Converter) and the process itself is called Analog to Digital conversion. It is however easy to understand the PCM process by explaining them as separate steps. The sampling and quantization process has also two methods-uniform and non uniform quantization. In the uniform quantization process the samples are converted to digital bits using discrete level that are uniformly spaced apart. In the non-uniform quantization process the samples are converted to digital bits with discrete level have non-uniform spacing between them. A-Law and m-law are two methods/ algorithm of non-uniform quantization.


Filtering
Filtering involves passing the analog signal through a filter called the anti-aliasing filter that allows selected portion of signal energy/ power to pass and stops other signal energy/ power to pass through the filter. This is done to avoid overlapping of samples and produce ISI(Inter-Symbol Interference) Effect.

Sampling
The sampling is performed on the output of the filter using sample and hold circuit. The sample and hold circuit consists of transistors that are turned on/off using periodic clocked pulse. On one input port the signal from the filter is fed into the sample and hold circuit and another another clocked pulse input is fed from another input port. The frequency of the clock is the sampling frequency that turns ON/Off the transistors. As a result of this sampled PAM signal is produced at the output of the sample/hold circuit. 

Quantization
The sampled PAM signal is quantized into discrete level. Each discrete level has a corresponding binary bit values. At each sampling instant, portion of the analog PAM signal is quantized to the nearest level and corresponding binary bit values appears at the output. In PCM, 8 bits/sample are used and sampled at the rate of 8Khz sample/sec (i.e sampling rate=Fs=8Khz).

The number of levels is \[L=2^{n}=2^{8}=256\] levels.

The PCM bit rate is-

\[R_{b}={sample/sec}\times{bits/sample}\]

That is,
\[R_{b}={8Khz}\times{8bits/sample}=64kbps\]

The sampling time is-
 \[T_s=1/F_s=1/8Khz=125\mu{s}\]

This means that every 125 micro-second, 8bits representing the analog signal sample comes out of the quantizer.

The bit duration of each of the 8 bit is then-

\[T_b=T_s/8=125\mu{s}/8=15.625\mu{s}\]

Encoding
Encoding process is the selection of and implementation of various available encoding format. Example includes Polar, Bipolar which may be RZ or NRZ types(see analysis of Analysis of Polar, Bipolar, NRZ and RZ baseband encoding), AMI Bipolar encoding, Manchester encoding etc. The selection of encoding format depends upon the application and requirement such as error detecting capability, synchronization capability, cost of circuit implementation etc.