Analogue to Digital Conversion (PCM and DM)

The advantages offered by digital modulation Performance Digital pulse modulation permits the use of regenerative repeaters, when placed along the transmission path at short enough distances, can practically eliminate the degrading effects of channel noise and signal distortion. Ruggedness A digital communication system can be designed to withstand the effects of channel noise and signal distortion Reliability Can be made highly reliable by exploiting powerful error-control coding techniques. Security Can be made highly secure by exploiting powerful encryption algorithms Efficiency Inherently more efficient than analogue communication system in the trade-off between transmission bandwidth and signal-to-noise ratio System integration To integrate digitized analogue signals with digital computer data

Digitizing Analogue Data Stallings DCC9e Figure 5.15 shows voice data that are digitized and then converted to an analog ASK signal. This allows digital transmission in the sense defined in Chapter 3. The voice data, because they have been digitized, can be treated as digital data, even though transmission requirements (e.g., use of microwave) dictate that an analog signal be used. The device used for converting analog data into digital form for transmission, and subsequently recovering the original analog data from the digital, is known as a codec (coder-decoder). In this section we examine the two principal techniques used in codecs, pulse code modulation and delta modulation. The section closes with a discussion of comparative performance.

Pulse Code Modulation (PCM) The simplest technique for transforming analogue data into digital signals is pulse code modulation (PCM). sampling theorem: “If a signal is sampled at regular intervals at a rate higher than twice the highest signal frequency, the samples contain all information in original signal” eg. 4000Hz voice message, requires 8000 samples per second strictly have analog samples Pulse Amplitude Modulation (PAM) assign each a digital value The simplest technique for transforming analog data into digital signals is pulse code modulation (PCM), which involves sampling the analog data periodically and quantizing the samples. Pulse code modulation (PCM) is based on the sampling theorem (quoted above). Hence if voice data is limited to frequencies below 4000 Hz (a conservative procedure for intelligibility), 8000 samples per second would be sufficient to characterize the voice signal completely. Note, however, that these are analog samples, called pulse amplitude modulation (PAM) samples. To convert to digital, each of these analog samples must be assigned a binary code.

Figure Components of PCM encoder

Figure Three different sampling methods for PCM Sampling called pulse amplitude modulation (PAM), the result is still analogue signal with non-integral value.

Figure Nyquist sampling rate for low-pass and bandpass signals According to the Nyquist theorem, the sampling rate must be at least 2 times the highest frequency (not the bandwidth) contained in the signal.

Example For an intuitive example of the Nyquist theorem, let us sample a simple sine wave at three sampling rates: fs = 4f (2 times the Nyquist rate), fs = 2f (Nyquist rate), and fs = f (one-half the Nyquist rate). Figure 4.24 shows the sampling and the subsequent recovery of the signal. It can be seen that sampling at the Nyquist rate can create a good approximation of the original sine wave (part a). Oversampling in part b can also create the same approximation, but it is redundant and unnecessary. Sampling below the Nyquist rate (part c) does not produce a signal that looks like the original sine wave. Telephone companies digitize voice by assuming a maximum frequency of 4000 Hz. The sampling rate therefore is 8000 samples per second.

PCM Block Diagram Thus, PCM starts with a continuous-time, continuous-amplitude (analog) signal, from which a digital signal is produced, as shown in Stallings DCC9e Figure 5.17. The digital signal consists of blocks of n bits, where each n-bit number is the amplitude of a PCM pulse. On reception, the process is reversed to reproduce the analog signal. Notice, however, that this process violates the terms of the sampling theorem. By quantizing the PAM pulse, the original signal is now only approximated and cannot be recovered exactly. This effect is known as quantizing error or quantizing noise. Each additional bit used for quantizing increases SNR by about 6 dB, which is a factor of 4.

Quantisation The results of sampling is a set of amplitude which can be infinite of non-integral values between two limits Vmin and Vmax. Divide the range into L steps, each of height Δ We approximate the sampled value to the quantised value (midpoint) (quantisation error)

Figure 4.26 Quantization and encoding of a sampled signal Δ=[20V-(-20V)]/8 = 5V

Quantisation level L, depends on the range of the amplitudes and how accurately we need. L= 2n, audio communication , L=256. less L more quantisation error. Quantisation error: Quantisation is a approximation process. –Δ/2 ≤ error ≤ Δ/2 The contribution of the Quantisation error to the SNRdB of the signal depends on the number of the quantisation level L, or the bits per sample nb. nb = log2 L signal to quantisation error (or noise) ratio SNRdb = 6.02nb + 1.76 dB Each additional bit used for quantizing increases SNR by about 6 dB, which is a factor of 4.

Example What is the SNRdB in the example of Figure 4.26? Solution We can use the formula to find the quantization SNR. We have eight levels and 3 bits per sample, so SNRdB = 6.02×(3) + 1.76 = 19.82 dB Increasing the number of levels increases the SNR.

Each quantised value can be coded to an nb bit code word. Encoding Each quantised value can be coded to an nb bit code word. Bit rate = sampling rate × number of bit per sample = fs × nb = fs × log2L We want to digitize the human voice. What is the bit rate, assuming 8 bits per sample? The human voice normally contains frequencies from 0 to 4000 Hz. So the sampling rate and bit rate are calculated as follows:

PCM bandwidth Bmin = nb × B analogue The minimum bandwidth of a PCM signal Bmin = nb × B analogue We have a low-pass analog signal of 4 kHz. If we send the analog signal, we need a channel with a minimum bandwidth of 4 kHz. If we digitize the signal and send 8 bits per sample, we need a channel with a minimum bandwidth of 8 × 4 kHz = 32 kHz. Maximum data rate of a channel Rmax = 2 × B × log2L Minimum required bandwidth

Delta Modulation (DM) PCM finds value of amplitude of each sample; DM finds the change from the previous sample. analog input is approximated by a staircase function can move up or down one level (Δ) at each sample interval has binary behavior function only moves up or down at each sample interval hence can encode each sample as single bit 1 for up or 0 for down A variety of techniques have been used to improve the performance of PCM or to reduce its complexity. One of the most popular alternatives to PCM is delta modulation (DM). With delta modulation, an analog input is approximated by a staircase function that moves up or down by one quantization level () at each sampling interval (Ts). The important characteristic of this staircase function is that its behavior is binary: At each sampling time, the function moves up or down a constant amount . Thus, the output of the delta modulation process can be represented as a single binary digit for each sample. In essence, a bit stream is produced by approximating the derivative of an analog signal rather than its amplitude: A 1 is generated if the staircase function is to go up during the next interval; a 0 is generated otherwise.

Figure The process of delta modulation Stallings DCC9e Figure 5.20 shows an example where the staircase function is overlaid on the original analog waveform. A 1 is generated if the staircase function is to go up during the next interval; a 0 is generated otherwise. The transition (up or down) that occurs at each sampling interval is chosen so that the staircase function tracks the original analog waveform as closely as possible. There are two important parameters in a DM scheme: the size of the step assigned to each binary digit, , and the sampling rate. As the above figure illustrates,  must be chosen to produce a balance between two types of errors or noise. When the analog waveform is changing very slowly, there will be quantizing noise. This noise increases as  is increased. On the other hand, when the analog waveform is changing more rapidly than the staircase can follow, there is slope overload noise. This noise increases as  is decreased. It should be clear that the accuracy of the scheme can be improved by increasing the sampling rate. However, this increases the data rate of the output signal.

Figure Delta modulation and demodulation components Stallings DCC9e Figure 5.21 illustrates the logic of the process, which is essentially a feedback mechanism. For transmission, the following occurs: At each sampling time, the analog input is compared to the most recent value of the approximating staircase function. If the value of the sampled waveform exceeds that of the staircase function, a 1 is generated; otherwise, a 0 is generated. Thus, the staircase is always changed in the direction of the input signal. The output of the DM process is therefore a binary sequence that can be used at the receiver to reconstruct the staircase function. The staircase function can then be smoothed by some type of integration process or by passing it through a low pass filter to produce an analog approximation of the analog input signal.

PCM verses Delta Modulation DM has simplicity compared to PCM but has worse SNR issue of bandwidth used for good voice reproduction with PCM: want 128 levels (7 bit) & voice bandwidth 4khz need 8000 x 7 = 56kbps data compression can improve on this still growing demand for digital signals use of repeaters, TDM, efficient switching PCM preferred to DM for analog signals The principal advantage of DM over PCM is the simplicity of its implementation. In general, PCM exhibits better SNR characteristics at the same data rate. Good voice reproduction via PCM can be achieved with 128 quantization levels, or 7-bit coding (27 = 128). A voice signal, conservatively, occupies a bandwidth of 4 kHz. Thus, according to the sampling theorem, samples should be taken at a rate of 8000 samples per second. This implies a data rate of 8000  7 = 56 kbps for the PCM-encoded digital data. But using the Nyquist criterion from Chapter 3, this digital signal could require on the order of 28 kHz of bandwidth. Even more severe differences are seen with higher bandwidth signals. For example, a common PCM scheme for color television uses 10-bit codes, which works out to 92 Mbps for a 4.6-MHz bandwidth signal. In spite of these numbers, digital techniques continue to grow in popularity for transmitting analog data. The principal reasons for this are • Because repeaters are used instead of amplifiers, there is no cumulative noise. • use time division multiplexing (TDM) for digital signals with no intermodulation noise, verses of the frequency division multiplexing (FDM) used for analog signals. • The conversion to digital signaling allows the use of the more efficient digital switching techniques. Furthermore, techniques have been developed to provide more efficient codes. Studies also show that PCM-related techniques are preferable to DM-related techniques for digitizing analog signals that represent digital data.